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GEOL7324 Rock Physics - Day4 10-29-2022 AM
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00:00 good. Yeah. So uh you this better than I did. You
00:06 how uh the two sayA equations fall along the mud rock line as to
00:14 free of formation equations. Okay, um In understanding these BPB. S
00:28 as I mentioned before, we start the VPMV. S for the pure
00:35 . So here are some examples of . So, cal side about 1.93
00:42 varies between 1.7 and 1.8. Uh is interesting because we'll come back to
00:52 because if you think of the rocks the gulf of Mexico that we deal
00:57 in in in the Mississippi Delta, example, where we have so many
01:02 . The main sentiments we deal with sandstone, shale and salt. So
01:08 be interesting to see where assault falls that muscovite. You know, if
01:14 had to take clay and uh cook all the way and you know,
01:18 the hardest clay imaginable. That would your, your muscovite. Um Well
01:25 more directly at clay minerals, Courts . PBS slightly less than 1.5 brain
01:34 , 2.65. Uh And hide right also has a high Vis PBS ratio
01:42 to court. So you see everything has a higher dp B. S
01:47 . Excuse me. So, and . Even those p wave velocity is
01:52 similar to courts. It could be based on its V. P.
01:57 . S ratio. Clays are a bit more problematical. Uh different clays
02:05 different uh what we call grain And here we have various extrapolations to
02:15 clay uh yielding be PBS ratios. could be high or low in light
02:21 similar to muscovite. Uh Usually clays some water in them and the more
02:29 is in them, the higher the . P. B. S.
02:35 . So um since rocks are aggregates mineral grains, we expect the velocity
02:42 highly lift ified, very low Rock to depend strongly on the velocities
02:48 the grains on the other hand. I'll show you from sphere pack
02:54 we also expect the velocities of unconsolidated to be weakly dependent on the grain
03:04 . So in a V. B. S. Cross plot for
03:06 monumental alec rock, water saturated, mineral velocity would be one endpoint uh
03:16 another endpoint would be the higher ferocity packed. So we would tend to
03:23 a trend between the velocity of the and go towards the velocity of water
03:29 ferocity goes to 100%. So let's for pure sandstone. So this is
03:37 give me something similar to the mud line, but slightly lower V.
03:41 . B. S. So this clean sandstone is very low shell.
03:47 have a V. P. S. Relationship and it's pretty
03:52 Um One reason it's so linear, know, we thought I thought about
03:57 for a long time. Why is linear? There's no reason why it
04:02 to be linear. But one endpoint a V. P. Which is
04:06 courts as a V. P. . S ratio of 1.5 or
04:11 Um The other at the other end just have sphere packs. And as
04:16 show you their B. P. . S ratio tends to be their
04:20 . Be PBS ratio also tends to 1.5. We add water and it
04:28 us right on this line. Lime for example, are curved. Uh
04:36 picket line we saw the PBS 1.9 a lot of the range. But
04:42 we go to very low velocity marine that are carbonate we send tend to
04:48 towards the the water point. So forces instead of staying along 1.9,
04:55 got to come back to increase the . P. V. S ratio
04:58 water. And in fact, that's sandstone is doing. Its at 1.5
05:04 . But it's approaching the water velocity shiro zero shear wave velocity. In
05:11 , it has to curve back towards . So there's some slight nonlinear parity
05:18 this end. Now for Claes, rare to have a pure clay.
05:27 , these are measurements on clay rich but we could extrapolate those values to
05:35 clay. And so you would get of an envelope here. So that
05:40 be our pure clay rock trend and . We have a much smaller range
05:51 measured dolomite velocities by the way these . P. V. S ratios
05:57 to be lower than pickets, 1.8 . S. So these are in
06:02 one point seven's. So putting them together, we get an important plot
06:10 Chevron did. We're gonna cross plot . P. B. S versus
06:16 a lot of important conclusions from this . So these are the trends for
06:22 different uh with Allah geez. And one thing you'll notice that for low
06:29 live stones, they're acting just like stones. Why? Because these are
06:35 of grades, right? So um you the theoretical modeling tells us,
06:42 I'll get back to that in a but tells us that the grains do
06:47 affect the V. P. S ratio of the sphere pack very
06:51 . On the other hand, 10, you know, pure clay
06:55 tend to be higher V. V. S. Ratios. Um
07:01 at gas sand though. A gas sandstone has a low V.
07:05 V. S ratio, irrespective of it's a hard rock or a soft
07:11 . So we'll try, we'll come to try to understand that by looking
07:15 measurements on dry velocity. So, gas Sanders, low velocity brian field
07:23 are high V. PBS. Gas or Lovie PBS. So there's a
07:29 distinction. So what this tells me in shallow rocks gas stands should stick
07:35 like a sore thumb with their P. B. S ratio.
07:39 that's why a video analysis works best in gas seeds. On the other
07:47 , when we get to harder The variation in length ology is a
07:53 bigger than the variation in the fluid . So in hard rocks, the
08:00 effect is covered up by little aaj variations. So just going just varying
08:06 Shelley nous of the sandstone. That be as big an effect as the
08:12 effect. And certainly mixing carbonates with stones can over print and be more
08:20 than the hydrocarbon effect. So there two regimes, low velocities where
08:26 B. O. Should work very as a hydrocarbon indicator. High velocities
08:31 A. B. O. Maybe of a lift ology indicator than a
08:36 indicator. Then there becomes the issue , uh what happens if we put
08:47 not only into a sandstone, but if we put gas into a
08:52 Uh This was work we did in mid eighties, this was before we
08:58 about shale reservoirs and we were seeing effects and shale. So this wet
09:04 line, something like the mud rock . Uh and then uh we had
09:11 gas sandstone line. We also did fluid substitution and filled the wet shells
09:17 gas and we get a gas shell and what we find is something in
09:24 . More recently we've studied uh P. V. S. Ratios
09:30 shale reservoirs and we found similar things the shale reservoir has a lower
09:38 P. V. S ratio than an inorganic shell. So this wet
09:44 line would be for an inorganic We put organic material and hydrocarbons into
09:51 shell and we lower the V. . B. S ratio. By
09:58 way, we have a trend for . This was a variety of coal
10:02 we had. And again, it's nonlinear relationship between V. P.
10:07 V. S in this case. and tend to have higher V.
10:13 . V. S ratio. So would be a B. P.
10:16 . S ratio to So all of uh have the PBS ratios higher than
10:23 . You start getting to the even here. B. P.
10:27 S ratio is higher than two. plotted Tufts, partially saturated Tufts from
10:41 radioactive waste sites uh that Lawrence livermore had and this was these were Tufts
10:48 in some prospecting up in Washington Um and again, Tufts tend to
10:56 their own V. P. S relationship. So, some conclusions
11:03 V. P. B. S . The lift ology discrimination is best
11:08 high velocities uh difficult to distinguish high shells from carbonates. For example,
11:19 , you have overlap between shells and by the way you also have a
11:25 with mixed with ology, suppose I sand and limestone. I could get
11:29 V. P. V. S equivalent to that for dolomite. So
11:35 the inverse problem going from the P. V. S ratio to
11:38 lethality is highly non unique by the , bringing back this plot, where
11:45 the mud rock trend be? It kind of be between the shell line
11:49 the sand line. Remember this is pure clay shell, This is a
11:53 court sand. So the mud rock would be in between. Um Now
12:05 we try to measure B. B. S. Ratios from seismic
12:09 , uh they're not very precise or . Um So there's a problem in
12:16 in estimating mythology using the PBS ratios . But we can conclude that the
12:25 between gas and brine saturation is large you have low velocities and it's small
12:33 you have high velocities. So soft , we can detect hydrocarbons more
12:38 Hard rocks. You have this strong over print, which gets in the
12:47 . Now this was a data set acquired um In the mid 80s and
12:55 was from a very special well logging . It had 40 receivers. So
13:01 velocities we were getting were very accurate uh these were measurements made in the
13:10 basin where you had a variety of , limestone sold dolomite, sandstone and
13:18 . And we draw a couple of on here for reference the mud rock
13:22 there and it pick its limestone lying and they kind of form envelopes.
