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GEOL7324 Rock Physics - Day5 09-10-2021
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00:00 this conference will now be recorded. , here we go. And as
00:09 mentioned previously, the V P B ratio is directly related to the Hassan's
00:17 , which is the fractional transbourse contraction the trans verse strain divided by the
00:26 longitudinal extension or the longitudinal strain. again here we've not included a minus
00:36 in the definition. That's all a of how you choose your axes.
00:43 I prefer not to use the minus because it becomes clearer that a positive
00:51 ratio. That means you have a if you change the length. So
01:00 with. So as I changed the , if I compress the length by
01:05 gets skinnier, if I take a uni actually compress it from one
01:11 the width. I'm sorry, the gets uh fatter. If I Pull
01:19 on one end, the wind gets . So that makes intuitive sense this
01:27 . And we also said there's a relationship between between Hassan's ratio and the
01:34 P B s ratio. So what persons ratio? Let me put it
01:41 way. What is the VPBS One person's ratio is .5. We
02:11 uh Hassan's ratio is .5. Uh both sides by the denominator there.
02:21 what you have is VPBS 2 -1 bp ds squared minus two. You
02:30 how that's an unreasonable relation. Uh the number that works for is if
02:39 p V s squared is equal to infinity minus one equals infinity minus
02:45 Right? The difference is negligible. you could say that the PBS approaches
02:52 As Pistons ratio approaches .5. Now we said that a person's ratio .5
03:00 to a fluid And that for a the shear wave velocity is zero.
03:06 , so again if persons ratios 0.5 means I have a fluid. That
03:12 VP over V. S. Is divided by zero. So VP over
03:17 . S. Is infinity. So go to the other end of the
03:21 spectrum. What is VPBS when persons is 0? If Parsons ratio was
03:33 when we cross multiply by the it it goes away because we're multiplying
03:38 zero. So then we have zero V. P. B. S
03:43 minus two. B P V s equals two. V P O V
03:49 equals square root of 21.41. And said that's a practical lower limit.
03:55 not aware of any rocks Where we verify a measured the PBS less than
04:03 square root of two. Okay, if we have length and length,
04:15 we strain, if we compress the vertically or uni actually it will shorten
04:23 it will also stretch. And the definition is minus transfer strain divided by
04:32 strength. And we could uh express in terms of the PVS a few
04:40 ways. Uh So Bp I like equation here, V P B S
04:46 equal to one minus the square root 1 - Watson's ratio Divided by .5
04:56 ratio here. So if you stick numbers in, if persons ratios .5
05:01 see this goes to infinity And if PBS um equals zero uh this goes
05:11 the square root of two And as showed last time there is a 1-1
05:19 between Soissons ratio and be PVS when get to very high vis Pds
05:27 Hassan's ratio doesn't vary very much. it becomes inconvenient to use the songs
05:35 . You're going into many decimal places persons ratio as the B p B
05:42 is very so as we approach the body bottom and be PBS gets very
05:48 Parsons ratio becomes nondescript. So it's convenient to use V p B s
05:54 the other side of the spectrum. a low VPVS ratios here we have
06:00 as our practical lower limit. You for a small change in V P
06:05 s, there's a big change in ratio. So it might be more
06:11 to use v PBS ratio. Now me say that there is no difference
06:16 sensitivity, there's no difference in signal noise ratio. As far as seismic
06:23 . Gobi PBS and Parsons ratio convey the same information. So if you
06:30 the information content of each using for , Shannon's information theory. You would
06:37 that you have exactly the same Uh There's no difference, It's the
06:43 information. It's a matter of which are you more comfortable with? Which
06:47 you do you prefer to use I to be uh most comfortable with the
06:54 . P. B. S ratio that's most directly related to what the
06:59 measurements we make and is most readily into our geophysical wave propagation equations.
07:08 I prefer V. PBS. But are others who used Hassan's ratio very
07:14 in geophysics applications. Now in the 60s, a famous paper paper by
07:25 in the Journal of Petroleum Technology show there is a relationship between mythology and
07:32 V. P. B. S . So in live stones we had
07:37 v. p. b. ratio of about 1.9 In Dolomites
07:43 And then sand stones lower one point 1.6, approaching 1.7 And above.
07:51 . Below 1.62 above 1.7. Um it's plotted in terms of sonic transit
08:00 or slowness one over VP and one V. S. So sonic transit
08:05 in microseconds per foot. So that results could be these are laboratory measurements
08:11 so that they can be directly Two. Mhm. Sonic log
08:20 Uh Pick. It also noticed uh V. PBS ratios. He has
08:28 limey sand here. So it's a with a lot of carbonate in
08:33 And it's got a higher be pds somewhere between calcite and the sands don't
08:42 . So for many years this was conventional thinking that D. P.
08:48 . S ratio was more or less for different mythologies and that you know
08:56 vis PVS ratio would be indicative So this is where things were when
09:01 was a grad student and I happened the time to be working on my
09:09 on some of the first shear wave logs that were being acquired. And
09:14 the same time I was working part while I was finishing my degree.
09:20 was working at a major oil company the video expert for that company came
09:26 me, he knew I was working shear wave velocities. I happen to
09:32 in our rock physics group at the . We actually are well walking group
09:35 the time. And he asked me simple question, he said what is
09:42 V. P. B. S for shell? And what would the
09:46 . p. b. s ratio Sam be at 5000 ft, 10,000
09:50 and 15,000 ft. And frankly I know the answer to that question.
09:56 I went around uh I was brand in the company. I went around
10:01 everybody what is the B. B. S ratio for shell.
10:05 the answer I got from it was , I don't know or greater than
10:13 . And but I was acquiring data the time and also why should the
10:20 velocity the depth dependent our radio experts to think it was but we had
10:29 um uh published data to suggest that be the case. So I started
10:36 at the data I was collecting and was started looking at data that was
10:41 in the literature. And the first I did was plot laboratory measurements on
10:49 pure minerals. So the major minerals deal with are in sedimentary rocks or
10:56 courts, calcite and dolomite. And grabbed the laboratory measurements that I could
11:03 and what we saw was that for uh You had pretty consistent results for
11:10 courts and It fell on a p. v. s. ratio
11:14 1.5 for courts for Dolomites. There a wide range of the PBS ratios
11:23 come back to that. There's a here at the PBS is 1.75.
