| 00:06 | So uh let me bring up your from last time. OK. So |
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| 00:23 | first one is from um uh Li and she has um two questions, |
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| 00:33 | concerning uh three questions, one con of them concerning presentation eight. |
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| 00:40 | So she says slide 28 we consider core fluid pressure is uniform within the |
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| 00:49 | , this implies seismic band waves, sonic or ultrasonic. And she |
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| 00:55 | why is it not Sonic? So you know, that's a very good |
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| 00:59 | . So let me ask you uh , what do you think is the |
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| 01:03 | to that question? Yeah. She she doesn't know her voice is very |
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| 01:09 | voice. And so I'm sure that folks uh uh online cannot hear |
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| 01:14 | Uh So let me turn to Carlos. Uh why, why did |
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| 01:19 | not include Sonic? Carlos? Are with me? Yeah. Yeah. |
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| 01:28 | . Professor. I just um I thinking but I am OK. So |
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| 01:31 | want you to think, I want to think out loud so we can |
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| 01:34 | you thinking. Yeah. Yeah. . No, no, I I |
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| 01:37 | know the answer. Professor. OK. So we want to consider |
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| 01:41 | case where the um where the uh pressure is uniform within each max, |
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| 01:49 | contains many grains. But it it's not as so big as a |
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| 01:53 | , but it's uh uh uh uh the size of a hand sample has |
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| 01:57 | grains. And we want to have uh consider the case where the pore |
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| 02:03 | is uniform on that scale. So gonna mean if we're doing this with |
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| 02:09 | uh waves of course waves are always . But uh if we consider low |
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| 02:15 | uh frequency, then we can uh can, we can um trigger that |
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| 02:21 | low frequency seismic waves satisfy the assumption local core pressure uniformity on every piece |
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| 02:32 | the formation, you know, the of a hand sample. So uh |
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| 02:37 | you know, uh it, it's good question because nobody really knows what |
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| 02:41 | means. A me we always assume that includes uh uh uh uh seismic |
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| 02:49 | . We always assume that low frequency uh it, it includes uh uh |
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| 02:54 | seismic band. And so we can these results at the uh at the |
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| 03:00 | for our seismic data. But think it, you know, doesn't that |
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| 03:10 | implies something about permeability. If the is very high, like in a |
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| 03:17 | , I, I guess it's probably that the seismic band is low enough |
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| 03:22 | . But what happens if it's a , if, if it's a |
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| 03:26 | you know, we have intrinsically low . And so maybe even in the |
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| 03:33 | band for shales, um that's not enough frequency, maybe because of the |
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| 03:40 | permeability uh during the passage of a seismic band wave in a |
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| 03:47 | maybe that's not enough time for the to equalize everywhere. This is |
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| 03:52 | a question, a problem which is uh well understood, hardly ever addressed |
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| 03:58 | everybody asked that question. And so question is, is not quite |
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| 04:03 | but it's, you know, it's . Now, let's turn to her |
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| 04:07 | about Sonic. So Sonic means um of about 1000 cycles per second. |
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| 04:14 | so, uh you can do some uh is that low enough or not |
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| 04:21 | enough? And so when you make estimates, you're gonna assume something about |
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| 04:25 | rotting. And uh it's uh uh think it's usually true that if uh |
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| 04:32 | have a permanent rotate like in a sonic frequencies are too high to really |
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| 04:39 | the, the uh um that the to really establish that uh uh ferocity |
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| 04:50 | that the uh pore pressure is uniform every hand sample. So that's a |
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| 04:54 | of a theoretical answer. Better answer come from our own rock physics lab |
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| 05:01 | in um here at the University of . Uh uh uh maybe do you |
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| 05:09 | uh long tang, you know? . So he's uh he's, he's |
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| 05:15 | graduate student. Uh uh is he postdoc? He's a graduate student? |
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| 05:21 | . And he's uh an advanced graduate and he's been doing a lot of |
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| 05:25 | of uh experiments in our laboratory And uh he's using a piece of |
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| 05:31 | which was developed by a, a uh student now graduated. And this |
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| 05:38 | is uniquely capable of doing experiments through wide band of frequencies as low as |
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| 05:46 | like 10 Hertz, I think. as high as um uh as high |
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| 05:52 | 1000 Hertz, I think. And I'll, I'll hi his experiments. |
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| 05:58 | , I'll verify, yeah, theoretical that I just gave for sandstones. |
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| 06:06 | I'm not sure what he has learned shale, but I suspect that for |
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| 06:11 | there might be a problem even so uh even a seismic frequencies just because |
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| 06:18 | preme is so low uh as the is passing through that there's not time |
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| 06:24 | for the fluid to squirt around inside rock to equalize out the core pressure |
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| 06:29 | the, on the local scale. But uh that's uh you know, |
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| 06:34 | are complicated and the best answer to like that is always experimental. And |
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| 06:40 | I, I just uh gave you , the results of the experimentation. |
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| 06:45 | let me uh now turn back to question um slide 35 and 36. |
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| 06:53 | could you tell the gas mine equation the wrong assumption from that plot? |
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| 06:59 | . So we've got to uh we've got to look at this, |
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| 07:01 | got to look at that. I am going to um bring up |
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| 07:10 | from, OK, here is um eight. Then I'm gonna go to |
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| 07:38 | 35. OK. OK. So am now going to show um |
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| 07:51 | uh I'm going to show this to goodly has. And so uh first |
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| 07:58 | all, put it at presentation mode then I'll um, I hope go |
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| 08:08 | the Z window and shoot her that and I'm gonna share a screen |
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| 08:20 | OK. So you should be able see this now, I think. |
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| 08:27 | . So uh uh this slide is taken from an early work by |
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| 08:32 | 1985. Wow, that was a time ago. That was for 40 |
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| 08:37 | ago, maybe before some of you were born. And so uh when |
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| 08:43 | did this work, I didn't had idea how it was gonna come |
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| 08:47 | Uh And so you can see we've here a lot of data, uh |
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| 08:54 | probably uh 50 or 100 data Uh And they're all uh uh and |
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| 09:01 | just uh uh the B let me my cursor. So it, it |
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| 09:08 | the bolt modulus K as a function uh pressure for lots of rocks. |
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| 09:14 | sort of the list of, of rocks. And uh I should |
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| 09:20 | out that I did not do make of these measures. My colleagues at |
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| 09:25 | did that and uh most of that already done. Uh uh before I |
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| 09:31 | joined the company and they had uh the, the data, you |
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| 09:36 | on uh pieces of paper in in the, uh uh in the |
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| 09:45 | . We didn't really have computers uh in the way of computers in those |
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| 09:51 | . Computers are very primitive, modern and we didn't have much in the |
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| 09:56 | of databases. So, uh as matter of fact that when I first |
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| 10:01 | the company, uh my boss OK, now we want you to |
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| 10:05 | build uh to build the database out all this data. So we can |
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| 10:12 | it out of the drawers and put in the computer so we can actually |
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| 10:15 | it. That, that was a plan. That was my first uh |
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| 10:19 | work at NFL. And out of came this project. So you see |
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| 10:23 | their observations around this axis and all predictions from um uh from gas mon |
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| 10:30 | on this axis. And if the were accurate, all the um observations |
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| 10:36 | lie on this line. But you , they don't, they lie to |
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| 10:40 | , to the right of this they lie at higher values of um |
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| 10:45 | is observed to be higher values of mean faster velocities than Gasman predicted. |
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| 10:53 | that's the uh uh the answer. how we can tell from this slide |
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| 10:58 | Gas Mont was uh uh was wrong uh I shouldn't say it this |
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| 11:05 | Uh uh I shouldn't say this proves is wrong. It proves that we |
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| 11:11 | not apply Gas Mon's theory to ultrasonic . And the reason of course is |
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| 11:18 | the ultrasonic data has some frequencies too . So that gas models assumption is |
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| 11:23 | met. So this uh graph does prove the gas model is wrong. |
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| 11:28 | just proves that we were, that were wrong to um apply his low |
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| 11:34 | theory to high frequency data. let's look at the next slide. |
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| 11:44 | . So uh this uh uh is single sample of Bria sandstone well understood |
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| 11:51 | as a function of pressure from uh friend Arthur Chang here, the function |
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| 11:56 | pressure here. And this is what measured and this is what the theory |
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| 12:02 | . It should be. What's the based on uh uh based on uh |
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| 12:07 | we measure um uh you measure the of uh the dry rock. And |
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| 12:14 | we uh we measure the properties of fluid and we measure the amount of |
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| 12:19 | porosity and we uh uh make sure properties of the core of are |
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| 12:27 | And so, uh then on the of that, that's all you need |
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| 12:30 | do for gas ma make this um prediction. And the prediction is way |
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| 12:38 | . You see the prediction is um a function. The the error is |
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| 12:44 | function of uh pressure. The error small here and it gets bigger and |
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| 12:48 | at lower pressure. And so what means is that uh uh uh the |
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| 12:55 | is at least in part due to closing of crack. So, if |
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| 13:00 | look in gas man's theory, there's mention of cracks. His theory is |
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| 13:04 | to apply to crack. And of , I guess man would have known |
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| 13:16 | there would be cracks in some but he, he must have assumed |
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| 13:21 | the effects of those cracks are um by the measurement of the dry |
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| 13:29 | And uh but here you see, not true. So this is |
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| 13:34 | this is a proof that uh that . Uh huh. Well, let's |
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| 13:42 | . Now, I guess it's not . Oh, th this again proves |
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| 13:48 | um uh we shouldn't be applying gasoline to ultrasonic. It could be that |
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| 13:54 | crack, this effect of the cracks you see here only happens at ultrasonic |
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| 14:01 | frequencies. And it, it could that if you did this uh uh |
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| 14:07 | experiment on the use of sandstone rock proper low frequencies, the gas mont |
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| 14:13 | be uh validate. Well, uh my knowledge that has not been done |
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| 14:21 | we have recent theory that says that mine is wrong even at the lowest |
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| 14:26 | . I think I talked with you that. So that answers uh Lena's |
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| 14:33 | question. So now let's look at third question. Rich and um my |
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| 14:43 | not. Oh And she asked her 35 what does the incidental reflector in |
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| 14:52 | mean when the real part is So uh uh let me go to |
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| 14:58 | slide 35 and you taste not like 35 it's not too far. |
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| 15:42 | 34 35. OK. So, , let me put this into presentation |
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| 15:56 | and then let me um share that the class particular stop sharing and then |
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| 16:41 | gonna start sharing again. Yeah, time I always have this problem. |
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| 17:08 | come over here and help me. , I'm not seeing that. |
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| 17:15 | Thank you. You're reminded. Now, um uh so uh a |
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| 17:26 | after the, the previous discussion leading to this slide, oops, let |
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| 17:33 | get the pointer. After the previous , we answer to the slide. |
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| 17:38 | , we uh discovered that the uh car efficient in the case of the |
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| 17:44 | or have attenuation in the uh has an imaginary part to it. And |
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| 17:50 | it shows explicitly and here's the real that you are accustomed to looking |
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| 17:54 | And, and uh all the rest your professional life is this uh jump |
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| 17:59 | uh impedance divide by twice. The impedance only thing is here. It's |
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| 18:03 | it says uh um the real part specified that we're talking about the real |
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| 18:10 | of that. Well, I'm sure whenever you looked at that normal |
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| 18:16 | reflection coefficient before ever in your you've always just assumed it was |
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| 18:21 | But now here we're saying, uh we're looking only at the real |
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| 18:26 | if there's an imaginary part to it's not inside here because this is |
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| 18:30 | the real part. And so now now what the sliders occur, we're |
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| 18:35 | at when the real part goes to . And so what Lily is asking |
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| 18:39 | what does it mean to have real ? Equal zero? Well, you |
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| 18:43 | , that, uh that is just , a matter of imagining lithology contrast |
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| 18:50 | that boundary uh where uh the, know, the fluids in the, |
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| 18:56 | fluids are about the same and the are about the same and um uh |
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| 19:03 | microstructure is about the same and everything about the same. So it's a |
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| 19:07 | weak reflection, that's all, all means. And so uh uh uh |
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| 19:11 | , I think it's easy to imagine um we have strong reflectors and weak |
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| 19:17 | and this is just uh uh to the issues in your mind, it |
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| 19:22 | imagine a case where we have a weak reflector and normal incidence. And |
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| 19:28 | uh and when I say we, mean, the real parts of uh |
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| 19:32 | , of OK velocity which go into are all are very similar on both |
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| 19:39 | of the, of the reflector, that doesn't say anything about the |
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| 19:47 | And so um uh the aeration is left when we assume this uh uh |
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| 19:53 | is pretty small, we're still left the imaginary parts and uh that implies |
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| 19:58 | the reflective wavelength uh is face shifted 90 degrees. Uh So what, |
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| 20:05 | sort of Ortho could we have Uh uh uh You, you could |
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| 20:10 | uh uh where it's uh uh sandstone e on either side of the, |
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| 20:17 | the reflector and uh um, one is gas and one side is uh |
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| 20:25 | brine. So you, you might thinking to yourself, well, how |