13:32 . Um All the data tends to uh on these trends or between these
13:39 . So what, what are these data points? Well, we had
13:45 lime stones fall right on the limestone . We had Dolomites would fall in
13:53 mud rock and limestone. Uh, have sand stones, many and
13:59 many of whom plotted along the mud line, some of them slightly
14:06 so maybe slightly cal Karius. we had lime stones with very similar
14:13 . P. V. S ratios Dolomites. These would be sandy lime
14:18 or Shelly lime stones with a lot classic material in them. And then
14:25 there were assault measurements here right on mud rock line. So that's very
14:33 if the only brine saturated with ology we have our sandstone, shale and
14:40 than anything that deviates from it to anomalous li lo V P V s
14:46 would be hydrocarbons. If we saw lee high B P V. S
14:51 , we might recognize that as a . Okay, same thing just plotted
14:59 terms of poison's ratio. So these our trends which were re plotted in
15:06 ratio. So, um being that the limestone trend has a curve,
15:17 fit into it with the polynomial. then for the other lithography, jeez
15:22 are straight lines. So you'll use trend curves later. So remember that
15:28 here these RV PBS relationships for pure , jeez all right now we want
15:38 try to understand what's going on? don't know how to, why isn't
15:44 turned off. Hmm, seem to covering things up. Okay, so
15:58 want to understand these trends. So we're gonna do is we're gonna start
16:02 dry sand stones. And if we block VP VS. B. S
16:09 dry sand stones, what we find a constant V. P.
16:12 S ratio of about 1.5. This would be a person's ratio of
16:21 So even as we approach zero shear velocity, the p wave velocity goes
16:26 zero. So there's no intercept, the mud rock trend has plus
16:32 Right? So mud rock trend would somewhere here. The dry trend is
16:38 constant V. P. V. ratio. So, we want to
16:41 to understand that. Why is it constant V. P V. S
16:45 ? Well, we know that courts here as the V. P.
16:48 . S of about 1.5. Why it stay 1.5 as we, as
16:54 velocities get lower. So, that's question. Usually this bar turns off
17:04 I don't know why it's not turning . Not sure dr top, what
17:13 that do? Well, okay, , so, we're gonna look at
17:20 V. P. B. S and these remember we talked about
17:24 the different packing of spheres. these are uniforms fears In various
17:32 We have the simple cubic packing. that guy 48% porosity. Hexagonal.
17:39 packing. And face centered cubic packing pretty similar in ferocity on the order
17:45 26%. We're gonna look at the . P. V. S ratio
17:52 the sphere pack versus the V. . V. S ratio of the
17:59 vs the poison's ratio of the Graves is around here at .1. So
18:07 is saying that if I had a packing of spherical spheres, I would
18:12 a v. p. b. ratio between the square root of two
18:16 about 1.45. So Lovie PBS Remember, sands aren't perfect uniform spheres
18:25 they have other materials in them. what we tend to see is about
18:30 , you see some of these higher ratios, grains will give you higher
18:35 ratio sphere packs, but it's very to the poison's ratio of the
18:42 The maximum range here is V. . B. S from 1.4 to
18:50 . As the grains go from P. B. S of square
18:56 root of two all the way to . Right, so you have a
19:01 change in the V. P. . S ratio of the grains,
19:05 very little change in the V. . B. S ratio of the
19:09 packs. So, for rock forming were in this range here and you
19:15 see that there's not gonna be a of variation in the V.
19:19 B. S ratio of the square . So we understand the low end
19:24 PBS should be on the order of . It would be low earth was
19:30 clean and perfectly uniform spherical spheres, in the real world on the order
19:36 1.5. And we said courts also a V. P. V.
19:41 ratio 1.5. So not surprising surprising the end points of this plot Have
19:52 . p. b. s. 1.5 at both ends. More of
19:55 sphere pack here, more of the mineral appear. Well, what about
20:00 between? Well, one thing we do is we could change the the
20:06 of Iraq and we did this before heat cycling it. So we take
20:13 sand stones and we measure their velocities two different pressures appear at high pressure
20:21 down here at low pressure and you there along the dry line and what
20:26 happened as we've increased the pressure from to here, we've closed the natural
20:31 fractures but we haven't changed the P. V. S.
20:35 So in the dry rock the P. B. S ratio is
20:40 to the degree of fracturing. And we proved that by heat cycling the
20:47 . And we uh we drop the by heating them up, quenching them
20:53 introducing microfractures. So uh this is high pressure, this is at low
21:01 . So the effect of changing pressure adding or subtracting micro fractures by heat
21:08 or by changing the effective pressure. change the D. P.
21:14 S. Treasure. So we have packs down here, we have pure
21:18 up here and then we have porous in between doesn't matter too much what
21:25 porosity is, what its shape is what it's uh how much it is
21:32 the ferocity or fracturing the rock puts up and down this line, Changing
21:38 , puts you up and down this but it doesn't move you off the
21:45 . And we could do inclusion So we're gonna mathematically add penny shaped
21:52 and we're adding any shape cracks with aspect ratio spectrum. So a variety
21:58 shapes that is similar to that of sandstone just to pick an example.
22:05 so what we find is that if draw line there's a V.
22:09 V. S ratio 1.5 most of way, we're pretty close to
22:16 So what we're doing is we're increasing ferocity here, adding those those ellipse
22:23 oblate spheroid inclusions to the two And again, we don't change the
22:30 . P. B. S ratio much. Now this is an interesting
22:40 because what we've done is we've connected measurements on dry rocks which are the
22:47 symbols which follow the dry line. some cases the symbols got filled in
22:55 scanning etcetera. But you'll see if on the low velocity side of the
23:02 . That's the dry measurement. And is the mud rock line here.
23:08 , you see the brine saturated rocks or less follow the mud rock
23:12 The dry rocks more or less follow dry line. This was a
23:16 a sandstone, low ferocity cal Correa . So the uh velocity, the
23:25 result in was actually above the mud line. Another ambiguity. The degree
23:33 cal curious cement can have an So, these timelines are interesting and
23:41 come back to this later when we about fluid substitution, but in most
23:48 , uh adding uh let's say, started with a brine filled rock and
23:56 added air to the brian field what should happen VP should reduce bes
24:02 change very much. In this We're seeing V. S increasing because
24:07 the density effect. All right, here we're seeing a huge drop in
24:12 . P. B. S. ? So, something else is going
24:17 . If I start with a dry and I add water, uh
24:21 S. Is actually increasing a tremendous . So that's very strange. But
24:28 whatever is happening to the frame, matter how strange that is. And
24:33 are two things that could be going . We'll come back to this and
24:36 ask you to try to think of could be causing these effects. Sometimes
24:42 wave velocity uh increases. Sometimes it as we uh as we replace the
24:52 with good air. Okay, now could theoretically try to predict the shear
25:05 velocity without using the V. B. S trend. And we're
25:11 use that using fluid substitution. And fact that for the dry rock the
25:22 . P. V. S ratio 1.5. So we're not gonna use
25:25 brine saturated trend. We're gonna use dry trend. We're gonna fluid substitute
25:32 and see what the bride saturated results be. So we can do this
25:40 can say let's estimate the porosity from p wave velocity. Now, I'm
25:47 to assume a frame shear module Uh It's not the right show module
25:53 I just assume any frame share I let the frame shear module is
26:00 the frame bulk modulates. That corresponds k overview of one which corresponds to
26:07 . P. V. S ratio 1.53, which corresponds to a person's
26:12 of 0.1. So conveniently are dry for sand stones, persons ratio
26:19 That means frame share modular equals frame module asse. So that's the rock
26:26 without support from the fluids. We then use gas mains equation comeback and
26:34 the saturated bulk module asse. Given I could predict the p wave
26:41 I can compare that to the original wave velocity. And um if they
26:47 I guess the right share modulates if don't match, I modify the assumed
26:53 modular and I select the sheer module that gives me the minimum error in
26:58 VP. So from the dry frame . P. B. S.
27:04 could predict the brine saturated V. trend. And we do this for
27:14 samples. We have the predicted shear velocity and the observed shear wave velocity
27:21 we're on a diagonal there. Which it's a it's a good measurement without
27:27 . So it's accurate because that's perfectly and we're close to the diagonal and
27:33 not a lot of spread around So it's a precise prediction and it's
27:37 accurate prediction. We haven't used our saturated V. PBS trend but we've
27:45 VP to predict Bs by assuming the share module, Sequels the frame bulk
27:55 . Okay, so um we could the same thing for our sphere
28:01 Right? So here are laboratory there's the mud rock trend there,
28:09 our simple cubic VP VS. S trend. So what we're doing
28:15 we're varying the pressure on that on sphere pack. And we're seeing how
28:22 . P. And V. Vary. And these were measurements on
28:26 stands and they fall pretty much on B. P. V. S
28:31 . If I had a denser packing grains face centered cubic or hexagonal close
28:41 . I would have that trend. you see how that trend is trying
28:45 work its way into those points So we kind of have an envelope
28:50 . Right? We have the simple face centered cubic and then something else
28:56 over because now these solidified rocks. all of these they start with the
29:00 rock trend. They go fall below mud rock trend and then they're going
29:05 come back to the mud rock So let's let's do inclusion modeling and
29:11 what happens up here. So we're take courts and we're gonna add pours
29:16 the courts to decrease the velocities and what happens there. Oh but I'm
29:25 I jumped ahead before we do We want to see how the PBS
29:30 versus depth. So uh we're gonna gonna take Gregory's P. Wave velocity
29:36 depth data for uh shales. And sand stones, I told you,
29:44 know in Gardner Gardner Gregory, they these trend curves with based on 17,000
29:51 in the gulf coast. So I the mud rock trend to the shale
29:56 and that gives me V. V. S versus depth. You
29:59 that's a nice continuous function. And sand stones, you have this
30:06 right? And we said this is point at which the sandstone is fully
30:13 , right? So we've reduced the as much as possible by rearrangement of
30:19 and defamation of grains. And then that it's mostly semente shin taking
30:25 Uh So ferocity czar decreasing with we saw that velocities are increasing versus
30:33 . And as the velocities increase, V. P. B.
