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GEOL7324 Rock Physics - Day4 09-04-2021
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00:00 this conference will now be recorded. can all see my slides.
00:09 Yes sir. Okay. All Just reviewing what we covered yesterday
00:16 Um We talked about with psalms ratio it's called an elastic module. I
00:23 it's a little bit different from the module i in that instead of a
00:31 between stress and strain, it's a of two strengths and these are the
00:39 strain. So why is the vertical X. And Z. Or horizontal
00:44 ? It's the ratio of the horizontal I'm sorry, excuse me. I
00:52 it back uh Z. Is the axis. So uh this is the
00:59 of the horizontal axis and it would equal in both directions horizontally. So
01:07 X. X. Or X. Y. Y. To the vertical
01:13 . That ratio. Trans verse two strength, his foot sans ratio.
01:19 as I said, sometimes it's represented this uh this creek v like it
01:26 like an italics. The um sometimes represented by sigma but you can see
01:32 sigma's representing stresses. So a different for persons ratio and exploration.
01:40 We typically use sigma for persons ratio in rock mechanics, sigma is stress
01:47 structural geology. Sigma is stressed. I mentioned this minus sign. This
01:53 all a matter of how you choose coordinates. Uh So I wouldn't worry
01:58 it too much. Um For our . Hassan's ratio is positive for Franny
02:08 rocks. We also talked about the wave modules. And this was a
02:17 situation for Hassan's ratio were laterally So if you squeeze the rock
02:25 it's gonna strain laterally for the P module lists. Um we constrain the
02:33 laterally, so we can't strain And this is analogous to what happens
02:40 you pass a compression all way through . Any volume elements, any infinite
02:47 , volume element of the rock is as you're compressing it and it wants
02:54 expand laterally but it can't because it's to another infant intestinal volume element which
03:02 is trying to stretch laterally. And those forces cancel out. And the
03:11 is that the volume element shortens but doesn't get better. So this is
03:18 kind of module, is that a wave? See and you can see
03:24 is equal to roe V. P also K plus four thirds view.
03:31 So uh this is the uni axial sigma's easy in the vertical direction and
03:42 the shortening in the vertical direction. strain in the vertical direction. And
03:47 the P wave modules is that uh constant. Notice that the other strains
03:57 all zero. Okay. So um huh. It is not allowed to
04:03 in other directions. And the difference the P wave module lists and Young's
04:10 list is uh the youngs module list unconstrained. Therefore the stresses the other
04:19 with the zero right? And it's to straight in the other directions.
04:28 , so all the other applied stresses zero. Now which brings us to
04:36 relationship between the essenes ratio and So we said Parsons ratio is the
04:45 verse strain to the ratio of the strain to the vertical strain with the
04:52 strain. And it could also be in terms of the velocity ratio.
05:02 we said that any elastic modules can expressed in terms of any two other
05:08 module? It well the B. . V. S ratio is has
05:14 modules and sheer modules in it. cancels out the density. So you
05:19 see we have bulk unsure module lists here. So actually this expression is
05:24 way of uh equating facades ratio to other elastic module. I both modules
05:31 cheer modules. And keep in mind is for an icy tropic material.
05:36 there are only two independent module I all the other module. Light Can
05:41 expressed in terms of those two. is an interesting expression because it relates
05:49 to rock properties. I mean to . Now if we measure Hassan's ratio
05:57 measuring VP and Bs, then that a what we would call a dynamic
06:05 modules. So it's derived from velocity . Now we can make some statements
06:13 Uh huh. What uh the relationship was songs ratio and the V.