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| 20:30 | that be? Uh surely the uh sides are sandstone then uh um uh |
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| 20:38 | , the, the um gas is cross the boundary or the liquid is |
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| 20:44 | cross the boundary. It's, it's homogenize. But think about this, |
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| 20:49 | suppose you have a, a AAA with a gas cap in it and |
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| 20:57 | the top of the uh uh reserve is gas and uh then um in |
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| 21:02 | lower part is uh brine or maybe . So, uh uh so |
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| 21:08 | in that case, the uh litho is plausibly the same above and |
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| 21:14 | but that gas cap, the thickness the gas cap was simply established by |
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| 21:19 | much gas got into the formation, ? Who says that the formation should |
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| 21:23 | be filled up with gap? So it's only half filled up a |
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| 21:27 | In that case, it's gonna be flat interface running right across the middle |
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| 21:31 | or somewhere in the middle of the and it's gonna be flat because gravity |
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| 21:36 | gonna make it flat. And uh of the uh uh the liquid below |
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| 21:42 | interface is gonna be uh denser than gas. So that's an easy example |
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| 21:47 | uh to imagine where uh uh uh real parts are uh small. But |
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| 21:55 | so the imaginary part is not We could say that, oh I |
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| 22:00 | , this is um this is for shale. But you could imagine also |
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| 22:04 | a gas for the uh uh the on one side has a Q or |
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| 22:09 | that's an ordinary Q and the Q uh on the other side uh is |
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| 22:14 | low because of the gap. Then put those numbers into here and you |
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| 22:19 | a pretty big um uh reflection of . It's a, it's a 4.5% |
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| 22:26 | so 4.5% is a lot less than . But, you know, normally |
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| 22:30 | deal with reflection coefficients with which are lot less than one. So normally |
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| 22:36 | a strong reflection would be a number 10% or, or, or |
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| 22:41 | Uh Those numbers are uh typical and here, we have 4.5% in the |
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| 22:48 | uh order of magnitude. But it's this eye here, which means that |
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| 22:52 | reflected wavelength comes back up is gonna phase shifted. And you can see |
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| 22:59 | in your data, if you see uh uh reflections from uh interfaces above |
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| 23:05 | horizon have uh uh are all uh pretty much the same shape of the |
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| 23:10 | . And then you suddenly see at , at this uh target horizon, |
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| 23:15 | see one that looks very different, you say ahuh that could be due |
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| 23:20 | this kind of effect, we don't assume this is zero. Just assume |
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| 23:25 | uh less than this. If this is comparable to this, then we're |
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| 23:29 | get a, a phase shift. example, let me just if |
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| 23:32 | if, if this imaginary part is the same magnitude as the real |
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| 23:37 | then you get a 45 degree phase . And so that makes a different |
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| 23:42 | coming out than coming in coming And furthermore, different from other nearby |
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| 23:49 | which don't have this large jumping tube . So, um oh to |
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| 23:55 | that's always been a very interesting possibility a new way to find gas. |
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| 24:02 | , and the reason it's so interesting we as, as a result of |
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| 24:09 | , we can call it AAA cue . We didn't lose any high frequencies |
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| 24:17 | the reflected wave never got down into lower medium with high um uh high |
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| 24:25 | , I mean, the lower meeting low Q. So it did, |
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| 24:28 | didn't have any extraordinary attenuation, but simply change, change its shape because |
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| 24:36 | this effect. OK. So let now go to uh OK. So |
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| 24:50 | is Carlos. So uh so he says uh is QP and Q |
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| 24:57 | incorporated in ci processing to compensate for attenuation. Uh So, uh I |
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| 25:04 | to tell you that the uh the to that is not very satisfactory. |
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| 25:09 | uh And the reason is because it's so easy to estimate what values to |
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| 25:13 | , to put in. So I would have to say that that |
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| 25:17 | current processing uses a lot of informal to uh deal with uh uh uh |
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| 25:26 | attenuation. Uh We're not doing and nobody argues that we're doing it |
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| 25:33 | . But what people argue, maybe uh uh it's good enough. |
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| 25:39 | um obviously, the most important thing do is to uh uh uh we |
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| 25:49 | gain correction. So you're accustomed to at uh your workstation and seeing maybe |
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| 25:56 | second or maybe several seconds of reflection there. And they're all uh um |
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| 26:02 | or less the same. But you that those long reflection times correspond to |
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| 26:08 | uh long distances of travel and those when they were coming into the |
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| 26:14 | they must have been a lot weaker the uh waves uh uh at the |
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| 26:20 | of your work station screen. And we correct for that by what we |
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| 26:24 | it, automatic gain correction. And we just uh uh uh we just |
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| 26:29 | boost up those long arrival times without the uh the frequency content at |
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| 26:36 | We just boost the amplitude, boost amplitude. Of course, we boost |
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| 26:41 | noise as well. As we boost signal. So we have to be |
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| 26:44 | little bit careful how we do But uh uh uh that's normally done |
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| 26:50 | in the process. So you don't who did that. Uh What, |
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| 26:54 | they were thinking was before it ever to you. There's been some automatic |
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| 26:58 | control using uh you know, details that um of that calculation which you |
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| 27:05 | not know about, but that's why can't ever uh uh believe and take |
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| 27:12 | the amplitudes at different uh well, at, at different times in your |
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| 27:21 | , which are uh separated a If they're only separated by half a |
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| 27:26 | , maybe it's not a big But if they're separated by two or |
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| 27:29 | seconds, then you know that the reflection got boosted up a lot and |
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| 27:34 | doesn't show on your screen, you , that's so you can see it |
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| 27:37 | that boosting hadn't been done, you not see it well with your |
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| 27:42 | So that's how we do it. uh there's an example of correcting for |
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| 27:47 | without correcting for frequency. And if know that that's true because uh uh |
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| 27:56 | can see with your eyeball that the arriving reflections have lower frequency content. |
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| 28:03 | whatever has been done with frequencies wasn't . Exactly. And of course, |
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| 28:12 | reason is it's, it's hard to and uh and uh it's hard to |
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| 28:18 | the cues. And then once you've the queue, there's a lot of |
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| 28:22 | and so on. So you uh we don't tax free. So that's |
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| 28:29 | remembering when you're doing a bo The question is from uh Rosa, she |
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| 28:44 | in the attenuation lecture S slide 13 14, why is the sign of |
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| 28:49 | imaginary part of m different for the elastic properties now? So let us |
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| 28:55 | at that. I have that slide . OK. So let's just go |
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| 29:05 | here. I forgot uh Rosada. , what uh number were you looking |
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| 29:21 | this? The slide 20. Um me check in slides 13 and |
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| 29:35 | That's what I'm gonna um go back . Uh Yes, I think it |
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| 29:45 | here. This is the one Um mhm Now, what is your |
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| 29:53 | ? Uh I uh I think it's the next slide. It says that |
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| 29:58 | new and for ps that it's similar these are positive but for VP it's |
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| 30:10 | . No sponsor. Oh So positive . And then it turned negative here |
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| 30:18 | this is for the inverse, this for the slowness. Oh The |
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| 30:27 | I wouldn't do that to you. these are not the Yeah. So |
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| 30:34 | I had in the case of the , but in the case of |
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| 30:37 | vs, it's the inverse of the here. And also in both cases |
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| 30:42 | the inverse of Q and I thought become the became the inverse because we |
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| 30:48 | the velocity as Well, you uh thank you for that. Uh |
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| 30:55 | What I did to you, there um uh a typo uh you know |
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| 31:01 | I, you know, obviously did I copied and pasted. And so |
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| 31:05 | should be the real part of vs the positive power, positive one are |
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| 31:11 | like this. So, uh uh you for that. You uh uh |
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| 31:17 | , you found a typographical error Ok, thanks. Ok. |
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| 31:24 | So I, I will uh correct and, and you should correct that |
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| 31:28 | your notes. I will correct it re and repost that uh file |
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| 31:37 | Ok. So that's good. So uh we finished. Oh, |
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| 31:51 | and I also had another question. . Get a free question because you |
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| 31:56 | so good on that one. You a free question, huh? |
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| 32:02 | I send you the, the question well in the email. OK. |
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| 32:07 | let, let me find that and it. Oh, yeah. Another |
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| 32:23 | . The strength of the squirt mechanism attenuation depends on the fluid and the |
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| 32:28 | b. What about the attenuation caused crustal heat? Hm, attenuation caused |
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| 32:35 | the crustal heat? What are the values on top of the crust compared |
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| 32:42 | kites and sediments or gas? I'm sure what you mean by attenuation caused |
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| 32:48 | the crustal heat. Tell me about . Uh Well, I've seen in |
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| 32:54 | images where the, where the crust very shallow, the sediments on top |
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| 33:04 | attenuated compared to the ones that are . And there's a more, |
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| 33:10 | more lithology underneath that they, where the cross is more buried. |
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| 33:17 | . So, uh yeah. uh of course, rocks get hotter |
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| 33:22 | we go down. Uh And that's you're talking about. The crustal heat |
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| 33:28 | the way. So, here's a uh uh oh, here's a puzzle |
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| 33:33 | you before I answer this question that ask. Let me pose to you |
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| 33:37 | question. So we have the the earth is hot. Everybody knows |
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| 33:41 | you got your volcanoes and everybody knows interior is hot and you know that |
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| 33:47 | heat flows from hot to cold. the heat is flowing out of the |
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| 33:51 | over the logic time um uh up the top and then uh radiated away |
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| 33:57 | outer space. OK. So that's obvious. Now, how many of |
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| 34:03 | have uh been out in the field gone into a cave whenever you go |
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| 34:10 | a cave, the cave is right? I think everybody here has |
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| 34:17 | inside of a cave, maybe a cave, maybe a small cave |
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| 34:20 | whatever cave you go into, it's strong. That means that we have |
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| 34:31 | full layer between the hot interior and outer space and, and the upper |
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| 34:40 | . So why doesn't the um heat from the interior? Why doesn't it |
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| 34:45 | up the um the cave? So cave is um uh almost as |
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| 34:53 | uh, you know, almost as as, as the interior. How |
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| 34:57 | we have a temperature minimum in the subsurface of the earth? Have you |
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| 35:03 | thought about that? Uh, I think it's related to the heat |
|
| 35:09 | the sun, right? Because, , very good. So the sun |
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| 35:14 | up the surface from above. and, uh, uh, |
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| 35:19 | so then, uh, if you're the desert, uh, uh |
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| 35:23 | and uh, the sun is beating , it's hot. But if you're |
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| 35:27 | to a cave, you go in cave and it's, uh, cool |
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| 35:32 | the heat from the sun has um, heated up the cave. |
|
| 35:37 | , uh, the heat from uh, the interior of the earth |
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| 35:40 | heated up the cave either. It out that this is a trick question |
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| 35:45 | , uh, it, it happens the, um, uh, the |
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| 35:49 | inside the cave is equal to, , uh, the, the, |
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| 35:55 | just a, a little bit warmer a little bit warmer than the |
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| 36:01 | And, uh, uh, that, that contradicts your, |
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| 36:06 | what I just said, you're standing the desert and getting heated from |
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| 36:10 | but at night time, the sun not there and in the wintertime it's |
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| 36:15 | . And so the, the, temperature in the cave is equal to |
|
| 36:19 | , uh, the, the yearly day and night of the, of |
|
| 36:25 | temperature at the top. And in , it's just a little bit warmer |
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| 36:29 | that because the heat is continually flowing of the interior through the cave to |
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| 36:35 | surface and then to outer space. there's no violation of the uh second |
|
| 36:42 | of thermos. OK. Now, to your question, what about the |
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| 36:46 | caused by the crustal heat also, makes you think that uh the crustal |
|
| 36:52 | was causing that attenuation that you saw the slides that you saw, you |
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| 37:00 | that it was uh uh some kind attenuation deeper than the shall or |
|
| 37:05 | But uh did uh uh did the who is uh presenting this material? |
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| 37:11 | he say anything about what caused the ? No, no, I, |
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| 37:17 | just saw that and I suspected that was because of the heat. |
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| 37:22 | I think I've read in some, some places but not deep. I |
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| 37:27 | , comparing the same section in portions the basement is shallower, very shallow |
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| 37:35 | compared to where it's deeper, it's of the, the sediments just above |
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| 37:43 | on top of that shallow cross or shallow basement are more attenuated. You |
|
| 37:51 | , I'm not familiar with this uh . Um Could you uh uh is |
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| 37:59 | written down somewhere? Is this uh in, in a paper somewhere? |
|
| 38:05 | . Uh Yeah. Uh So it's for me to uh so, uh |
|
| 38:09 | know, um if the attenuation was , more generation means lower Q just |
|
| 38:18 | the basalt than elsewhere in the uh . Uh surely that would have attracted |
|
| 38:24 | attention of the expert and he would said something about that. You don't |
|
| 38:29 | what he said. No, nobody said anything about it. Yeah. |
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| 38:35 | , if you come across that slide , I'd be happy to, |
|
| 38:38 | if you send it to me and me think about that. I |
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| 38:42 | I can imagine that there might be , not for that, but it |
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| 38:46 | be complicated because, you know, subsurface is complicated and I, I'd |
|
| 38:52 | to think about uh first, I think in my mind if you showed |
|
| 38:57 | that slide, I would think I would say, is that really |
|
| 39:00 | ? How did, how did he about this? What makes him think |
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| 39:04 | the attenuation is high above the He better give a good explanation. |
|
| 39:09 | ought to tell me how he came that conclusion. And then once he |