30:36 ratios decrease at a given depth. could have a difference in the
30:43 P. V. S ratio of versus sand. But most of that
30:47 because the shell is lower p wave than the sand and that gives it
30:52 higher than PBS ratio. So here comparing multi component data. So this
31:04 a p wave surface seismic shear Surface seismic measuring the interval velocities from
31:12 . And comparing the VP VS. . S. Of those interval
31:17 Uh So uh let's see, I I described I said somewhere what these
31:24 are. Uh I haven't so I'll to remember. So uh 67 and
31:33 were shells. Five was a uh I'm sorry 456 and seven.
31:43 got it backwards. 56 and seven shells. They plod along the mud
31:49 line. I should have had a here. 42 and one were
31:55 So they plot close to the mud line. Three was a gas
31:59 It fell along the dry line. this is field verification of what we've
32:06 happen in the laboratory and with sonic . But here we're seeing it with
32:11 seismic data. Now we could convert velocities to a ratio of Ma July
32:26 density. So if I have my trend in my mud rock trend.
32:34 That shows me the relationship between shear bulk module asse for the dry
32:40 remember we said they're equal. so a diagonal. And for the saturated
32:46 , the bulk module is is higher the share modules. We could plot
32:54 in a different way. We could poison's ratio versus compression velocity. So
33:00 looks a lot like one of the I showed you before, where you
33:04 the saturated rock increasing the poison's ratio as we go shallower or to lower
33:13 . Whereas the dry rock, the ratio doesn't vary. Okay, now
33:20 gonna do the inclusion modeling. this was a little bit out of
33:24 . What happened here. Uh It's little bit out of sequence.
33:30 so we have our mud rock we have our laboratory measurements uh superimposed
33:38 the mud rock line. And what gonna do is we're gonna do inclusion
33:44 . And we're gonna take this uh uh what we call the aspect ratio
33:52 . That's they call this the concentration each poor shape. Right?
33:57 we have equal pores. We have on the order of .1. We
34:02 cracks with an aspect ratio very small , et cetera. And we're only
34:10 to include the poorest that are larger .1. And when we add those
34:17 , mathematically, we get this red . Now we haven't extended it further
34:24 there is a theory, theoretical limitation how far we could go. On
34:31 other hand, when we include the spectrum. So we include the very
34:36 pores. That gives us the green , which is almost perfectly on the
34:40 rock trend? So you can see sometimes sand stones will be on or
34:46 the mud rock trend and sometimes And it has to do with the
34:49 shape. But these are minor details to the precision and accuracy of the
34:57 data. Right? As a initial , for example, if you're gonna
35:02 do seismic inversion to refine the guests the at the V. P.
35:08 . S ratio. The mud rock is a good starting point. It
35:13 you in the ballpark. Uh, if we take the time average equation
35:25 we do the same exercise, we fluid substitution with the time average
35:31 It gives us this line and it through most of those points there,
35:42 brings up the idea that if there I have rocks that obey the time
35:47 equation, then they should for p , then perhaps they'll obey a time
35:53 equation for shear waves. That would a huge advantage because share waves are
36:01 affected by the poor fluid. So you were using sonic logs to estimate
36:06 , you could do it better with waves. If you have hydrocarbons in
36:11 system. So, if there are hydrocarbons, your p wave ferocity estimate
36:17 going to be wrong, whereas the wave is gonna work. So is
36:21 a shear wave time average equation? the answer is yes. I remember
36:26 time average equation has the fluid transit as the parameter what is the uh
36:33 transit time for share waves? It's , right? The velocity is
36:40 So but you still get and I'll you if I if my rocks obey
36:47 P wave time average equation, they'll obey the share wave time average
36:52 So it's the share wave transit time the share wave transit time of the
36:57 , plus what's supposed to be the transit time minus? The share
37:01 The transit time of the solid. that's just an empirical coefficient. We
37:07 call it an effective transit time, it's not meaningful in in physical
37:16 So here I have rocks, I've I've cherry picked the literature, I
37:23 laboratory measurements where the p wave velocity the P wave time average equation.
37:31 then for those same rocks where share were measured, it turns out they
37:38 the shear wave time average equation. , kind of a nice thing.
37:49 let's take it one step further. We like the Raymond Gardner equation for
37:55 most liquefied rocks. So let's take Raman Hunt Gardner equation literally and let's
38:03 the values for share waves. So of the p wave velocity of the
38:08 , I'm gonna put the share wave of the matrix in And instead of
38:13 p wave velocity of fluid, I'm put zero in there and that gives
38:20 a share wave time, Ray martin equation and it's a very nice
38:27 Well, does that work? Remember said that for fluid substitution? The
38:32 behind Gardner equation isn't too bad. gets you in the vicinity or at
38:37 in the direction of the right Whereas the widely time average equation you
38:43 use to do fluid substitution. So investigate this guy a little bit more
38:51 the way. Um it's non trivial solve the p wave time average
39:03 I mean Ray martin Gardner equation for and you have never seen it in
39:07 literature and actually I solved it And here you have the ferocity
39:18 right? Uh using the Ramayana Gardener . So now I could put ferocity
39:27 here, I have ferocity, I substitute ferocity and I get this.
39:35 I could get uh the shear wave is equal to all of this
39:44 Yeah. Are you changing the Because I think the slides frozen showing
39:51 figure 19. I'm not sure which 19 is, can you describe figure
39:57 . So you with it's like the spectrum and the again from here it's
40:09 moving your laser pointer is like stuck the top and everything. So Oh
40:16 , that's because I paused it So let's resume share. Yeah,
40:26 paused. So could you see Yes. Okay, so what I
40:35 saying is that the time average equation through a lot of these points in
40:39 vs. V. S. So we did is we took the p
40:44 time average equation. We did fluid . We calculated what it would be
40:49 shear waves. Remember I showed you sequence back here where we could predict
40:56 S. Right. So given a versus ferocity trend, I could predict
41:02 versus ferocity. So doing that, get a shear wave time average equation
41:10 this process this effective transit time in fluid is not infinity as it should
41:17 for share waves. It's just a . It's just a number. It
41:21 no physical meaning. But if I points that match the p wave time
41:30 equation, they will match the shear time average equation. Okay, now
41:39 gonna go one step further and we're go to the Ramayana Gardner equation.
41:43 we're going to just assume the equation correct. And therefore these are not
41:49 empirical coefficients. They have physical If that's the case, I could
41:54 substitute shear wave velocity in here and shear wave velocity of the matrix of
42:01 solid material and for the fluid velocity becomes zero. That gives me a
42:07 simple equation for sheer waves. So good is that? Well, one
42:13 we did is we solved it We said here is solving the Ramayana
42:21 equation for ferocity and as I you don't see that in the literature
42:26 it's a pretty complicated equation. In you have to solve a quadratic
42:32 but there's only one real root. now we could eliminate ferocity from uh
42:41 shear wave equation and we could express shear wave velocity just in terms of
42:47 p wave velocity and other things which known velocity and the fluid and velocity
42:54 the matrix and the shear wave velocity the matrix. Right? So we
42:59 um compute the V. P. . S relationship without ever explicitly solving
43:06 the porosity. So we don't have know the porosity. Some aspects of
43:10 equation. Uh as VP goes to . P matrix, uh then
43:18 S goes to V. S. , V p B s Goes to
43:26 VPBS ratio of the matrix as the velocity goes to zero. So or
43:34 VP goes to vP matrix. So a gas field rock we get the
43:41 P V s ratio equals the P V s of the matrix,
43:45 is exactly what was the critical porosity assumes. So somehow, with the
43:52 martin Gardner equation, even though it's an empirical equation with no theoretical
43:59 there are aspects of it which are . It'd be nice if someday somebody
44:07 derive that equation theoretically. Okay, whereas the time average equation was linear
44:17 ferocity, the Ray martin Gardner equation nonlinear and um same for the shear
44:26 for a martin Gardner equation. So are very similar for p waves and
44:32 waves, although notice they cross at , slightly different points here. All
44:41 now, um really, if we the time average equation to predict
44:54 what we find is that if we're deep under very high pressure, we
44:59 a good prediction of ferocity. So we divide the true ferocity by the
45:05 ferocity, that would value would be to one as we get deeper and
45:16 . Now, if we take the of true ferocity to predicted ferocity,
45:20 is called a lack of compaction Right? So you have to to
45:27 the true ferocity. Uh you have uh correct with this compaction factor.
