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GEOL7324-Rock Physics - TuesNov2
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00:01 this conference will now be recorded. me come back to this line because
00:13 misled you a little bit when I talking about this slide and then realized
00:19 I had it wrong and I corrected . So just so there's no
00:25 I'm going to repeat myself again. the idea here is that we have
00:32 , that's approximately 15% ferocity. So , 15% ferocity. But we see
00:41 relatively a significant change of velocity with . What they're not factoring into account
00:50 the ferocity would be changing with This is just taking into account the
00:56 difference. So if on the You know, one cm per foot
01:03 for a level of depth. And you could Using a poor pressure gradients
01:10 .465 p. S. I per . Then you could calculate approximate number
01:16 C. Associated with the death. actually this axis was probably the horizontal
01:24 was probably converted from a differential axis in the laboratory. Now as you
01:30 this is as you're pressuring or rock in the rock frame up in the
01:35 story. So what's not being considered the fact that when you pressure up
01:41 rock frame, the ferocity is going change slightly as you compress the
01:46 you'll reduce the porosity. But the is that that small porosity change in
01:53 hard rock, like a dolomite. might be a very small percentage of
01:58 , but you're closing the most significant , you're flattening grain contacts,
02:06 You're closing the very low aspect ratio which had the biggest influence on the
02:13 . So you can change the velocity lot without changing the ferocity. Very
02:20 so the prostate is approximately constant. the other thing that's being done
02:27 So the frame module I versus depth determined and then using geothermal gradients versus
02:35 and pressure versus depth, the fluid for different fluids are being considered and
02:41 being added to the rock. And you see that there's a cross over
02:46 between heavy oil and water. The effect is making that heavy oil from
02:52 compressible than water to two more compressible water at depth. Then there was
02:59 light dead oil and then a light oil and in each case the velocity
03:06 reduced. And of course gas has lowest philosophy. So I hope that's
03:13 . Now. Now let's go into laboratory and look at how velocity changes
03:21 saturation and what we're going to find things in the laboratory don't do exactly
03:27 I've been telling you. They In fact, they don't follow gas
03:32 equations. Um Now there are various to get different partial saturation in the
03:41 and how you do it matters. were measurements on a beach sand and
03:47 are ultrasonic measurements and the dry sand here is measured and then gas mains
03:56 are used to predict the velocity versus . So you see this is doing
04:03 we've been uh seen with gas Bruce . and then you get to the
04:09 brine saturated velocity, so the density , because his velocity to decrease as
04:16 add brian. But then when we're brian saturated, the module lists the
04:21 off switch on Woods equation takes over you quickly go to brian module list
04:27 and you have a brian velocity. , oddly enough, what you measure
04:32 the laboratory is not that curve. so there there are differences in
04:38 at low saturation, there are tremendous and you could argue this is
04:46 Uh but that is because these are frequency measurements versus gas lose equations,
04:52 low frequency, zero frequency, but is this is a huge difference from
04:59 to here and we have not observed kind of dispersion, where we had
05:05 high frequency and low frequency measurements, that's enormous. Um so there's something
05:15 going on. There is this but there is also, there was
05:22 the fact that BEOS equations are the frequency equations, whereas gas means equations
05:29 the low frequency limit of the O theory. So there is some
05:35 , but there's also something else going and it has to do with how
05:39 gas is distributed in the rock. already looked at the patchy saturation
05:45 And so we see that if there patches of different saturation that could cause
05:52 to deviate from gas loose equations, there is also how the gas is
05:58 in the pore space. If the tends to have the highest concentration in
06:07 big round spherical pores, uh they have a small effect on the velocity
06:13 if they were concentrated in the low ratio for us, if the gas
06:18 in the low aspect ratio for it's going to make those pores infinitely
06:24 compared to those pores being fully bride . So how the gas is distributed
06:31 the rock matters uh, at high , we don't think that low frequency
06:37 that is very important. But to sense of a laboratory measurements, we're
06:43 to have to consider that. I'm sure we'll have time in this
06:47 If we do, when we talk this version, we'll come back to
06:52 . I'll show you a little bit it today, but we'll talk about
06:55 in more detail if we have time also that the shear wave velocity is
07:04 . I think when we have partial or full brian saturation, than would
07:10 predicted that just from the density So, there are other things going
07:20 . All right. So, as mentioned before, this is the form
07:24 gas music equations that I that I uh to remember the equation easily and
07:32 conceptually think about, it's less convenient calculation because the variables occur in multiple
07:42 . Right? So this is not for case saturated, but if you
07:48 at this ratio of cas saturated versus solid module list minus the saturated
07:56 If you look at this ratio, can notice a few things first of
08:02 , If ferocity goes to zero, term blows up. So a very
08:11 um conclusion. The lower the the bigger the fluid substitution effect.
08:20 is the fluid substitution effect? Is the difference between K. Dry and
08:26 ? So this term is the difference these two terms. Right? So
08:32 this term blows up the change in lists due to the change in fluid
08:40 is the change in saturated modules, that change in fluid modules becomes greater
08:46 that's counterintuitive. And the reason it's is because we've held K dry constant
08:54 we've changed ferocity and we can't really that. If we're going to drop
09:00 uh to make this term blow we're going to have to increase K
09:06 As ferocity goes to zero. Uh dry goes to K Stalin. So
09:17 draw wrong conclusions. If we hold variables constant when they're truly not independent
09:25 each other. But this is The fact that the lower the ferocity
09:32 a given K drive to lower the , the bigger fluid subsume fluid substitution
09:39 . We're going to be able to some conclusions about the poor shapes uh
09:48 on that. Now, another counter thing, let me this is the
09:56 substitution equation. I have K I'm adding a fluid and I get
10:02 saturated. But I could write this twice for two different fluids.