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| 39:14 | me that's uh uh really the then I'm prepared to think of. |
|
| 39:19 | how could this happen? And so just think here uh situation where the |
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| 39:26 | sediments are normal except that they're Well, so maybe the fluids are |
|
| 39:35 | , you know, maybe um oh , I would have to be motivated |
|
| 39:42 | some data to, to think about uh what to think about explaining this |
|
| 39:51 | in a, in a complicated scenario that. So I'm gonna beg off |
|
| 39:55 | tell you uh send me uh uh slide that you're remembering. OK. |
|
| 40:02 | your next question is what are the values on top of the crust compared |
|
| 40:07 | Q values in sediments of gas? , that's a really good question. |
|
| 40:11 | , at the top of the let's uh think about the, the |
|
| 40:14 | 100 m. Sorry, it's mostly it's dirt. And uh we |
|
| 40:21 | uh figure is always almost always less consolidated than in the deeper segment |
|
| 40:29 | So it's gonna have a larger value and it's gonna have less cementation. |
|
| 40:35 | so uh uh uh furthermore, at uppermost part of the earth, it |
|
| 40:41 | be partially saturated with air, Why not? And so all of |
|
| 40:47 | things are going to increase the attenuation to the deeper cost. So we |
|
| 40:53 | be happy that uh uh we get seismic energy at all through the upper |
|
| 41:00 | because uh uh the upper, the worlds layers are really highly attenuated. |
|
| 41:05 | And uh luckily, uh our uh don't have to execute very many wavelengths |
|
| 41:12 | this highly attenuated, highly attenuated material um they would never make it to |
|
| 41:20 | receiver. Now, the sediment uh uh the rest of your question asked |
|
| 41:30 | two values in sediments or gas. , all those are also high, |
|
| 41:34 | high generation two numbers like five or less are you can expect in a |
|
| 41:44 | saturated sediment even if, even if well consolidated. Oh And I'll tell |
|
| 41:50 | this Yeah, this is a good to bring this up. Why do |
|
| 41:55 | explore with uh uh for oil and using c why don't we use some |
|
| 42:03 | thing? Why don't we use, example, a radar? So, |
|
| 42:09 | anybody have any ideas, why we explore for oil and gas? The |
|
| 42:15 | ? So Utah, you have an . Say it again. That's |
|
| 42:24 | So what Utah says is the seismic can travel deeper than the radar |
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| 42:28 | And why is that? Because the wave has attenuated very high, |
|
| 42:34 | it uh it's got a Q value than one the radar. OK. |
|
| 42:41 | let me then uh pursue you on uh uh Utah radar uh has high |
|
| 42:48 | to it. And so why don't overcome this objection by doing low frequency |
|
| 42:55 | exploration? So have very low And I can tell you that the |
|
| 43:01 | who do this for a living, look at frequencies of about uh one |
|
| 43:06 | or less lot, a lot lower than radar. And so such low |
|
| 43:13 | do penetrate a lot deeper than the . Uh They can go a couple |
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| 43:20 | kilometers down. And so uh uh yeah, the uh the attenuation is |
|
| 43:33 | . Uh uh The, the ainu by the time the signal goes down |
|
| 43:38 | uh uh a kilometer in back, has uh decreased in amplitude by over |
|
| 43:43 | million uh over a factor over you can barely detect it, but |
|
| 43:49 | have good detectors, they can detect . And so, um I can |
|
| 43:56 | you that that oh so electromagnetic wave inside the earth in a way is |
|
| 44:05 | similar to psychic waves with two One, the attenuation is very high |
|
| 44:12 | electromagnetic wave and it was very low of Q. And so they overcome |
|
| 44:17 | with uh uh low frequency. And and you know, from our uh |
|
| 44:23 | last time that since, since the is high, the dispersion is also |
|
| 44:31 | for electromagnetic waves. So different frequencies uh uh electromagnetic waves uh travel uh |
|
| 44:40 | different velocities depending uh on the And so that means that the uh |
|
| 44:47 | analysis of electromagnetic wave is very different the analysis for. Uh so for |
|
| 44:52 | sound wave, even though they're pretty the same, the only differ in |
|
| 44:57 | electromagnetic waves have higher generation and highest but if you account for that, |
|
| 45:04 | you can do a pretty good job , of e exploring with electromagnetic |
|
| 45:09 | And we had, I had a here uh a few years ago at |
|
| 45:13 | University of Houston who basically taught us to do that. And uh uh |
|
| 45:19 | ideas have been slow to uh penetrate the community of uh uh electromagnetic |
|
| 45:28 | But I think they're right. And think that uh uh eventually his ideas |
|
| 45:33 | become mainstream and uh and uh his will be famous. It hasn't happened |
|
| 45:42 | . So that's it. So, , the same way. But it's |
|
| 46:07 | uh um yeah, in, in same way. Yeah. Yup. |
|
| 46:16 | . So thanks for these questions, . And now let us go to |
|
| 46:30 | let us go to the anti oxy that dancing coldest we lost. |
|
| 47:04 | Uh So, uh uh I will uh uh sir, uh Utah was |
|
| 47:08 | happy with my, was, was satisfied with my previous explanation. And |
|
| 47:15 | goes further and he says in um waves, uh the the waves are |
|
| 47:21 | the particles that makes friction and the makes heat. And so that makes |
|
| 47:25 | heat. Well, so in the place, that's not a good uh |
|
| 47:29 | uh a good description of the way waves make heat. Uh When we |
|
| 47:36 | friction, it's uh uh uh sort meaning moving stuff like on a, |
|
| 47:42 | a table top that that's friction. uh So uh we don't have that |
|
| 47:52 | of friction in uh sound waves at amplitude, right? We, |
|
| 47:59 | we're not scraping atoms past other atoms all as uh in a sound wave |
|
| 48:06 | uh were uh sort of maybe uh uh putting a, an infinite testable |
|
| 48:12 | on the, on the grains and fluids in the rock. And uh |
|
| 48:18 | that is naturally making. Uh uh basically, I think the, the |
|
| 48:22 | to do the way to say it because of the second law. The |
|
| 48:26 | law of thermodynamics says that whenever you a change, uh uh it's gonna |
|
| 48:32 | increase the entropy uh meaning uh uh of uh of energy to heat. |
|
| 48:40 | so, when you think about it way, basically, it's the same |
|
| 48:43 | electro magnetic waves in electromagnetic waves, moving electrons around in uh inside the |
|
| 48:50 | the cloud of electrons surrounding each nucleus the rock. And uh uh |
|
| 48:59 | the second law of thermodynamics is going apply that we, we are going |
|
| 49:03 | uh convert some of that energy to the entropy. And oh no, |
|
| 49:13 | we assume that let's back up early in this course, we assumed that |
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| 49:19 | sound propagation was adiabatic. That, , that none of the oh energy |
|
| 49:29 | lost the heat got outside of the . So we're here, we're not |
|
| 49:35 | about using heat outside the sample. uh the heat is still inside the |
|
| 49:41 | , but it's changing from uh mechanical to random uh vibration of the atoms |
|
| 49:48 | heat uh in, in every little tiny piece of the rock. And |
|
| 49:54 | think you can say the same thing electromagnetic uh uh uh exhortation uh uh |
|
| 50:02 | is, is uh deforming the cloud electrons around every nucleus in the |
|
| 50:10 | And that also is gonna lead to local increase of entropy according to the |
|
| 50:16 | law. OK. What's up? Now forgot where we left off. |
|
| 50:28 | we begin the talk about anisotropy last we met. So we didn't. |
|
| 50:35 | so we did a, a few . Uh uh So let me see |
|
| 51:01 | . OK. So, uh did see this slide before? OK. |
|
| 51:07 | , uh uh let me uh go and we saw these slides, I |
|
| 51:19 | it's maybe a good idea to uh repeat this. OK. So, |
|
| 51:27 | start here. And um so the slide says that uh Helmholtz, let |
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| 51:37 | get it. Um uh Let me them. OK. So Helm's uh |
|
| 51:43 | uh theorem is still valid. It means that uh for these anisotropic |
|
| 51:48 | you can separate uh into a curl part and uh uh divergence free |
|
| 51:54 | But each of those is a mixture PNS. So you can't separate with |
|
| 51:59 | uh PNS uh uh in that But let's, let's uh um uh |
|
| 52:08 | the Helmholtz thing and uh uh look the scalar potential which uh is uh |
|
| 52:14 | curl free and it obeys this expression . So, uh uh uh uh |
|
| 52:25 | see what it's got here, it's lots of derivatives. See we got |
|
| 52:28 | derivatives with respect to the space and to space. And over here, |
|
| 52:33 | the left side, we have one with respect to uh space and two |
|
| 52:40 | respect to time. That's not the equation, isn't it? And, |
|
| 52:44 | we did use this expression and go uh um in your earlier notes, |
|
| 52:49 | see where we use this expression to uh a separate P and wave. |
|
| 52:57 | so we're gonna have to uh uh more clever. Yeah. So first |
|
| 53:03 | we do is we look at uh the stiffness a tensor. And remember |
|
| 53:10 | is not a tensor. This is matrix which represents the uh tensor, |
|
| 53:15 | has four indices. This has only indices. They count from 1 to |
|
| 53:20 | . And uh so here they And uh uh a as it turned |
|
| 53:25 | , the simplest case that we are with in geophysics ha has this |
|
| 53:31 | we call that polar an isotropic. it's for uh un fractured shale and |
|
| 53:38 | bedded sequences where all the dias directions the same, but they're different from |
|
| 53:43 | vertical direction. And so when you a stiffness element, a stiffness matrix |
|
| 53:49 | this, and we're only showing the triangle and lots of zeros and |
|
| 53:57 | Uh Yes. So uh this uh is sort of pointed in the wrong |
|
| 54:03 | . Uh uh This, th this , it should be pointing here and |
|
| 54:09 | point should be pointing here. I know exactly how uh it got messed |
|
| 54:15 | , but uh the 11 direction and 22 directions uh uh are the same |
|
| 54:21 | from the three directions. So we uh uh have this pattern. This |
|
| 54:30 | the same as this different from And down here in the shears, |
|
| 54:34 | have the same thing. This one different from this, the same as |
|
| 54:38 | and different from this, that makes different parameters. Here's 1/5 1. |
|
| 54:43 | then this one up here is calculated ones already developed five independent criminals. |
|
| 54:50 | one is gonna govern the vertebral P velocity. And you know that because |
|
| 54:57 | uh the, the first three uh we got two subscripts, |
|
| 55:01 | 1st subscript means that uh it's a unit area and the unit area is |
|
| 55:08 | in the three direction means it's uh the, the unit area is, |
|
| 55:13 | horizontal with its normal vector pointing in three directions. And the way was |
|
| 55:19 | going in three directions uh in the section that governs the uh P, |
|
| 55:25 | P wave vertical Vasi this governs the V these two go horizontal here. |
|
| 55:34 | got two different sheer moduli uh And one a a the main uh uh |
|
| 55:39 | also. And uh what we uh from this on the last time was |
|
| 55:46 | we can't uh separate the uh the modes and the sheer modes from each |
|
| 55:54 | using these two curl free parts and divers free parts uh uh Helm doesn't |
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| 56:02 | us, heals is still correct, he doesn't help us. So what |
|
| 56:08 | do is we go back to this full tensor equation of motion. |
|
| 56:13 | we have this placement here. This almost like the wave equation, doesn't |
|
| 56:18 | , it's got uh two derivatives with to space two days, we respect |
|
| 56:24 | time of the displacement here and here our, our stiffness, a stiffness |
|
| 56:31 | with four indices. And uh we how to deal with this. |
|
| 56:37 | uh uh we've done this so many . We guess the answer and then |
|
| 56:40 | verify the guest. So we're gonna that the displacement uh vector is given |
|
| 56:48 | a con uh uh uh an amplitude in front. It's a vector. |
|
| 56:53 | you got a subscript I oh And see this is not very clever of |
|
| 57:00 | . I see this is not clever me because here I've used ISF as |
|
| 57:05 | hope effective component here. I used as a over to minus one. |
|
| 57:13 | this means that I there is another where I need to go back and |
|
| 57:20 | the uh materials so that you don't confused when you read about this |
|
| 57:26 | But for now you uh you'll recognize it, when we have a subscript |
|
| 57:30 | a vector component. And so the is a vector pointed in the uh |
|
| 57:36 | the I direction. And then uh have AAA re SOTO factor or phase |
|
| 57:45 | of either the IO mega plus or K dot X. So since we |
|
| 57:50 | K dot X, this can be way of going in any direction. |
|
| 57:54 | can we say about the wave vector ? Well, if it has a |
|
| 58:02 | length is given by this and if has a length uh uh related to |
|
| 58:07 | frequency, this gas is gonna solve equation. So that's what we, |
|
| 58:12 | put this into the equation of And uh now you see no more |
|
| 58:17 | because we've got make, we've executed derivatives and we got an omega |
|
| 58:25 | we got a minus omega squared on left side and we got a minus |
|
| 58:29 | on the right side and I'm not the minuses here. I don't need |
|
| 58:34 | anymore. OK? And you can that there's um a subscript I here |
|
| 58:41 | three paired subscripts JM and net. we, we're gonna sum with |
|
| 58:48 | And so that means there's gonna be term on the right side of this |
|
| 58:54 | when you spell out all the sums equations like that. So now we're |
|
| 59:01 | divide by the square of the length the wave vector both sides. And |
|
| 59:09 | uh this omega square to a case that is gonna be equal to the |
|
| 59:16 | velocity square, we know that and are gonna be what we call direction |
|
| 59:21 | that it gives the and the case of the wave factor in a ratio |
|
| 59:30 | the amplitude uh the magnitude of So that's AAA what we call a |
|
| 59:36 | cosine. And so when we put change of notation um uh into the |
|
| 59:44 | expression, we get this expression which in matrix notation looks like |
|
| 59:53 | And this is called an eigenvalue This is a famous equation in mathematical |
|
| 60:00 | . Turns turns up in lots of in mathematical physics. And what, |
|
| 60:05 | is this um operator L here? , here is the definition of uh |
|
| 60:11 | the operator here, you know, got that just from here. And |
|
| 60:16 | mm so uh the eye uh and uh the L, the L operator |
|
| 60:24 | two indices and that comes from or over uh uh J and N here |
|
| 60:33 | over J and N. And uh , more notation here, this identity |
|
| 60:40 | is just like the Chronicler delta it's a zeros off Dagle and one's |
|
| 60:47 | the Nagle. So this is a way to write that expression. And |
|
| 60:54 | we know how to solve this Physicists have learned, you know, |
|
| 61:00 | century ago how to solve an eigenvalue this equation. And uh uh immediately |
|
| 61:07 | and what are the unknowns here? The unknowns are the uh uh the |
|
| 61:13 | the three components of displacement. And immediately you can see uh a solution |
|
| 61:21 | this solution to this is U equal because it, if uh the displacement |
|
| 61:27 | zero automatically um salt, but we want a nontrivial solution. And |
|
| 61:36 | uh it's been known for, you , 100 years that uh you only |
|
| 61:41 | a nontrivial solution when the determinant of matrix is zero, not the L |
|
| 61:46 | , but the L minus V squared . So um do people know |
|
| 61:54 | what we mean by a determinant, . Do you, do you know |
|
| 62:01 | it? What you mean by a ? Yeah. So, um, |
|
| 62:07 | and I'm guessing that Carlos doesn't know . He, he's keeping a discreet |
|
| 62:13 | over there because Carlos is a geologist uh uh this is really, |
|
| 62:19 | this is really a, a AAA idea. And so, since it's |
|
| 62:25 | mathematical idea, let us uh go the support slide. So what I'm |
|
| 62:30 | do is stop sharing this and I'm to bring up. Um Thank |
|
| 62:41 | What I wanna do it. I think what I wanna do because |
|
| 62:53 | wanna go to uh the glossary. Well, I'm, I'm in the |
|
| 63:05 | file. You, you can't see but OK. Hm. Sorry about |
|
| 63:36 | looking for where I talk about the . OK. So I just, |
|
| 63:58 | uh I thought I just turn that . I know who's trying to call |
|
| 64:14 | there, but I don't want to to him here. So I'm gonna |
|
| 64:17 | close this and I'm gonna go to 101. Yes. OK. So |
|
| 65:22 | gonna show you this slide right Cool. Let's start off with this |
|
| 65:54 | in the spring. OK. Um you see this, this is um |
|
| 66:16 | M 60 from math one or And so uh we're, we're dropping |
|
| 66:25 | the middle of the discussion about tensors and here's an example of a |
|
| 66:31 | A three by three tensor. And it says is you can rotate, |
|
| 66:35 | can always find a principal coordinate system uh uh the uh stress tensor looks |
|
| 66:42 | this simple form like this. Remember all of these, these uh indices |
|
| 66:47 | , they refer to directions in the system. The coordinate system is something |
|
| 66:51 | decide on. And so maybe the doesn't know or care what you're |
|
| 66:56 | It has its own cord system, matter what it is, you can |
|
| 67:00 | find a cord system where the, tensor reduces to this diagonal form. |
|
| 67:07 | that's called the principal current system. these are called the principal values of |
|
| 67:13 | , the 10. Now, next , I said principal values, the |
|
| 67:21 | word is eigenvalues. And so we use that word eigenvalues even today, |
|
| 67:28 | those of us who are not And then it says the new directions |
|
| 67:33 | after we rotate from this uh into , the, the directions of the |
|
| 67:38 | system which leads to this diagonal form called the eigenvectors. OK. So |
|
| 67:48 | , what can we say um uh this uh in what we wanna do |
|
| 67:56 | say, what independent of the court ? What can you say? |
|
| 68:01 | uh uh look at this is the one says that if you sum the |
|
| 68:06 | eigenvalues, that's the same, that's uh uh that's invariant to the uh |
|
| 68:12 | court system. Let me back up . So the sum of these three |
|
| 68:17 | is exactly the same as the sum these three terms. So you don't |
|
| 68:21 | to find uh uh that magic court and then sum them up, you |
|
| 68:26 | sum these things up uh from the data because we know that uh um |
|
| 68:35 | , the, it's called the That's some of the, of the |
|
| 68:39 | terms. That's the same independent of cord system. And then here is |
|
| 68:45 | um quantity which is independent. But is still, I used, I'm |
|
| 68:50 | go directly to this third one. called the de the, the |
|
| 68:55 | And so what is this determined? the uh mhm trying to the three |
|
| 69:08 | back up here. So if you these three together, that uh is |
|
| 69:14 | number all the determinant. And uh uh so that's not the same as |
|
| 69:22 | these three together because these are not uh these are not the eigenvalues. |
|
| 69:28 | what is the de determine? Uh can you calculate the determinant from this |
|
| 69:36 | finding this first? OK. Without the principle value, the deterrent is |
|
| 69:42 | by this. So in two it's uh uh it's simple. It's |
|
| 69:47 | uh a 11 times a 22 minus product was mixed. These are |
|
| 69:53 | the off diagonal. If we had two di uh uh uh two by |
|
| 70:01 | matrix, then the determinant is, defined in this way in three |
|
| 70:07 | Uh It's more complicated. So let's uh make three column vectors, we |
|
| 70:14 | them a one vector, a two and a three vector. And then |
|
| 70:19 | determinant is, is defined by this product A one dot A two cross |
|
| 70:26 | three. So uh you can see uh number one, it's complicated and |
|
| 70:34 | um two, it's uh uh uh , the determine is gonna depend for |
|
| 70:40 | 3d uh tensor. It's gonna uh uh depend on uh it's gonna have |
|
| 70:46 | dimensions of uh of the cube, cube of, of the individual |
|
| 70:53 | And uh this is a CIC OK. So that's all I wanna |
|
| 71:00 | about the determinant. You can go to math 101 and uh uh uh |
|
| 71:06 | that again at your leisure. So now I'm gonna stop sharing and I'm |
|
| 71:12 | to oops. So c me up to um I'm sorry, what, |
|
| 71:53 | . This is where I'm gonna come , this is where we left |
|
| 71:57 | And so share that again with It's great. Very true. |
|
| 72:41 | OK. So this is where we off when we first introduced the term |
|
| 72:46 | determinant. So, uh when, we made the excursion back to math |
|
| 72:54 | , we found that the determinant is property of any matrix uh um uh |
|
| 73:00 | tensor. And we can take this the tensor inside this, this double |
|
| 73:08 | oh double bars here. And here means we're gonna make the determinant of |
|
| 73:15 | tensor and I gave a formula for , but it was complicated. And |
|
| 73:21 | what it says. In fact, here, it says it's a |
|
| 73:28 | it's a cubic equation in the elements this um uh of this 10, |
|
| 73:37 | right here. And since it's a equation, there's gonna be three solutions |
|
| 73:43 | uh uh uh the solutions, uh is the unknown or the unknown is |
|
| 73:48 | squared. It's not V itself, the square of V. So we |
|
| 73:52 | three different uh three different solutions for squared. And looking ahead, you |
|
| 73:57 | think now, OK, this is be V squared for PV squared for |
|
| 74:02 | and then another V squared for S be fast and slow values for S |
|
| 74:07 | , that's what we're gonna find, there yet. So those three solutions |
|
| 74:11 | called eigenvalues and the eigenvectors are gonna the polarization of those three ways. |
|
| 74:20 | that's what it says here. In case, we got velocity squared and |
|
| 74:25 | corresponding polarization directors. So uh let uh pause here for a moment and |
|
| 74:32 | well, thank you very much and a little quiz. So it says |
|
| 74:39 | um uh I'm gonna start with So pay attention here. It says |
|
| 74:45 | an isotropic elasticity is more complicated than elasticity because A B uh C or |
|
| 74:53 | and notice the last one is none the above. And so, um |
|
| 74:58 | let me ask you about um uh is a, a legitimate answer to |