45:35 here the true ferocity, the predicted , if we're shallow, would be
45:40 higher than the true ferocity. that ratio becomes small. And for
45:48 time average equation, the compaction factors different for the p wave velocity time
45:56 equation in the sheer weight of time equation. Right, So that is
46:01 satisfying. It's not nice that you different compaction factors. For the
46:10 On the other hand, if we to the wily time average equation and
46:16 look at the ratio of compact compaction . P wave to s wave.
46:21 we find is that with the Rain Gardner equations. Uh that compaction factor
46:27 the same every place, whereas with time average equation, you have this
46:32 difference. And in fact here you're at a ratio on the average of
46:40 , right, So one more aspect the Ray martin Gardner equation, that
46:47 very satisfying. You don't have to about the degree of compaction. Let's
46:57 it one step further and let's predict vs. B. S. So
47:05 take the Raman Gardner equation, the wave and shear wave rain martin Gardner
47:11 . We get a line, we PVP get ferocity from the Rainman Gardner
47:20 . Then we do fluid substitution. sorry. We go the other way
47:25 , Start with B. S. the other way to do fluid substitution
47:29 predict VP. We get the same and both of those lines happen to
47:36 with our sandstone trend from observations VP . V. S. So everything
47:44 holding together the remote and Gardner gas mains equations and the empirical trend
47:51 all giving us the same VP V. S relationship. And we
48:03 compare this then to sonic log So here's a mars rock line,
48:09 is our uh roemer and Gardner And here we've included both branches
48:17 So we've gone to very low Remember Raymond Gardner had two branches,
48:22 I showed here was just the uh porosity branch. Now we're gonna go
48:29 the very high porosity ease and you see it's, it's kind of obeying
48:35 our measurements were doing. These were c. Two velocity measurements. And
48:40 see there between the Raymond Gardner equation the gas sand or the dry sand
48:47 . So the idea here is we have partial saturation in these gas
48:57 Yeah. And this is comparing the martin Gardner equation to our laboratory
49:03 And you see it does a very job of matching the sand packs and
49:09 , kind of forming a lower bound rocks will tend to be on or
49:18 this remote Gardner line. And so gonna be interesting. We're gonna come
49:24 to to this lower bound on P. B. S. We're
49:29 look at that again when we talk fluid substitution But it's time for a
49:35 . So we'll take a 10 minute and reconvene at 11:15. Now remember
49:48 velocity versus depth data. So if take all those data points predict shear
49:56 velocity and plot on a VP V. S plot, we find
50:03 we follow pretty closely to the roemer Gardner equation. Okay, one more
50:20 of um the PBS ratios is the with ology determination problem. So I
50:30 to go in the inverse direction. , I want to start with
50:36 P. V. S. And and invert for the rock type.
50:46 let's do a little experiment. We're go to the laboratory and we're gonna
50:52 V. P. B. S density measurements on a wide variety of
50:56 log types sand stones, lime Dolomites and mixed with Allah jeez,
51:02 gonna leave shells out because they complicate even more but suppose were restricted to
51:09 little, ala jeez, could we predict the lethality? And what we
51:17 is that we can if We have for all of our sand stones,
51:27 of the time they were predicted but there was a finite amount of
51:34 where they were misidentified as lime stones Dolomites. So if the sand stones
51:41 Shelly or if they were felt empathic if they were abnormally high B.
51:47 . B. S ratio because of poor shape they might be mistaken for
51:53 carbonate. But the situation gets worse lime stones, Lime stones, they
51:59 correctly identified only 80% of the time 20% of the time they were misidentified
52:07 Dolomites. And we understand very well this happens, when you have a
52:12 or Shelly limestone that will suppress the . P. V. S
52:16 Make it look like a dolomite. worst case scenario are Dolomites because they're
52:23 lime stones and sand stones. So they're misidentified as lime stones and sometimes
52:33 were misidentified as sands. So the is not unique. Okay so we're
52:43 do a few more exercises before we on to the next topic. So
52:50 first thing is to plot Watson's So you have the equations for poisons
52:59 , Let's make a plot of poison's versus K. Over mu. So
53:07 go to I'll stop sharing, you share your screen, we'll go to
53:11 spreadsheet. You'll find equations uh that to assassins ratio to the ma july
53:23 make that cross plot. Okay. exercise here and I don't understand why
53:37 is not collapsing. You see my of options here, don't you?
53:44 we don't. Oh you don't see . Oh. Oh good. All
53:49 okay so it's only bothering me. . Alright so now uh plot VP
53:59 . V. S. For these minerals and compared to the mud rock
54:11 . So I'm gonna give it back you. Okay. Yeah so the
54:25 then are that and hydrate Musca by dolomite, pull you off the mud
54:33 trend. Everything else is on the rock trend. Okay. Another exercise
54:43 up and these are VSP measurements in . Two VSP measurements. And so
54:51 can take these measurements and we can Vp VS. V. S.
54:56 this will be relatively easy because these we can plot this as one series
55:02 just V. S. In the hand in the first column. Vp
55:06 the right column. So I'm gonna it back to you and we're gonna
55:12 this then we're gonna take a break lunch. So let's do this
55:20 So you can see that first point is in the upper 10 ft right
55:28 zero depth to 10 ft. So right at the surface. So there's
55:34 good chance that that is above the table. You see my point.
55:40 those are partially saturated rocks. And did we say we would do would
55:46 if we added gas to Iraq we lower the V. P.
55:52 S. Ratio. So that first plus below the mud rock trend.
56:09 now let's do this. We have sand and shale equations. I get
56:17 them to you in that table for shell and sand. Let's use use
56:25 trends and then plot V. B. S versus VP for those
56:33 with Allah jeez into the mud rock . You see what I'm saying?
56:41 I'm gonna stop sharing now and give back to you. So how would
56:52 affect your use of the mud rock ? Um Well it's it's not gonna
56:59 totally representative of a clean shell, really a quartz rich shell at the
57:06 the high velocities. There we Okay now we're gonna look at the
57:18 of pressure on these V. B. S. Trends. And
57:22 do this we're gonna look at dr relations. This was his PhD thesis
57:30 stanford and he came out with this after my paper on mud rock.
57:36 I couldn't use his values. So would be very interesting to compare since
57:43 did his work independently. So we're to plot the V. P.
57:48 . S trend at different pressures. , so these are his values and
57:58 I want to make sure about one and that is um actually this came
58:06 a report that I wrote at arco it was typed by a typist.
58:13 what I found was that there were typos in the report. So I
58:20 have the type written values from the , but I also have uh a
58:30 clip of hans table from his paper you notice the signs are different,
58:39 the values are the same, Are the values the same? Uh
58:46 , pretty much. Right. So happening here is uh that his
58:56 he put the minus sign in the , I left the minus sign in
59:02 coefficient. Right. So don't get by that. But so what we're
59:09 to, what we have to do we have to vary porosity and uh
59:17 very V clay, we could do we did last time. We could
59:20 it for v Clayton. Well, and V Clague were one. So
59:25 could have plots of p wave velocity s wave velocity at different pressures.
59:32 , I'm saying I'm sorry, mvp ferocity ves versus porosity. And that
59:38 give me vp vs V s at pressures. You see what I'm getting
59:44 ? I think so, yes. , so um yeah, use the
59:50 at the top and leave the minus in the coefficients and do it for
59:57 clay, you know, very porosity arrange say 0 to 30% and let
60:05 let X clay be zero and then X clay B one and C.
60:12 plot vP vs V s at the pressures. So there are four common
60:19 . You don't have to do effective of 50 because we don't have shear
60:24 for that one. But we have we have it for 400,
60:31 200 and 100. So I'm gonna it back to you and let you
60:41 with it. Okay, now the one is uh for carbonates, this
60:56 uh some laboratory measurements that were made M. I. T. And
61:04 is the equation. So don't get off by the partial derivatives here,
61:11 are just the coefficients. Right? , you know, when we had
61:15 , B and C. So this C and uh B and a.
61:22 , well actually this would be the of courts and plus eight times this
61:29 B times that. Right. So are the same types of equations were
61:34 at before. So they're writing this a partial derivative. It's the change
61:41 shear wave velocity with calcite content or change in shear wave velocity with
61:49 So those are the same types of we were using before. And and
61:56 same thing for p wave. So am I asking you to do
62:01 Plot vp vs V S for pure and pure calcite. And uh let's
62:10 do it at one pressure right So let's do it at one killer
62:16 . Right? So how are we cross plat V. P. And
62:19 . S. We're gonna let ferocity say 0-30. And we're gonna let
62:27 B. Zero. That will give sandstone. And let calcite be one
62:32 will give us limestone. You see I'm saying? I think you already
62:38 because my table doesn't look like your but it has the same information.