11:30 course if I'm cross plotting VP V. S. A constant
11:34 PBS ratio is just a slow All , it's just a lot. And
11:39 the calcite points were up there on average slightly below two. And I
11:46 plotted a value. I had it clay. This value for clay was
11:54 from some sonic log measurements that we made at Arco. And so we're
12:02 a higher of the PBS ratio. remember uh the answer people would give
12:07 over to, Well we had a clay point here which had a
12:14 P. B. S ratio over . So remember pickets data, Limestone
12:22 1.9 Dolomites, 1.8 which would be here. Sand stones lower below 1.6
12:30 1.8. But there is kind of correlation of higher V. PBS ratio
12:37 the calcite rich rocks, intermediate for rich trucks and lower for courts rich
12:43 rocks. So certainly composition is a here being that the observed the PBS
12:52 are correlated to the mineral the PBS . So it seems like that is
12:57 component. It's not the whole game sand stones, we pick it,
13:02 its GPS ratios for sand stones were than for the mineral. By the
13:08 Later on we explain this variation in properties as being related to the iron
13:18 of the Dolomites. So you do cast iron substitution in Dolomites. And
13:24 B. P. V. S for the mineral varies by the
13:28 what do we mean by mineral P. V. S ratio.
13:33 fact the remember these minerals are all psychotropic. So in rock physics,
13:40 we talk about the velocity for a , the velocity purports what we really
13:46 is a zero porosity poly crystalline aggregate course where the crystals are randomly
13:55 Uh And the same thing for the minerals. And the way that's usually
14:02 this poly crystalline aggregate. We take Royce bound. We take the boy
14:09 and we average those values. And is called the hill average which tends
14:15 be an exact average if the sure the virginity of the minerals is the
14:21 . So if the rigidity is not far from being the same. For
14:26 , here you can see dolomite and are pretty similar rigidities but at least
14:31 the same order magnitude for these So that's probably a reasonable way to
14:39 the mineral properties. Uh Then we looked at a bunch of laboratory measurements
14:48 we were able to acquire. There a lot in the literature and most
14:53 these were pulled out of the at the literature. And what we found
14:59 that from this count from this clay that we observed this kind of saw
15:05 spread of values between clay and courts is at the upper end over here
15:13 would be there. Clay would be , cal state would be off the
15:18 . But if we just drew a connecting claiming cal state we'd be
15:24 And you see the clay calcite line kind of an upper bound so presumably
15:30 cement is pulling you up towards that calcite line we have a whole spread
15:35 values as the courts, clay mixture varying but we go below the clay
15:41 line. So it's you know this we're just connecting graphically. It doesn't
15:47 that that's the correct bound. We be using probably a Royce mountain would
15:52 something more like this. And then have points below the clay lines suggesting
15:58 we have some ferocity, some trapped or bound water. So the effective
16:07 associated with the clay. So the play point would be moving down.
16:12 this is a dry clay point. well to extend this? Uh What
16:25 did was I took my uh look at mud rocks And I extended it
16:35 lower velocities by looking at NC two . We had some shear wave velocity
16:41 . We had some inverted two graves DSPs. We had inversion of some
16:47 waves. We had uh huh a shear wave measurements. Vp and and
16:57 . S shear wave uh BSP I didn't have a lot of die
17:06 clock measurements at the time. And didn't have conventional sonic wave form measurements
17:14 you can't get shear waves from directly a conventional compression away sonic log.
17:23 you're velocities are very slow, you won't create sheer headways in the
17:29 and so back in the early 80s we were doing this, This was
17:35 the compilation of the few laboratory We had some c. two
17:40 Uh and uh from sonic logs. then a bunch of other geophysical
17:47 And what we found where things were along the line and basically fit a
17:55 by I I selected a court's point and I changed my ruler such that
18:03 minimize the mean squared error there. don't use regression because these were
18:08 these points can all equally be waited we didn't know how to wait the
18:12 . So I just, I just a visual line here and I ignored
18:18 point that was very shallow and later I decided that was above the water
18:24 . So this was the line that found for brian saturated mud rocks.
18:32 what is the mud rock? It a rock composed of mud. The
18:38 being particles that are silt sized and . We didn't call this a shale
18:44 because not all of these rocks were by definition. A shell is a
18:52 mud brock and we knew nothing about facility of these. And we had
18:59 combination of silk stone shells, clay , uh, so we didn't
19:05 We just called it the mud, rock line. And to our
19:12 we found that this line tended to fairly universal. So pretty much it
19:18 matter where you got went in the , you weren't too far off of
19:23 line. And we got a lot verification, a lot of feedback after
19:29 published this um, internally within the . I was working for, it
19:36 our co at the time. No man is a prophet in his
19:41 country and I was in our well group are seismic analysis group was
19:48 you know, making a point of what we were doing and well logging
19:54 they had to be the source of knowledge. And we had a group
19:59 scientists come in as advisers and I these results to them and one of
20:06 top scientists, world famous guy by name of lee Silver ask our manager
20:12 seismic analysis, are you using this ? And the answer was, we're
20:20 it. Um, and they valued so little that they let me publish
20:26 . And then suddenly we got all of feedback from the rest of the
20:32 that this equation works And we I would bump into people in the
20:38 and they would congratulate me. I've had people say uh, oh you're
20:43 Castagna. I expected you to be , these kinds of things. So
20:49 equation at the time, I was interested in a video analysis and I
20:55 that this equation would be important for and else. So this is known
21:00 as the arco mud rock trend and one of the most cited papers in
21:10 . Uh we also did our own . So this was the line and
21:17 dug out Any other kinds of measurements could find as they became available.
21:22 this was work was being done in early 80s. Well. Um there
21:27 a paper uh in the transactions of society of professional, well log analysts
21:35 early on before they were even recording wave forms, they were taking the
21:40 scope pictures in the field of sonic forms. And then they were manually
21:46 arrival times off of these oscilloscope pictures they came up with a table bp
21:54 values at a number of different depths the wellbore. And I plotted these
22:00 they were precisely on the mud rock . I then plotted um picket sandstone
22:13 and to my surprise it fell almost on the mud rock line. Now
22:19 are sand stones, not just mud . So the fact that sand stones
22:26 close to the mud rock line is important because in the classic section,
22:32 large majority of Iraq's our brian saturated stones and shells. And if they're
22:38 following along the same line that gives a background against which you can find
22:45 behavior. Either anomalous pathologies or as see later hydrocarbons. These were some
22:57 our own uh measurements. These were I made at the University of texas
23:03 milo bacchus and Ray Gregory. And fellow who wrote, you're one of
23:09 review papers you're supposed to read of Gregory and Gardner and here we had
23:15 poorest water saturated sand stones. They to plot slightly below the line.