06:20 . B. S pressure for when Parsons ratio is zero, what
06:26 the V. P. V. ratio? You'll have to you'll have
06:29 do the math here. So I'm you to do the math And calculate
06:40 PBS when persons ratio is zero. when persons ratio is zero, B
06:57 B s squared minus two is equal zero. B P B s squared
07:02 two. Then So be PBS is square root of two. So 1.41
07:12 the minimum practical. VPBS remember we zero is a practical limit of Hassan's
07:18 although it can be negative. Um , I'm not personally aware of measurements
07:26 Hassan's ratio below or V PBS below root of two on rocks. Not
07:33 , I believe. So that is our minimum be PBS ratio for our
07:42 . Let me ask you a more question. Uh what is Hassan's
07:49 When the V pDS ratio is In other words, when the shear
07:54 velocity is zero, the B P s ratio is infinity. So in
07:58 fluid, what is Hassan's ratio No, for fluid With sons ratio
08:10 .5. And we could uh if use a crazy math, if you
08:16 my kind of math, that's easy see We have the PBS. This
08:22 . So we have infinity squared -2 infinity. And we have twice in
08:27 infinity squared -2. That equals twice . Infinity Divided by twice infinity is
08:35 . of course that mathematics doesn't You have to take the limit of
08:40 expression as V P B s approaches . And you'll find that as being
08:47 approaches infinity for persons ratio approaches point And as I said, if you
08:54 to convince yourself of that, an math problem would be to maintain the
09:02 constant and try to compress a If you compress it is gonna spread
09:11 because it's not constrained. And if go through the math holding the volume
09:17 that new shape equal to the volume the first, the original shape,
09:22 get a possum's ratio .5. So can do that easily by compressing a
09:28 or a a square or something like , not a square, a
09:35 something like that. And and work the volumes as you compress the rock
09:40 cetera. So from that equation between ratio and be PBS We see that
09:50 is a 1-1 relationship between persons ratio the v. ratio. If I
09:57 B P B S, I know ratio and vice versa. Um Now
10:04 V. P B s ratio. I if I take Vp over bs
10:10 , that's K plus quarters mu That's K over mu plus four
10:17 So you can see that the BPBS also has a 1-1 relationship with Ko
10:23 you. And so, okay, is k overview is uh conceptually and
10:31 or or an easy concept to So it helps you conceptualize in terms
10:38 in compressibility and rigidity What the different ratios. Me now, it turns
10:45 that there's a a bit of a in the literature about the use of
10:53 ratio in geophysics. Uh There's a of workers prefer to use the
11:00 P. B. S ratio and personally prefer to use the V.
11:05 . B. S rations. Um fred Hiltermann on the other hand,
11:12 works in very similar areas that I prefers to use those odds ratio.
11:17 you can see there is no except if you're dealing with Heidi PVS
11:24 . Hassan's ratio varies very little, it might be a little bit clumsy
11:28 deal with with sons ratio, which why I prefer the B.
11:31 B. S ratio similarly at very since ratios. Uh huh. You
11:39 there's a lot more variation with sons than in the B P.
11:42 S friction. So if you're dealing very low uh with songs ratios,
11:48 the V. P. B s is uh more interesting. I'm
11:53 the person's ratio shows a greater range it's somewhat easier to deal with,
12:01 really there's no difference. Uh The reached a point where leon, Thompson
12:09 a paper in the leading edge entitled was not a geophysicist. In other
12:15 , he argued that there is that should not use Hassan's ratio. Uh
12:21 geophysical equations. I'm agnostic on I prefer V. PBS but I'm
12:28 to use persons ratio. In fact has come up with some useful approximations
12:34 Hassan's ratio. So my school of is whatever works for you, whatever
12:40 you through the night. Use Yeah. Now there are theoretical limits
12:51 the elastic module. It uh for , rigidity is greater than zero.
12:57 module is is greater than or equal zero. Similarly, the both modules
13:01 greater than equal to zero, Lama is constant can be negative because
13:10 you work through the math Land of two, you over Third is greater
13:16 or equal to zero. So you lambdas is greater than equal to minus
13:23 mu over three. So lambda can negative. But also interestingly, Hassan's
13:30 from the negative. We said the lower limit Of uh four songs ratio
13:37 zero but the theoretical limit is So I'm going to ask you to
13:44 a little exercise right now and calculate B P B s ratio When persons
13:52 is -1. So what is the theoretical, the minimum theoretically uh smallest
14:02 . P. V. S ratio you can have. So I'm gonna
14:06 recording and ask you to make that . This conference will now be
14:19 I was muted. Uh so the is square root of 4/3 1.15 is
14:28 . Yeah or pretty close to square theirs. And so this is
14:37 We have two minus two V. . B s squared on this
14:42 Um Is that right to minus to P B s squared. And so
14:50 we have uh subtract two from both . So you have uh two
14:57 P. B. S squared and . P. V. S.