62:45 all crazy. It looks like the as the last one. Oh
62:52 But you know what happened? There typos. Okay. Yeah there were
62:57 on my table. And so I the actual publication to my my table
63:08 there were some typos. So if could use these numbers maybe you should
63:13 these numbers. Let me take a real quick. Hold on. And
63:17 I can just airdrop it to my . Okay. Got it.
63:45 So basically I'm just doing the same . Yeah. But we're going to
63:50 what happens when we're very side instead clay. Okay. You can stop
64:21 . Oh okay. I didn't understand question because I didn't realize it was
64:27 to this set of data. so this is gonna take a
64:34 Um Because what what we're gonna The first thing you're gonna have to
64:39 and I should have I have to this table in digital form. The
64:44 thing you're gonna have to do is all these values in. So that's
64:49 tedious task which will have to And then uh we'll calculate the densities
64:58 the case of brian sand and shell use water density and the case of
65:03 , sand will use a water mixture . So first order of business is
65:10 type these values in. So type in. Let me know when you're
65:14 . We'll take a 10 minute break that point, give you a rest
65:19 then we'll come back and you'll share screen and we'll do the calculations only
65:25 take a screenshot. I think it's next one. Yeah. So unfortunately
65:33 don't have a digital one of these , I'm gonna have to make sure
65:37 have it digitally. But you're just have to type those values.
65:43 Am I typing all of them? ? Yeah. Yeah, you're gonna
65:48 all of them. Okay, thank . Okay, so here we have
66:07 two plots, one is VP Porosity and one is V. S
66:14 porosity. And there are values for and wet for both. And so
66:23 me if you see anything unusual about velocities, there's something that strikes you
66:34 odd. I mean they're pretty much same dry and wet. Right.
66:41 think for a shear wave velocity, think you could see that the the
66:46 tends to be a little slower. the density effect. But for the
66:52 wave velocity, there's really no Right? So what's going on?
66:59 these are synthetic sand stones, so fused quartz beads. And so the
67:10 are very equal, they're very spherical as a result, the modular exchange
67:18 fluids is not very strong. These rocks are so strong. Also
67:23 the trend of that curve. It's a straight line, straight line,
67:28 concave up like none of the velocity curves we've seen other than the critical
67:34 model. So this is kind of what the critical porosity model would
67:40 It's going trying to go towards, know, the critical porosity down
67:46 So, the moral of this story if you happen to have a rock
67:51 very spherical pores, um the velocity care, even the p wave velocity
67:57 care what's inside those pores, because pores are so strong that they don't
68:04 and they don't. So the fluid the pores don't have to help resist
68:09 compression. Okay, asking you to on this one, we think that
68:24 measurements overestimate the change of velocity with . I mean, they're measuring the
68:31 in velocity versus pressure on the lab . I'm not saying it gets that
68:37 , but I'm saying that's not representative what's really happening in the earth as
68:43 change pressures in the earth and I'm argue that laboratory measurements way overestimate the
68:54 of pressure? So the change in versus the change in pressure, especially
69:00 low pressures is much higher then I happens in the earth. Why do
69:08 say that? Why would I think would it be because laboratory measurements maybe
69:16 assume perfect conditions, whereas like when actually get into the field and get
69:22 data, like there's just a lot things that come into play that you
69:28 really measure in a lab. Um let's say we have a piece of
69:37 the same piece of rock. So velocity is not a matter of how
69:43 measured, The velocity is a rock , isn't it? So that piece
69:49 rock has a velocity. And I'm to study how that velocity changes with
69:57 . And I'm gonna argue when I that piece of rock out of the
70:01 , bring it to the laboratory and the change of velocity with pressure,
70:08 at low pressure, that change in with pressure is going to be very
70:13 . Remember some of those curves, saw a velocity versus pressure at first
70:18 have a very rapid increase in velocity pressure and then it levels off.
70:23 that And I'm saying that I don't it happens that way in the
70:30 I think if I have a I leave it buried two miles down
70:35 I reduce the effect of pressure. change in velocity is not gonna be
70:41 as large as it is in the . Why do I say that?
70:50 it? Because when you measured uh rock in the lab, your uh
70:59 stresses exert on the sample body measure the velocity under the ground.
71:10 trace is where we separate two adjacent . So the effect of Well,
71:17 mean keep in mind that in the we're trying to simulate the EMC two
71:22 . So we're trying to put it the same stress conditions as in the
71:28 . So let's say I've done let's say in the laboratory, I
71:33 vary sigma one sigma to sigma three I can make it exactly the same
71:39 it is in the earth. now, usually it's close to that
71:45 in the laboratory I use confining it's hydrostatic, whereas in the earth
71:50 are tectonic stresses. But ignoring that , the effective pressure I could create
71:58 the laboratory is the same as the of pressure I could create in the
72:04 . So I'm not gonna I'm gonna , let's assume the pressure conditions are
72:10 the same. And even in that , I would say in the
72:15 we have a bigger increase of velocity pressure at low pressures than I do
72:23 the earth. I'm gonna let you on that one. Keep throwing out
72:39 is just uh other uh condition the like temperature. Yeah. So I'm
72:49 the conditions the same. Let me you what happens when I core the
73:30 , take it out of the earth bring it to the laboratory. What
73:34 to the rock because it generated the inside the Yeah. When I when
73:44 bring it from it's in Setu There's going to be stress relief when
73:50 bring it to the surface and the is gonna fracture. And any,
73:56 know, incipient microfractures will open So when I first put it in
74:02 laboratory, when I first increased the , I start closing those fractures.
74:09 as I increase the pressure more and more of those fractures close at some
74:15 , I've closed all the fractures that gonna close and the velocities level
74:22 So I'm gonna argue that drilling, of drilling induced fractures, that initial
74:28 of velocity with pressure in the laboratory much greater. Then what would happen
74:35 the earth? Okay, so we dry a sandstone trend, Right?
74:53 have K over μ equals one and have a bride saturated sandstone trend for
75:04 Bride. So could you cross plot the dry rock and for poisons
75:11 I mean for the brine, saturated poison's ratio versus bulk module asse.
75:25 uh one of the easier ways to this is to use the V.
75:32 . B. S. Ratio. . P. B. S.
75:36 squared, right? So for the saturated rock you have the Bride saturated
75:42 between V. P. And S. So you could calculate uh
75:48 . P. B. S. you vary the shear wave velocity.
75:53 from that you could get lessons ratio the V. P. B.
75:59 . You could get um from the . P. You could get
76:04 Plus four thirds mu for V. . You could get mu so you
76:11 calculate both modules. Right? So should be able to cross flop Wilson's
76:17 versus both modular. So let's go and do that. Okay, so
76:34 have to start with the V. . B. S relationships.
76:38 Let's make it easy. Let's say for the dry rock V.
76:47 equals v. p. divided by . And for the brine saturated rock
76:57 have a brine saturated equation. And that was where did I give
77:03 to you? Right there? All , So let's vary v. from
77:11 to 6. So in your first you could vary v. from 1.5
77:18 6. And let's do that in of .1. Okay, We can't
78:23 you anymore in case you're talking. , So I forget where we
78:32 So you're gonna cross plug bp versus and compare that to the trend
78:38 So compare that to ray martin Gardner to Wiley and later we'll do VP
78:44 . B. S. But the thing you have to do is type
78:47 these values, unfortunately. So you'll in V. P. Observe es
78:54 ferocity. So let me know when got that typed in. Okay.
79:04 gonna have mercy and I'm gonna skip one unless we have time at the
79:08 of the class. Uh These are bitch's equations and they include a stranger
79:16 of with Allah jeez they have uh one a two a three a four
79:23 five. Weird stuff. And having dry have calcite and quartz. These
79:30 in Dolomites. So we're going to we're gonna skip all of this.
79:40 . So I'm gonna ask you to at a well log cross plot and
79:46 a bunch of data points on here there are some trends shown.
79:52 So I want you to interpret these points. See if you could see
79:56 groupings of data points and see if could uh explain uh what these data
80:04 are or explain why they're there by them to the trend curves for example
80:36 by the way they flipped the axes us. So uh V.
80:43 Is the vertical axis. Do you the different groupings of points? Like
81:04 have this grouping? We have this we have this grouping. So um
81:15 if you could try to explain. also have some courts measurements. They
81:25 some calcite measurements. So it would this thick middle section. Would that
81:34 be because that's the limestone line. those are probably lime stones.
81:41 Unless unless you have it's a very mix of other stuff. Right.
81:48 this is the sandstone line. So think maybe those are sand stones?
81:55 about these things here? They have higher be PBS than everything else.