23:21 this is a trend. We're going see, we're going to see many
23:24 stones plot on the line. But you have very clean, very poor
23:29 stones, they will tend to plot below the line still gives you the
23:35 rock line still gives you a rough , but it puts you close.
23:45 are was a paper from slumber Now it's 1982 were starting to get
23:50 recording of well, logs and these again, measurements and sand stones,
23:55 see many fall right on the some fall slightly below the line.
24:04 were some of my own full waveform measurements and a lot of scatter more
24:12 less along the mud rock line. but a bunch of these points that
24:17 very far below the line, we I had to do with the presence
24:24 some gas saturation. Uh you might yourself why points will fall above the
24:31 . So we're saying gas saturation being clean courts being uh as you have
24:39 , clean court sand stones, these will move you below the line,
24:43 are on the line. But if were above the line and uh we'll
24:48 in a bit what might bring pull above the line. Anyway, this
24:56 a collection of all the laboratory measurements could find. These included measurements we've
25:03 in our rock physics lab at And again, we're seeing a similar
25:08 . These are for sand stones. sand stones are spread along mud rock
25:15 here, there's a tendency and these all brine saturated. Now we know
25:20 because these are laboratory measurements, um see a tendency for some to plot
25:27 the line, some to plot below line. And there are a few
25:32 then that are significantly above the And we're going to have to understand
25:37 that happened. Now, looking in formations, what we find is for
25:50 for the same rock sample and various is a very clean sample. Uh
25:58 . When uh the sand is We get a constant D. PBS
26:05 of about 1.5 independent of pressure. points are all on the same sample
26:12 there are measured at different pressures. had four points for a fully Brian
26:18 sample. And most of the points on the mud rock line are slightly
26:24 , but one point fell above and point was at low stress, it
26:30 at the lowest pressure. And what see later is that if I'm at
26:36 effective stress there is the potential for fractures to open and we could show
26:42 that microfractures will move you above the . So that's one perturbing factor which
26:48 give you a in a brine saturated abnormally high vis PBS ratio is the
26:55 of a lot of microfractures and presumably low pressure. The microfractures aren't closed
27:02 that will pull you above the As I mentioned, we got a
27:08 of verification from uh the industry. was work that came out of chevron
27:16 they plotted be PBS versus P wave and the green curve is the mud
27:22 line. And they were pretty impressed the correspondence of the B.
27:29 B. S ratio measured in different . Cross hole law, Well walked
27:36 and laboratory measurements for shells. And noticed some sand stones here and very
27:43 velocities are flooding above the trend by way, these velocities higher than the
27:51 of course. So presumably there's carbonate in these sands towns. And as
27:58 see, carbonate cement is one of things that could increase the B.
28:03 rations. Mhm. They also compared from the mud rock trend. Two
28:14 velocities measured using multi component seismic And so here we have a p
28:21 sonic log here is the predicted shear sonic log using the mud. Right
28:28 . And these were interval V. ratios measured by looking at the time
28:35 of shear wave on the shear wave to the time difference on the p
28:41 section. So delta T. S delta Tp from seismic data and they
28:48 a very good correspondence to the mud trend. Okay, so uh we've
28:58 these equations before, uh we have measurements that we made full way from
29:07 measurements in the frio formation and we a multiple regression on these. And
29:13 seen these equations before. We're p velocity is equal to a constant minus
29:23 constant times porosity minus some constant times of clay. So if Iraq if
29:29 start with pure courts, I have ferocity, zero volume clay. These
29:34 be the implied course values if everything perfectly linear. Uh and increasing ferocity
29:42 the velocity, increasing the clay, the velocity but percentage wise uh they
29:50 a bigger effect on the shear wave than they do on the p wave
29:55 . So if I increased ferocity, V P B s ratio goes
30:00 If I increased volume clay, the P B s ratio goes up because
30:06 reducing shear wave velocity by a larger . By the way these equations were
30:13 a specific formation free information happened to a gulf coast. Geo pressured brian
30:21 sands and shales. These were the that came from stanford early measurements by
30:28 SIA on shells. And she got amazingly similar empirical trend from her from
30:38 sandstone measurements. So we're seeing completely data. These were from hard
30:47 you know, well lit defied shells you could make laboratory measurements on our
30:55 were on poorly lit defied geo pressured and sands. And yet they're showing
31:02 similar relationships. Okay, now, I have these equations either ours or
31:10 equations, I can see what these and fly in terms of the B
31:16 . B. S ratio by holding example the clay constant and changing ferocity
31:26 holding ferocity constant and changing the And I can see what that does
31:31 my VP VS. V. S . So uh here we're going to
31:40 compare these trends. So draw these on a V. P.
31:48 S cross plot. Do it three for pure courts. That means zero
31:56 . Clay for pure clay. zero , 0 porosity. I'm sorry the
32:06 at one because you're going to be porosity And halfway in between. So
32:15 course 50% play. So be clay .5 and plot these trends only VP
32:22 . V. S cross plot. if everybody understands the question, I'll
32:28 recording now and this conference will now recorded anyway just to recap. So
32:37 on, it's on the recording the trends here uh for precisely on the
32:45 rock line. Uh huh. If said be clay equal to zero,
32:52 we're up here and if we said equal to one, we're down here
32:59 both cases as we change ferocity, move down the line. So ferocity
33:05 you up and down the line right these these trends. And buying with
33:12 moves you up and down the This is zero courts. This is
33:17 courts. So it seems like improving saturated shales. It doesn't matter too
33:25 what we do. We stay on mud rock line. Okay, I
33:32 mentioned the cross blood of minimal properties this is just the reference. These
33:38 some values out of the literature. are values That are extrapolated to 100%
33:53 . And what we're getting is some B. P. V. S
33:56 here. If we had Purell I would have higher VP higher V.
34:02 . And the lower the PVS These other plays are mixtures of clay's
34:10 of spec tights and delights. And you find is that you get lower
34:16 wave velocities, lower shear wave velocities higher V. P. V.
34:20 . Directions. And the argument here that you've got bound water that's being
34:28 into the clay property. So the being calculated is not the true total
34:36 volume, fractional, bulk volume of , right? It is putting some
34:41 the water into the clays Uh and counting it as porosity. So then
34:47 you extrapolate 200% clay, you actually some ferocity included here. So by
34:54 way, so the next thing I'm to ask you to do is on
34:56 previous plot. Put these points on previous plot. So, uh on
35:04 , put those individual points and see you get and I'll stop recording
35:37 Conference will now be recorded. The was that all of these data points
35:44 essentially on the mud rock line. if we're talking about decreasing bound water
35:52 we move from the lowest velocity to highest velocity. That's essentially a variation
35:59 total porosity because we're counting bound water trapped water all as part of the
36:07 porosity. So you can see porosity uh as we move from one clay
36:13 to another. Okay so some conclusions rocks are aggregates and mineral grains,
36:26 expect the velocity of a highly liquefied porosity rock to depend strongly on the
36:33 of the grains. We also expect of unconsolidated rocks with higher porosity to
36:41 weakly dependent on the grain velocities more on the packing of the grains.