10:08 I could write case at one equals . Dry term and the K fluid
10:14 . And then I could write the again, case at two same K
10:20 Que fluid to write two different modular. It could be two different
10:25 . It could be the same fluid two different temperatures for example.
10:30 now I have two equations and if subtract one from the other, I
10:36 cancel out K dry. And that me here. Okay, I have
10:44 saturated one Kate fluid one case saturated and kate fluid too. And you
10:51 , I canceled out K. So, hopefully by now I've discussed
10:57 this idea enough that you will be to exercise. Uh Uh you'll be
11:04 to answer 9.1. All right. just tell me this equipped this fluid
11:14 equation seems to indicate that I don't to know. Okay, dry,
11:19 could do clued substitution without K. . And that's true. Uh
11:26 But you could conclude that then the substitution is independent of the dry frank
11:33 . You could argue you don't uh effect is doesn't depend on the dry
11:40 pre modules because it canceled out in two equations. So, tell me
11:45 that argument would be wrong. Now, if you've read Gregory's
11:54 I hope you have Gregory 1977 Great paper. He gives tables of sandstone
12:02 versus death and we can play the fluid substitution game on. He gives
12:09 , he gives velocities, he gives uh huh the 10th series. So
12:17 could do the fluid substitution game with kinds of velocity versus death trends.
12:23 if this these are my fully brian rocks. Again, the same thing
12:28 saw with the dolomite things are more here because porosity is varied with
12:33 We haven't held ferocity constant here. , So the ferocity is decreasing with
12:39 and the velocities go up. You the spread between brian and uh the
12:45 and oil increases as you get shallow the frame modules gets weaker and weaker
12:51 as you get deeper and deeper they closer together. The other significant thing
12:58 , the light live oil is getting and more like gas, so it's
13:05 depth because as temperature increases, it to the modules becomes more similar to
13:12 of gas, Which brings us to 9.2. Uh what is unusual about
13:24 shear wave velocities and explain the p velocity. So we have dry and
13:29 velocities. Notice at 2000 psi differential , there are pairs of measurements being
13:40 . One is as you're increasing The other is as you're decreasing
13:45 So that shows you the histories is on. And um you have the
13:53 in general increasing with pressure here is mains equation. So starting with the
14:00 P wave velocity and adding the brine you two here and in fact,
14:08 we observe in the laboratory is it's more and similarly dry shear wave
14:17 . And these are the observed excuse me, saturated, fully saturated
14:22 wave velocity. So uh describe the , tell me what's unusual about the
14:29 wave velocity and explain the p wave . How can these uh huh velocities
14:38 faster than what gas mains equations predict the dry module. All right
14:49 for the next couple of weeks, hope you will be doing fluid substitution
14:57 your own. And so I've laid a few workflows. You're going to
15:03 to do this in order to answer questions to come. Uh So,
15:09 you can do it in that you can do it in a
15:11 I don't care how you do but but this is uh kind of
15:17 . So suppose I give you vP in density. Give me the martial
15:26 . And so this is following some they stole from tad smith who has
15:32 review paper into physics. And you it's in your reading list. You
15:37 should be reading that. Now. calculates the bulk modules and share modules
15:42 G for share modules instead of Uh So uh the sheer module is
15:49 row the S squared, you've measured . Now, if you don't have
15:53 Bs, you'll predict it from a . P. B. S
15:57 assuming you're fully brian saturated. If not fully brine saturated, that gets
16:03 complicated. And we'll talk about how do that later with the uh we
16:09 an algorithm called the Greenberg Castagna algorithm that enables us to do this fluid
16:17 , starting with any saturation. But , let's assume that we're fully brian
16:25 at this point. So we either VSO we predicted from the V.
16:31 trend. We calculate the sheer modules the sheer module lists and in the
16:37 wave module is roe V. P . We calculate the saturated modules.
16:44 to do fluid substitution, we need few other things. We need the
16:50 module list. So I'm gonna uh that from the mineralogy, we'll talk
16:56 that in a bed. But let's just let's just assume for the moment
17:04 we're dealing with a homogeneous material. it's all courts, for example.
17:09 we'll just use use the solid modules courts. Actually gets, we have
17:15 make some assumptions if we have more one mineral. But right now we're
17:20 going to assume we have course. You need to know the fluid
17:25 So we have the bats long We need to we need to know
17:30 saturation. So from temperature and pressure fluid compositions, we calculate their module
17:36 and we get the fluid module lists uh from the baths along equations and
17:44 applying Woods equation. So all the faces get mixed together and we get
17:49 fluid modules. Um Now that's not to do includes substitution. We need
17:56 know the frame markets. And fortunately we have all of these parameters
18:04 we can calculate the frame modules. here ted smith calls K star,
18:10 like to call it K frame or skeleton. A lot of people call
18:14 K dry. It's just easier notation to use K. D. Or
18:19 dry. So, you'll often see try later. I'll talk about why
18:24 don't like that term que dry. anyway, you rewrite gas and this
18:29 just Gaspar's equations. That L should a subscript. Sorry, this was
18:36 , you know, I stole this . I hadn't noticed that that L
18:39 be sub scripted. Right? So cape fluid there. And I also
18:45 you this equation elsewhere on a previous . So, we can now solve
18:50 K dry. Now we have K . We could rewrite Gaskins equations.