|
| 75:05 | question. You think? Me? do you think DD for David? |
|
| 75:29 | ? Well, OK. So uh let's do this one at a |
|
| 75:33 | Uh uh But you, you think not a, yeah, we, |
|
| 75:36 | don't care about uh uh uh hetero or, or not uh on the |
|
| 75:42 | of a hand sample, we can uh anisotropic rocks, of course. |
|
| 75:48 | also you could have anisotropic crystals we it's on a slider though. So |
|
| 75:53 | not a matter of heterogeneity. Um And it's uh uh but uh do |
|
| 76:00 | think it's b um the C uh obviously not? Right. Uh I'm |
|
| 76:06 | go uh with, I'm gonna go B, I'm gonna uh uh agree |
|
| 76:10 | you on that. OK. Next , it comes to you Carlos. |
|
| 76:16 | need to use different mathematics for the case because ABC or none of the |
|
| 76:23 | . So, uh uh how about Carlos? Uh uh Do you agree |
|
| 76:27 | a Carlo? Yeah. Yeah, , I, yeah, I, |
|
| 76:44 | am not sure is valued for isotropic so that I wouldn't agree. I |
|
| 76:49 | that. Well, uh actually we about this a lot. Uh And |
|
| 76:54 | Helmholz was a mathematician and he didn't or care about um elasticity. Uh |
|
| 77:02 | isotropy is, but he didn't know of that. He just did his |
|
| 77:07 | and it's valid for uh any uh factor which varies as you know, |
|
| 77:15 | as function of position like displacement. uh And uh it doesn't uh uh |
|
| 77:22 | , he doesn't know or care whether rocks are isotropic or what. |
|
| 77:27 | uh this answer is not correct. Yeah. OK. So, um |
|
| 77:36 | , um turning to you, uh you got BC or D, |
|
| 77:47 | . 1st, 1st, let me you about C I is C |
|
| 77:52 | No, no, that's, that's not correct that the anti equation of |
|
| 77:57 | we showed you uh before uh it's , it is linear. It has |
|
| 78:01 | uh the unknown UO only appears to first B. So, so C |
|
| 78:07 | wrong um Now, but uh how B it says the scalar potential is |
|
| 78:13 | a solution to the equation of Yeah. Uh uh I'm gonna say |
|
| 78:36 | this is, I would say this a big of an ambiguous question. |
|
| 78:40 | , what do you mean by the of motion of? We did show |
|
| 78:44 | equation there as it had the uh potential in there. But was that |
|
| 78:48 | equation of motion? You know, think we meant we described that in |
|
| 78:53 | um uh in different ways. In past, I'm thinking this term is |
|
| 78:57 | bit, this one is a bit ambiguous. So I'm gonna put on |
|
| 79:02 | one And I'm going to uh uh you bra you don't have to answer |
|
| 79:08 | one because it's uh ambiguous. So get the next one which is, |
|
| 79:16 | this true or false. The so-called equation is just a special case of |
|
| 79:22 | simultaneous linear equations in three unknowns which homogeneous since all of the terms contain |
|
| 79:29 | of the unknowns. So there, , yeah, that's true. There's |
|
| 79:35 | in that equation like we had in uh uh uh I in the inhomogeneous |
|
| 79:42 | equation with the source term, there's source term there. Uh uh All |
|
| 79:46 | terms contain one of the unknown. that's true. Yeah, very |
|
| 79:50 | So now let's look at the So this is the uh the uh |
|
| 79:58 | stiffness matrix corresponding to the stiffness tensor the simplest case which we call polar |
|
| 80:05 | . The old fashioned name for this transverse isotropy. Uh I guess it |
|
| 80:10 | vertical, transverse isotropy because you can that it's this vertical direction which is |
|
| 80:17 | and the one direction and two direction equivalent. So what it says here |
|
| 80:22 | this uh uh uh cubic equation for determinant is too complicated and nobody here |
|
| 80:29 | how to solve a cubic equation. in instead of uh uh you |
|
| 80:33 | uh slogging our way through the uh uh let's try to be more |
|
| 80:38 | . And so um uh we know in isotropic seismic sh waves play a |
|
| 80:46 | role and, and they, they , are simple, they're decoupled from |
|
| 80:50 | P and SV waves. So, this as a clue, let's look |
|
| 80:56 | sh waves in this anisotropic case. . So uh we're gonna have, |
|
| 81:02 | gonna look for a solution which propagates the 13 plane, but it has |
|
| 81:07 | a cross line component U two So when we have these six, |
|
| 81:13 | this is gonna be now the, square velocity for the second Eigen value |
|
| 81:20 | here. And it's gonna be multiplying only the U two um component |
|
| 81:27 | of displacement because that's what it says . We're gonna be looking for solutions |
|
| 81:32 | only have a U two component. so, uh we're gonna call this |
|
| 81:38 | eigenvalue vsh square. And in terms the uh uh the uh right side |
|
| 81:46 | the equation, um all we have do is put into this position and |
|
| 81:52 | uh distance matrix two. And then have to perform all these songs. |
|
| 82:00 | ? And also we're gonna ha uh gonna insist that the wave is traveling |
|
| 82:04 | the K two direction. Uh That uh uh uh it's in the 13 |
|
| 82:10 | . So there's no uh wave vector the two direction. So that's gonna |
|
| 82:15 | things. And so, uh uh did we give this uh uh that |
|
| 82:20 | essence, you know, by OK. Now, putting uh we're |
|
| 82:26 | turn out to be a uh uh we do all these sums, it's |
|
| 82:30 | simplify a lot. So, uh , when we assume that K |
|
| 82:39 | we're gonna sim simplify the previous song to only these four terms. And |
|
| 82:57 | , here we go because of the and the stiffness sensor, some of |
|
| 83:02 | get uh um uh eliminated. So left with only a 2121 term and |
|
| 83:07 | 2323 term. So, for uh here we have a 23, |
|
| 83:12 | can't have M equals two. uh uh uh We can't have a |
|
| 83:16 | uh a 21 number here because that element is zero. And so, |
|
| 83:23 | of the zeros uh uh that's gonna it like itself. And so uh |
|
| 83:32 | changing to the two index notation, is gonna be a 66 and this |
|
| 83:37 | gonna be a 44. And will recognize these are um direction cosine? |
|
| 83:43 | so, uh uh the great thing it is because of these simplifications, |
|
| 83:49 | is only um on the right hand , we have only uh the twos |
|
| 83:56 | uh um uh you, you one and you three disappeared, we have |
|
| 84:01 | you two. And so we can by you two out. And so |
|
| 84:06 | get this solution directly divide out the two. And so immediately we find |
|
| 84:13 | the, the square of the uh , of the vsh velocity times the |
|
| 84:21 | is given by C 66 times sine plus C +44 times cosine square. |
|
| 84:28 | uh what are those angles, those are from the uh from the |
|
| 84:34 | OK. So, Um Let me at, at, at this figure |
|
| 84:40 | carefully, remember X two is out the plane in the plane is X |
|
| 84:45 | X three. So K is in plane and so this is the angle |
|
| 84:48 | right here. So uh uh and this is our Eigen Valley, we |
|
| 84:56 | very clever to um uh simplify the problem by recognizing that maybe sh is |
|
| 85:05 | play a special role. And sure it did. And what is the |
|
| 85:10 | for find this eigenvalue? Well, the displacement vector uh in the uh |
|
| 85:15 | the two directions. And we're gonna uh take it it, it's a |
|
| 85:21 | is one. OK. So that us the other two components coupled |
|
| 85:30 | Mm oh I see lots of um of um uh mistakes in the uh |
|
| 85:42 | of these slides. Obviously, this should all be on one line. |
|
| 85:47 | But uh when you look at these equations, it has the form of |
|
| 85:51 | two by two eigenvalue equation which is easier to solve than the three by |
|
| 85:57 | equation because the, the, the the determinant of this two by two |
|
| 86:03 | is uh so much easier. So it is the determinant of that two |
|
| 86:08 | two matrix is uh uh the product the, of the diagonal terms minus |
|
| 86:16 | product of the off diagonal terms. , you know, the, the |
|
| 86:22 | diagonal term is gonna have an L and A V squared times one that's |
|
| 86:30 | times the other diagonal term, which L 33 minus V squared times |
|
| 86:36 | That's this one. And then we to, to get the determinant, |
|
| 86:40 | have to track off the product of uh off diagonal terms. And so |
|
| 86:45 | is a diagonal uh excuse me, is a quadratic equation in the unknown |
|
| 86:50 | square which is um um uh a easier to solve than a cubic |
|
| 86:57 | And what it says here is uh second row of L has tensores |
|
| 87:04 | So the, the three showed up even though it's a two by two |
|
| 87:08 | because uh uh we eliminated um the right. OK. Now, the |
|
| 87:15 | to this are uh easy to work . Everybody knows how to solve a |
|
| 87:20 | equation. And so here are uh squared and VS squared uh uh |
|
| 87:25 | in terms of, you know, from uh the solution and it looks |
|
| 87:32 | easy, doesn't, it, it looks like it's some simple trigonometry |
|
| 87:36 | uh something called D or uh what that down here for? Sh look |
|
| 87:41 | this. Um It's the same simple here. And the only difference between |
|
| 87:49 | two is the minus instead A plus D is. Well, here's |
|
| 87:54 | wow. D is complicated. He complicated. It's got square roots, |
|
| 88:02 | got squares and fourth powers and uh a mess. And so these are |
|
| 88:10 | solutions for VP and VA SV. we still have a solution for |
|
| 88:16 | So uh that's simple. Now, want you to notice that each one |
|
| 88:21 | these expression and of course, we're be mostly interested in this one. |
|
| 88:28 | one of these contains four distance Let's count them 123. And the |
|
| 88:35 | one is inside D where is Here? It is out here. |
|
| 88:42 | , this should be a um a bit disturbing to you here is the |
|
| 88:48 | for VP. And you know that for vertical traveling VP, it's gonna |
|
| 88:54 | on C 33. you know how gonna work. But look right in |
|
| 88:58 | is AC 44, that's supposed to a sheer wave modules. But it's |
|
| 89:06 | here in the VP solution also down by the way. And also by |
|
| 89:12 | way inside here, let's take a here. So here it is inside |
|
| 89:19 | . OK. So let's draw now picture of these three solutions. That's |
|
| 89:23 | they look like. So here is uh the, the symmetry axis and |
|
| 89:28 | symmetry plane and those three waves all down at the same angle theta. |
|
| 89:35 | um the P wave here's the P traveling in the KP direction. |
|
| 89:40 | And here is the um um And can you see this he wave |
|
| 89:50 | not exactly in the longitudinal direction is exactly the displacement is not exactly pointed |
|
| 89:59 | the direction of propagation as you it is for isotropic meeting. So |
|
| 90:05 | can assure you that this little angle is not a, a graphical |
|
| 90:09 | That's um uh that's real and the of that little angle there uh depends |
|
| 90:17 | um uh the amount of anisotropy. course, uh I can tell you |
|
| 90:22 | nobody has ever uh figured out a to make any use out of |
|
| 90:27 | So we're gonna call the, we be calling these quasi P waves, |
|
| 90:31 | we're gonna call them P waves OK. So the next one is |
|
| 90:37 | SV wave, it's going down in same direction, it's following the KSV |
|
| 90:43 | , same direction here. Uh but different velocity. So uh uh it's |
|
| 90:49 | vector and this one is polarized in plane perpendicular to the direction, but |
|
| 90:57 | closely. Does that look to you it's not quite perpendicular? Yeah, |
|
| 91:02 | true in the same way. This not quite large Jual, this one |
|
| 91:07 | not quite perpendicular, but again, has ever figured out how to use |
|
| 91:13 | at all. And then the third is the sh wave. And you |
|
| 91:18 | see right here there is another one those uh font errors that uh mysteriously |
|
| 91:24 | up. That's supposed to be a The same size as this dot Just |
|
| 91:29 | dot Indicating that it's polarized out of plane. So I'm gonna have to |
|
| 91:34 | back and uh and repair this I don't know how it got me |
|
| 91:39 | up. Actually, I do know , uh, last, uh, |
|
| 91:43 | , a few months ago my hard , um, fail. It |
|
| 91:50 | no. And so I had a struggle to recover all my data |
|
| 91:57 | um, uh, download new powerpoint and everything. It's a big |
|
| 92:04 | Uh, no, the reason I you this is to, uh, |
|
| 92:08 | , warn you, you all should up your computer re religiously, |
|
| 92:13 | like every week on a schedule Friday before you go to bed, you |
|
| 92:18 | up your computer to an external uh, actually should, uh, |
|
| 92:23 | , back it up to an external so that if your hard drive fails |
|
| 92:29 | a Saturday morning or on Thursday uh, you won't have too much |
|
| 92:34 | to do to, uh, recover of your previous work. Especially those |
|
| 92:38 | you who have a lot of, , of, uh, uh, |
|
| 92:43 | on your computer which is gonna support supposed graduation from University of Houston. |
|
| 92:49 | , uh, I, if your drive were to fail you t you |
|
| 92:54 | be shit out of luck. So particularly, you back up your stuff |
|
| 93:00 | , with an external drive, you buy it for 50 for 50 bucks |
|
| 93:04 | , at the store, uh, , and back it up so that |
|
| 93:08 | your hard drive on your computer fails mark you're not lost. Otherwise I |
|
| 93:14 | you would be lost. You, lose a year of your life, |
|
| 93:18 | , uh, uh, you wouldn't able to graduate in time, |
|
| 93:22 | uh, you know, and, , maybe lose your scholarship apart. |
|
| 93:27 | would be terrible. You probably wanna your throat. Uh, uh, |
|
| 93:31 | do that drive and invest in in a, an external hard drive |
|
| 93:36 | back up your stuff every week or day. Probably not every hour, |
|
| 93:43 | a week is probably good enough. I learned that the, uh, |
|
| 93:47 | hard way, uh, you we, we get accustomed to technology |
|
| 93:52 | and when it works it's wonderful and it doesn't work, it's terrible and |
|
| 93:57 | broken heart cars are terrible. So was able to replace my hard drive |
|
| 94:02 | only about 50 bucks. It was cheap part. I lost all of |
|
| 94:08 | information of it. Terrible. So, uh, and the reason |
|
| 94:14 | I was struggling because I had been about backing up. So, whatever |
|
| 94:19 | I had were way out of date a lot of work to recover. |
|
| 94:24 | , don't let it happen here. . So, um, uh, |
|
| 94:30 | , your, uh, your turn says anisotropic rocks have ABC D or |
|
| 94:37 | of the above. Uh, or yeah. D all of the |
|
| 94:43 | . Ok. Uh Now, what's mean? It says at least five |
|
| 94:48 | constants. You got it. Why it say at least we showed five |
|
| 94:52 | , in, in the example we that was five. It has at |
|
| 94:58 | F what does that mean? here's the answer we considered only the |
|
| 95:06 | case of anisotropy. Only the simplest fractured shales well over anisotropy that has |
|
| 95:13 | . But uh the real earth is turn out to be more complicated than |
|
| 95:18 | . So it's gonna have more elastic system content. OK. Now, |
|
| 95:22 | you do, do have I, uh uh if it's a fractured |
|
| 95:27 | say with a single set of single set of vertical fractures in an |
|
| 95:32 | un fractured chair that makes it orthorhombic nine constants. Wow. And uh |
|
| 95:40 | now tell me this, look at no point, it says two |
|
| 95:44 | sheer body ways, not one, it doesn't say at least two sheer |
|
| 95:48 | . So if we got nine different constant, how many different waves are |
|
| 95:52 | gonna be having? Still only Because it's three dimensions, right? |
|
| 95:58 | the three comes from three dimensions. uh uh So for these orthorhombic |
|
| 96:06 | we're gonna have three wave types and one is gonna be complicated. |
|
| 96:13 | So um let us turn to the of weak polar anisotropy. So um |
|
| 96:32 | three people in history have understood the implications of the equations that I just |
|
| 96:38 | you five minutes ago. So the guy was a guy named Maurice |
|
| 96:44 | he was a professor in Polo and died 100 years ago. And, |
|
| 96:51 | he was the first uh person to call himself professor of geophysics. And |
|
| 96:58 | , um his research specialty was anisotropy which he understood polar anisotropy and he |
|
| 97:06 | everything. Uh and then he And, um the next guy who |
|
| 97:12 | it up was Klaus Helbig, who still alive, still with us. |
|
| 97:16 | is about 90 now, maybe over . So, um uh full of |
|
| 97:21 | , lives in Germany, mostly retired , but not completely. I hope |
|
| 97:25 | see him this summer. And uh he took it up and then |
|
| 97:31 | uh the uh the cause is smarter most of us. And so he |
|
| 97:36 | things in a way that nobody else understand only clout. And then the |
|
| 97:41 | guy who understood it was my Amaco , Joe Dellinger. You might know |
|
| 97:46 | name Joe is younger than me. uh uh joined uh Amaco when I |
|
| 97:52 | there. Still, he still works BP. He was the smartest guy |
|
| 97:56 | Amaco and now is the smartest guy VP. Uh And uh uh uh |
|
| 98:03 | those three guys are the only ones own, who understood those equations. |
|
| 98:10 | you might know that uh uh that am well known for anisotropy, but |
|
| 98:14 | not on this list. This is these three guys because they understood, |
|
| 98:20 | a lot more than I did. I found was that uh what I |
|
| 98:28 | was the following because these solutions which just showed you are so complicated. |
|
| 98:35 | gonna have to make uh some approximations for man, most of the last |
|
| 98:42 | , the popular approximation was to uh what so-called elliptical an isotopy. We're |
|
| 98:48 | talk, define that later. But not a good assumption because most rocks |
|
| 98:53 | not like that. A better approximation that the anisotropy is weak. Think |
|
| 99:00 | it. We have um we have found an awful lot of oil and |
|
| 99:06 | by assuming the anti start to be zero, assuming that isotropy is good |
|
| 99:13 | . And it was good enough to enormous amounts of oil and gas. |
|
| 99:16 | by the way, it was good to find uh all the features of |
|
| 99:20 | deep interior of the earth that we about, but maybe we can do |
|
| 99:27 | today. And so if you look at the previous expression, you can |
|
| 99:33 | the following combinations, two of them have the dependence of velocity, you |
|
| 99:39 | , was the the square root of modulus divided by density and uh three |
|
| 99:46 | dimensional parameters. You can see how three are non dimensional. I don't |
|
| 99:51 | , I, you know this one this one, obviously this one also |
|
| 99:56 | dimensional. And furthermore, you can that in the case where the anisotropy |
|
| 100:02 | zero, this one's gonna be a because if the anti be a |
|
| 100:08 | this element is the same as this . So this is a zero, |
|
| 100:12 | way this is a zero. And they think about it a little bit |
|
| 100:15 | see that one's also a zero in case of isotropy. So now let's |
|
| 100:23 | the case of weak anisotropy or we that these terms are not zero, |
|
| 100:30 | they're small compared to one. And we're gonna put that assumption into |
|
| 100:35 | um uh we're gonna rewrite the previous in terms of these things, you |
|
| 100:41 | there's five and all. So uh is as you look at it as |
|
| 100:45 | a rep parameterization of the exact So now we're gonna assume that these |
|
| 100:50 | small compared to one. And we're do a first order tailor expansion returning |
|
| 100:56 | terms which are linear in these three . And so when we do that |
|
| 101:03 | happens suddenly those three equations um are simple that anybody can understand. You |
|
| 101:12 | , for example, for the, P wave velocity, it's got |
|
| 101:16 | a reference velocity and too simple, isotropic, um too simple trigonometry, |
|
| 101:23 | isotropic um contributions. And these are be small, see we assumed uh |
|
| 101:32 | that uh delta is small and epsilon small. So uh uh these are |
|
| 101:36 | make uh only small changes here. how did uh uh how do we |
|
| 101:41 | them or the tailor of approximation? , before we go on, let |
|
| 101:47 | point out to you that this combination you see right in here, which |
|
| 101:52 | repeated right in here, that's gonna that a new name SMA. It's |
|
| 101:57 | uh not a new parameter. It's AAA new name. So now with |
|
| 102:03 | new notation, let us look at policy, these are the three equations |
|
| 102:08 | we just um uh looked at and apply to waves traveling at some uh |
|
| 102:15 | angle in the subsurface. Now, vertical propagation oops for vertical propagation, |
|
| 102:27 | we have sate equals zero. So term goes away and this term goes |
|
| 102:33 | and we're left only with the VP . So it wasn't that clever for |
|
| 102:37 | to call this reference velocity VP zero indicates uh uh propagating at the, |
|
| 102:43 | zero angle vertical. OK. look the neck at VSV. So |
|
| 102:50 | signed eight equals zero. So we're with VS zero just like it says |
|
| 102:55 | , FVSH against Syed equals zero for propagation. Again, we left with |
|
| 103:01 | it's the same sheer velocity for both modes for propagation. OK. So |
|
| 103:11 | us um look at horizontal propagation. there we have uh a sine theta |
|
| 103:20 | one, but course the equals So this term goes away again, |
|
| 103:26 | term becomes a one. And here have epsilon plus one times VP |
|
| 103:30 | That's the horizontal V velocity. Let's look at SV uh at, |
|
| 103:37 | , at the SV mode. Uh got um cos state equals zero. |
|
| 103:43 | this term is zero again. So left with only the VS zero for |
|
| 103:49 | it's the same as vertical. But course, in between, it's different |
|
| 103:52 | between is given by this. And on to the sh mode, we |
|
| 103:58 | scient eight equals one. So we gamma plus one times vs zero equals |
|
| 104:05 | . So uh you can see that waves traveling horizontally, uh shear waves |
|
| 104:11 | horizontally have different velocities whereas shear waves vertically have the same. Now, |
|
| 104:19 | of our data is P wave. let's concentrate on that. Here is |
|
| 104:24 | wavefront emanating from here and I'm showing two D cross section. And uh |
|
| 104:32 | so um it says that the wavefront a homogeneous layer is not a |
|
| 104:37 | So here's the wavefront and um uh is an isotropic circle to guide your |
|
| 104:45 | . And both of these have the vertical velocity. So this is a |
|
| 104:49 | taken after uh a certain number of . So the wave can go down |
|
| 104:54 | far, doesn't necessarily mean 2000 It just means it went down that |
|
| 105:00 | . You see the circle uh uh you have the 2000 on there. |