36:49 we'll show that later in a VP . V. S cross plot a
36:54 mineral alec, water saturated rock is to be have uh one endpoint while
37:03 higher porosity, unconsolidated rock would show V. S. Trending towards zero
37:10 VP trending towards the velocity of So you see that v.
37:15 is going towards zero While VP is towards the porosity of water which is
37:22 1.5 km/s. Now it turns so burying ferocity moves you up and
37:31 the line, varying courts content moves up and down the line, also
37:37 pressure, varying pore pressure, varying pressure, effective pressure. All of
37:44 things just move you up and down line, that's why the mud rock
37:49 is so universal. Okay so now try to break things out a little
37:59 more by with ology, the mud , you know, we didn't say
38:04 about the composition of the mud Well let's go to a to pure
38:10 . So these are sand stones, we call clean sandstone, some are
38:14 porosity, some of her high They all have very low clay
38:20 And you see they fall along the and I probably should have drawn the
38:24 rock line here for comparison. Uh But you'll see that these plots slightly
38:32 the mud rock line. Clean court . By the way, here's limestone
38:39 pickets trend was the dash line. you see at low velocities you deviate
38:47 the velocity of water. So actually trend of a constant v. PBS
38:53 1.9 is only good for a high livestock as you get to plastic,
39:03 porous, unlit defied lime stones, or poorly lit defied limestone as you
39:10 , you veer off of this constant PVS ratio. So this is a
39:18 we got for uh shells. It's trying to slightly higher than the then
39:28 courts trend. And uh we could that we can move it up to
39:35 an envelope to say, you because these shells are all a variety
39:39 mineralogy is in them. We could the The most extreme values and hypothesis
39:48 okay, this this would be pretty to 100% pure clay. And so
39:53 have a pure clay line depending on application, we may want to use
39:59 hypothetical pure clay line or we may to use a line which fits the
40:06 uh clay mineralogy. So if we're trying to predict the velocity of a
40:13 and I don't know anything more about . I could use this trend or
40:18 could use the mud rock trend. one would be pretty close, but
40:22 I had composition, I'm going to able to be a little bit more
40:28 about predicting the shear wave velocity. if I know the amount of courts
40:35 amount of clay, I'll be able move myself up or down, you
40:39 from below the trend to above the to to bury the volume of
40:46 And we'll see that later. this is a dolomite trends, not
40:54 different from uh pick it now, them all together and plotting a different
41:03 plotting V. PBS versus VP is useful way to look at things because
41:10 is a proxy for death. As get deeper, I lower the
41:15 I increased the pressure. I moved the right. So you can think
41:18 depth moving to the right here and velocity is increasing with death. And
41:24 , so then we can see what PBS does versus death. And you
41:29 at shallow rocks, pure shell limestone pure clean sandstone all plot with increasing
41:40 PVS as you lower the P wave By the way, the velocity of
41:45 here is about 1.5. So these all going very high as we go
41:51 low velocities. Now, where is mud rock trend here? The mud
41:56 trend is actually more or less halfway the shell line and the sandstone
42:01 It's in between in there. So we've broken things out a little
42:07 and I could contrast that the same wave velocity. I could contrast to
42:12 sand versus a very rich clay Now keep in mind that shells could
42:20 mostly course, in which case they'll closer to sands town and sands can
42:26 feldspar in them, they could have in them. They could have some
42:30 in them. Uh in terms of and that would increase the B
42:35 B. S ratio. So we'll some variation percent for sands, depending
42:40 the composition. So here we're just by with Ology. But later on
42:45 be able to be even more precise we have a volumetric log analysis in
42:51 what the V. P. S ratio is. Notice that gas
42:56 is a constant be PBS ratio independent depth porosity, etcetera. Uh And
43:06 investigate why that might be the case gas sands. Yeah. Now,
43:14 terms of the use of the PBS a mythology indicator. Uh if I'm
43:28 , all the lethality, jeez have high B P. B. S
43:32 and there's, you know, the stones and sand sounds at the same
43:39 have similar Heidi PDS ratio. Now I'm in a pure shell, I
43:44 have even higher, but the overall is V. P. B.
43:48 . Is high. I add gas is very low compared to that high
43:53 . P. B. S So shallow the presence of gas is
43:59 to be very detectable from seismic responses the D. P. V.
44:03 ratio. So a video analysis, stacking version, multi component analysis all
44:12 be effective at distinguishing brian saturated rocks gas saturated sandstone reservoirs. But as
44:21 get deep, he sees something different going on. Uh the hydrocarbon effect
44:28 relatively small. A gas, sand brian sand have very similar the PBS
44:36 and the carbonates in the pure shells hired the PBS not necessarily able to
44:43 the carbonates from shells based on the . P. V. S ratio
44:48 , but the carbonates are often higher . So, if I have a
44:52 Vis. PBS ratio and a high , meaning a high p wave
44:57 probably it's probably carbonate. If it's lower VP, it could be a
45:05 or it could be a very porous . The lower V. PBS would
45:10 indicative of sand and but the length effect is much stronger than the hydrocarbon
45:16 at high velocities. So V. is more of a with ology indicator
45:24 a hydrocarbon indicator in hard, well defied rocks. Now it turns out
45:38 this was work we had done in late 80's and we realized that shells
45:45 have gas in them and gas saturated could have abnormally low V.
45:51 V. S. Ratio. So this is the wet shell line later
45:58 , I'll show you how we do substitution. We can theoretically predict given
46:03 wet shell line. We could predict the gas gas saturated share line should
46:09 . And we get that line and we observe is something in between.
46:14 the reason we get something in between because the theory is not exactly right
46:19 shales. But uh what we were was that we do have log shells
46:27 BPD S ratios and are lower than should be according to our empirical
46:34 And it turns out nowadays we realize are our shale reservoirs there. We
46:39 just thinking about it in terms of it affects the seismic response. It
46:44 occur to us back then that we actually produce these shells. But
46:48 this would be what today would be an unconventional shale reservoir. Uh We
47:01 have other VPs relationships. For coal is a little bit different than
47:08 others. And the V. B. S. Ratio. And
47:11 could be quite high even though this uh this here is a a pretty
47:19 liquefied coal. Uh huh. More a towards an anthro site, it
47:26 has a very high B. B. S ratio. So
47:30 which relative to uh most sedimentary rocks abnormally low impedance, low density,
47:40 velocity. His V. P. . S ratio is very high.