15:04 so uh that's K. Plys four view over twice. K plus four
15:13 view. And and so You work it and you get square root of
15:26 . Ok, so coming back to velocity equations, if we look at
15:32 ratio of BP over bs quantity you could see that's K plus four
15:39 view. Uh huh, overview. is k overview plus four thirds.
15:46 you could ask yourself what is the possible V. P. B.
15:50 ratio. Uh Well uh we'll have problem if we send you to zero
15:57 then we'll be dividing by zero. won't work. Uh But uh we
16:03 let both modulates the zero. And then we have the density cancel
16:09 When we take the ratio rigidity cancel . When we take the ratio.
16:14 what we got get is VP over . S. is the square root
16:17 4/3. So that's when you can the minimum be PBS ratio is when
16:24 have some rigidity but you have both of zero. Again, we know
16:29 no rocks uh that that are obey mm Okay, so as I mentioned
16:39 , there are various different ways to uh the elastic module I and
16:46 psychotropic material. Given any two, can calculate the others, for
16:51 in this case, uh they're calculating is constant from young's modules and persons
16:59 . And they're arriving at the solving the V P. B. S
17:06 here. So, or the V . V P ratio, beta is
17:10 . S. Alpha is VP on slide. Uh So you can uh
17:18 for the V P V S ratio terms of innocence ratio. Another interesting
17:25 , llamas constant over here Is equal K -2/3 of you. Now,
17:32 been another controversy in the industry where claim is that lamb is constant is
17:40 sensitive to hydrocarbons uh than other things the D P. B. S
17:46 or so forth. And you can that the change in LAMAs constant due
17:54 high departments is exactly the same as change in the bulk modules. The
18:00 is not affected by hydrocarbons. Lamb constant. Then can be no more
18:08 than the bulk modules. So that put that idea to rest like somehow
18:15 constant has some magical properties that are to other. All right.
18:23 quantities. Mhm. Okay. I want to talk about History
18:31 History says uh but can have different in different applications, but in our
18:41 it means that the velocity you measure the module is that Iraq has,
18:48 depends on its history of deformation. which then depends on the history of
18:55 apply to it. So in other , the velocity of Iraq doesn't depend
19:01 on the current day pressure regime, depends on pressures that were fired in
19:09 past. So history says at least has the word history in it.
19:17 . So it means the properties of rock depend on its history. And
19:25 here we have some velocity measurements. are on uh uh igneous metamorphic
19:33 And so these, you can presume to be very low porosity rocks.
19:39 you see that when the the death here is as you're increasing the
19:46 so you're increasing the stress um pressure an omni directional stress. So uh
19:56 pressure is increasing the velocities go but then they level off, why
20:02 they level off? Because essentially you've all the cracks that that might exist
20:09 the rock or the large majority of . And then the mineral itself is
20:16 in compressible. It's relatively insensitive to pressure. It's mostly the closing of
20:22 space and forcing grains against each other firmly. Right, so you you
20:30 up the rock and let me get point, you're here, you pressure
20:37 the rock, you close crash, force brains against each other more strongly
20:43 then you've done all the work, know, most of the work.
20:46 then just a very minor increase things leveled off, you then deep pressure
20:52 rock, which is the solid So we're going from high pressure to
20:57 . And you see that uh, a certain point, you start opening
21:01 cracks again that had previously been But the final velocities are higher.
21:08 when you were pressuring the reason the velocities are higher is because you permanently
21:17 the rock, you've closed them for that then won't open up again or
21:23 changed certain grain contacts. Maybe you've those brain contact stronger and um,
21:34 don't completely go back to the original . So here we're seeing some kind
21:41 irreversible deformation. Right? So there's plastic behavior here. Uh, and
21:48 resulting velocities are higher. And we that here, we see that
21:55 uh, we see no history since this case. So all the fractures
22:02 opened or or that closed when you pressuring then opened back up as you
22:08 deep pressure. Now, interestingly, of these curves come back to the
22:15 situation. So, at intermediate some of the cracks have not opened
22:23 . Some of the cracks that have closed, don't open back up.