18:56 is a convenient form and we have dry. We have the solid module
19:03 . We have the porosity that must known. And so we would have
19:08 that from the density um fluid module . We calculate the saturated module list
19:16 the new fluid modules. All So, this fluid module is to
19:20 paid dry, was the institute fluid ? This now is the fluid modules
19:26 the new conditions. By the one thing you can do in gasolines
19:31 is calculate the effect of changing right? I can't hold everything else
19:37 and just change the ferocity and see my new case saturated is. That
19:43 won't work. In fact, that works in a backwards direction.
19:48 don't try doing that. The reason warn you is because I've seen people
19:52 to do that. You can't prosperity the same at both saturation conditions and
19:59 , ferocity is the same and dry module. This is the same.
20:07 right. Do influence substitution on the density is easy. That's just the
20:12 balance equations. So, we need need the new fluid density and then
20:17 calculate the velocities from our velocity equations the way, the sheer module
20:23 He doesn't mention what happened to shear . Well, that stays the
20:27 We're going to ignore chemical interactions between fluids and the solid. So,
20:33 is purely mechanical and all the cores in communications such that poor pressures can
20:39 great. So, the sheer this is the same at any saturation
20:47 . So, uh, pretty straightforward . And so you're going to be
20:52 that. All right. So, just run through it again.
21:00 there are different flavors, different ways set things up. So,
21:07 I thought this was a pretty clear of explaining things. So, I'll
21:11 through it one more time and I'm that I have V. P.
21:17 porosity. Those are the only uh I have at 100% water saturation.
21:27 it's A. S. H. equals 100%. Now calculate The velocity
21:34 50% saturation and with the other fluid whatever could be gassed air,
21:42 you'll use, the appropriate fluid module for those different fluids. So we
21:49 with uh well this is the equation need to solve right? We need
21:54 predict the d. p. at saturation. So we need, the
21:59 module is at 50%. The share is at 50 And the density at
22:06 . Alright, so that's easy So let's go after these one at
22:10 time. So getting you 50% is easy part. Uh We started at
22:19 saturation. So we could use a . P. V. S trend
22:22 we can calculate bs right? And uh we get you 50% from roe
22:29 . S squared. We'll need density do that. And so we're 100%
22:34 saturated. So if the fluid density one and the solid is 2.65.
22:40 gives me the density and the sheer list is roe V. V.
22:44 squared and I'm going to hold that . It since virginity is not affected
22:49 the fluids. So the rigidity at is equal to the rigidity had
22:56 So I've got one out of What's happening? The ferocity,
23:01 And then you said it was sub . But are you supposed to substitute
23:07 number and what? The mass balance there? Now ferocity stays the
23:13 It's what was given or maybe density well, log analysis and a ferocity
23:21 predicted. You need to hold that throughout the fluid substitution process. It
23:27 change in the fluid substitution. If change the ferocity, your predictions are
23:32 to be wrong. Okay, and me, I've seen people do
23:40 Okay, so um The density at saturation. That's an easy one.
23:48 could just use mass balance again, need the fluid density at the new
23:54 . So we could apply mass balance the fluids and it would be the
23:59 saturation times the density of water plus minus the water saturation times the density
24:04 the other thing. The other fluid , oil hair, whenever drilling
24:09 whatever it is. All right. um so that's pretty easy. Um
24:17 it's gas, you know that density going to be quite low. But
24:22 , you can't assume that zero under conditions, but it is going to
24:27 low. So but this will be new density will be on the order
24:33 of half of the uh well, with the a ferocity, but the
24:41 saturation is have so the fluid density be half. And then you get
24:46 new density. Okay, now, need uh The bulk modules that
24:55 And so, right, Gaston's equations way where K dry, I'm using
25:03 . Right. And uh we can this equation and you've got it in
25:09 couple of places or to make life . We could just use a bulk
25:17 versus share modules trend. And it out the sands down, they're about
25:24 . So it's for a clean sandstone stake to assume that the frame both
25:32 is equal to the share modules. won't work if it's the Shelley
25:36 But for clean sands and in the world, that will be a safe
25:43 . All right. And the of course modules has to come from
25:48 that's a long equations and uh we'll pick a number and put it in
25:54 . We now need the fluid mixture. Right? So, we're
26:00 to use with the equation and we're based on the water saturation, we're
26:04 mix the module, list of water the modules of oil or gas.
26:09 this all comes from battle and And we get the new fluid modules
26:15 that's all we need. We have new fluid modules, uh we have
26:20 dry frame modules, we stick these in and we can Now we have
26:26 we need to calculate the velocity Okay, is that clear? You
26:40 be doing it. So you know you have problems, please get started
26:46 that as soon as you can and me know. Uh huh Because they're
26:53 to be a lot of exercises in chapter and it's going to take you
26:58 time. Yeah. All right. you can see that this is a
27:04 of a clumsy uh I should say but it's involved procedure. You
27:11 you need to uh write a program a spreadsheet or have to have used
27:18 else's program to do it. It's like the kind of thing that you
27:21 make a simple calculation on the back an envelope. But if you're in
27:25 rush or you you don't have the for that stuff. Uh There are
27:31 couple of ways you can proceed if just looking for a ballpark figure and
27:36 going to make the argument later that practical purposes maybe a ballpark figure is
27:43 to be close enough. But uh that's the philosophical debate. But let's
27:49 at a couple of alternatives. And is an empirical trend I published many
27:55 ago and another was a paper from where he came up with another heuristic
28:02 which gets you in the ballpark as . So of course coming from me
28:10 purely empirical and so I don't have data to show you. But this
28:16 a polynomial fit between the gas stand and brian sand philosophies. The data
28:22 all proprietary. It was internal to company I was working with, but
28:27 let me publish the equation. And that's what you really need. And
28:31 showed you this one before where if just plot brian stand velocity versus gas
28:37 velocity pretty much follows that trend. there's a lot going on here.