|
| 105:03 | circle comes out here at also horizontal , the wavefront is ahead. |
|
| 105:11 | So, uh uh so it's the VP zero here, but uh the |
|
| 105:17 | is not traveling along this line. is a, the isotopic circle. |
|
| 105:22 | is um oh this is the So the wavefront has gotten out ahead |
|
| 105:32 | horizontally. So it's not a circle it's not an ellipse. So just |
|
| 105:38 | show you that I'm gonna put on a perfect ellipse in green and this |
|
| 105:43 | an ellipse drawn um with the uh given to me by Mr Bill |
|
| 105:50 | And you can see it as um , it's tied to invertical and it's |
|
| 105:55 | at the horizontal, but in between doesn't tie. So we call that |
|
| 106:00 | epsilon ellipse. Why it's because it the uh el electricity is epsilon. |
|
| 106:09 | an ellipse has uh has only one of uh a degree of elliptic and |
|
| 106:15 | epsilon in this case. Now, so what we establish is the wavefront |
|
| 106:25 | not an ellipse. I'm gonna show now another ellipse which I call them |
|
| 106:31 | ellipse. That's this one in You see it's also matched here and |
|
| 106:36 | the original of the initial variation is to the wavefront. So you see |
|
| 106:45 | , the wavefront stays closer to the ellipse than it does to the green |
|
| 106:52 | . And then eventually at large enough it crosses over from the red ellipse |
|
| 106:57 | the green ellipse and ends up on green ellipse. So what is this |
|
| 107:03 | ellipse? It's an ellipse with elliptic . That's what it is. So |
|
| 107:09 | have the right to ask, uh . So um uh uh what does |
|
| 107:14 | mean here? Uh The, it's obviously not the same as the |
|
| 107:19 | velocity because that's what, what velocity this correspond to the real velocity is |
|
| 107:25 | here? Right? The, the wavefront has come and gone. |
|
| 107:29 | uh it's out here. And what this point here? Well, for |
|
| 107:34 | , it's simply the, uh the the notional idea that's where this d |
|
| 107:41 | comes out at the uh at the , the delta lips is defined. |
|
| 107:47 | it matches the wave front here, matches the initial uh variation, |
|
| 107:52 | an isotopic variation from the circle. the wavefront stays close to the delta |
|
| 108:00 | and then crosses over and ends up the absolut curve. So that's um |
|
| 108:05 | way the way P wave funds So now, before we, uh |
|
| 108:14 | we pass on, let me just uh ref refer this in two |
|
| 108:19 | One is gonna, I'm going to the and uh two dimension identities, |
|
| 108:27 | know, the cosine squared equals one sine squared. So just put that |
|
| 108:31 | here and we're left with this expression where, where the delta, |
|
| 108:35 | the minus delta sine fourth is now here. So we're gonna have, |
|
| 108:39 | have a name for this. Uh And so this arrow is supposed to |
|
| 108:44 | pointing here. That's the so-called an elliptic. A, an ecliptic |
|
| 108:50 | departures from elliptic. So if uh , if epsilon equal to delta, |
|
| 108:58 | would be a zero and we'd have these two terms and that would be |
|
| 109:02 | perfect ellipse. So the, a estimate uh uh uh measures the departures |
|
| 109:12 | electricity. OK. Now, another we can do is if we restrict |
|
| 109:19 | to small angles. The then we write this in this way only and |
|
| 109:25 | this term whether or not the coefficient a zero, if we restrict ourselves |
|
| 109:30 | small value. The because for small of data sine squared is small but |
|
| 109:36 | of the fourth is even small. we can uh simplify this for a |
|
| 109:41 | angle, you know, near vertical . So now let's ask ourselves, |
|
| 109:46 | does this show up in reflection move well. So here is the canonical |
|
| 109:52 | reflection problem. We got a homogeneous layer. You've seen pictures like this |
|
| 109:58 | , but those are isotropic pictures. so uh uh this is an an |
|
| 110:03 | layer. So here is the hyperbolic out equation and uh uh it has |
|
| 110:09 | Taylor series coefficient which is one over NMO velocity square. See the Taylor |
|
| 110:16 | efficient uh is, is the, derivative of the unknown here with respect |
|
| 110:22 | the small parameter. In this a small parameter X squared evaluated at |
|
| 110:29 | uh where the small parameter zero. that's the uh uh the Taylor series |
|
| 110:36 | . And so when you um uh this uh that you do this derivative |
|
| 110:43 | on the previous equations, you learn the move out velocity looks like |
|
| 110:48 | It is the vertical velocity times one delta. You recall that was the |
|
| 110:55 | mysterious uh uh uh velocity that we out before. That's this. |
|
| 111:06 | Now we're gonna go back for Now, here's a, here's a |
|
| 111:12 | good question for you. This equation evaluated near the origin for very small |
|
| 111:20 | , right? That's what it says here. So since this picture is |
|
| 111:25 | quite right, this uh uh this the wave vector should be coming down |
|
| 111:31 | almost vertically, right? Because we right here in the limit of small |
|
| 111:37 | . So the question is why isn't short spread move out velocity equal to |
|
| 111:42 | vertical velocity? You know, the the waves are going down like |
|
| 111:55 | I in the limit of small they should be traveling with uh |
|
| 112:01 | the, the a the vertical average . Well, this is a homogeneous |
|
| 112:05 | . It should be traveling with the vertical velocity, not the average |
|
| 112:17 | . This is a similar question to we had uh before uh uh uh |
|
| 112:22 | talk about uh from isotropic layers. said why is it the move out |
|
| 112:28 | A V zero vertical average instead of MS average? And we found out |
|
| 112:33 | that time, the reason is because what we measure is not this, |
|
| 112:38 | measure the horizontal move out that leads us the, the R MS |
|
| 112:44 | And so uh uh for the isotropic the anisotropic case, you're uh |
|
| 112:50 | homogeneous, no layers, homogeneous A isotropic case. Again, we're measuring |
|
| 112:56 | the horizontal move out right here. the uh uh as the delta in |
|
| 113:03 | and it's not just the vertical even though it's it's defined in terms |
|
| 113:08 | very short offsets, the figures uh in a bit uh defined in terms |
|
| 113:14 | very short offsets. But it it the delta in it right there is |
|
| 113:20 | delta in. So let's look at implications. This short spread move out |
|
| 113:29 | is hyperbolic even though the layer is isotropic. So when you see hyperbolic |
|
| 113:34 | out on your workstation, that does mean that the media is isotropic because |
|
| 113:42 | isotropic, meaning even if it's homogeneous leads to hyperbolic move out for short |
|
| 113:50 | set. So uh where is the I said, well, it's hidden |
|
| 113:57 | the move out velocity because what you on your workstation screen is this |
|
| 114:03 | You don't measure these two things you measure this so that uh the |
|
| 114:09 | ectopy is hidden inside what you And uh now it gets nasty. |
|
| 114:16 | you want to use the velocity, pick from your workstation to convert time |
|
| 114:22 | depth, you're gonna get the wrong because the depth is equal to the |
|
| 114:28 | time which you know times the vertical , it's not equal to the vertical |
|
| 114:35 | times the move out velocity. This is gonna be different than this one |
|
| 114:39 | of delta. So if you've ever a time to death this time, |
|
| 114:45 | only two explanations, you're screwed up or uh oh let's assume you didn't |
|
| 114:52 | that. The other assumption you might have made is you might have assumed |
|
| 114:56 | the media are isotropic. And we from the, from this argument that |
|
| 115:03 | media has uh has an isotropic move velocity even when we consider only very |
|
| 115:15 | offsets. Now here is a real right here. This one, it |
|
| 115:22 | the magni the an isotopy is magnified the move out because of the argument |
|
| 115:27 | . So follow me on this, the vertical velocity as a function of |
|
| 115:31 | angle uh offset. And let's consider only small angles. So we're |
|
| 115:37 | use the small angle approximation. And consider a case where uh the value |
|
| 115:43 | delta is 10% and consider a case all the angles are less than 30 |
|
| 115:50 | . OK. So uh the sine 30 degrees is one half square is |
|
| 115:56 | . So that means that all of rays in our gather are traveling with |
|
| 116:02 | um uh which differ from the um velocity by less than 2.5% which is |
|
| 116:10 | times 1/4. All the velocities are within 2.5% of the vertical velocity |
|
| 116:18 | So the move out velocity differs from vertical velocity by that full 10%. |
|
| 116:25 | see, because there's no signs for here science square time makes this a |
|
| 116:31 | uh uh uh uh small variations. even so the move out velocity differs |
|
| 116:39 | the vertical velocity by the full 10% them. So, you know, |
|
| 116:47 | out is our primary observable. The is a secondary observable primary observable. |
|
| 116:54 | one we use to make our images , that's the arrival times and the |
|
| 116:59 | of arrival time with offset which is out and in that move out the |
|
| 117:05 | shock we use in there right And it's magnifying in a magnified |
|
| 117:13 | So we derive this assumption, this using the assumption that the HM layer |
|
| 117:22 | uniform. Of course, that's not very good assumption. So let's uh |
|
| 117:25 | at a case where there's uh an uh uh layer one D me laterally |
|
| 117:35 | , but they're an anisotropic and then the move out velocity then is given |
|
| 117:41 | the R MS average of the vertical with an anti and I should copy |
|
| 117:49 | which I have a subscript on Uh It says RM SS uh a |
|
| 117:55 | on this average here. So that's in the lecture notes. Uh Not |
|
| 118:02 | election notes that you have. but uh in the, in the |
|
| 118:06 | notes which I published uh um for seg this is defined in detail. |
|
| 118:16 | I don't wanna say anything more about . But uh what I want to |
|
| 118:19 | is if you use, uh if say I want to know what are |
|
| 118:24 | interval losses. So if you find the interval VLO is either by migration |
|
| 118:32 | analysis or by diggs differentiation of of of velocities. Like we. So |
|
| 118:39 | before you find that you get for interval layer, you get an interval |
|
| 118:46 | which is contaminated by its own So for example, this subscript out |
|
| 118:52 | referred to the whole thing as the velocity for this third layer times one |
|
| 118:57 | delta for the third layer and the with all the other. So you |
|
| 119:01 | get away from the anisotropy by going velocity. You can't escape by computing |
|
| 119:11 | general velocity. And you can't consider escape it by considering only short offsets |
|
| 119:18 | in there. So you can measure by comparing VNMO with VP zero, |
|
| 119:24 | you obtain from a vertical VSP. to do that, you have to |
|
| 119:29 | drill a well, you have to 100 $100 million to drill a |
|
| 119:35 | the different, you always find differences which uh vary by the way uh |
|
| 119:40 | for delta as a function of de And uh because delta is right with |
|
| 119:50 | . So I think I I is possible for us to make this correction |
|
| 119:55 | growing? Well, that's all that's real bummer. Spent $100 million to |
|
| 120:00 | find the value of delta. Let's at longer offsets and observe the non |
|
| 120:05 | move out and use that to estimate Z. So I think I showed |
|
| 120:10 | this um uh picture before. Uh here we have a real data and |
|
| 120:16 | it's a, a common midpoint E gather flatly geometry. And a velocity |
|
| 120:23 | has been chosen to flatten the gathers short offsets, but at long |
|
| 120:28 | it doesn't uh work. So the thing to do is uh let's do |
|
| 120:36 | a higher order tailor expansion. This , have another term in tailor expansion |
|
| 120:41 | like we considered for the many layer case. And we'll make a physically |
|
| 120:47 | correction factor put in here uh so we're gonna choose this value a in |
|
| 120:53 | clever way. So that um at offsets uh this square and offset cancels |
|
| 121:02 | two of these. So again, end up uh T squared varying as |
|
| 121:06 | squared with the right velocity instead of wrong philosophy. Well, the way |
|
| 121:10 | do that is to define a in way uh uh uh but that requires |
|
| 121:16 | you need to know the horizontal So um when uh uh I, |
|
| 121:28 | was a co-author on the paper that uh pointed this out and I was |
|
| 121:33 | happy at the time, very pleased myself at the time, but I |
|
| 121:37 | realized that nobody else was particularly impressed that work. And the reason is |
|
| 121:42 | it required for processors to um determine each vertical arrival time. A, |
|
| 121:50 | short spirit move out of velocity, quantity, a four and a quantity |
|
| 121:56 | in order to flatten the gather. normally most um data does not allow |
|
| 122:02 | enough flexibility to determine three parameters at uh time. So a few years |
|
| 122:14 | , um uh somebody else made a invention and, and he found out |
|
| 122:19 | for a single polar ANAs tropic homogeneous polar, polar ANAs tropic |
|
| 122:25 | the previous formula simplifies to this. it looked the same as um I |
|
| 122:31 | showed you except that look here, got one, we renamed that uh |
|
| 122:36 | four to be a minus two A and look the same to a and |
|
| 122:42 | here. So for this formula, only have to uh uh determine at |
|
| 122:48 | uh uh vertical travel time it has travel uh to determine a short |
|
| 122:55 | move out velocity and an a, parameter and every uh there's two instead |
|
| 123:00 | three. So that suddenly makes it . And so what is this |
|
| 123:05 | in terms of quantities we already Well, it's epsilon minus delta over |
|
| 123:11 | plus two T one plus two delta um uh I should uh tell you |
|
| 123:22 | this guy uh Falcon uh was a at Amaco and then he went to |
|
| 123:27 | Colorado School of mines still there. his first student was this guy and |
|
| 123:33 | uh uh probably his best student was guy. And so al Khalifa came |
|
| 123:39 | with this and uh he, he that uh uh only have to make |
|
| 123:44 | single approximation, which is a good and you convert the previous column which |
|
| 123:49 | impossible to this one, which is . And uh uh uh compared to |
|
| 123:55 | with what we did before, uh is uh uh in lecture four, |
|
| 123:59 | came up with a similar equation, a similar notation. Uh But all |
|
| 124:05 | stuff that uh you don't need any this stuff. Now, the animal |
|
| 124:11 | don't need any of this stuff. so, uh uh uh now, |
|
| 124:17 | yeah, well, I th th shows the comparison between is should topic |
|
| 124:22 | we did before and I should topic we doing. Now here, we |
|
| 124:25 | two different parameters here to determine what's third one here, here, there's |
|
| 124:29 | two to determine that's really good OK. Now, that uh simplification |
|
| 124:41 | strictly valid only for um a sing single layer. But people use it |
|
| 124:46 | the time by putting in here an uh the parameter. And we determine |
|
| 124:52 | empirically by flattening the together for our of uh uh uh as a function |
|
| 125:01 | time. So when you do you, you uh transform the previous |
|
| 125:06 | uh problem. This, it's still perfect at, at f, at |
|
| 125:11 | offsets, we still have problems. before this major reflector was departing from |
|
| 125:18 | right about here. And now it flat for a lot further. |
|
| 125:26 | So I think this is a good time to stop for a break. |
|
| 125:33 | uh resume at 330 Kirsten type. uh uh I will see you |
|
| 125:41 | Uh Yes. Uh So, um Carlos, are you there? |
|
| 125:52 | Professor Rahi and uh Bruce, are there? It's not back yet. |
|
| 126:01 | Well, OK. So we're gonna anyway. Uh So this is where |
|
| 126:07 | left off. And uh what I to show you next is the effect |
|
| 126:12 | anisotropy on images. So, let show you what we have here. |
|
| 126:23 | We have some uh 2.5 D You know what I mean by 2.5 |
|
| 126:27 | modeling in the computer means that the the um the wave propagation in the |
|
| 126:33 | is three dimensions, but the model only two dimensions. So this model |
|
| 126:38 | into and out of the a figure variation. So that's called 2.5 D |
|
| 126:44 | . And so here is a, representation of the model in four different |
|
| 126:50 | . And you can see uh the shades of gray, we have the |
|
| 126:54 | the p the vertical P wave velocity in the upper left corner, you |
|
| 126:59 | see sedimentary layers here and you can see uh a salt body. And |
|
| 127:05 | so the uh uh the, the point of this uh study was |
|
| 127:13 | understand the effects of uh anisotropy in imaging. So now let's see what |
|
| 127:22 | we have uh we have here uh the upper right corner, we have |
|
| 127:27 | uh the representation of the model parameters delta. So you see insult of |
|
| 127:34 | zero. And then we have uh uh uh uh measures of uh uh |
|
| 127:39 | various values of delta in the very layers. Over here, we have |
|
| 127:46 | the uh the same parameters uh the the, the model representation for the |
|
| 127:53 | epsilon also in layers also zero in salt body. And here we have |
|
| 127:59 | a and you know that ADA is calculated from delta in epsilon. So |
|
| 128:04 | want to show you now uh uh these uh people on at all uh |
|
| 128:10 | for forward modeling, I think it Kirkoff Migra for forward modeling through this |
|
| 128:16 | 2 2.5 year model. And then Kirkoff migration of the results trying to |
|
| 128:25 | the uh uh an image of the model. Now, I think the |
|
| 128:29 | would have been the same if they'd a more modern form of migration. |
|
| 128:34 | That is to say it would be same an isotropic effects. Excuse |
|
| 128:42 | Can you repeat again, what is meaning of 1.5 a model? I |
|
| 128:49 | understand that part. What is uh mo why these are 2.5 models? |
|
| 128:56 | . Uh So the definition of 2.5 modeling 2.5 D is when you have |
|
| 129:01 | two D model, but you do wave propagation in there. So you |
|
| 129:06 | a source right here and the energy out in three dimensions. OK. |
|
| 129:10 | model is only two dimensions, so can call that 2.5 D. |
|
| 129:15 | Thank you. OK. That's uh that's kind of cute. But uh |
|
| 129:19 | I think that's a, a well uh notation. OK. Now I'm |
|
| 129:24 | show you the results of uh of in this um for this model with |
|
| 129:31 | different styles of migration case, I 44 different cells. So the first |
|
| 129:36 | here is uh uh an isotopic Uh uh So, and where they |
|
| 129:43 | they know what are the, the values of the parameters because they made |
|
| 129:48 | model that they know what they put . And so it's a pretty good |
|
| 129:51 | uh image and it's only for part the model, I'm gonna back |
|
| 129:56 | And uh uh it's this part of model right here, this part of |
|
| 130:01 | model right here. That's what we're on here. And I want to |
|
| 130:06 | out to you uh where this um uh this intersection between the fault and |
|
| 130:13 | betting is happening here. So, normally you cannot do this kind of |
|
| 130:19 | because you don't know what are the parameters can only do that uh uh |
|
| 130:26 | a uh kind of modeling environment. then they model in a different |
|
| 130:31 | then they model with an an isotropic and they fed the uh um uh |
|
| 130:39 | , they fitted the migration parameters uh with the uh move out velocity uh |
|
| 130:45 | you know, the flatten the gathers and find the best fit velocity for |
|
| 130:51 | the Gers. And then from flattening far offsets, they determined ADA and |
|
| 130:56 | they couldn't, they did not know is um uh the vertical velocity because |
|
| 131:02 | is uh fitted from the surface seismic . So they had to assume |
|
| 131:07 | So they assume that delta equals And so if you look at this |
|
| 131:12 | , uh it looks like a pretty image except that, and you see |
|
| 131:17 | intersection between the uh the uh the bed and the fault is happening at |
|
| 131:24 | lower level. This is a perfect line here. So because they uh |
|
| 131:30 | the model delta was not equal to , but here they were forced to |
|
| 131:34 | it's zero. So they got the bits. That's what I talked about |
|
| 131:39 | with you before you're gonna get the depths if you have the wrong |
|
| 131:45 | So the next time they uh next they did, they did isotropic migration |
|
| 131:50 | where they found the best fit but they assumed it's all isotropic |
|
| 131:56 | And you can see that this image uh distinctly inferior to this one. |
|
| 132:02 | It's got, got, you uh uh artifacts here and there. |
|
| 132:07 | uh uh uh so this is what state of the art at that time |
|
| 132:13 | the year 2000, I would say this is close to the state of |
|
| 132:18 | art today. Although they might make different assumption, you know, |
|
| 132:22 | if you're gonna assume delta, you have to assume delta equals zero, |
|
| 132:26 | could assume delta equals 5%. Why ? Maybe you have a nearby, |
|
| 132:32 | , uh that where they have a VSP in that. Well, and |
|
| 132:35 | actually measure uh uh delta. And , uh maybe from there, uh |
|
| 132:40 | you measure delta equals 5%. Uh why not just assume uh the same |
|
| 132:46 | uh nearby, probably wrong, but better than assuming delta equals zero. |
|
| 132:55 | in the fourth instance, what they was they drilled a well right here |
|
| 133:00 | they uh measured uh vertical velocities in well. And so it's a terrible |
|
| 133:06 | . You looked at this image and see all kinds of, of uh |
|
| 133:11 | uh uh uh artifacts in there because vertical uh uh well, it vertical |
|
| 133:17 | of uh velocity uh is uh is uh correctly known now because of the |
|
| 133:25 | uh from the VSB. But it's wrong velocity for imaging for imaging. |
|
| 133:31 | wanna use the NMO velocity. And that's why this image is wrong. |
|
| 133:37 | bad, but you see it's at proper depth, you see it's the |
|
| 133:42 | depth, whereas this one is deeper this one is deeper, this one |
|
| 133:46 | drilled in the right depth. It's bad image, but at least it's |
|
| 133:49 | right depth isn't that interesting. um uh these four panels give an |
|