22:27 if you take enough pressure off, all open back up. And so
22:32 come back to your original velocities. a few cases, we talked about
22:45 void and Royce pounds previously. And , we're going to go over that
22:51 because essentially Boy and Royce bounds are medium theories. What is an effective
22:59 theory? It says given for a arrangement of constituents. I'm going to
23:09 the module lists of the composite from module I of the individual constituents.
23:16 if I know exactly how the constituents arrays are arranged, I can do
23:22 calculation analytically. And so I can the properties of the effective media.
23:29 we said that for the void this would be the highest possible module
23:34 you could get would be when uh you're applying uh you have columns here
23:43 you're playing compression is parallel to those . So, imagine a plate
23:49 Imagine I have this thing in a and I'm applying a uni actual stress
23:57 along an entire plate. And what is the strong columns. Say they're
24:03 dark grey. We'll do we'll provide of the resistance to that compression.
24:12 And the resulting module, this is volume weighted average of the module I
24:19 the individual constituents. So, here have to constituents. Is the module
24:25 . So, the effective module lists the medium? Is the volume weighted
24:31 ? Where f is the volume The volume weighted average of the individual
24:37 are, by the way, is is the screen cutting off? Are
24:42 not seeing the very bottom of the ? Um Yes, so the screen
24:48 customs. Oh, okay. That's . Um So maybe uh you may
24:56 to follow me on on your Um Yeah, I don't I don't
25:02 an easy way to fix this right . Let's see if I. All
25:10 , Well this is one way. , so we'll try going doing it
25:22 this for a while. We'll see that works. Um Now that does
25:28 , but now you have the whole , right? So for the void
25:35 , it's a linear volume weighted So we've seen this before, on
25:41 other hand, we have the Royce where we have parallel layers. And
25:46 course, you can just imagine uh planks or metal place with foam rubber
25:53 between. Right? If I sit that stack of layers, the foam
26:00 is going to compress much more than planks of wood. Right? So
26:07 foam rubber is a common, is for most of the strength. So
26:14 in the Royce configuration, that's the compressible situation. The softest layer is
26:21 uh most of the work, it's reciprocal bye and weighted average here.
26:32 the strain in each layer is different this configuration. You can see that
26:37 that played if I have, if have a plate here compressed against those
26:43 vertical columns, all the columns are to strain that are going to shorten
26:48 same amount, right? The week or the highly compressible columns can't compress
26:56 than the in compressible columns and that's columns are used in architecture, in
27:03 . You have the column, which be stone or cement? And you
27:07 the weak layer being air in between columns. Right? Um So the
27:14 layers can't compress any more than the layers, whereas here the strong layer
27:19 hardly compress and the weak layers accommodate of the compression. Okay,
27:30 I think we could go full strength this one. So um here we
27:38 we've seen a lot of velocity versus and we're seeing some things that we've
27:44 before. So the Royce equation is lower bound that's also called woods relationship
27:52 devoid averages. The higher bound. you can't have velocities higher than the
27:57 average or lower than the Royce When you do get velocities below
28:03 There's either experimental error or you don't that. You haven't calculated the bound
28:12 properly. Probably using the wrong constituent . Right? So, but we're
28:20 far wrong here. All right. we have a variety of data
28:25 there seems to be an upper bounds , a practical upper bound which is
28:30 from the void average. And this be described by the critical porosity
28:37 But then in between uh that upper and and the voice pounds, you
28:42 a lot of points in between. why would you get points? You
28:47 if these are the most fully liquefied ? Why do you get these points
28:53 between the upper bound and the lower ? What would be a hypothesis if
28:59 see points down in here, what you guess about the rock? Why
29:20 rock spot closer to the lower You see out here past the critical
29:26 , you have a bunch of, know, these are ocean bottom sediments
29:30 at the lower bound or slightly These are completely unlit defied rocks.