82:03 we need a hypothesis to explain those . No, Let's see the yellow
82:23 goes more than 25%. Could that like doll online? Well, dolomite
82:43 to be between limestone and these are V. P. B. S
82:49 than even our limestone line. So those are hard to explain.
82:56 one way you could explain it is you had liquid filled fractures.
83:02 that was my next thing. So the liquid filled fractures could give
83:08 abnormally high V. PBS ratios. skip this one. This is more
83:20 a homework essay problem. So that us through with V. P.
83:26 . S. Ratios. So I to then move move on to my
83:37 section which is fluid properties. So going to start talking about fluids.
83:46 go to desktop or physics professional Where is it? And we want
83:59 fluid properties. Okay, so what the properties of the fluids? Well
84:14 elastic ones that we need for a , oh that's because I didn't
84:25 It's good to share. So let share a screen and it's not showing
84:35 here for some reason. Let me out of here. Let me share
84:38 screen again. There we go. we have it. Now, you
84:44 see it, Yes. Okay. what are the elastic properties of
84:52 Their bulk modulates and density. if we start thinking about fluid flow
84:59 we think about the in elastic like the visco elastic properties of rocks
85:05 need to consider the viscosity of the also. But right now it's gonna
85:12 just simple mechanics. And we're going look at the elastic properties of the
85:17 . So the bulk modules and the and the density of the fluid.
85:23 of these will vary tremendously. They vary with temperature, temperature and pressure
85:31 they will vary with composition. Oil . The modulates and density depends on
85:38 is called the api gravity of the . Uh, the lower the gravity
85:46 some reason, the lower the gravity the, of the oil. Uh
85:51 more long chain hydrocarbons you have. the dense stir the oil and the
86:01 less compressible the oil. The other property in oil is the gas oil
86:08 . How much gas is dissolved in oil. We're not talking about free
86:13 , free gas. We can calculate effects using Woods equation, but this
86:19 the dissolved gas in the oil and more gasses dissolved with the oil.
86:27 lighter the oil is the lower the and the uh, the lower the
86:34 modulates the more compressible the oil is Royals. Now gasses also have something
86:42 gas gravity. So the properties are on again, the composition of the
86:48 , you could have very light gasses are pure methane, but you could
86:52 a longer chain gasses, right, , butane, things like that.
86:58 , um again, uh, the of the gas will determine its
87:05 and with brian's a major factor is of the brine, dissolved gas in
87:12 brine is also can be a but you don't get a lot of
87:17 gas and brine. So we're not worry about that at this point.
87:23 the fluid properties then vary all over place. They depend on temperature pressure
87:30 they depend on on the composition. , now we want to put that
87:38 into a rock and at low frequencies use what are called gas Men's
87:45 I've been alluding to those. I you the results of those in the
87:50 section. We'll go through them. is what the industry has used for
87:54 years, and leon Thompson has just a paper proving that gasman made an
88:02 in his derivation, and these equations actually wrong. So, if we
88:07 time, I'll elaborate on that a , but most important that we understand
88:13 standard practices. So we will go gas men's equations. Now, gas
88:19 equations are the low frequency limit. , and so, uh, we
88:28 that they're applicable at seismic frequencies, frequencies are relatively low, but we
88:33 know that for a fact. And fact, there is no direct hypothesis
88:41 of gas mains equations. Um they've been validated um in any way.
88:48 we know is that, uh, the use of gas men's equations,
88:54 have not been terribly offended when they them under the proper circumstances, so
89:01 haven't been heavily falsified uh in Right. But as far as scientific
89:10 or valid scientific hypothesis testing doesn't exist . There are other equations that are
89:21 to be applicable at high frequencies, these are called the B.
89:25 Equations. And gas mains equations are to be the low frequency limit of
89:31 equations, and that's what we've thought these years. But leon Thompson just
89:36 they are not the low frequency limit BEOS equations. In fact, BEOS
89:42 are correct, and gas mains equations not quite right, but they can't
89:48 terribly wrong otherwise, people would have . So, we're going to continue
89:55 gas men's equations in the next Okay, so in gas mains
90:05 we need to know the bulk modulates the fluid. We need to know
90:12 bulk modules of the solid material. bulk modulates of the frame in this
90:18 , we're just going to worry about fluid also. We're going to use
90:21 mass balance equation when we compute the of changing fluids on velocity. We
90:27 to predict how the density of the changes. So, we have to
90:33 predict, or we have to take account the dense how the density of
90:38 fluids change. So, in this we'll look at the gas, I
90:44 , the fluid module asse and the density primarily versus temperature and pressure,
90:51 also against composition. So, here a plot of gas module asse versus
91:01 . And the module asse is in pascal's, Right? So you have
91:07 put a decimal point in front to these giga pascal's. Okay,
91:13 you know, we were, you , we've been talking about the modulates
91:17 courts being 40 giga pascal's and here on the order of half,
91:25 half a giga pascal. So, the gas is 80 times more compressible
91:32 the courts. So, you can why the presence of gas has a
91:37 effect on the compressibility of the If if the rock frame is at
91:42 compressible now, at low pressure, even less at low pressure, it
91:48 be uh, you know, down virtually near zero. Okay,
91:56 measurements in the laboratory and surface the gas effects are really enormous
92:05 So, as we increase the the modulates of the gas increases,
92:10 , by the way, this is pressure. Right? As we increase
92:15 pressure of the gas, what does mean? What is what causes high
92:19 pressure, What causes high pore pressure when the container has a high external
92:27 and it's pushing in on the gas it's pushing the gas molecules closer
92:35 The gas molecules are closer together. want to spread out. Gas molecules
92:43 vibrating. They're moving around and they to they want to spread out as
92:51 as they can. So, if enclose the gas and I push them
92:57 , those gas molecules will literally be against the wall of the container and
93:05 be pushing out on the container and more I push them in, uh
93:10 more of them they're closer together, more of them are pushing out on
93:14 container, right? So, um hire poor pressure is a result of
93:22 gas molecules being closer together. So are higher pore pressures. Uh the
93:29 molecules are closer together and the closer get together, the harder it is
93:35 push them further together. So, the gas molecules are very far
93:42 it's easy to push them together. at low pressure, they're very far
93:47 . It's easy to push them in high pressure. They're packed up against
93:52 other and they don't like it, vibrating. They're pushing out. They
93:55 to get out so it gets harder harder to push them together and they'll
94:02 repel each other. Um So hire pressures. The fluids are less
94:11 Now, We also talk about the of gas, but this is opposite
94:18 the gravity of oil. I don't why uh you know low gravity oil
94:25 very dense. I mean it's backwards they get it right with gas.
94:29 gas .6 gravity is a light 1.2 gravity is a heavy gas.
94:37 makes the gas heavier? It means got more longer chain hydrocarbons in
94:45 So a very light gas would be and a very heavy gas gas would
94:50 more long chain components. Okay, what do we observe? The modular
94:59 with temperature. Why does the module decrease with temperature? Because as we
95:05 the temperature, the gas expands. , the gas so the gas
95:12 the molecules are further apart. We the pressure, the molecules are closer
95:19 . So we increase the modules. , so you want to understand the
95:24 there And the heavy chain. The chain hydrocarbons are harder to push together
95:31 the light chain. Well, at pressure, it's the same for
95:35 But at low temperatures, the heavy , the longer chains are harder to
95:42 together. You increase the temperature enough you have to get up to pretty
95:48 pressures to see a difference. And see pretty much the same thing with
95:56 density. So the gas density and gas module asse are very related and
96:02 makes sense. The more you push gas molecules together, the more dense
96:07 gasses. Right? So as you the pressure, you the gas gets
96:13 . You push pushing the molecules closer . So you have more mass per
96:19 volume, you increase the temperature the expands and you have less molecules per
96:27 therefore less mass per unit volume. , so here comparing this is straight
96:39 a very famous paper paper battle and who came up with these relations showing
96:50 density versus pressure. And here the are in uh mega pascal, so
96:56 relatable to the numbers we like to . Whereas the previous plot was in
97:03 . So that got changed. Which nice. And uh a quick question
97:11 the previous slide. So in the , how do we measure the density
97:16 gas? Well, you know the because it's in a container and you
97:24 weigh it. Okay, so the is fix and yeah, so if
97:43 increase temperature, the weight will Yeah, you have to adjust for
97:50 . Right? So um you you could use a P.
97:55 Equals NRT if you have ideal gasses you could use Boyle's law and Charles
98:02 . But you know, these are ideal gasses. So it starts to
98:07 very complicated. Um and these are measurements as opposed to uh or based
98:15 direct measurements as opposed to based on of state. Uh So if you
98:22 the equation of state, you have worry about whether you're a diabetic or
98:25 a thermal for the module. I starts getting very complicated. So battles
98:31 is uh is very empirical actually uh battle and I worked together at Arco
98:39 he was doing this and he had a graduate student, a graduate student
98:47 stanford who came for a summer and wound up staying all the way till
98:53 to do all this work. And he turned out to be a fellow
98:58 the name of Z. W. has three textbooks in rock physics.