| 133:57 | nice uh uh uh demonstration of the of uh including or uh ignoring anisotropy |
|
| 134:06 | imaging. And so, um uh days uh uh normally, what we |
|
| 134:12 | the best we can do is the of this is where we have the |
|
| 134:17 | fit parameters which we fit from the . But we know that's not enough |
|
| 134:21 | it doesn't give us doubt. So anticipate time to death. Misty, |
|
| 134:28 | you see here. So let's uh have a quick uh survey, uh |
|
| 134:35 | quick uh quiz here. Um For see, I think uh Carlos, |
|
| 134:41 | think it's your turn uh P velocity weakly or anti formations depends upon how |
|
| 134:50 | elastic parameters. All these choices. do you say, read it |
|
| 134:57 | We're talking about P wave velocity and an isotropic uh formations or assume it's |
|
| 135:05 | an isotopic. It depends upon how elastic parameters. No, Professor, |
|
| 135:13 | not sure. I be guessing. me hm. Well, I would |
|
| 135:20 | I would say three. But then , that's, that's exactly right. |
|
| 135:23 | . You saw the formula, it's zero, delta and epsilon. So |
|
| 135:28 | is exactly right. OK. So Brice, are you in, are |
|
| 135:31 | here now? Yes, yes, am. OK. So this one's |
|
| 135:36 | . Um uh It says here as statement, anisotropy effects move out velocity |
|
| 135:43 | uh ABC or D notice the D only about So, for, for |
|
| 135:49 | , I'm gonna do only a uh this is true no matter how short |
|
| 135:53 | the maximum of thats and the determination uh velocity. Is that true? |
|
| 136:00 | true. That's right. OK. uh uh le le uh how about |
|
| 136:05 | b also a trip? Um And Carlos uh uh c we got two |
|
| 136:16 | true. So uh uh either this , yeah, this better be |
|
| 136:20 | And in that case, we got of the above. OK. So |
|
| 136:25 | back to you uh Merce um says affects non hyperbolic move out. Now |
|
| 136:32 | this one is about a hyperbolic move . Uh So this is non hyper |
|
| 136:38 | of these statements describe this anisotropic Uh You see there's no, none |
|
| 136:43 | the above. So um uh it here, the non hyperbolic term depends |
|
| 136:49 | the near offset anisotropic parameter delta. two from sorry, I was |
|
| 137:05 | I think it is false. It be CC yeah. A better answer |
|
| 137:12 | C it de it depends on delta not delta alone, depends on |
|
| 137:17 | but not epsilon alone. It really upon a and has nothing to do |
|
| 137:21 | SV. So that's very good. . So my professor, I have |
|
| 137:27 | question if you can. Do you back to the example of that the |
|
| 137:32 | images uh migrated with different? So in four, the, the |
|
| 137:43 | in the isotropic migration and the isotropic , the velocities are the same. |
|
| 137:48 | the, but the anisotropy parameters are ones that are missing. That's |
|
| 137:55 | Here, we assumed the, anti zero, but here we found |
|
| 138:00 | best fit ada from the, non acro bolic move out. But |
|
| 138:06 | means that also the, the VN changes. Right. Well, the |
|
| 138:13 | is gonna be, uh, it's gonna depend upon, uh, |
|
| 138:16 | zero and delta in a certain combination you can measure this one from the |
|
| 138:22 | offsets uh independently of your measurement of one from the further offset. See |
|
| 138:28 | that works. OK. Yes. . No, no, I unders |
|
| 138:32 | understand now, thank you. now this is a very important topic |
|
| 138:38 | . Uh poor anisotropic A VL. so, uh uh I should tell |
|
| 138:43 | that I uh I should remind you I am the inventor inside chemical of |
|
| 138:49 | L back in 1981 I think. uh uh uh shortly after uh uh |
|
| 138:57 | and the way we discovered that is , we learned from uh a mistake |
|
| 139:02 | by uh uh uh uh uh a partner that they revealed to us |
|
| 139:08 | it could be done that a vo be done. And then uh |
|
| 139:12 | the uh well, hm the assignment figure out how to do it came |
|
| 139:18 | me, I was new in in-house . And so I quickly uh figured |
|
| 139:23 | how to do it and then we realized that that could be a great |
|
| 139:27 | way to reduce risk in drilling. uh we scrambled a tiger team of |
|
| 139:34 | experts uh went off site, looked a lot of data and confirmed that |
|
| 139:39 | vo you know, as you currently it, a vo uh we already |
|
| 139:44 | it basically in the same terms back 1981. And it can be a |
|
| 139:49 | way to reduce risk and prove that looking at a bunch of historical |
|
| 139:54 | So as soon as the importance of vo became obvious to everybody, they |
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| 139:59 | the project out of my hands because was new hire and they were gonna |
|
| 140:04 | this over to one of those people have more confidence in. And so |
|
| 140:09 | what they did. Uh But I um um I continue to think about |
|
| 140:16 | unauthorized and I thought here's what I to myself. We're, we're doing |
|
| 140:20 | analysis. Um amplitude offset, amplitude with offset, it means amplitude variation |
|
| 140:30 | incident bank. But we're assuming that we're doing this, we are assuming |
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| 140:35 | the rocks themselves are isotropic, shouldn't be considering a vo in the presence |
|
| 140:43 | velocity variation with OIE an iso? so I, I uh asked myself |
|
| 140:51 | question and I did a little, followed a little bit and decided the |
|
| 140:57 | was probably so, but I couldn't out how to do it. I |
|
| 141:01 | not figure out how to do a excuse me, could not have figure |
|
| 141:06 | how to do an isotropic a So um uh and now we |
|
| 141:11 | so I'm gonna uh uh talk about next. So here is the uh |
|
| 141:17 | that we had from before. Uh recognize the oh yeah, a vo |
|
| 141:26 | intercept great and character in terms of isotropic ideas jumps across the horizon of |
|
| 141:35 | is. So I think you're all with that, we analyzed it in |
|
| 141:40 | of this diagram here. And we the fact of uh uh by looking |
|
| 141:47 | the cross plot and seeing where these board um uh events, this is |
|
| 141:53 | uh intercept versus slope in the cross . And uh we saw these uh |
|
| 141:58 | things that look like uh on the on the cross pot, they look |
|
| 142:02 | like noise, but they all come the top of the structure. So |
|
| 142:05 | says they're not noise, they're telling a pattern and all that was very |
|
| 142:09 | good. And uh we, and by doing that, we learned how |
|
| 142:13 | use cross spots like this to find and to find fluid anomalies at your |
|
| 142:23 | greatly reduced risk. But there's something with this analysis. The slope of |
|
| 142:29 | green curve in here, which is taken from off to the side for |
|
| 142:34 | are no anomalous fluids in here and too steep. Something is wrong. |
|
| 142:40 | uh what I decide what I learned in 1981 is that if you do |
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| 142:47 | same analysis with an anisotropic half but an an isotropic half space, |
|
| 142:52 | get a formula that looks pretty much same accept that in the gradient |
|
| 143:01 | And in the curvature team, you to find explicit and isotopic terms. |
|
| 143:07 | as soon as I saw it back 1980 I thought to myself, you |
|
| 143:12 | , this is big trouble because these here are gonna be small compared to |
|
| 143:18 | , but there's no one in the . All of these terms are |
|
| 143:23 | So this term which we always ignore be just as big as these other |
|
| 143:29 | terms could be, it could actually the sign to change the algebraic sign |
|
| 143:35 | the result if we include these So how to include that? Uh |
|
| 143:42 | , uh we can't solve uh uh problem like we did before because when |
|
| 143:47 | uh are trying to solve for these , which we, the vertical velocity |
|
| 143:52 | density, we got the uh the in anisotropy on the right hand |
|
| 143:58 | three equations in five unknown. So just to show you how crucial |
|
| 144:08 | is, we're gonna do an exercise now. So let's get out of |
|
| 144:14 | . So I'm going to stop And while I'm fumbling around here, |
|
| 144:20 | bring up the classroom exercises which I shared with you before via canvas. |
|
| 144:28 | what I'm gonna do right now is , I have to search around a |
|
| 144:37 | bit. Hold on a second. , I'm browsing for the file that |
|
| 144:58 | want and you should be browsing for , for the classroom exercises, trial |
|
| 145:10 | I provided to you for. it's called a VO exercise. |
|
| 145:20 | So, uh, now I am to show this, iii I don't |
|
| 145:27 | it. I cannot find it in en canvas. Oh, you don't |
|
| 145:32 | this. Uh, it might be I forgot to give it to |
|
| 145:36 | OK? I will check because this probably the most important part of this |
|
| 145:42 | course right here. And so uh uh um what I'm showing you here |
|
| 145:49 | an a vo exercise and uh I'm share with you, share the |
|
| 146:05 | I'm gonna have to stop sharing the screen. Can you see this um |
|
| 146:57 | the spreadsheet with the red and yellow and some curves at the bottom and |
|
| 147:03 | at the right? Yes. So let me show you what we |
|
| 147:08 | here and you can see my cursor around. OK. So never mind |
|
| 147:12 | tables to the right. They just the cursor. So what we're gonna |
|
| 147:16 | is some forward modeling of anisotropic A . And so we have uh uh |
|
| 147:21 | model parameters here in color. So got a red formation over a yellow |
|
| 147:27 | in the red formation. We got uh VP zero, which is gonna |
|
| 147:31 | you're gonna s uh set it to own parameters by uh using this |
|
| 147:36 | So you put your cursor on top the slider, depress the left key |
|
| 147:41 | then move it and you see the uh the number here changes. And |
|
| 147:46 | that's a different velocity that I OK. And the same thing with |
|
| 147:52 | ratio down here, same thing with , same thing with delta and |
|
| 147:57 | So what the first thing I'm gonna is I'm gonna put the uh the |
|
| 148:01 | delta equals zero here and I'm gonna epsilon equals zero. And I'm gonna |
|
| 148:09 | the same thing down here in the body. OK? Now you're gonna |
|
| 148:17 | down a little bit, scroll down little bit. And so the curves |
|
| 148:24 | here and that's a typical a VO that's exactly calculated from uh the model |
|
| 148:30 | I showed here. Um I shouldn't exact. So this is uh the |
|
| 148:35 | , the, the linearized approximation, A VO and this is what you |
|
| 148:39 | calculate for uh uh uh for a that you do. And uh so |
|
| 148:45 | can see that there are three values uh three curves plotted on top of |
|
| 148:50 | other. OK. So uh and I'm just gonna uh change uh |
|
| 148:55 | the, the velocity ratio up here you can see how the curve |
|
| 149:01 | OK? As soon as you let of the, of the slider, |
|
| 149:05 | the curve changes. OK. Now gonna put in here uh oh and |
|
| 149:09 | the way uh we got here. correct. Thank you. OK. |
|
| 149:33 | I got 0% here and this is um uh delta and epsilon. And |
|
| 149:39 | are the values which are uh reported this uh field from the, from |
|
| 149:45 | slider. OK. And down here the calculator value for A. |
|
| 149:51 | So now let's um iii I put here just oh And by the |
|
| 149:56 | these are, are um uh uh scale here is exactly given from |
|
| 150:05 | So, so it shows a 6% um uh P wave reflection coefficient. |
|
| 150:14 | are exact numbers. So let's put here a little bit of delta. |
|
| 150:18 | I'm gonna put in, so imagine is a shale above. So I'm |
|
| 150:22 | put in a little bit of delta . OK. So I put in |
|
| 150:31 | of delta. Suddenly we see different uh um different curves. And so |
|
| 150:39 | got this uh uh the uh the in dark blue, that's the same |
|
| 150:44 | as we had before. That's the curve. And here we see down |
|
| 150:49 | two curves on top of each One showing the uh uh the Avio |
|
| 150:55 | uh e exactly calculated properly. And other one calculated not quite right. |
|
| 151:02 | um Never mind the difference from And o off here to the |
|
| 151:08 | you can see the uh uh the gradient which is 2.9% for the isotropic |
|
| 151:16 | and 1% for the an isotropic. uh by putting in just a little |
|
| 151:23 | of delta right here, I changed gradient by a factor of three. |
|
| 151:30 | you see that changed the gradient by factor of three? But only changing |
|
| 151:36 | uh the delta but a little And if I, you know, |
|
| 151:43 | the epsilon, I'm gonna put in a little bit of epsilon make it |
|
| 151:51 | because usually epsilon is bigger than And now things are changing some |
|
| 151:57 | So that did not change the um gradient. If you look back at |
|
| 152:01 | formula, uh uh the gradient depends only on delta uh the jump in |
|
| 152:09 | in uh upper and lower. And I just now changed uh uh |
|
| 152:13 | And now you can see that these two anisotropic curves are a little |
|
| 152:18 | different. Uh So, uh since is short, I'm not gonna talk |
|
| 152:23 | these differences only. Uh but uh only about these differences only. And |
|
| 152:28 | main thing is that by um wi imagining just a little bit of |
|
| 152:36 | we changed the gradient by three times change in. And uh and, |
|
| 152:44 | , but we do this all the , we ignored Dal. And what |
|
| 152:49 | exercise proves to you that delta could very important depending on the other |
|
| 152:54 | So let me just uh change uh up here and I'm gonna uh change |
|
| 152:59 | of these down here. This is change the velocity ratio in the reflecting |
|
| 153:05 | . I'm just gonna move it And when I let go, uh |
|
| 153:11 | see that everything changed, of Uh um but the um it's what |
|
| 153:19 | see and you see, I didn't the anti I just changed the, |
|
| 153:23 | ratio of vertical velocities in one And again, there's major differences in |
|
| 153:29 | gradient and uh now they're both negative major changes in the gradient. Uh |
|
| 153:36 | uh uh that uh comes from uh uh all these terms work together and |
|
| 153:43 | relative size of the anti soy terms depends upon the uh the size of |
|
| 153:50 | isotropic terms. So I just changed isotropic term and you saw the |
|
| 153:56 | So this uh spreadsheet exercise. Uh very simple but it should uh give |
|
| 154:06 | uh grave concerns if you're interested in ll because you can see that the |
|
| 154:12 | that when we ignore an isotropy, could be making major major errors in |
|
| 154:22 | analysis because the anti soy term might just as big as the terms that |
|
| 154:28 | calculated. Yeah. So think about and talk about it with your other |
|
| 154:35 | in your company who are uh um vo expert. And think about |
|
| 154:39 | I will then I will upload this um spreadsheet to you tonight. |
|
| 154:46 | I should have done that before. And you should talk to your um |
|
| 154:51 | talk about this with your friends and uh guys, we could be making |
|
| 154:56 | big mistake by ignoring anti. So since time is short, I'm gonna |
|
| 155:03 | at that. And since I didn't uh give it to you, I |
|
| 155:06 | to spend some time uh allowing you play with this, but I will |
|
| 155:10 | it to you tonight and that you uh uh play with it and then |
|
| 155:18 | it to your colleagues and say what about this? And now I'm |
|
| 155:24 | go back to the, as a . Yeah. Are you with |
|
| 156:09 | OK. So this is where we at office. And so uh I |
|
| 156:15 | um discovered this effect back in 1981 before you all were born and uh |
|
| 156:26 | I could not figure out how to determine that parameter delta delta from the |
|
| 156:35 | . Easy to imagine what it could . And, and uh the uh |
|
| 156:41 | allows you to put in some plausible and immediately you're shocked by how large |
|
| 156:47 | contribution it makes, but I couldn't out how to evaluate it from the |
|
| 156:54 | . And so that's what I'm gonna you next. So uh uh he |
|
| 157:03 | our problem to uh analyze the A data while accounting for the anti and |
|
| 157:09 | issues are that logs have high spatial and accuracy. They don't measure |
|
| 157:16 | Why is that? Because they have array pads, only vertical ray pads |
|
| 157:25 | . Now the surface seismic travel times low special resolutions. So uh you |
|
| 157:31 | that um delta is uh in the out velocity and you can uh um |
|
| 157:38 | can deduce it by comparing with vertical from the VSP. But even if |
|
| 157:43 | did that, it's gonna have low resolution. So uh uh so you're |
|
| 157:50 | gonna be able to find a local in delta from surface seismic travel |
|
| 157:59 | You can find trends in delta, you can find uh uh you |
|
| 158:04 | uh uh of course, averaging of , but you can't find a local |
|
| 158:10 | in delta from surface travel times. when you look at surface amplitudes, |
|
| 158:17 | figure it's in there. But uh the amplitudes are affected by many |
|
| 158:22 | We talked about those uh of those you can imagine them uh before you |
|
| 158:27 | , there's uh uh uh geometrical there's a generation, there's uh uh |
|
| 158:35 | effect of trans uh transmission uh coefficients the overburden. There's uh uh uh |
|
| 158:43 | as with, there's a angle variation the source strength. So many things |
|
| 158:48 | you can think of uh affect the a and basically, it's simply not |
|
| 158:55 | to make deterministic corrections. So this was solved uh with a uh master's |
|
| 159:02 | here at the University of Houston um 10 years ago by now. And |
|
| 159:06 | this is the uh the um the you can uh if you're scribbling it |
|
| 159:12 | . Well, yeah, you have in in your files. But uh |
|
| 159:16 | you have to do is look up these names in the seg annual |
|
| 159:21 | Expanded abstract from 2013. And so is our, this is what we |
|
| 159:28 | thinking. Remember this convolutional description of propagation from less than four uh uh |
|
| 159:36 | our seismic data. Uh uh uh start right here. So here is |
|
| 159:41 | source strength uh uh parameter. Here's initial wavel, then it propagates down |
|
| 159:47 | some complicated uh operator that we don't what it is, then we have |
|
| 159:52 | reflection and then it propagates back upwards a complicated operation. And we have |
|
| 159:58 | summation over many different reflections. And the data then gets involved with an |
|
| 160:03 | response and uh something is going on the computer. And then on top |
|
| 160:08 | that, we have all this So uh most of these different effects |
|
| 160:13 | result in angle dependent variation of amplitude is offset the penetration that but most |
|
| 160:21 | these a vo effects accumulate gradually as go down a little bit and down |
|
| 160:29 | little bit more and down a little more. All of these things change |
|
| 160:34 | . And the only thing that changes is this one right here, their |
|
| 160:40 | . So when you go to f to Hampton Russell, they are going |
|
| 160:46 | uh compare the surface seismic data with uh uh reflect a synthetic reflectivity which |
|
| 160:54 | get from the law. And so surface agent data has units, you |
|
| 160:58 | , plus or minus 1000 or so . But the the the reflectivity are |
|
| 161:03 | probably less than plus or minus So what uh Hampson Russell is gonna |
|
| 161:09 | ? They're gonna normalize the surface seismic intercepting curvature to an isotropic synthetic reflectivity |
|
| 161:17 | which is constructed from logs and doesn't any of the propagation effects in there |
|
| 161:27 | these normalization. The slowly growing parts good acquisition and propagation effect. And |
|
| 161:33 | only part that varies rapidly with vertical time is the reflectivity. So what |
|
| 161:46 | Thompson did in 2013 was to realize what this uh these normalization factors, |
|
| 161:53 | functions of time, there are functions uh uh vertical travel time. And |
|
| 161:59 | so um uh that means they have , a fourier um spectrum and the |
|
| 162:07 | spectrum has uh high frequency parts and frequency parts. So if you don't |
|
| 162:12 | this, if you don't um uh e everything, all the surface uh |
|
| 162:18 | seismic through this uh synthetic gather, only um only use the slowly varying |
|
| 162:28 | , slowly running parts of the app factors use that only to normalize thereby |
|
| 162:38 | correct for the propagation acquisition effects leaving rapidly changing reflectivity unchanged. So here's |
|
| 162:46 | example of that, this is from uh master's thesis 11 years ago. |
|
| 162:53 | here we have a vertical arrival time milliseconds from 400 milliseconds to 700 mills |
|
| 163:00 | there, we had um uh 123456 reflectors. And so uh um we |
|
| 163:11 | uh and we have had a seismic uh at the midpoint right at that |
|
| 163:19 | right at that at the where the was held. And from the seismic |
|
| 163:24 | , we took all those seismic um uh received gradients and divided them by |
|
| 163:32 | factor of uh 10,000 so that we plot them on this graph. And |
|
| 163:37 | slopes looked like this for each of various uh reflectors. Understand how you |
|
| 163:44 | take a surface seismic ganner. Look each, you know, you flatten |
|
| 163:48 | and do everything. All the magic Hampson rustle uh gives stretch and squeeze |
|
| 163:53 | and everything. So it get to the well. Exactly. And then |
|
| 163:58 | look at the um um uh at uh the gradient for that seismic data |
|
| 164:09 | you normalize it to the isotropic reflectivity or for you, you normalize it |
|
| 164:16 | by 10,000. So you can put on this block and then uh look |
|
| 164:20 | the syn syn synthetic isotropic reflectivity slope the logs. So from the |
|
| 164:26 | you can calculate R two and R and R zero everything. And that's |
|
| 164:31 | red curve. And now if we this to this, we get |
|
| 164:37 | we learn nothing. All we and what Hampson muscle does. It, |
|
| 164:40 | teaches you how to model surface seismic terms of sy synthetic. Um And |
|
| 164:47 | terms of isotropic changes, we want find the local job in delta. |
|
| 164:53 | what we do is we low pass that normalization function and only normalize with |
|
| 165:00 | low pass fraction of the normalization So you see we did not get |
|
| 165:07 | normalization. Matter of fact, uh every one of these is wrong and |
|
| 165:11 | wrong by a little bit. You this one is wrong by 30%. |
|
| 165:16 | one is accurate. This one has opposite side. So these, we're |