28:42 the relationship between frame modules and brian velocity. There is the effect of
28:50 properties. Now, you know, is old gas ban properties are not
28:57 same, right? Uh this is a rough average to get a
29:02 And uh as you get deeper your , your temperatures go up. So
29:06 have different module. Ivan shallow, . You'll notice that the diagon if
29:11 draw a diagonal here from 5 to . All right. five here to
29:17 . The change is much bigger, and smaller. Deep. And that
29:24 intuitively makes sense. So, this get you in the ballpark. What
29:32 it? A measure up If you your data using your little equation to
29:38 use equations. Yeah, I should should I show you that in a
29:44 . Of course with jasmine's equations, were variables right? You can
29:50 you can have an unusual ferocity for frame module. So, you could
29:54 oil module I instead of gas module Right. So, you could get
29:58 could get a lot of deviation from trend. Uh huh. And I'll
30:03 you that I'll show you the deviations you can get from this trend,
30:07 can be enormous, but we're going put that in a context, in
30:12 practical context, I'll come back to . I'm saying this is better than
30:18 a number out of the air, ? It gets you in the
30:21 Um and maybe you don't, maybe kidding yourself. If you can
30:28 if you can think you can do than that, we'll come back to
30:34 . Okay, This is what Mafco , and his point was,
30:38 to get both modules, I need wet philosophy. Well, what if
30:45 don't have shareware philosophy? So he , let's just use the plane wave
30:49 . That's just robp bro, V squared And concluded that the error in
30:55 predicted velocity is no more than Right? So, a student of
31:01 wrote a paper kind of rebutting this and he was, the paper was
31:11 and uh the reviewers said it was diabolical attempt to mislead the public,
31:18 I I assume that the reviewer must been a stanford graduate. But
31:25 this implies that the PBS relationship, you actually work at all backwards?
31:31 if no having of the PBS ratio U. K. Well then assuming
31:37 you could use them here is the as uh using uh the PBS
31:44 It therefore implies that the PBS So you can work the math,
31:49 math backwards and you know, it show you that the B.
31:54 B. S relationship you get, get has no relationship to reality.
31:59 no rock that gives you a the ratio like that. But the more
32:05 point was the 3% error that was . Uh that's the percent of the
32:12 velocity, right? So um it the error, if the change in
32:21 to the hydrocarbons is 10%,, Then 3% error in the velocity is like
32:27 30% error in the fluid substitution, you see what I mean. So
32:33 get by sliding the error as a of the total velocity. Uh it's
32:41 that you're doing a better job with fluid substitution than you really are.
32:46 again, maybe in the end it matter. That's a philosophical point.
32:50 we'll come come back to that. , so, uh, this is
32:56 uh Mapco heuristic equation. Oh, so you could compare it to gas
33:07 equations for uh these are clean sand theoretical as you change ferocity uh,
33:19 the error compared to the gasman And what you find is that my
33:30 is uh similar to Mapco at high ease. It's more corrected, intermediate
33:38 . The men goes really bad at porosity is below 10%. But as
33:43 can see, they both go they both got really bad and uh
33:49 these intermediate ferocity is you know percent here, you know, enormous
33:55 And so The approximations relative to gasoline really bad, less than 10%
34:04 But I'm going to make the argument gas men's equations themselves are essentially unusable
34:12 10% porosity, that there's too much in the input parameters that gets propagated
34:20 that the result you get is pretty and I'll show you that in
34:37 Few other points remember that the I do depend very much on the
34:44 pressure and this was from a paper our own dr Hahn and Mike basil
34:51 impedance, congressional and champagne and simple ratio versus water saturation, shallow and
35:00 in this case the precipitous on off and deep where it's a more
35:07 more well behaved change. Um so keep that in mind. This part
35:23 a way of looking at Gaspar's equations I haven't seen anybody else. Uh
35:29 things in quite this way. And think it's really, you can conceptually
35:37 some important insights from this plot. we're going to remain with this plot
35:43 a while and what it is is cross plot of the frame module lists
35:50 the saturated modules where the saturating fluid this case is brian, so or
35:57 giga pascal's and there is no there also for hypothetical courts. So I
36:07 a known fluid module list, I a known solid module us Uh
36:13 So gas mains equations, I could the porosity and I could have lines
36:20 constant ferocity and calculate the saturated module associated with any frame module C.
36:27 I haven't had to make any assumptions , This is just purely gas mains
36:35 . And what you find is uh , for one thing, at low
36:48 , ma july, the difference between porosity and high porosity is enormous.
36:55 ? As we approach the Stalin module the effect of ferocity becomes quite
37:01 but at low frame ma july the of ferocity dominates. And you can
37:07 that as this term goes to This term is going to dominate,
37:13 if you have low porosity is this is going to get magnified. So
37:17 our 1% ferocity And you can see even with a zero frame modules at
37:26 porosity, this would have to be very strange rock right in the frame
37:32 totally compressible. Uh But it's got ferocity. Right, so how would
37:40 accomplish that? These would have to extremely low aspect ratio for us?
37:46 you would have to have a lot them. Right. So you have
37:49 rock totally fractured to hell and you've approached the critical porosity where the parts
37:58 the rock is just not. It's it would be low essentially plates of
38:05 material that are floating with a film water between the plates basically that
38:14 that's what it would have to And you have a saturated module
38:19 that is almost that of course. right. So, you see from
38:26 we are on this plot, we make inferences about the poor structure.