47:47 here we're toughs. And here with laboratory measurements on tops, here were
47:53 log measurements, a trend from lock . Uh and uh so we're also
48:00 that toughs can have follow a P. B. S trend.
48:08 , to summarize mythology, discrimination is at high velocities. VPBS is 1.6
48:17 stand stone, 1.8 per dolomite, for live stone. Uh However,
48:27 high velocity shells and carbonates can look . Measuring the PBS from seismic data
48:36 difficult and not very accurate. So could be error in predicting mythology from
48:43 . The Pds rations also at high is the difference between gas and full
48:50 saturation is relatively small. So the ology effect hope arises. Uh
49:04 This was a data set that we , where we had a variety of
49:12 Allah, jeez, and we're measuring B. P. And B.
49:17 . From full waveform sonic data. here we're going through many different
49:22 limestone salt, dolomite, sand stones shells. And we have our limestone
49:28 here and we have our mud rock here. What do we find?
49:34 sand stones and shells flat along the rock line. Some of the carbonates
49:41 along the limestone line. And that kind of forms an upper bound
49:46 of the carbonates plot in between and would be your Dolomites, which are
49:51 open squares, but there's also the that you have sandy lime stones.
49:56 picket observed the lower the PVS ratio sandy or Shelly limestone. So that's
50:04 what's happening here interestingly. Salt, have a couple of salt measurements which
50:12 to fall precisely on the mud rock uh which in some ways is a
50:20 because in the gulf of Mexico, know, what are the dominant
50:25 Well, sandstone shell, there's some uh in the Mississippi delta. Not
50:31 lot, but there's some, but also a lot of salt. And
50:37 what we're finding is that all the saturated rocks and salt fall along the
50:42 trend. And uh that's convenient because anomalous things, we'll be distinguishable from
50:55 pathologies by the way, the same trends. Of course there's a
51:04 to 1 relationship between V. V. S and Hassan's ratio.
51:09 for those that prefer Hassan's ratio, can look at the same trends and
51:16 same conclusions can be drawn. And for your reference and for you'll use
51:25 in your exercises here is the table the PBS trends. So uh they're
51:34 as pollen out meals. The next to a business is to try to
51:45 these trends in particular. We're going try to understand why we have well
51:51 VP VS. V. S relationships we're going to start with dry sand
51:59 . Remember these V. P. . S. Friends are for brian
52:04 rocks. So adding fluid complicates So let's start with dry rocks.
52:12 if we understand what's happening with dry . And remember I said that for
52:17 gas sector in Iraq you of the . P. B. S ratio
52:20 1.5. A Clean Sandstone has a ratio of 1.5. Well um that
52:29 to persons ratio of about .1. so here we have dry sandstone
52:35 These are clean sand stones. And see they all fought along a.
52:42 Where the v. p. s ratio is about 1.5. Now
52:53 should the D. P. S ratio be around 1.5? Well
52:59 if we have perfect spheres? And if they were made of?
53:04 And we could look at what the are made of. We can look
53:08 the mineral composing the sphere so we make theoretical calculations. And these are
53:15 these equations were worked out in the . They're very well known very
53:19 very solid, very well defined. remember we talked about different kinds of
53:27 we had loose packing like simple And we have the densest packing here
53:34 is face centered cubic. And we predict the V. P.
53:40 S ratio of the packing of uniform versus the person's ratio of the
53:50 Remember courses around .1. And what find is that for a wide range
54:00 Hassan's ratios, you don't get a different change in the dry rock.
54:06 PBS ratio Remember .5 is a So to get a simple cubic arrangement
54:15 year fluid materials You get a maximum . PBS ratio here 1.73. Or
54:24 . But our grains are not fluids , grains are going to have much
54:28 concentrations so you can see no matter we packed the spheres and no matter
54:33 the spheres are made of, we a similar dry rock, be
54:39 Maybe it's not surprising then that, know, low velocity wine stones and
54:47 velocity sand stones have similar V. ratio. But anyway this is the
54:52 rock and uh what you find is to mineralogy. Remember we said if
55:02 have a zero porosity rock there should a strong dependence on mineralogy,
55:08 It's it's zero Porosity. Rock is poly crystalline aggregate of the mineral randomly
55:14 mineral grains. So that gives us mineral velocity. And so therefore the
55:21 porosity rock will have the bpd s of the mineral but a very unconsolidated
55:28 poorly consolidated rock that is some kind packing of grains is going to have
55:34 low V. PBS ratio when Okay, now let's see what other
55:45 do to the V. P. . S ratio. Again, we're
55:51 rocks and we have a variety of stones here. And a number of
55:58 made on those samples and we have circles which is the original sample.
56:07 then we have the filled symbols here are the sample after heat cycling and
56:15 microfractures by heat cycling. Um We have the measurements at low pressure and
56:25 high pressure. And what do you what you see is it doesn't matter
56:31 we do to the rock, no how we we injure the rock or
56:37 matter how we try to damage it the point where we heat it and
56:41 we cool it rapidly and cause the to fracture. We haven't changed the
56:46 . P. B. S ratio much in this dry rock. The
56:51 . P. B. S ratio still About 1.5. And if I
56:56 the pressure, I am a low here, I increase the pressure.
57:02 still maintain the the PVS ratio relatively . This is fundamentally different behavior than
57:09 a brine saturated rock. Remember we as we bury things, we move
57:15 and down the mud brock trend. , what I failed to mention
57:20 we go back to any of these . Let's go to the mud rock
57:25 there as I move up and down mud rock trend. My V.
57:29 . B. S ratio is varying Here. v. PBS is 1.5
57:36 . V. PBS's infinity. So I stay on the same trend
57:43 I changed my V. P. . S rations. So all these
57:46 ferocity, you know, buying the pressure, all of these things move
57:54 up and down the trend. But they all are affecting the V.
58:01 . B. S ratio. On other hand, in a dry
58:08 these dry sands towns, they're not the BPD s ratio. So uh
58:19 can see that the fluids are having big impact on the variation of the
58:25 ratio. We could do the same numerically. And uh we talked about
58:37 modeling. So if we add penny pores to a rock and this happens
58:45 be using a poor aspect ratio spectrum Boise sandstone and we add more and
58:55 poor's to we start with courts and we add ferocity to the courts.