|
| 165:24 | attribute these differences to uh uh the term of delta delta in the |
|
| 165:30 | which we could not measure in the . And we could not measure in |
|
| 165:34 | seismic. We're gonna estimate it by procedure and we just noticed that these |
|
| 165:39 | are not small as a um a percentages of the total. So now |
|
| 165:47 | do we know whether or not we're a reasonable answer? What are we |
|
| 165:52 | use for ground truth? Well, we're estimating here is delta delta. |
|
| 166:00 | no way for us to measure no matter what we do uh in |
|
| 166:06 | earth for delta delta. But what can measure is delta. And so |
|
| 166:13 | we're gonna do is we're going to that at the uh uh uh we're |
|
| 166:18 | assume a value for delta at one these layers and then add up all |
|
| 166:23 | delta deltas and get an absolute value delta itself. In each of the |
|
| 166:28 | , you know how that works. , and uh uh uh the, |
|
| 166:36 | , so that process is very Let me show you the next slide |
|
| 166:40 | . This is the deltas that we uh uh came up with. And |
|
| 166:46 | gonna start off here with delta equals right here. And you can see |
|
| 166:51 | the maximum here is about d equal 20% possible. But who knows whether |
|
| 166:59 | right or not, where did we up with this zero right here? |
|
| 167:03 | , you can see the gamma ray over here and uh uh uh right |
|
| 167:08 | is a uh as a player with gamma ray. So we uh gone |
|
| 167:14 | decades. What that means this is sandstone, but let's assume that in |
|
| 167:19 | sandstone layer, delta is zero. that's why we come up with this |
|
| 167:23 | here, then we add delta delta we add delta delta again. And |
|
| 167:27 | , and we come up with this . And so uh uh what you |
|
| 167:31 | here is that uh um uh that's calibration sandstone. And you see that |
|
| 167:38 | we've uh calculated uh uh high values delta, those correspond to high values |
|
| 167:45 | the gamma log, gamma ray So everybody knows that this means that |
|
| 167:49 | is here. And so that's what predicting uh with high values of |
|
| 167:54 | That means uh we're expecting high values delta in a shale. And sure |
|
| 167:59 | , we see a shale here. I si sent Miss Lynn into her |
|
| 168:05 | thesis uh examination with this. I it was pretty good. The other |
|
| 168:12 | here at the university who were much hard nosed than me. They said |
|
| 168:17 | ? You've only got two successful predictions you call that a success. I |
|
| 168:22 | , well, she said, you , she was answering their criticisms on |
|
| 168:26 | feet in front of the examination committee she said, well, you |
|
| 168:31 | I I only had this much log . So one of the professors here |
|
| 168:35 | named Professor Stewart, you probably know . And he said, well, |
|
| 168:40 | uh I know where this data set from. This comes from Canada, |
|
| 168:44 | from Canada. I know that We have lots more logs from that |
|
| 168:48 | field here in our files at the . You go into our database, |
|
| 168:53 | University of Houston, find some nearby from nearby wells in the same |
|
| 168:59 | same formation and add them in. that's what she did. And so |
|
| 169:03 | a nearby log, she was able extend the um uh the logging, |
|
| 169:09 | gamma ray logging uh higher up. you see, here's another sandstone and |
|
| 169:13 | shale she predicted low. And then said that's five out of five so |
|
| 169:18 | they gave her a pass. So uh uh uh graduated with distinction with |
|
| 169:23 | master's thesis in 2013. And I hoping she would come back for a |
|
| 169:31 | thesis but she said I'm tired of poor. I wanna go get a |
|
| 169:38 | . And so she went out and a job. Well, she made |
|
| 169:43 | money but she wasn't having a lot fun. So she, uh, |
|
| 169:47 | back to the university a few years and got a phd and now she |
|
| 169:51 | a good job somewhere else. I forgot, uh, where, |
|
| 169:55 | I'm sorry to report that she uh, do for her phd. |
|
| 170:00 | did not do an extension of this . On the other hand, what |
|
| 170:03 | means is an extension of this work still available for, uh, U |
|
| 170:09 | H students uh, today. And can think of lots of ways |
|
| 170:13 | uh, uh, do better. , this was, uh, |
|
| 170:20 | just a master's thesis and I think are several phd thesis, uh, |
|
| 170:26 | to be, uh, pursued along same lines. Well, the University |
|
| 170:33 | , uh, Houston has a betting this and we hope to find something |
|
| 170:37 | make some money out of that. I would have to say that so |
|
| 170:40 | . We haven't made any money. , so that is all, |
|
| 170:46 | uh, uh, to the point this study limited as it is shows |
|
| 170:53 | the, uh, the, uh, the plausible, um, |
|
| 171:03 | that the neglected effect on polar an coming from polar an iso could be |
|
| 171:09 | important. And this data set shows , in fact, in this |
|
| 171:13 | it, it was and, and is. And uh what I think |
|
| 171:18 | does is it, it uh goes doubt every single isotropic a analysis ever |
|
| 171:26 | . So every such isotropic Avio analysis in your company could be rethought and |
|
| 171:34 | it needs to be revised or maybe depending on uh what are local values |
|
| 171:39 | delta delta in that area. And this algorithm, you can figure it |
|
| 171:47 | and furthermore, you can probably um out improvements to this algorithm. So |
|
| 171:59 | leave that to your imagination. So do a little quiz here. Here's |
|
| 172:03 | quiz. Um because the anisotropic effect A VL was not recognized for so |
|
| 172:10 | , 30 years now. ABC or or all of the above it |
|
| 172:15 | Uh um So um uh li how about a, is that |
|
| 172:26 | Yeah, that's true. Uh uh B is this true when you look |
|
| 172:32 | seismic surface seismic amplitude, they contain variations, but those are due to |
|
| 172:39 | propagation and reflection. Is that I think it's true, professor. |
|
| 172:46 | true. And we just named it of them and you can think of |
|
| 172:49 | probably um uh uh versa. How you? This, the seismic altitudes |
|
| 172:55 | on you usually normalized using logs and normalization forces the agreement with isotropy. |
|
| 173:03 | can't learn anything about anisotropy. yeah. Uh uh uh uh uh |
|
| 173:10 | including I think, and uh uh sha software uh so that means I |
|
| 173:16 | read all of the above. So uh back to you le le |
|
| 173:21 | says uh anisotropy effects which coefficient um gradient curvature B and C are all |
|
| 173:30 | the above. What would you Mhm OK. And we talked about |
|
| 173:45 | , we didn't talk about C but uh we didn't talk about it much |
|
| 173:48 | we showed it. Yeah. So uh so it's B and C and |
|
| 173:52 | we didn't talk about C A curvature that's usually very noisy. Mhm |
|
| 173:59 | So um uh Carlos since the anisotropy usually weak when it's, you |
|
| 174:06 | measured as a rock property, uh effect on the AC O intercept is |
|
| 174:11 | small, true or false. I it's false professor. And why is |
|
| 174:18 | false? Because I mean, the can be uh yeah, you, |
|
| 174:22 | , you were actually showing that we have a significant significant impact in |
|
| 174:28 | in the response of the A what said is true, but you did |
|
| 174:32 | read carefully. See it says the Aviel intercept, you were thinking |
|
| 174:38 | the next quiz question, which is A vo gradient. So that was |
|
| 174:43 | trick question. So when you um uh do the uh the um |
|
| 174:51 | the alert for trick questions like OK. OK. Please. So |
|
| 174:59 | here's the next, even though um anti sci is usually its effect on |
|
| 175:05 | Avio curvature is usually large. Is true or false? I think it's |
|
| 175:17 | . Well, I'm gonna say it's because it says usually, and, |
|
| 175:21 | , you know, we don't have experience. The set on Avio curvature |
|
| 175:26 | usually very noisy. It's usually so . We don't even look at it |
|
| 175:29 | all. So, uh who, to say, what is the effect |
|
| 175:33 | anisotropy? I would say that this uh uh uh uh this is false |
|
| 175:39 | stated, although it might be we might learn this true. And |
|
| 175:44 | fact, there's another set of phd to learn how to use a vo |
|
| 175:49 | uh curvature. OK. Next as you uh uh if you have |
|
| 175:57 | estimate of delta as a function of for move out, you can use |
|
| 176:02 | to estimate the jump at the target . Is that true or false? |
|
| 176:12 | gonna say that's false because usually this is has low spatial resolution, |
|
| 176:18 | So you can't get a local jump a smoothly varying curve. Mhm |
|
| 176:24 | So uh uh uh from uh you , you just don't have the spatial |
|
| 176:31 | , you need to get D and we could, I would have solved |
|
| 176:34 | 3040 years ago. But uh uh , that's a false. OK. |
|
| 176:40 | , so much for polar and So now we're gonna deal with auth |
|
| 176:46 | symptom. And the first thing I is that uh you're probably familiar with |
|
| 176:54 | model called tilted polar on isotopy. here's a picture of a tilted mat |
|
| 177:03 | you can be sure that there beneath joints, there's um uh uh layers |
|
| 177:10 | uh the layers are uh uh uh originally polar an isotropic. And now |
|
| 177:16 | whole thing has been tilted. So common uh a model even today is |
|
| 177:23 | uh uh you assume tilted or an pain. But that's probably not plausible |
|
| 177:32 | you know, the same stresses which the dip probably opens up fractures. |
|
| 177:38 | so these uh here, here's a example, these fractures, these joints |
|
| 177:44 | parallel to the strike of the uh of the structure. That's not an |
|
| 177:50 | . These uh joints were opened up the same stresses which caused the structure |
|
| 177:56 | the same in the um in the place. If they were oriented at |
|
| 178:01 | random angle with respect to the you'd say that it's just an |
|
| 178:06 | but these are lined up parallel to strike. So what that shows is |
|
| 178:12 | uh uh probably a a common So these beds here have tilted or |
|
| 178:18 | symmetry. Why? Because these joints on the strike of the structure. |
|
| 178:26 | furthermore, so-called ht I horizontal transverse is never plausible in the sedimentary |
|
| 178:34 | And here's the reason hori horizontal transverse could come from a single set of |
|
| 178:43 | circular fractures in an otherwise isotropic And in fact, this was the |
|
| 178:49 | that we first had when we first thinking about is and isotropy uh uh |
|
| 178:55 | years ago. Uh But here's the , there are, there aren't any |
|
| 179:00 | fractures anywhere. In fact, the ones which we're most interested in are |
|
| 179:04 | shaped like we showed in the previous . Uh Furthermore, in the segmented |
|
| 179:10 | of the background is always an So both of these um assumptions are |
|
| 179:17 | . So uh we should um not a TIHT I assumptions today. Uh |
|
| 179:24 | were suitable when I first came into business, but not today. |
|
| 179:32 | it would be nice if we could fracture. But it turns out that |
|
| 179:38 | we can't, the most realistic approximation that of orthorhombic an isotropy. So |
|
| 179:46 | is an air photo of a, place in um uh Southwester United States |
|
| 179:51 | you can see the laying. you know, it's gonna be uh |
|
| 179:55 | a uh uh gonna have floss which with the incident angle and you can |
|
| 180:03 | the joints and you can see the other joints and there's a good |
|
| 180:07 | why um uh uh we commonly see joint sets like this. I don't |
|
| 180:15 | we have time today to talk about that is, we don't always see |
|
| 180:21 | , but we frequently see it. it's obvious in this uh in this |
|
| 180:26 | here. So in that case, uh the, the symmetry matrix is |
|
| 180:33 | complicated. This is what we had polar symmetry for un fractured shells and |
|
| 180:39 | bed sequences. Now, for uh Robic Symmetry with uh which shows fourth |
|
| 180:48 | sequences with either one or two vertical crack sets. If the Ortho to |
|
| 180:55 | other, then that's gonna lead to Robic Symmetry. And that has um |
|
| 181:00 | uh an anti stick. This matrix like this still has a lot |
|
| 181:06 | of zeros off here. But we nine independent quantities in here instead of |
|
| 181:15 | . So for orthorhombic formation, it the, the symmetry of a |
|
| 181:22 | So uh uh it has three different symmetry directions uh perpendicular to each |
|
| 181:30 | And uh notice in this graph, have right handed cord system, uh |
|
| 181:36 | lay your uh your pro your index along the X one direction, curl |
|
| 181:41 | over uh to make it into the two direction and look which way your |
|
| 181:46 | is pointing. It's the downward X direction. If you're using your right |
|
| 181:52 | , if you're using the left it would be the opposite. So |
|
| 181:58 | the equations for velocity in terms of two angles, which is the angle |
|
| 182:08 | incidence here and the an angle those equations are very complicated. If |
|
| 182:16 | uh uh the first time I, looked at them on the screen, |
|
| 182:19 | printed them out and it, it about 10 pages of output for one |
|
| 182:24 | for the, for the P wave . It was just amazingly complicated. |
|
| 182:31 | I gave up but my friend, , whose name you saw earlier uh |
|
| 182:37 | than me. And he didn't give . And he looked at those equations |
|
| 182:41 | thought about them and he realized that in the case where um if you |
|
| 182:50 | at only um propagation in this one it, it has the those complicated |
|
| 182:59 | reduced to the equations for polar anisotropy approximation. And that's true, not |
|
| 183:08 | for that this plan, but you know, a any plan parallel |
|
| 183:11 | this one has the same um um same polar anisotropic media. And we |
|
| 183:18 | how to handle all ANAs tropic And furthermore, if you looked at |
|
| 183:23 | o this other plant also true, it's a different uh uh uh a |
|
| 183:29 | um or an a striping medium for plant than for this one. So |
|
| 183:39 | um look to see what we uh uh let's parameterize that system for those |
|
| 183:46 | planes. So in the one plane have uh uh a P wave, |
|
| 183:51 | P wave velocity, vertical shear wave and three antihero parameter here. And |
|
| 183:58 | the other plane, you have the vertical uh uh P wave velocity and |
|
| 184:04 | a different shear wave velocity and uh more anisotropic parameters. And it looks |
|
| 184:12 | you have nine parameters in all you know, four plus four plus |
|
| 184:17 | makes nine what you need, but is smarter than that. And he |
|
| 184:21 | carefully at these at the, and saw that within this set of, |
|
| 184:26 | uh parameters. You uh don't have 13, excuse me. Uh You |
|
| 184:34 | have AC 12, here's AC here's AC 23, but you don't |
|
| 184:39 | AC 12 anywhere. So he made uh out of whole cloth, he |
|
| 184:44 | up another parameter which has a functional like this except it's got AC 12 |
|
| 184:50 | there. And so now we have set of nine parameters with uh which |
|
| 184:56 | all the hook in parameters in And we know how to uh handle |
|
| 185:02 | anisotropy. In that case, for , the P wave anisotropy has a |
|
| 185:08 | phone. It looks exactly like we before. Except that look here, |
|
| 185:14 | anti sucky parameters have an auth dependence them. So if you want to |
|
| 185:20 | what that is and this is what is. Here's the azimuthal dependence of |
|
| 185:26 | , it's got delta in one of planes, one of the vertical planes |
|
| 185:30 | the trigonometric factor and uh plus delta uh the other plant, another trig |
|
| 185:36 | factor, et cetera. And so here that this is the uh the |
|
| 185:43 | , this is an elliptical as an variation. So if we look at |
|
| 185:50 | offsets, we're gonna see only elliptical of delta. OK. So let |
|
| 186:00 | show you this in real data. is data from Lynn et all in |
|
| 186:06 | . This is a different len than one that I just mentioned that uh |
|
| 186:10 | L also a woman um was an colleague of mine back in the |
|
| 186:15 | And then she left Amaco and she up a company of her own called |
|
| 186:20 | Incorporated and you can hire them. uh I think you can just uh |
|
| 186:26 | Lyn Incorporated and you can hire them do specialized data processing even today. |
|
| 186:34 | , um for all this time, , she's made a living uh as |
|
| 186:38 | boutique processing house. And so this set comes from a study she did |
|
| 186:44 | 1999 for the, the US Department Energy and it's a wide Asus |
|
| 186:51 | And so uh uh look what she's . She's taken the data over |
|
| 186:56 | this is a time series of data she's made a narrow and it's wide |
|
| 187:02 | data with all asthma represented in this set. And she took uh uh |
|
| 187:08 | took a small uh asthma sector around degrees of asthma from zero from 0 |
|
| 187:15 | 10 degrees. And that's plotted here is uh about 20 degrees, 30 |
|
| 187:21 | and all around here to 180 And then of course, it wraps |
|
| 187:26 | the rest of the compass, wraps it because of the scalar theorem of |
|
| 187:32 | . And so I want you to on this um uh uh event down |
|
| 187:39 | and I want you to ignore the hyperbolic move out here and, and |
|
| 187:45 | on, she, she's applied uh short spread hyperbolic move out to this |
|
| 187:54 | only that she wasn't interested in And so uh you can see that |
|
| 188:01 | right here, the uh layers are . But if you go around the |
|
| 188:09 | , you see they get to be and less flat and here they're, |
|
| 188:13 | straight but they're not flat. And you keep on going around the circle |
|
| 188:18 | it gets better and better and then comes back here to flat. So |
|
| 188:23 | all of these are flattened with the uh same applied velocity function. And |
|
| 188:31 | see the difference between where it's most and where it's most incorrect, that's |
|
| 188:37 | degrees difference. That's the 90 degree in move out velocity as a function |
|
| 188:46 | as muscle uh variation right there in of your eyes. Nothing difficult was |
|
| 188:53 | to uh to show this. Uh know, if you do some kind |
|
| 188:58 | migration with uh uh anti sci beer , or whatever, it's always a |
|
| 189:04 | business and you never know where you have made a mistake. But |
|
| 189:08 | basically, all she did was sort data into subsets and apply an AIC |
|
| 189:14 | out. And so she couldn't possibly made a mistake. So because of |
|
| 189:21 | previous formula about um uh let me up here. Yeah, because of |
|
| 189:28 | formula here, we can use that to understand what happens between the |
|
| 189:38 | Now, let's uh uh think about . We're looking at um of the |
|
| 189:46 | of orthorhombic anisotropy on seismic. It's complicated, I would say, not |
|
| 189:54 | . But it's uh um uh it's when I was a boy back in |
|
| 190:01 | 19 forties. My father was finding in uh East Texas as a geophysicist |
|
| 190:10 | he didn't know about any of He didn't know about Ortho Robic |
|
| 190:16 | He didn't know about anti soy. didn't know about anything and he was |
|
| 190:20 | finding a lot of oil and it's our while to ask ourselves, how |
|
| 190:24 | those guys back in the day? they were so ignorant, they weren't |
|
| 190:29 | but they were ignorant. So many we know today, they didn't know |
|
| 190:36 | and yet they found a lot of . How could that be? |
|
| 190:39 | here's a partial answer back in that , he was doing one D size |
|
| 190:44 | , excuse me, two D size of a single line of acquisition. |
|
| 190:49 | furthermore, he was using short So he didn't know or care about |
|
| 190:56 | hyperbolic. Move up. Here's a of his line, his uh two |
|
| 191:04 | acquisition line right here, going at angle uh uh with respect to the |
|
| 191:09 | Ortho IIC symmetry. He knew nothing this, but he's assuming that the |
|
| 191:18 | are gonna go down flat line country East Texas. So he's assuming the |
|
| 191:24 | are, are uh uh uh the are gonna lie in the vertical plant |
|
| 191:29 | this acquisition line. Well, he know it, but you see the |
|
| 191:33 | rays were actually bouncing off to the and coming back into his receivers. |
|
| 191:39 | didn't know that all he knew was had uh uh to the light flat |
|
| 191:45 | uh geometry and he said, I for sure. Now the rays are |
|
| 191:49 | be lying in this same vertical plank that whatever images he was making were |
|
| 191:58 | , not too much, but some bouncing at the mid, near the |
|
| 192:03 | here. Um And he had uh hyperbolic move. So here's this hyperbolic |
|
| 192:15 | and he didn't know it. But there was a uh an anti parameter |
|
| 192:20 | which he didn't know about. And was the, the one which is |
|
| 192:25 | for this angle. So he didn't about this separation between anisotropy and uh |
|
| 192:32 | . He didn't know about the Asmal variation because it wasn't, it wasn't |
|
| 192:36 | his data and he used these numbers convert time and depth and he got |
|
| 192:44 | this time, but everybody got his . And so he didn't worry about |
|
| 192:48 | much uh found a lot of This is why guys in, in |
|
| 192:56 | era were so successful despite the fact they were so ignorant compared to what |
|
| 193:01 | know today. Now, let's think azimuthal variation of Avio uh uh move |
|
| 193:10 | , move out velocity, not move out velocity. Uh This is |
|
| 193:13 | problem that we, that I showed uh that Lynn was uh looking |
|
| 193:18 | So this is our one D tailor . And for uh if we have |
|
| 193:23 | variation, if we have uh uh 3d survey, uh oh uh land |
|
| 193:30 | , uh marine survey, whatever we're have uh uh offsets in the X |
|
| 193:35 | , offsets in the Y direction. uh Mr Taylor says that if we |
|
| 193:40 | first order Taylor expansion in small X small Y, you get three terms |
|
| 193:46 | this, you get an X square , you get a Y square term |
|
| 193:49 | you also get an xy term. that is the equation for any lips |
|
| 193:55 | in the uh in the plane of map. So you never get any |
|
| 194:02 | in the vertical plant. But you got any ellipse in the plane of |
|
| 194:05 | map if you restrict your yourself to offsets. And you can see there |
|
| 194:11 | 123 parameters to be determined here and , they've been given uh complicated names |
|
| 194:18 | , but you can see there's only parameters to, to determine. So |