38:31 I'm here, these have to be low aspect ratio pours. There's no
38:37 way to get a low frame What happens as the frame modules as
38:44 ? This is interesting. What's Well, that happens to be the
38:50 bound. So that's woods equation. these numbers are precisely with equations.
38:56 let let k dry go to And what you find is if you
39:01 with the algebra, you'll wind up with these two terms, you'll wind
39:05 with woods equation again. So, that is so, you can think
39:11 the frame module list has helped of the deviation from the Royce average.
39:19 , now, so here we From 1% porosity to 5%. An
39:24 amount 5 to 10 10-20%,, 30, 30 to 40, 40
39:34 100% ferocity. Well, what kind rock is that? 100% ferocity.
39:40 got to be like, you bubbles, right? Like soap
39:47 Right? Uh it's got to be very thin film, like a
39:55 right? The walls of a building the rooms have, you know what's
39:59 ferocity of a building? It would extremely high. Right? So we
40:05 a structure a house of cards basically lots of poor spacing between and what
40:11 find is a pretty linear relationship between material of the walls and the material
40:22 the rooms. Right? So here have fluid modules here, here we
40:28 solid modules. You can see as dry goes to zero the saturated
40:34 ghost cake fluid as K dry goes K solid case, that goes to
40:40 solid. Right. So it's a linear relationship between the two. But
40:47 is the, I think the amazing which is, rocks usually don't occur
40:57 here. What you see is that lines are all converging against This lower
41:06 like so this is a 40% So that's uh nor ours.
41:13 critical porosity line right there. 40% ferocity. You see how the
41:21 converge on that. So, you know, when I have a
41:27 frame module list or the lines are and as I get to lower frame
41:32 I the ferocity is that exist in pulled me there. So I could
41:39 the extreme view and I could say 100% porosity line is the theoretical lower
41:48 of the saturated module. It's versus frame models. And I could then
41:56 what that lower limit gives me in of Hassan's ratio to do that.
42:02 going to have to take the relationship both modules and share modules for the
42:08 frame. So we'll use the dry and remember according to the critical porosity
42:14 , that's equal to the courts, p b s ratio or persons
42:20 Um and equal bulk, unsure modules the same thing. Uh So let's
42:28 let's go to the next plot where well before we do we will eventually
42:35 this to velocities. But before I , I want to look at the
42:39 is change at different Torosidis. So, uh consider this a volume
42:48 , not volume fraction. Consider this fractional change. So you multiply by
42:54 to get percent here. And as measure the for different frame, ma
43:02 at low porosity as you can see the variation of the change in the
43:11 us with a tiny variation in ferocity enormous. So if I'm in this
43:20 , I have to know the porosity , very precisely. Or I start
43:26 large errors here, I'm up to 50% error right in the in the
43:32 of the modules. So again, is going to tell me and this
43:36 just ferocity. I have to know ferocity very accurately now from well,
43:42 how accurate do I usually know the I'd say a good number is plus
43:47 -2%. So, you know, can see the difference right if I
43:54 Uh two vs 6. Right? can see, especially at the low
44:01 module I that's going to be can an enormous difference. Okay,
44:07 we're gonna take this plot and we're to assume a frame Hassan's ratio.
44:14 that allows us to plot V. versus V. S. So,
44:18 the same thing below abnormally low Rocks for a given velocity deviate
44:28 But you can see this 100% line is now acting like a beautiful lower
44:35 on the V. P. S ratio. Right? So,
44:39 no points are low of the PBS than this started line. And,
44:47 enough, here I go from 40% 100% ferocity. So, you can
44:53 there's just not a lot of sensitivity the porosity unless I have in
45:00 abnormally low porosity is for these Right? So, I have shareware
45:06 going towards zero And yet I am low porosity here, 5%. The
45:12 way to do that is to highly Iraq. So, what do these
45:17 low porosity? These mean in terms the PBS ratio, you could see
45:22 they're abnormally high the PBS Russia. , here we are approaching the
45:27 V. S ratio of infinity All right. So, these are
45:33 high B. P. V. . Operations. So, the conclusion
45:37 if I have liquid saturated fractures. we saw that the fractures don't change
45:45 dry sandstone B. P. S ratio. But they will change
45:51 water saturated or oil saturated B. . V. S ratio. So
45:56 might be an avenue to detect If you were in a liquid
46:02 you would look for abnormally high the rations. Now this lower bound,
46:12 mean, I'm going to make the that nature is going to try to
46:19 you towards that lower bound. So or geology over geological time is going
46:26 try to uh close up these low ratio for us. So you could
46:32 that there are disturbance that has been by tectonic stresses for example. But
46:38 time the tendency is for them to . The tendency is to push you
46:44 this line. How do I know ? Well, let's look at the
46:48 the PVS trend, which is this Well, remember we saw before the
46:55 Garden of the PBS trend uh is on top of our empirical be PBS
47:02 And this is the 100% ferocity gasman . So just from the fact that
47:09 frame poison's ratio is bulk and share I are equal forces you onto this
47:18 . So this lower limit is, the fact that our empirical trend occurs
47:26 on that lower limit tells me that is trying to push you towards that
47:32 limit is trying to minimize the P. V. S ratio by
47:37 the low aspect ratio force. Okay, now, to compare to
47:50 measurements, remember Gaston's equations are zero . Laboratory measurements can be uh kilohertz
47:59 megahertz. Hundreds of kilohertz. We're . We need to use equations that
48:06 into account uh the fact that the and solids maybe moving out of phase
48:15 each other. In other words, inertia if I if I move,
48:22 I move the rock frame, the of the fluid in a permeable rock
48:28 lag behind somewhat right. There are , but they're not perfectly in phase
48:35 each other. Plus there's, I'm the rock. I'm going to uh
48:41 crush of gradients in fluids are gonna around in the rock. So the
48:46 theory takes over to listen to account gets more came along and reduced his
48:55 to equations people understood. Uh but is one parameter that we we don't
49:03 what it is or it's called a coupling factor. And uh, the
49:11 is the degree of coupling between the and the solid. When the mass
49:17 factor is infinity, that means perfect and when the mass couple in factor
49:23 one, it means they're completely All right, so, presumably at
49:30 frequency, that would be uncoupled in zero frequency they're perfectly couple. So
49:36 can play that game. We could the mass coupling factor the infinity.