59:01 get this line here, we can't any further than this because we violate
59:08 the limits of the theory. But least down to this point, we're
59:14 much staying on the dry line. this is the computer line for a
59:21 porous rock as I increase the So we think there's a strong fluid
59:35 and we're going to come back and going to look at this diagram again
59:40 on when we talk about fluid but there are tie lines here between
59:47 saturated measurements and dry measurements on the rock. And you can see the
59:53 line is here and the dry measurements or less fall along the dry line
59:59 for this guy which was cal Karius lot of limestone which is moving you
60:06 to hire the PBS treasure uh that is cemented with limestone. Uh But
60:14 have our dry line here and then we move from one at one end
60:19 the tie to the other, it you off the dry line and moves
60:23 towards the mud rock line and we'll back and we'll look at these different
60:29 in detail and we'll try to explain different points in detail. But anyway
60:36 see the tendency here so the fluid moves you off the dry line to
60:41 mud rock line. Remember the dry is a constantly PVS ratio the mud
60:46 line has increasing be PVS ratio as lower the velocity. Okay, now
61:05 if we take this assumption that my . P. B. S ratio
61:11 this line, I could come back I could do a theoretical fluid substitution
61:19 we'll show you how to do Uh Probably tomorrow. And well we'll
61:26 able to take this dry line and predict what the fully brian saturated line
61:33 look like. So we can do . Uh For example suppose I have
61:43 p wave velocity. I could uh estimate the ferocity from the p wave
61:52 . I could assume a frame share . Let the frame sheer module is
61:58 to the frame both modules by frame modules? I mean the share modules
62:02 the dry rock? I could let dry rock sheer module is equal to
62:07 frame both modules that puts me right this line. Okay, maybe Pds
62:14 1.5 and a person's ratio of about means the bulk modules equals the sheer
62:22 and that's what's happening in dry Um I can now predict the saturated
62:31 module issues using gas mains equation and is the question. We'll learn how
62:36 use later. I predict VP I compare that to the original P wave
62:45 and I could modify the assumed sheer until I've matched until the predicted and
62:53 observed P wave velocity match. So could then predict the the wet be
63:02 trends from the dry trend. Everybody me there, by the way,
63:11 me just say one more thing about dry trend. We sat down
63:17 we're sphere packs up here, we're mineral and in between we're adding ferocity
63:24 the mineral. Why is this a line. Well, I think it's
63:32 because the sphere pack has a Pds ratio of about 1.5. The
63:37 has a V. P. S ratio of 1.5. And we're
63:41 going to deviate much as we move the two as the rock becomes less
63:47 packing of grains and more mineral you know, probably crystalline mineral aggregate
63:53 inclusions in it. We move up down this line. But what if
63:58 were limestone and that sandstone? Well it. The sphere pack doesn't care
64:05 the V. P. V. ratio of the of the grains
64:12 But up here I would have a higher v. PBS ratio. I
64:16 have a v. p. s. ratio at 1.9. So
64:20 number would be closer to eight or closer to closer between seven and
64:27 Right, It will be one B times 1.9. So what would that
64:32 ? 7.6. I would have a higher velocity here. So this curve
64:38 line would have to curve. It have to go from along here and
64:43 curve up towards the mineral. but let's go come back to sand
64:51 and let's just accept the fact that frames or the dry rock share module
64:57 is about equal to the bulk So this is a bunch of sandstone
65:06 and I had the observed shear wave and the predicted shear wave velocity in
65:12 way, going through gas Men's equation using this thinking and what I find
65:19 I could predict the shear wave velocity well. All right. So uh
65:25 don't necessarily have to measure the shear velocity and as we'll see later,
65:33 could predict the VP VS. Bs . Um Now something else I can
65:42 . I could look at these fear And here we have a simple cubic
65:47 pack and I have a face centered hexagonal close packing. And by the
65:52 they both give the same V. . D. S trend. How
65:56 this accomplished? Well the dry uh huh. Simple cubic packing,
66:04 , velocity was predicted as a function pressure. Do you get a range
66:11 VPs and VDs? Is those those from theory? And then I I
66:18 this approach and I predict VP here predict VP using gas men's theory.
66:28 so I get this line and these to be measurements on beach sand.
66:36 see they're giving a B. B. S relationship very similar to
66:41 fear pack equation. Of course beach are not perfectly spherical sphere packs,
66:48 there be PVS is similar. And notice that. Okay, so this
66:53 as I'm varying pressure. This is being done theoretically and this is as
66:58 burying pressure. You notice that it's to pull me below the mud rock
67:04 . And it's pointing at this trend here is trying to come back
67:10 These are leveling off. You can about these Osama automatically approaching a
67:16 Right? So what would that line ? We'll come back in a bed
67:21 we'll see what that line is. . Now we could go back to
67:30 paper, his review paper and he shell bP versus depth and sand sound
67:35 . P versus death. And we do the same thing. We could
67:39 the mud rock. Friends predict the . And we could use the technique
67:45 mentioned to predict the s in the stones. So then we could predict
67:50 pds versus death. This is in gulf coast. You see shells is
67:56 smooth compacting curve and sand stones have knee in that. So this is
68:05 we have a, you know, rearranged and we've uh we've consolidated the
68:13 to the point where it's fully consolidated we have less porosity variation with
68:21 And we saw this in the velocity . We also see it in the
68:25 . P. B. S. . On the other hand,
68:28 it's a more continuous deformation that keeps going as you get deeper. So
68:40 were multi component results. Uh So are interval velocities from multi component seismic
68:52 . And so seven stations where we measured V. P. V.
68:57 . Ratios from multi component data and , Station three was a reservoir at
69:07 interval. Uh These were morrow sandstone . Sometimes we were on the mud
69:14 lines. Sometimes we're on a lower which we're going to talk about some
69:19 but station free was well below all other lines and that was a
69:28 So uh the V. P. . S ratio can be used to
69:32 hydrocarbons and that's basically that's the basis a video analysis. Yeah, we
69:43 also see what the implied uh elastic ir of course, from velocities
69:53 we can't get the module i directly knowing the ferocity or the density.
69:59 we could calculate sheer module is divided density in both modules divided by
70:06 knowing that the density is varying along lines. But what you find is
70:13 for dry rocks, if the bulk is equal to the sheer modules,
70:18 the bulk modules divided by density is to share modules divided by density.