|
| 194:23 | me show you um OK, some to illustrate this point. So this |
|
| 194:33 | a cartoon showing a single uh uh with hyperbolic move out from wide estimate |
|
| 194:39 | . And uh uh uh you can there's some uh a scatter about the |
|
| 194:44 | the best fit hyperbole here. And offsets are represented here uh uh |
|
| 194:49 | in this and the little pics here give the, the tops of the |
|
| 194:55 | peaks in the survival. And so looks like a bunch of scatter. |
|
| 195:01 | We did the best job we could this uh average. But if you |
|
| 195:05 | closely, you can see that the from the north and the south come |
|
| 195:12 | early and the arrivals from the east the west come in late. And |
|
| 195:18 | can you see that the uh uh discrepancies at far offsets are bigger than |
|
| 195:24 | discrepancies at their offsets. So these patterns in the data. And whenever |
|
| 195:29 | see patterns in the data, uh you know, your uh patterns in |
|
| 195:34 | residual, you're look, you you're looking at uh data, not |
|
| 195:39 | noise. So when you remove uh that hyperbola is flattened uh |
|
| 195:48 | then it looks like. So and this looks like it's been uh |
|
| 195:53 | doesn't it? But I move out uh NMO stretch, but it's |
|
| 195:57 | you see, uh it's the thighs basically of my um uh pointer. |
|
| 196:03 | Now I'm gonna go back and you see it's the same size here. |
|
| 196:07 | we haven't stretched anything. We that's just the way it looks. |
|
| 196:12 | more clear to the uh to the to the eyeball. Now that we've |
|
| 196:18 | the best fit hyperbole. No, good way to look at this kind |
|
| 196:28 | data is to look at the not by mixing all the offsets |
|
| 196:33 | So this is what we call an uh ordered gather. But uh uh |
|
| 196:42 | I'm gonna show you what happens if sort the gather by aims. |
|
| 196:47 | before we sort it by. So is from 0 to 100 and 80 |
|
| 196:51 | . So the, so before we it, we wanna limit the |
|
| 196:54 | So we have only uh uh say um uh uh 9000 m and 10,000 |
|
| 197:03 | offset between 8500 m off. So somewhere range limiting and then an animal |
|
| 197:12 | it. And then the uh you're likely to see a um uh |
|
| 197:16 | variation like this. These are the uh picks that you saw before, |
|
| 197:22 | off of the um uh top of peaks. And number one, you |
|
| 197:28 | that not all asthma are recorded, always happens that when you do a |
|
| 197:32 | asthma survey, some as are not . So here we have some blank |
|
| 197:39 | then the events here are not uh . And so all the uh we |
|
| 197:47 | then uh uh we, we see the, the peak right here is |
|
| 197:52 | degrees away from the trough. So can remove it with the uh with |
|
| 197:56 | parametric residual move algorithm. Um All do is uh uh estimate those uh |
|
| 198:04 | uh uh the amount of that ellipsoidal move out velocity. So we uh |
|
| 198:13 | the uh slow velocity, the fast and the, and the auth uh |
|
| 198:19 | the orientation of the ellipse. So me show you what this looks like |
|
| 198:23 | real data. This comes from Val , talked about that before val. |
|
| 198:32 | comes from um off to the side the uh the data quality is |
|
| 198:36 | This is the top of the, the reservoir right here. It's obviously |
|
| 198:42 | flattened very well. But if you closely, can you see this jitter |
|
| 198:49 | arrival times, nearby traces have uh uh have uh differences in arrival times |
|
| 198:56 | several uh milliseconds, maybe a 10 or more. Uh uh And then |
|
| 199:03 | changes back and forth in a, rapid variation. You can't imagine any |
|
| 199:10 | of, of one B structure which gonna uh do, do that. |
|
| 199:17 | so um this is a conventional gather , sorted and flat. So now |
|
| 199:25 | gonna do it as, as sorted limited and now we put it as |
|
| 199:31 | function of Asma. So number you see that there are some places |
|
| 199:35 | there's no data. And number there are places where there's early arrivals |
|
| 199:41 | then nearby places which are later So I thought this, this uh |
|
| 199:47 | ship, I'm gonna go back, , look at this place when I |
|
| 199:51 | off the white LA. And so heavy black um uh uh first arrival |
|
| 199:59 | in right here and then it comes later over here. And so um |
|
| 200:12 | what this the 90 degrees difference in . This is exactly what Lynn showed |
|
| 200:19 | her data set. Now, I'm show you as a mostly an isotropic |
|
| 200:25 | vo. So to make it we're gonna use only the two terms |
|
| 200:31 | , of a Avio gradient and uh 22 terms of, of Avio |
|
| 200:37 | We're gonna ignore this four third term here. The gradient term has all |
|
| 200:42 | features in it. It has a jump in uh uh vertical uh |
|
| 200:48 | VP it's got um uh a jump vertical shear modulus. This is modified |
|
| 200:56 | from what you're familiar with. And it's got a jump in delta as |
|
| 201:00 | function of asthma. And uh so is where the differences are from uh |
|
| 201:08 | what we saw before. So in of verbals, let's just uh um |
|
| 201:13 | all the uh um the terms which as smoothly verbal out of here and |
|
| 201:20 | them um uh into one term. inside here is a bunch of stuff |
|
| 201:25 | I don't want to show you But uh uh it's at a single |
|
| 201:30 | with this kind of as a little . So now let's look at some |
|
| 201:37 | um white as with data. And this is uh uh an amplitude data |
|
| 201:47 | uh uh this is the kind of curve that you're accustomed to looking |
|
| 201:51 | You see, it's got an intercept and curvature and you also see a |
|
| 201:57 | of scatter. So this is a three term a curve. So uh |
|
| 202:05 | here up here is another data point not part of the legend, that's |
|
| 202:10 | data point way out. Uh uh uh couldn't possibly think of any physical |
|
| 202:17 | for it to be there. So is a procedure in statistics called robust |
|
| 202:22 | fitting, which enables you to eliminate like this. But don't eliminate terms |
|
| 202:28 | this using uh a standard uh criteria deciding uh we're gonna just throw this |
|
| 202:35 | , but we're not gonna throw out one. So you ask your |
|
| 202:41 | the question is this variation of the or is it signal the way you |
|
| 202:49 | that question is you look for pattern the variation patterns in the residual. |
|
| 202:56 | one way to look for it is rep plot this data in terms of |
|
| 203:01 | . So now you see it has coherent variation with asthma, there's still |
|
| 203:07 | lot of scatter, but you can see a high zone and a low |
|
| 203:11 | here. And now let's look at same real data set. Well, |
|
| 203:15 | is real data from elsewhere in, uh vowel hall. But uh uh |
|
| 203:20 | look at the same vowel hall data that we just saw uh asthma, |
|
| 203:25 | range limit range with asthma. And you see that uh the uh uh |
|
| 203:33 | bright amplitudes are here and the dim are here 90 degrees separation. That's |
|
| 203:44 | , that's a, an, an variation in amplitude just like the elliptical |
|
| 203:51 | and move out that we saw a slides ago. So let me look |
|
| 203:57 | a map view here um of that effect. So this is about 505 |
|
| 204:04 | m by about 2000 m, something that. And you see in, |
|
| 204:11 | each pixel of the map, this a map view in each pixel, |
|
| 204:16 | a little arrow without an arrowhead. so the length of the arrow tells |
|
| 204:20 | the amount of Ainu variation ra and orientation of the arrow indicates the uh |
|
| 204:29 | hot direction. And so the same variation is uh shown in color. |
|
| 204:35 | uh all the the bi the big variations is in red, see down |
|
| 204:40 | in blue hardly in amplitude variation and scale is given over here. So |
|
| 204:44 | red, it's 200% variation. We're talking about small effects here, 200% |
|
| 204:53 | in Avio gradient. So maybe you see a right angle here or maybe |
|
| 204:59 | . Uh I always look for right in this kind of data and I |
|
| 205:03 | see them, maybe I'm fooling who knows. Uh But you can |
|
| 205:08 | that if you were doing a two survey along this line here, as |
|
| 205:12 | as you came into this area, would immediately realize that you had entered |
|
| 205:18 | Avio anomaly area just looking at the . But if you came at it |
|
| 205:26 | this direction, you wouldn't see a because all because uh these gradient is |
|
| 205:31 | vector quality, all the variation is this northwest direction. There's no, |
|
| 205:36 | no variation in the northeast direction. this, this line is perpendicular to |
|
| 205:42 | these other, these other lines So uh uh let uh this is |
|
| 205:48 | small study. Let's look at a study. This is about five kilometers |
|
| 205:53 | about uh 15 kilometers. And uh can see uh it's the same uh |
|
| 206:01 | bar basically, except that the maximum white instead of red. And also |
|
| 206:06 | can see a dark area in here no variation that's corresponds to these navy |
|
| 206:14 | um uh colors down here. and it means that where we didn't |
|
| 206:20 | the data or the data was so that we didn't believe it, we |
|
| 206:25 | colored it navy blue and that was clever thing to do because it took |
|
| 206:32 | away from our eyeball and then all colors that we can see are gonna |
|
| 206:36 | things that we do believe in. , I gotta tell you about these |
|
| 206:41 | uh red lines here. This is uh uh BPS, um permanent insulation |
|
| 206:50 | receivers. They buried 10,000 seismic receivers the sea floor about a meter down |
|
| 206:58 | the mud along these lines, connected all up to uh uh uh to |
|
| 207:03 | central platform which lies right here. then from there, they sent a |
|
| 207:08 | cable to the shore. And so they were a uh able to uh |
|
| 207:14 | these things, uh uh uh you , analyze them onshore. The reason |
|
| 207:21 | this expensive installation of c ocean bottom was so they could do cheap rapid |
|
| 207:34 | D reshoots. Think about this, you do an ordinary um uh four |
|
| 207:39 | survey, you do a 3d survey . That's your baseline survey. It |
|
| 207:44 | cost to five or $10 million and uh you come back about a year |
|
| 207:49 | and do another one. And it costs five or $10 million and you |
|
| 207:53 | never do a third one because it's expensive. But BP had the idea |
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| 207:59 | if they spend uh um uh $20 up front to bury the receivers in |
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| 208:06 | sea floor, then the seismic reshoots be very cheap, maybe $50,000 |
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| 208:12 | Why is that? Because they they come back with a uh uh |
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| 208:17 | of a large um uh uh seismic vessel with 10 kilometers of cable stretching |
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| 208:24 | uh you know, on a small with only a source behind it because |
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| 208:28 | receivers are already in place. And with that, that's a very good |
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| 208:33 | . And they've done now over 20 and over this field uh uh using |
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| 208:38 | clever idea. And uh furthermore, can see that the, the, |
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| 208:44 | can see that platform is right here the middle and we, we only |
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| 208:51 | instrumented half of the field, there's half of the field over here that |
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| 208:55 | didn't instrument, obviously, the platform gonna be in the middle of the |
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| 209:00 | . Uh uh Because when we did , it was expensive and we, |
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| 209:04 | weren't sure whether it was gonna be . So then after we operated this |
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| 209:09 | of the field for a while, we realized this is very useful. |
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| 209:13 | we uh instrument the other half. that happened actually that after I left |
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| 209:19 | , so I don't have any, data from there. Um So notice |
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| 209:26 | values are only shown where the consonants the fit is more than 95% of |
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| 209:32 | . So let the maximum values show than 100% difference in um avi O |
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| 209:40 | as a myth damage. So let's in on the northwest corner. You |
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| 209:43 | see areas here like in here where little pixel is calculated independently and uh |
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| 209:50 | has a consistent direction, then there's no uh a narrow uh transition zone |
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| 209:56 | then another area over here again, uniform with a different orientation. And |
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| 210:03 | that's typical of uh of uh fractures my experience is that they're locally uniform |
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| 210:12 | a narrow transition zone. And then else on the other side of the |
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| 210:16 | zone, let's look at the southwest , the same corner thing, a |
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| 210:22 | at some data that goes into So this shows the set of uh |
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| 210:27 | a for a given point a given in that map. This shows the |
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| 210:32 | of asthma and, or, and offsets that went into that. And |
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| 210:36 | shows uh the uh Avio gradient. can see there's lots of scatter but |
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| 210:43 | a, a well defined uh sinusoidal that goes through there here shows um |
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| 210:54 | another case with lots of scatter. uh here, the scatter is so |
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| 211:01 | that you don't believe the variation. is not a straight green line |
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| 211:05 | This is a, a Sinusoidal um uh variation with a small uh a |
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| 211:12 | variation, but we don't whatever it , we don't believe it because of |
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| 211:16 | large value amount of scatter. And are standard statistical ways for you to |
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| 211:22 | when you want to believe it or . This is the same data set |
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| 211:28 | a different color bar. And we that to our colleague in Norway, |
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| 211:33 | knowing what these colors meant. And sent us back this curve. The |
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| 211:38 | thing about this curve is that he also the uh central platform with the |
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| 211:45 | uh the boreholes. They go straight from the platform and then they deviate |
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| 211:49 | all corners of the field. And see that every bore hole ends up |
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| 211:54 | a patch of color and there are patches of color except for this one |
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| 211:59 | don't have a borehole in them. what that means at the uh what |
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| 212:05 | means uh you gotta realize at the of this bore hole, they perforate |
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| 212:09 | lining and they, I inject um fluid or they produce uh hydrocarbons from |
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| 212:16 | perforated part. So that when you the perforations in like, so that |
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| 212:21 | that this, these colors were caused those operations in the field over the |
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| 212:27 | 15 years, uh uh uh uh activity. And when we see them |
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| 212:34 | in the uh uh surface seismic now take a look at this |
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| 212:42 | Oh I should be operating uh by way, this is the sec. |
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| 212:45 | is the second survey. This is first one, they tell mostly the |
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| 212:48 | story. We'll come back to the between those in a second. You |
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| 212:52 | this one missed it. It's very that uh when they put this well |
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| 212:56 | here, they were expecting to produce a, an area around like. |
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| 213:00 | instead all the changes came from What that means is that there's a |
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| 213:05 | barrier in here somewhere which which prevents fluids from this side of that bore |
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| 213:11 | from entering this uh borehole. uh what that means is there's an |
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| 213:16 | uh uh there's a exploitation opportunity put another borehole right in here and |
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| 213:22 | produce this all multimillion dollar uh uh to be made from high tech to |
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| 213:30 | f So now let me show you difference between these two. So those |
|
| 213:35 | were taken uh uh with uh uh uh the time difference here between uh |
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| 213:41 | three months time, between here and . And you can see that these |
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| 213:44 | the differences, see the differences all in the same places where the uh |
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| 213:50 | just grow over time. So that the uh one of the uh the |
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| 213:55 | fidelity that you can measure these things high temporal resolution as well as vertical |
|
| 214:02 | . So folks, there are many topics uh uh that I uh would |
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| 214:08 | to talk to you about, but have run out of time today. |
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| 214:12 | so um I wanna uh uh uh leave you with this with this result |
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| 214:21 | this study. Uh uh Today, have talked about three sorts of oo |
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| 214:29 | uh effects from seismic anisotropy. All which come from the fact that the |
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| 214:36 | and the kinds of rocks we explore is usually small, less than |
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| 214:42 | Nonetheless, there are three types of you might see of that we call |
|
| 214:49 | second order effects, which are also of the order of 10% like in |
|
| 214:53 | move out velocity, they're not but they're small of the order of |
|
| 214:57 | . And there are large effects, effects. That comes for example, |
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| 215:04 | ale from the same small anisotropy block , you get a 100% you can |
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| 215:13 | a 100% effect difference in the A Grady and then there are zero order |
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| 215:19 | which are completely new, completely um seen at all in isotropic seismic. |
|
| 215:25 | that includes, for example, sheer splitting, which we didn't have a |
|
| 215:29 | to talk about today. So with , what we're gonna, what um |
|
| 215:34 | we're gonna do is uh um all to uh uh close and uh since |
|
| 215:42 | is out, I have to hurry of here. Um But what I |
|
| 215:47 | do when I get home tonight, gonna send you all uh in the |
|
| 215:52 | exam and it will be in the of a, of a, of |
|
| 215:58 | attachment to an email. And so you do is uh uh you, |
|
| 216:03 | look at that and set it aside you have three hours to look at |
|
| 216:08 | and then you uh uh uh open attachment. My advice is to print |
|
| 216:15 | off on paper so that you can in your answers by hand and you |
|
| 216:19 | have to worry about the logistics of uh entering stuff by the computer. |
|
| 216:25 | then once you uh uh and it's be uh unlimited time, open |
|
| 216:30 | open notes. Um So it's a of your understanding, not of your |
|
| 216:37 | , not of your mathematical, not of your understanding. You will turn |
|
| 216:42 | this in before midnight on Wednesday, turn it into Utah when he collects |
|
| 216:48 | all, he will send them to . Um your eye um you will |
|
| 216:54 | unlimited time, open, book, notes, no consultation with anybody |
|
| 217:00 | So se uh, set aside some at least three hours when, |
|
| 217:06 | you, uh, will be undisturbed you did this. So, with |
|
| 217:11 | , uh, I should say there's a pleasure to teach you and I |
|
| 217:15 | you do well on the exam uh, I will grade the exams |
|
| 217:21 | them and get them back from your . Oh, stop sharing right |
|
| 217:27 | And uh, so for you folks remotely, uh I'm going to say |
|
| 217:33 | and thank you. I will um uh expect to hear from you via |
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| 217:40 | by Wednesday from me later tonight after . Thank you. Thank you. |
|
| 217:46 | you. Bye |
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