49:41 what you'll find is this reduces to lens equations. Beta is K dry
49:46 k solid. The seas are one the both module is these are
49:52 So that's CB is the frame compressibility this notation. So this is
49:59 B. There is rigidity. Uh if you look at this pretty
50:04 it reduces the gas mains equations um you know, there are a
50:09 of additional terms which we have to into account also because of the inertial
50:17 , we no longer have destiny in denominator. Actually, the mass coupling
50:23 into play in the denominator at well well. And so uh I heard
50:29 Chesnokov talked about frequency dependent density. , it's an apparent density from the
50:34 propagation, right? So you have you have to change the density term
50:39 , and it's the same for both waves and share waves. So,
50:45 the mass coupling factor goes to this becomes zero. So you reduce
50:50 low frequency, that just becomes This term goes away and that terms
50:56 , goes away. So this is minus beta squared in the numerator at
51:00 frequency. Well, that's guess those , if you if you remember by
51:07 way, this does not take into squirt flow doesn't take into account squeezing
51:15 poor and opening up a nearby for that the fluids squirt from porter
51:20 this is a more macroscopic phenomenon. Anyway, we'll we'll look at this
51:29 more detail when we talk if we to dispersion, but there's something else
51:36 exciting that comes out of this Uh huh. We'll come back to
51:42 in this unit. But first I to talk about uh huh laboratory
51:50 Right? So we talked about this we have transducers either side of the
51:56 , we put a pulse in, measure the travel time, right?
52:00 here's the trigger pulse here is away . And you can measure an arrival
52:06 here or maybe an arrival time there whatever. Um And you can do
52:11 while you're varying the saturation and it's to matter whether we use a flow
52:19 or drain it, what it's called , where you let the fluid you
52:22 the Rockwood fluid and then you let fluid drain out, or you imbibe
52:28 the fluid into a dry rock. the fluid gets sucked into a dry
52:33 . We could call that inhibition or could let it drain out of a
52:38 saturated rock, and we would call drainage. We'll come back to that
52:43 a bit. Okay, and we at this chart before. The question
52:51 , is that the story, is uh dispersion, if its dispersion.
52:57 video theory should be able to predict curve. And what we find with
53:04 out theory is we can't predict the . So here we have mass coupling
53:11 okay, brian saturation versus velocity. we have a mass coupling factor of
53:17 . That would be Gaskins equations here have a mass coupling factor of one
53:23 it's predicting dispersion. You can see if you let the mass coupling factor
53:30 dependent on frequency, then you could a variable dispersion here. Uh but
53:38 all happens if you have homogeneous lee Gas in the pore space, it
53:46 happens very close to 100% saturation. what got measured was this,
53:52 So anyway, there's something else going and we'll come back to that.
54:02 one of the keys in understanding what's on is comparing the drainage versus imposition
54:10 . And in this case they didn't it all the way to 100%,,
54:15 I'm not sure exactly why they went to 90% saturation. Yeah, but
54:21 they started with the dry rock and imbibed the liquid into the rock,
54:28 know, baby basically capillary effects sucking into the rock. Maybe they couldn't
54:35 all the way, they might have a vacuum or something to pull on
54:39 . But anyway, uh they went to 90% saturation. You see,
54:46 was until you got to very high that you got a big increase and
54:52 you let the fluid drain out and a much more gradual change,
55:00 Eventually it comes to a similar but there's this big difference and the
55:08 are that here we have a homogeneous and here we have an in homogeneous
55:18 of gas throughout Iraq and between the and we're gonna, we'll try to
55:24 this better later, but it's more just a discretion effect. And we
55:31 going to assume that as far as frequencies are concerned, the rock acts
55:39 you have a homogeneous distribution. So going to use gas masks equations.
55:50 one more really exciting thing comes out B. O theory and I expect
55:57 to memorize these equations for your I'm only kidding here. They're uh
56:03 equations. But I want to point that in the equation for P wave
56:11 at infinite frequency, there's a plus minus sign here, That means there
56:20 two solutions. So, according to B. O theory, there's a
56:27 P wave and there's also a slow wave and no one had ever noticed
56:34 slow P wave before. They're actually than the shear waves. And
56:41 what is the slope P wave? , it's kind of, it's a
56:45 it way through the fluid. And slow P wave attenuate. It's very
56:56 . So, if you set off source at the surface, you would
57:01 if you set up a slow P at the surface, you would never
57:04 a reflect its slow P wave. called evanescent propagation. It just attenuate
57:11 rapidly. You would never see But in the laboratory uh huh they
57:22 able to find these events. uh here we have velocity versus porosity
57:30 a few different samples. So this the fast the wave. This is
57:36 shear wave and this is the slow way and the solid line is predicted
57:46 Vo theory. So this was just amazing outcome. And we've never really
57:55 one in the field. But if take my D. H.