70:23 we're along that line there with equal and share modules. Yeah. So
70:33 if I had water going to you mentioned earlier in the last lecture
70:41 engaged the rock rescued you. They yeah, they can at very low
70:49 . Yes. But there was a you asked the question. I remember
70:58 want to upset this one back for . Okay, we'll make this mission
71:02 share with you introverts. We all spices, Right? Yeah. And
71:10 what you find for the saturated rocks is for the same bulk modules as
71:17 dry rock. The wet rock has much lower share modules, but really
71:24 of the effect is at the same modules. The dry rock has a
71:30 bulk module lists and the saturated So if I were to draw lines
71:36 this, these two, most of happens is horizontal here. That adding
71:42 increases the bulk modules and that's the effect more so than decreasing the share
71:55 . And you can look at, example, pistons ratio versus compression of
72:02 for the saturated rock versus the dry . And if the V.
72:06 B. S ratio is constant versus of velocity, so is the person's
72:13 . And you see prisons ratio increasing , going towards .5 as the compression
72:20 velocity approaches the velocity of water. here we are at the mineral persons
72:28 . Here we're at the person's ratio water. Okay, now if we
72:38 the sandstone points, they tend to along the mud rock line, but
72:46 a tendency to go slightly below the rock line. And we could do
72:52 same kind of modeling that we did the dry rocks and we could predict
72:59 huh The velocities as we add ferocity different types. So the green
73:09 I've added a ferocity with the spectrum aspect ratios of Boise sandstone and what
73:18 find is that doesn't drop you very below the mud rock line. On
73:25 other hand, if I close all microfractures and I keep only The aspect
73:32 is greater than .1. That gives the red line. And you see
73:37 heading towards the majority of points here fall below the line. So we
73:42 our sphere packs are coming in this and now our equant ferocity is coming
73:49 in this direction. So what pushes up towards the bedrock line? It
73:54 be a few things, it could mineralogy, could be adding,
73:58 could be adding court feldspar, could adding calcite, it moves me
74:06 but having big ground pours moves me , we're having a spear pack in
74:14 would move me down. So uh is explaining the points below the mud
74:22 , kind of course gas would also me down. But we're assuming everything
74:26 is fully brian surgery. What moves up? Well, we said
74:31 remember that causes me to veer towards calcite velocity. So there would be
74:37 sight line here and we could also other minerals on there and explain those
74:48 . Uh huh. Now I could a velocity ferocity transform and I could
74:54 the same approach in predicting shear wave and what I get the time average
75:01 gives me this line here and you see how the time average equation is
75:07 a lot of the data points So, and my spear packs are
75:13 in here and trying to join up the time average equation. And then
75:17 time average equation takes over. Now I have a p wave time average
75:26 , ferocity is moving me up and this line right. Uh Then I
75:32 also have a shear wave time average because ferocity is moving up and down
75:39 line, A shear wave velocity is . So we could write a time
75:43 equation for shear waves and predict ferocity shear wave velocity. Uh This is
75:51 equivalent fluid velocity for shear wave Of course, we know that's
75:57 We know the shear wave velocity is , this is the observed shear wave
76:03 time, this is the sheer weight transit time for the solid material.
76:09 Now, why would we bother to this? Is there an advantage in
76:13 sheer ways to estimate ferocity rather than waves anybody? Um When you were
76:30 a certain, remember you saying, think people is moved around to the
76:36 condition. Do they have a lot respect? Oh, you go
76:42 It's found Bill. Yes, So because they do that, uh
76:53 volume changes and so the fluids are . So the flu is have a
77:00 effect on the p wave. on the other hand, the shear
77:05 shear waves are rotational waves, They distort the shape, but they
77:10 change the volume. So the shear are much less affected by the
77:15 So, if I'm trying to predict , uh if I'm in a
77:21 the shear waves won't be affected by hydrocarbons to the extent that p waves
77:27 . So the shear wave time average would be a better way of estimating
77:36 . Okay, so we can let's take a 10 minute break at
77:41 point. So I'll stop recording and proceed this conference will now be
77:51 So remember that there is not a velocity porosity transform. And we said
78:00 you know only certain rocks will obey particular equation. So not all of
78:06 obey the time average equation. But select rocks that do. Let's select
78:12 stones. Yeah, follow the P time average equation. So we've selected
78:20 points and they fall on the P time average equation. Now let's for
78:28 sand stoves let's then take their shear velocities measured in the laboratory. And
78:37 compare them to the shear wave time equation. And the conclusion is if
78:44 degree of consolidation and lift, ification pressure conditions are such that you are
78:51 the p wave time average equation, you will also obey the sheer weight
78:56 time average equation. If you Uh huh. A fully brian
79:09 Now, we also talked about the Gardner equation kind of being applicable to
79:17 most liquefied rocks. And just to the p wave velocity is equal to
79:24 minus prosperity squared times the p wave of the matrix plus porosity finds the
79:32 velocity. Now this is purely an equation, there's no theory behind
79:40 but let's assume for that. It theoretically correct for the moment. And
79:47 say okay if somehow we don't know but somehow this is theoretically correct then
79:56 I put shear wave velocity of the of the solid grains here and I
80:03 put the shear wave velocity of the here, then I should get this
80:08 right shear wave velocity, the fluid zero. So that goes away.
80:14 , I have no more fluid effect worry about and now my share wave
80:18 is just one minus porosity square at the velocity of the matrix. Um
80:25 how does that work? Oh by way, um from this p wave
80:38 average equation I could solve for ferocity I'm you should be thanking me that
80:44 not giving to you this as a problem, but that's the solution for
80:52 . And what this tells me is given the P wave velocity I could
80:58 out the porosity. So this is measure P wave velocity. I could
81:04 out the ferocity and now I could ferocity here. That means from p
81:13 velocity, I could predict shear wave . So there's the equation with ferocity
81:20 as a variable. So, given wave velocity and the fluid velocity,
81:26 matrix p wave velocity in the matrix wave velocity, I could predict the
81:33 right philosophy. So that gives me D. P versus V. S
81:39 . Right? Similarly, I could done this in the spreadsheet sheet quite
81:43 by varying ferocity and computing B P G. S. So either way
81:49 could get a trance, this is analytical solution, but you could just
81:54 it numerically in your spreadsheet Now, few things, looking at this equation
82:04 as the sheer able as um the wave velocity approaches the matrix philosophy.
82:12 as VP becomes VP matrix, what find is that Bs approaches V.
82:21 matrix and then VP over Bs also BP matrix over V. S
82:33 So this happens as BP goes to VP matrix, which should also happen
82:38 BF goes to zero. Okay, now remember I had the two time
82:49 equations uh for p waves and s . Now I could plot roemer and
82:56 equations for p waves and S So at any porosity I can predict
83:02 and Bs. So that gives me V P B. S trend.