99:04 You know, went to Chevron, a top manager at Chevron. Uh
99:11 so he had a very successful career I have to say he was the
99:16 productive graduate student that I ever had my group at at Arco. And
99:24 this was fantastic work and it's used the industry. Okay, so here's
99:33 gas bulk modulates again in mega Same same as I was showing
99:45 Oil module asse very similar. Now a 50 degree A P.
99:52 Is a light oil, whereas 10 is a heavy oil. So,
99:59 thing, there's a variation with The heavier the oil, the greater
100:04 module asse, the greater the the greater the module asse, the
100:09 the temperature, the lower the So these are some of the uh
100:18 that mike battle made. And this on a very viscous oil sand on
100:24 north slope of Alaska. So this a very heavy oil and uh they're
100:30 uh the velocity as we're increasing uh confining stress. Hold on, let
100:40 I'm sorry, as we're increasing the . So uh and this is the
100:46 of the rock and this is feet where oil is about 5000 ft per
100:54 . I'm sorry. This is directly the oil. So this is a
100:59 oil. Water would be 5000 ft second approximately because it's gonna vary with
101:04 and pressure. So here we're varying pore pressure. So as we increase
101:09 pore pressure, the velocity of the increases and notice that this is one
101:14 those rare cases where the at low , the heavy oil can be faster
101:22 brian at high temperature, then it's than brian. Also, you'll see
101:27 viscosity change at low temperature. It's viscous at high temperature. It's less
101:40 . Alright, well, oil density to be more linear with with temperature
101:48 again, the same exact kinds of . Light oil is lower density than
101:56 heavy oil. Uh notice that all these are less dense than water.
102:03 , uh all of these will float water as was true for all the
102:12 , but the higher the poor the greater the density of the oil
102:18 the higher the temperature, the lower density of the oil. Okay,
102:29 305. It's time for our 10 break. So let's reconvene 3 15
102:39 , some more measurements we made at ? This was a live oil.
102:45 , what's the difference between a live and a dead oil? A live
102:51 . There is dissolved gas in the oil in C2, in the subsurface
102:57 pressure. But as that oil comes the surface, that dissolved gas exalt
103:06 of the oil and becomes free And what's left behind then is called
103:13 dead oil. An oil with a of dissolved gasses is called a volatile
103:21 . And uh you know, it's very readily exalt gas. And uh
103:29 know, there's gas could be very , very volatile, right? If
103:35 was any friction, any sparks, flame, the result could cross an
103:41 very easily. So you bring this oil that's in C2 in the
103:47 you bring it to the surface and coming out of solution. You have
103:52 all over the place, uh so you have to put it through
103:57 separator to separately correct collect the gas collect the oil. Uh So,
104:06 remaining oil is called dead. in the early days of thinking about
104:12 effects of fluids, uh people would measurements on oils and say, you
104:17 , oils aren't too different from So you shouldn't be able to see
104:20 big difference between oil and brine in of its properties. And so there
104:28 a paradigm that the effect of oil not produce amplitude anomalies. And that
104:33 because people were making measurements on dead , but actually you dissolve a lot
104:39 gas in the oil and you reduce you make it much more compressible,
104:45 you reduce its velocity. So these velocity measurements on oils. And there
104:52 a couple of things going on Uh number one, they make they
104:58 the velocity at a couple of different . So dead oil, 72°
105:06 That's pretty hot, 23° C is more like room temperature. And what
105:13 they find is that in fact it's a little bit opposite, but the
105:22 dead oil seemed to have a higher at higher temperature. So something weird
105:31 going on there. And also there the calculated temperatures. These are these
105:40 from battles empirical work. He came with uh ways to predict the velocities
105:48 oils versus poor pressure. And um got these measurements. I'm sorry,
106:00 have this backwards, don't I? have this backwards. So ignore whatever
106:05 said, about 72° being higher. was already formulating hypotheses to explain that
106:12 . But in fact, these are different situations that the top two pairs
106:19 curves are dead are the dead oil this is a live oil.
106:24 uh for the dead oils, the drop with temperature and for the live
106:30 , the velocities drop with temperature. please disregard what I said previously.
106:37 my brain is turning to mush after hours of this. That's my excuse
106:41 . My brain is always mush. that that's another matter. So,
106:46 , you can see that for the oil battles, empirical equation gets you
106:53 , but it doesn't get you precisely . And the reason is you could
106:58 dead oils of the same gravity, different compositions and so and their properties
107:06 vary even though they have the same . So Bachelor's equations are based on
107:10 gravity of the oil. So they're a rough average for the oils that
107:16 was sampling. Right? So, empirical calculation is slightly off. Not
107:23 too bad, but slightly off, it's predicting the variation with pressure quite
107:31 . And it's it's it's predicting the with temperature quite well. But oddly
107:37 , you have lie boils and now empirical equations are essentially perfect fits to
107:44 data. Uh they show the the velocity is increasing with pressure at
107:53 right rate and they show the change temperature. And they predicted perfectly.
107:59 is that? Well, it's because dissolved gasses such a dominating effect on
108:06 result that it really overwhelms the compositional of the oil. So dissolved gas
108:15 a huge impact on the velocities of oils. Now, something funny is
108:22 down here, we lower the pressure the velocities increase. So now I'm
108:30 ask for a hypothesis to explain why velocities would increase as we lower the
108:45 . Is it because the guests come ? Yeah, it comes out of
108:52 . And so that means the remaining that's left after the gas comes out
108:57 solution has less dissolved gas. So the gas oil ratio of the liquid
109:04 dropping and the velocities are increasing. other possibility is that when you have
109:10 bubbles, um then the the attenuation quite high and the velocity measurements can
109:18 unreasonable, but it looks like they it off. Uh Probably below
109:24 they couldn't make the velocity measurements, they cut it off where they thought
109:28 were getting valid velocities. And these well behaved enough, they're similar enough
109:35 it looks like these velocities are So, it's it's primarily the effect
109:42 gas coming out of solution leaving behind oil, which is less compressible.
109:56 , so, um yeah, if look at the density of oil,
110:08 could see the density of the oil dependent on the gas oil ratio.
110:13 , if there's no gas in you have relatively high density and here
110:19 a very heavy oil, 10 degree P. I. They're approaching the
110:24 of water there. Um Now, you get to higher oil gravity's uh
110:33 low gas oil ratios, uh the decreases with increase with increasing lightness of
110:42 oil increasing a P. I means less dense oil. So as you
110:49 the api gravity of the oil, velocity drops for whatever gas oil ratio
110:54 have. But eventually you reach a where the gas oil ratio is so
111:00 . The velocity then becomes essentially independent the gas oil ratio, I'm
111:10 independent of the oil gravity. Strong on the gas oil ratio. So
111:17 of these numbers, you could see a very volatile oil can have a
111:22 almost half that of water, so could be quite significant in a porous
111:28 . Uh That alone could potentially give a in impedance contrast. But of
111:36 , the modular, if the oil less dense, it's also going to
111:39 more compressible. So now here are the velocities of different fluids um as
111:53 function of pressure, but also as changing. So here's a heavy oil
112:00 its velocity at high enough pressure. even in high temperature becomes higher than
112:07 of water, but at uh at temperature, it's this heavy oil is
112:13 uh less compressible than the water water um is a little bit backwards because
112:26 you increase the temperature, the the water becomes less compressible. So
112:32 is contrary to what I was saying about you increase the temperature, you
112:37 expansion and the molecules are further So you're more compressible. Water is
112:44 very strange liquid. It's an exception the rule. And that's because the
112:48 molecule is Is not symmetrical. The have H20 and the oxygen is in
113:00 center and the hydrogen atoms are at angle to each other, they're not
113:08 a plane, there actually is some angle Or I'm sorry some angle less
113:17 180°. Right? So there, so means you have an imbalance in the
113:23 . You have the oxygen side of molecule and the hydrogen side of the
113:29 . So that creates all kinds of type of effects. And so water
113:36 order itself in a very strange And that's why water again, unusually
113:42 when you freeze it right? It becomes less compressible at at low pressures
113:49 low temperatures it becomes less compressible as heat it you get up to high
113:55 pressure and temperature and then it starts like the normal fluid. It um
114:00 over those effects overwhelm the charge imbalance the water molecule. Okay, and
114:09 we have a light dead oil. shallower it tends to be more compressible
114:14 water. But if you get the high enough um you know, you
114:20 uh you can cross with water but light live oil, it is always
114:28 to be more compressible than the So here is the brian modulates and
114:38 the previous two cases, gas in , the modular versus temperature was a
114:46 a tonic decrease as you increase the . Well brian is different at first
114:52 increases before it decreases. So there's temperature range where it acts backwards.