58:00 Class, I'll show you many examples what we call low frequency shadows that
58:07 think are locally converted slow p But that's beyond the scope of this
58:20 . Let me uh Rail against term frame modules one more time. Uh
58:29 dry frame, a dry rock is from Iraq in contact with fluids.
58:36 mean, just walk around a patch dirt in the rain right? Far
58:45 solid when dry, very slippery when . Right? So um the idea
58:54 we want to talk about a dry by the way, and it's not
58:58 clay's where this impacts even at low a loose sands when you wet
59:06 it forms a silica gel and the will actually repel each other slightly.
59:14 reason you can make a sand castle not because of that is another effect
59:22 takes over which is even stronger. the capillary forces capillary pressures uh that
59:31 the sand together. So uh you see that the wedded sands can have
59:38 different frame property than the dry And this most severe. The silica
59:46 are very are overcome a great uh under the kinds of pressures that we
59:52 with. They might be important at very near surface. But that
59:56 the silica chemical interactions are not that . But the client interactions are extremely
60:05 . And it can work in both because the frame to be lower rigidity
60:09 the friends would be more compressible. it could help bind the frame.
60:13 variety of different things can happen. um the dry frame measured in the
60:21 is not necessarily representative of the institute . So the proper way the proper
60:28 frame modules to use is that which extracted from gas fruits equations uh in
60:36 . Two. So it's derived from N. C. Two measurements.
60:41 So I prefer other terms like frame lists, skeleton module lists or drained
60:49 . Um That would be the module where you squeeze the rock and you
60:52 the fluids leave. Um I'm not about drains because maybe the flu is
61:00 get out right. But so I frame and skeleton modules. It is
61:08 we'll call it K. D. K. Drive. But keep in
61:11 it's not totally dry. It's the in contact with the NC two
61:21 All right. And so again, of chemical and interactions could be weakening
61:27 frame where it could be hardening the . Gas mains equations don't take that
61:32 account. Remember the dry frame is same? Uh for both situations.
61:38 , gas mains equations are mechanical, mechanical. All right. So here
61:52 persons ratio for the frame for a of different sediments. Um And the
62:00 sediments would be at the .1. you start getting too dirty or cal
62:05 sentiments goes up 2.2. So it's , pretty small range, But you
62:11 , a difference of .12.2 in the can have a big effect on the
62:17 substitution. So you certainly have to into composition. Uh This is from
62:23 and wang basically a linear straight line . No, pretty much on the
62:31 between frame share and bulk modules, there is significant variation. All
62:41 Leave some words for you to read tad smith. And then the question
62:47 . How does the rock frame module very with ferocity. And this comes
62:52 to our velocity density relationships that are over the place. Right. It
62:58 on how uh liquefied the rockets. here we have some uh synthetic sam
63:08 with an almost linear relationship between the is more like the critical ferocity type
63:14 model. Here we have so, are convex. Up here, we
63:19 concave up. This is both modules sheer modules. Uh here we have
63:26 millennial relationship here. It's more of exponential type relationship. So, a
63:34 variety of different relationships between uh ferocity frame modules. So, it's hard
63:41 build the rock from scratch. It's to say, okay, I have
63:44 sandstone on his ferocity is such and . Therefore it's frame modules will be
63:48 and such. It's much better to the frame module us from the institute
63:54 if you can now a couple of models to use um to look at
64:06 relationship between frame module and ferocity. the Creek model and there's the critical
64:14 model. We already talked about this . Both of these models. You
64:19 that the ferocity term here cancels out I take the ratio K over
64:26 So both of these have persons ratio the frame equal to Parsons ratio of
64:33 grains. And that just happens accidentally be true for courts, clean coarse
64:40 stones. But it's not true for mythologies. Uh There's another way to
64:45 this. You could just uh say the frame property is equal to uh
64:52 . The uh the uh mineral both time say one minus ferocity to some
65:01 . Right? It could be depending the rock how liquefied the rock
65:06 It could be different powers. increased model. He has a specific
65:11 here. Uh But you could just this exponent to uh to match the
65:19 . And you could do the same with the critical process. The
65:23 You can add an exponent here. And I did that in one of
65:28 papers white and Castagna in geophysics. uh this way you can fit rocks
65:36 don't obey the critical ferocity model. huh. These two models are very
65:44 . Uh the the critical ferocity models of catastrophic. Right? The frame
65:51 goes to zero precisely at the critical , where's the creep model? Uh
65:57 a more gradual change, Right? it takes more time to go to
66:03 to zero. Um So um but very similar when you're not at the
66:16 process. So for the times the , rocks we're dealing with, both
66:20 these models give similar results. so here we have some uh measurements
66:29 stands from a variety of authors and these particular measurements are a linear
66:38 Uh And these have a different linear . These rocks are interesting because these
66:45 synthetic crops. These are synthetic sand . And uh they've either been
66:52 you know, the grains have been to the point where they fused together
66:58 otherwise cemented to each other. So is as highly liquefied as you can
67:04 . and you notice they're even at . They're hanging in there,
67:08 because of the grains being bonded Whereas these rocks are losing it at
67:16 40%. So you can fit various to these things. So here we
67:22 two different critical porosity models which is with the critical porosity model, you're
67:29 to change the critical porosity for different types. So it would be appropriate
67:34 change it for these rocks. Um Anyway, so the creep model
67:42 not work for these most highly liquefied rocks and as I said, the
67:49 porosity model tends to be a practical bound now to get more precise in
68:02 predictions, we have to start taking account the poor shape and there are
68:09 different ways to do this. Um to be very honest, none of
68:14 techniques are really useful for predictive Um However, they are worthwhile for
68:24 us conceptual understanding of what's happening. uh here K. M. Is
68:31 mineral module lists. And we're looking the custard tuxedos effective module lists.