83:10 we'll come back to the, to predicted trends in a moment. But
83:17 notice something that's more satisfied about the and Gardner equation than the time average
83:27 . So remember the time average equation take into account degree of lift,
83:38 so often what is uh used is correction factor for a degree of lipid
83:47 and the compaction factor is equal to troop ferocity divided by the ferocity you
83:53 have predicted from the time average So you can then multiply this by
83:59 predicted ferocity to get your estimate of true ferocity and sometimes that compaction factor
84:08 estimated from the shale velocity because remember compact very regularly. So you could
84:15 come up with a relationship between the factor in the shear wave velocity and
84:22 approach is used by a log analysts using the time average equation in poorly
84:30 defied rocks. So, knowing the ferocity versus that, which would be
84:37 by the time average equation. We calculate the compaction factor and we could
84:44 that for Gregory's data that he remember has velocities versus death for thousands of
84:53 in the gulf coast. And from wave data, as we said
84:58 we could predict the shear wave data gas men's equations. So we could
85:04 at this compaction factor and what you in the gulf of Mexico all but
85:10 very deepest rocks need a compassion And when we're fully compacted, you
85:22 say but not fully lit defied. compaction factor is small, but then
85:27 becomes very large as we get But the unusual thing here and the
85:33 that's not satisfying about the widely time equation is that for share waves you
85:41 a bigger compaction factor. Much bigger factor for p waves. But that
85:49 make sense. The rock has the degree of compaction for both P waves
85:56 share waves. P waves and share should have the same compaction factor.
86:01 yet they don't indicating an imperfect relationship compaction. Now let's look at the
86:12 of the P wave to the s compaction factor for um um the rain
86:22 Gardner equation. Well compare it for roemer and Gardner equation and the time
86:29 equation. Mhm. So, so the Ramayana Garden equation for p waves
86:35 then doing our Gassman substitution, that's called Vo theory. It's the low
86:41 limit of video theory. So doing that way. What we find is
86:49 factor for P waves and S waves the same. The ratio of the
86:53 factors are the same for P waves S. Ways whereas for the time
86:59 equation, the S waves need a bigger correction, shallow. So again
87:07 roemer Gardner equation is not a theoretical , but there's something satisfying about
87:17 Coming back here according to the reverend equation. I could do a fluid
87:26 just by changing the velocities here and we find that it's not correct,
87:33 it is much more correct than the time averaging questions. So I could
87:39 a crude kind of ballpark fluid substitution from the Ramayana Gardner equation that I
87:46 do with the wildly time average Yeah. Now what about the implied
87:58 VVS relationships? Well remember I could vary porosity calculate VPN Ds and I
88:09 cross plot bp versus V. S I could do that for the ray
88:15 Gardner equation and I could compare that the sandstone line. That's this empirical
88:25 that we see here and they lay on top of each other Or I
88:31 take the empirical trends, take a for any velocity ferocity transform. And
88:38 could predict versus gas mains equations as did before the shear wave velocity,
88:45 of these trends overlay. So doing Gardner for P waves and shear
88:52 Doing Raymond Gardner for P waves and fluid substitution to predict the shear waves
88:58 the empirical V. P. S. Relationship. They're all
89:10 Now, here was real shear wave uh measured in a gas sand reservoir
89:22 there's a mud rock line. There's Ray martin Gardner line, which happens
89:28 agree with our empirical trends. And the dry line. And what you
89:34 is that the get reservoir has lowered PBS ratios on the average than predicted
89:43 the rain bird garden a lot. hydrocarbons are still a factor here.
89:49 we could have varied the VF. this is the brine saturated trend.
89:55 could have pulled that trend down. lowering VF. Right. If we
90:03 back here, we lower VF, don't change V. S. We
90:09 VP. So we lower the P. V. S ratio.
90:13 what would happen here from the remote garden of line? If I reduce
90:21 , I would move the point, don't change shear wave velocity and I
90:24 move the points down. And we see that this rain behind Gardner
90:38 wow, it follows kind of a bound here and even matches those lose
90:49 down at that end. And this just matching the uh huh Data from
91:02 predicted shear wave velocities versus what would predicted from Ramer and Gardner. So
91:09 giving the same V. P. . S strength. Okay, now
91:16 more point on shear wave velocities. I want to predict mythology from P
91:27 velocity, shear wave velocity and Let's just for the moment assume that
91:37 don't have the complication of fluids. . So everything's brian saturated just to
91:42 life easy. Can we invert if just take a few 100 measurements of
91:49 PBS and density, can I correctly mythology from just V. PBS and
91:57 ? If we come back to these here, this is like a balkan
92:03 of material. Here we go. look at these trends here. Well
92:12 multi valued here, right? I have Uh huh. The same same
92:18 ratio for different mythologies. I could the same velocity for different mythologies.
92:25 and even with density, what you is that the solution is non
92:31 There are different little logic combinations which give me exactly the same p wave
92:37 shear wave velocity and that's so if try to do that inversion sometimes we're
92:44 to be wrong. And so then just a matter of how often if
92:49 take a random sampling of measurements, often will I be wrong? Okay
92:54 here I have the actual mythologies that known from looking at the core
93:03 Um is the actual ethology, Is it actually limestone? Or is
93:09 actually dolomite? Um or is it mixture? So, um if it's
93:20 a sandstone, It will be predicted be a Sandstone, of the
93:28 Notice we don't have shell here. would greatly complicate things, But relative
93:35 limestone and dolomite, I could correctly us and 92% of the time.
93:41 I would call the sand dolomite if truly a limestone, 80% of the
93:51 , I'll call it a limestone 20% the time. I'll call it a
93:55 might almost never call it a If it's a dolomite, that's more
94:03 . Remember it's in between limestone and . So some fraction of the time
94:09 call it a sandstone. Very often call it a limestone and only about
94:16 the time will I correctly identify? has a double mind and in the
94:21 of a mixed pathology here, it , there weren't many of these,
94:26 it was always called the limestone. , so we're ready for the next
94:37 . And so that has to do persons ratio. So cross plot with
94:45 ratio versus the ratio of the bulk share modules. So you'll have to
94:50 a little bit of algebra there, have the you know that the velocity
94:56 . So, you know, the between VP over B. S and
95:01 over me. You know, what that relation? VP Over V.
95:06 . Is equal to square root, from you plus four thirds. All
95:11 . So you could do that algebra then, you know, persons ratio
95:15 terms of the PBS. So you then cross plot with sons ratio over
95:21 overview. So let's start there with exercise, and I'll stop
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