115:00 other than that, it's pretty The higher the pressure, the less
115:06 it is. And by the at high pressure and high salinity,
115:11 could get extremely high module i uh the old days we used to assume
115:17 on the order of 2.5. Uh that was not considering what it would
115:24 if you had a lot of dissolved in the in the brine. So
115:31 variation of brian modules can be quite . And here we have brine
115:43 brine density is a little bit better . But you do get, you
115:49 , a different behavior for a pure versus a saline brine. Right?
115:56 it starts to get a little complicated these lines actually can cross. So
116:07 kinds of salinity ease do we have worry about? Well, um look
116:13 , uh these are gulf coast salinity versus depth And you get some very
116:19 salinity ease. The smack over formation up around over 300,000 parts per million
116:27 the way. Um the smack over being used to mine lithium. Uh
116:35 do I mean by that mining Well, they produce the fluids out
116:39 the reservoir. And if you have saline fluids, there's a lot of
116:45 dissolved lithium salts are dissolved as So you could actually extract lithium from
116:53 very deep waters. Now, this something that worried us. uh we
117:09 an equation in an engineering handbook back 1945 and there was an equation from
117:19 we could predict the variation of bulk asse of a brine with temperature depending
117:28 the dissolved gas. And here you that the more dissolved gas you can
117:36 a big drop in the brine modules we were concerned then that fresh water
117:42 give you amplitude anomalies. Well, turned out to be false. Uh
117:49 was something we didn't have to worry because you can't get uh gas oil
117:55 of this kind in uh in I'm sorry. Um you can't get
118:04 kinds of gas oil ratios and brian's only place this can happen is in
118:10 . So that's still a concern. here's a empirical plot showing uh calculated
118:27 velocity versus measured velocities uh And bats Wong's uh equation here. Well,
118:37 this this is the trend for the velocities. This is calculated usually in
118:41 and long and again, that's because given gas oil gravity, not all
118:47 with the same gravity are the So there can be differences from the
118:54 and Wong equations. And these equations published in 1992. It's 30 years
119:00 . And uh the the equations have updated ever since to make them more
119:07 more precise. And in fact there's program offered here by our rock physics
119:12 . It's called flag dr han and battle collaborated for many years and we
119:19 have this program and dr zeng and students are updating the program to handle
119:26 . 02, which has become very uh nowadays for carbon sequestration. So
119:35 , the point is in the flat there are updated uh equations all right
119:45 , up until now, I've been much talking about individual phases, but
119:53 we have liquids in the subsurface under pressure and temperature conditions, they can
120:01 as different phases. So this is phase diagram for a typical uh single
120:11 molecular fluid like water. And you see that there are regions where its
120:18 if the temperature is high enough and pressure is low enough, you get
120:23 , you get a boundary between liquid gas. So there's uh you lower
120:27 temperature in, droplets will come out solution, you lower the temperature more
120:33 it will freeze. Right? You also get, for example, with
120:38 ice, you could go directly uh solid C. 022, gas
120:44 02. Right? So there's that to not go through the liquid
120:50 Now, there's a point if the and pressure get high enough, there's
120:55 sharp distinction between what's a gas and the liquid. So, if you're
121:00 here, it's called a supercritical your pressure and temperature are higher than
121:07 critical point. The supercritical fluid acts a liquid at lower temperatures here,
121:15 it acts more like a gas at temperatures, but it doesn't exist as
121:21 , as different phases. It's more a continuum between the two. But
121:29 oils being that these are complex mixtures a variety of a variety of different
121:36 compounds. These phase diagrams get very . And instead of a critical
121:44 we talk about a pseudo critical What happens at this pseudo critical point
121:53 very complex. But if I'm at temperature and pressure than the pseudo critical
122:00 , I will have a bubble The bubble point separates the supercritical fluid
122:06 like a black oil from the two region In this region, it will
122:14 as gas and oil. So, this is called the bubble point,
122:22 drop the pressure below the bubble point you increase the temperature above the bubble
122:28 and gas will start to boil out solution. So gas will come out
122:34 solution. So in this case this the liquid volume. What, what
122:40 in the two phase region will be liquid, but 20% of it by
122:47 will be gas as we increase the and more and more of it is
122:55 . If I increase the temperature oddly enough, I could go back
123:00 a supercritical fluid and this is a fluid with gas like behavior. But
123:08 I drop the pressure, say from down here or I drop the temperature
123:13 here to here, bubbles. I'm sorry, droplets will come out
123:18 solution. So you may get a bit of do. And so this
123:24 called the dew point. Right? oil droplets will start coming out of
123:30 and you'll have a two phase region is gas and oil. Okay,
123:39 if my reservoir is here and I the pressure, gas bubbles could come
123:45 . That could be really bad for reservoir because those gas bubbles can impede
123:51 flow of oil through the reservoir. could start to clog up the pore
123:59 . So you want to try to having that kind of situation? Um
124:05 sorry not gas bubbles, liquid bubbles coming out of solution as we cross
124:10 dew point line, liquid droplets are out of solution and clogging the pore
124:16 . Now, what happens as the as the oil comes to the
124:22 as it comes to the surface, dropping the temperature and pressure.
124:27 So, um there's the potential in something uh at the reservoir out here
124:36 we start bringing this supercritical fluid which acting like a gas. We bring
124:42 past the dew point and then liquid will come out. This is called
124:49 . So that condensate is oil. some gas reservoirs produce a lot of
124:58 and the oil can be very valuable compared to the gas. Anyway,
125:05 point is, this is a super region. Supercritical fluid acts like gas
125:12 acts like a liquid here, uh drop things below the dew point and
125:18 come out, you drop things below bubble point and gas bubbles come
125:28 So we have liquid like behavior Gas like behavior here. So what
125:34 you call the supercritical fluid in the ? If you're above the two phase
125:43 , if you're in the supercritical well, if the temperature is very
125:48 , you would call it a dry here. You would call it a
125:53 because you dropped the pressure and uh of oil come out. This would
125:59 a volatile oil and this would be black oil. So depending on the
126:15 of the oil, you'll have different diagrams. Right? So here I
126:23 a reservoir at a certain condition. , depending on the composition. This
126:32 diagram at those reservoir conditions would mean have a dry gas. If I'm
126:41 this phase diagram based on the composition the fluid, then this would be
126:45 condensate. Right, I'm above the point here, I dropped the pressure
126:50 come out. If I had this diagram, this reservoir would be a
126:55 oil. It's near the critical, slightly below the critical point and lots
127:00 gas could come out of solution. If I had this phase diagram,
127:06 would be called the black oil at reservoir and not a lot of gas
127:11 gonna come out of solution. So we come back to this plot
127:17 I have a little bit of But as I get close to the
127:21 point, I have a lot of coming out of solution. So this
127:25 be volatile. This would be more a black oil. Now, what
127:38 in this two phase region? a couple of things happened. Gasses
127:43 out of solution, so the remaining is gasses left, and so if
127:51 an oil and gas is left, becomes compressible, its bulk modular
127:59 So that's one thing that happens. now I want the effective modular of
128:04 two phase region, say I have phases in the reservoir. How do
128:10 compute that modulates? Well, we woods equation and Woods equation is exact
128:17 gas bubbles in a liquid, it be gas bubbles and water, it
128:21 be gas bubbles and oil. It matter, it's this reciprocal volume weighted
128:28 of the module, I and remember, this is a Royce
128:33 So uh the smallest module asse is to dominate. So that two phase
128:40 , if there's any significant gas at , it's gonna have properties similar to
128:47 . On the other hand, the is linearly related to the saturation.
128:58 depending on the oil and gas module this mixture or well, this happens
129:05 be a water gas mixture. say this is the modulates of
129:10 This is the modulates of gas. as the the poor pressure increases,
129:16 gas modulates increases such that this becomes more gentle curve, but at very
129:23 pressure. This is a very sudden . It's like an on off
129:28 It doesn't take any gas at all reduce the compressibility because that gas is
129:33 compressible, it's going to take all strain. I have gas bubbles in
129:40 and I compress this and the entire in volume can be accommodated accommodated by
129:48 those gas bubbles because they're so So anyway, here I've written a
129:55 equation for gas and water uh with volume fractions represented by the water
130:02 So water saturation is the fractional volume of water one minus the water saturation
130:09 the volume fraction of gas. And think with that, I think we're
130:20 break. So, are there any before we go? I've thrown a
130:29 at you. So we'll pick it again next week. You'll have time
130:33 review. You might want to read in Wong's paper 1992 in geophysics or
130:39 could read the rock physics tutorial. it's on basic, it's on a
130:48 . Uh and I talk about fluid there as well. So, if
130:54 are no questions, we'll see you friday then. And we'll pick
130:59 we're right here. All right, you. Bye,
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