68:38 ? So this is what we're going be trying to predict. And uh
68:44 algebra is such that these are the ma july of the individual constituents.
68:50 you're summing these terms for the different uh that we're going to build up
68:58 rock. So here we are making rock from scratch. So um we
69:03 the bulk module lists of the individual , but there's a term here which
69:12 um related to the poor geometry. so you can stick different ports that
69:19 can put different equations into these Um If we get to it,
69:25 look at some of the different not in this section though. Um
69:30 just keep in mind that the poor is accounted for there and we could
69:39 different kinds of predictions. So look the predicted dry bulk modules versus ferocity
69:50 um you can assume spherical forests. we've seen this before, we looked
69:57 these kinds of calculations And we're looking aspect ratios at .1. So all
70:03 ferocity has an aspect ratio .1. you see there's a stronger dependence of
70:09 modules on ferocity when you have a aspect direction uh in a lower aspect
70:16 would even have a stronger dependence. problem with the custard tacos model is
70:22 works as only ballad for dilute concentrations pores. So you can have
70:30 zor concentrations of that particular aspect ratio than the aspect ratio. So this
70:37 has to start stop at .1 hysterical . You could go all the way
70:43 one. Right? But here you but you can get to higher porosity
70:52 if you let a lot of if look at a mixture of aspect ratios
70:55 you let a lot of the ferocity accommodated in the spherical porous and then
71:02 add cracks to that porous material. so that's what's being done here.
71:11 it's 6% sparkle. uh Uh huh which gives you a which would give
71:19 a certain velocity And then you add ratios of .1 or .01. Uh
71:28 this is a crack concentration 4.01. could not go to more than
71:35 So you can't go to more than correct concentration, .01 aspect rations.
71:42 this is a very low uh huh of ferocity contained in these cracks having
71:51 dramatic effect on the velocity. Okay um that is why uh huh when
72:04 put a rock under pressure you get big change in velocity without changing the
72:09 . The very much because you're closing very flat cracks. Now, interestingly
72:19 at the D. P. S. Ratios. Uh This is
72:24 , remember this is we're talking about rock frame here and uh we're seeing
72:29 very small difference with aspect ratio. You know in some cases you can
72:36 the B. P. V. ratio With other aspect ratios. You
72:40 decrease the aspect ratio but not by much right? So we're all on
72:46 order of 1.5. And this is with them with what we've seen
73:00 Now another thing that is often assumed the dry frame is that the dry
73:08 is not disperse it. So beyond , as we go from zero frequency
73:15 ultrasonic uh Or really there are a of different theories which would predict a
73:23 , there would be a low frequency . High frequency limit in some kind
73:27 ramp in some theories that ramp is fast over a small frequency range.
73:34 If you combine a bunch of different you can put that over a wider
73:40 range but there's some low frequency limit some high frequency limits and the difference
73:46 the low frequency limit and the high limit would be uh the dispersion,
73:51 see or the difference between zero frequency met the frequency of your laboratory measurements
74:00 be the observed dispersion. Uh dry are assumed to have at very little
74:10 . And the reason this is so because dispersion is linked to attenuation.
74:18 I have frequency dependent insinuation, I must have dispersion. The two go
74:24 in hand, we'll have, you , I hope we do have
74:28 We should have time to talk about more in this class. But in
74:34 rocks, the solid solid friction is small. So the attenuation is much
74:42 in dry rocks than in uh saturated . Or partially saturated rocks usually have
74:49 highest attenuation. Ok, so coming to BEOS high frequency equations, I'm
74:59 uh repeating this here. Um If going to try to pull out the
75:09 frame modular, so remember be as over K matrix and we have Katie
75:15 . So, if we want to this for the dry frame frame matrix
75:19 for the matrix frame, If I with ultrasonic measurements and I use gas
75:28 equations, I'm going to get the frame modules. So for ultrasonic
75:34 I should use a mass coupling factor the order of one closer to
75:41 Whereas gas mains equations are assuming So let's see what happens. Let's
75:47 if we if we extract the bulk shear module I from laboratory measurements on
75:54 rocks, uh, saturated rocks extracted gas mains equations or from saturated rocks
76:04 BEOS equations. So I'm going to right to the results here because we're
76:10 out of time. But as we've before, if we cross quad,
76:15 modules versus sheer modules for sand dry sand stones, uh you get
76:22 trend, we're pretty much on the , right? So both modules is
76:29 equal to share modules And our intercept of a regression trend is zero.
76:34 ? Both modules equal share modules. a constant ratio as one goes to
76:40 , the other goes to zero. the other hand, if I make
76:45 measurements on brian saturated sands, towns I use gas mains equations to extract
76:53 bulk of share module I, what I find the bulk modules, I
76:57 always significantly greater than the sheer I have an intercept here. So
77:04 with songs ratio is much higher. cave amuse hire persons ratio is
77:09 The DPD s ratio is higher. this could be a lead you to
77:15 wrong predictions, if you assume that is the uh, the dry frame
77:23 that you're going to use using gas equations on the other hand, if
77:29 go ahead and do the same but assume a mass coupling factor of
77:34 embryos equations. You come back to , unsure module have been approximately
77:42 So this is, I think, important result, and It gives me
77:50 to be able to uh, use , Hassan's ratio .1 for the dry
77:57 . I see it in dry sand . And when I use the correct
78:01 , I could back that out from video theory. Okay, well,
78:06 all I have for tonight. any
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