29:37 if the Royce band represents completely unlit and the critical porosity model represents fully
29:48 , why might you be in between up? You know, the just
30:04 different degree of uh speaking. exactly, that's it. So,
30:11 these rocks maybe slightly cemented or slightly , such that they're interlocking and have
30:18 rigidity, but they're not fully liquefied this. Yeah, okay, so
30:29 we have various velocity versus pressure And as we're increasing the pressure,
30:35 velocity goes up. But you we have different uh degrees of velocity
30:44 . Notice that at high pressures, of these rocks tend to level off
30:52 presumably with this stance down. If want to even higher pressure, it
30:56 level off. So at some point level off and we talked about Uh
31:03 initial increase in velocity as having to with four space and cracks closing.
31:11 here we see different relations, for , this limestone has a very small
31:17 . This dolomite has a very big . So, could you hypothesis differences
31:25 the poor structure of this limestone and dolomite. Can you make an inference
31:33 as to why you have this different to pressure, if you don't
31:42 Um they were going to their So we'll be seeing some key points
31:47 the Dolomites compared on some micro porosity the limestone. Okay, so,
31:53 gonna say it's it's in this I'm gonna say it's exactly the
31:57 Right? I'm going to say, , buggy pores have a poor,
32:02 velocity is poorly sensitive to the buggy . If I increase the pressure,
32:08 buggy poor is still going to be buggy poor. Right? So,
32:11 going to close it a little I'm going to close some micro
32:15 maybe some fractures, but I'm going suggest that this limestone is dominated by
32:21 poor. So, it doesn't have big change. The dome light on
32:26 other hand, when you re crystallize into dolomite, you can introduce inter
32:34 ferocity between the faces of the dolomite and those pores, uh, will
32:43 to be very flat and act like . Or it's possible that this particular
32:50 happens to be more fractured than that . So, generally, when I
32:55 a large sensitivity to pressure at low , that generally indicates cracks closing.
33:03 , I would argue that this dolomite more low aspect ratio ferocity in
33:10 like step, the sand pack, are no fractures here. It's a
33:16 pack. Right, but I'm suggesting the grain context themselves are acting like
33:24 . So they have the effect of same effect as cracks. And as
33:30 mentioned uh stand stones tend to have aspect ratio an effective aspect ratio on
33:37 order of .1. Their velocities behave if They were composed of ellipse soil
33:45 with an aspect ratio .1. Now I'm the stand pack as I increase
33:51 pressure, how do we how is done in the laboratory? They take
33:56 sand pack and it's jacketed, it's . So as as I increase the
34:03 pressure on this thing, I really have a lot of opportunity to rearrange
34:08 grains, especially if it's a high confining pressure from all directions. So
34:17 I increase that confining pressure, the are going up. Not because I'm
34:23 the grains and reducing the ferocity very I don't change the ferocity very much
34:28 all in this experiment. But the do level out. And what that
34:35 is underperforming the grain contacts and I'm the grain contacts flatter and flatter.
34:41 the grain context becomes stronger and stronger you platinum. So a point contact
34:48 easy to compress, but once it's compressed it gets harder and harder to
34:54 it as the contact area between the increases. Okay, so here we
35:06 Iraq where the shear wave velocity is increasing very much with pressure and we
35:13 have that sharp increase in p wave . Uh This is a dry
35:21 We do have and it's apparently not fractured because we're not having a very
35:26 change of velocity By the way, is the compressibility of the rock.
35:32 compressibility is one open the bulk module . Uh So you can see that
35:38 I compress the rock, it becomes compressible. So the bulk modules is
35:44 and again it levels off. so again here you can see pores
35:50 , maybe grain contacts, maybe micro ferocity, maybe fractured ferocity, these
35:58 are closing and then things level Okay. We already talked about the
36:08 between static and dynamic module I and just to review um dynamic module I
36:15 derived from velocity static module, I a constant stress and these these values
36:23 typically different um dynamic module I are larger than static module. I uh
36:33 reasons. Uh First of all dynamic I involved much smaller variations in
36:44 Uh in in in stress you are small changes in uh devia turyk
36:51 So the strain amplitudes are much smaller static stresses tend to be a lot
36:58 . So there's the ability with a stress to uh induce some plastic deformation
37:07 will cause the rock to be more compressible or less rigid as you plastic
37:13 to form the rock. Whereas a measurement, you're strictly uh in the
37:19 range also dynamic measurements you have an , You have a frequency of the
37:27 and the higher the frequency, uh higher the velocity. And you can
37:33 that fundamental laws of physics. Using Cramers chronic relationships, you can prove
37:40 if you have a frequency depend if have attenuation, you have frequency dependent
37:47 . And for body weighs, the frequencies are faster than the low
37:53 So for both reasons, the dynamic I tend to be larger than the
37:58 modules. So here is just a of dynamic to static Young's module lists
38:08 a variety of rocks. And this the ratio. And you can see
38:12 most rocks the dynamic module is is than the static modules. But there's
38:21 rock here where the dynamic modules is then the status modules. Uh
38:28 do you want to offer a hypothesis explain that? That seems to be
38:33 unusual behavior. So what if there no porosity? And uh and you
39:00 a material that was so uh competent under the static compression it's not going
39:07 undergo any kind of plastic deformation. statically is going to be elastic and
39:14 . It's elastic then. And there there's no porosity. So there's no
39:21 negligible attenuation. So you don't really anybody wave dispersion. In that
39:27 the ratio would be one to get ratio smaller than one is difficult to
39:34 theoretically. So, I personally do have a reasonable hypothesis to explain this
39:41 . And I've asked students for many to offer hypothesis. I've not come
39:46 with a reasonable one. So this a case where I'm going to resort
39:52 experimental error. Maybe a the module are measured on different rocks. Maybe
40:01 the sample if it's the same maybe there's history sis effect. Um
40:08 for the two measurements, um maybe just experimental error. I I hate
40:18 resort to experimental error as an but uh it's possible that that is
40:24 we're dealing with. That's always the resort. Always try to come up
40:29 a physical explanation first. Okay, here we're looking at the ratio of
40:40 dynamic to static modules and similar to we were seeing with cracks,
40:48 affecting the velocities. Uh it also to affect the ratio of static to
40:55 module. So the greater the the smaller the ratio and at low
41:02 you have where you have more open space, more open microfractures and more
41:11 contacts between brains. Uh Your ratio dynamic to static module i is
41:23 Now, uh here's an important rock lesson. Uh here we have relationships
41:32 different quantities, all velocity related So these are quantities that can be
41:38 from seismic data. So, uh I cross plot and these are for
41:45 and shells. The sands are the points and the shells are the red
41:54 . And if I cross plot, wave impedance versus shear wave impedance.
41:58 see that I break out into two and that there is a line more
42:03 less dividing those populations. If I flood the D. P.
42:08 S. Ratio versus and penalize you I now have a horizontal line separating
42:16 looks like I have more spread. remember the I can get the elastic
42:24 I by multiplying the velocity times So I could calculate uh lambda times
42:35 and I could calculate new times So what would a new times density
42:42 uh if I take uh Rovi Rovi squared uh and multiplied by density.
42:51 rho squared rho squared B. Square. So that would be
42:56 And I could do a similar thing lamb because remember uh it's lander plus
43:03 plus two. I'm sorry. lambda plus two U. Is equal
43:09 K plus four thirds view. So I could get euro from V.
43:14 . M. V. S. could also get lambda rho from
43:17 P. M. V. And again, lambda rho has been
43:23 as being more sensitive to the rock uh than uh these other parameters.
43:33 uh why don't you look at these cross plots and tell me which of
43:41 has greater sensitivity to live theology to type. I would say the d
44:01 the V. P. V. . Against the PM opinions. Well
44:06 V. P. B. S PM penis in my mind is the
44:09 convenient one because I don't even need impedance to distinguish fill apologies. Apparently
44:15 straight from the V. P. . S ratio. I could distinguish
44:19 anthologies here. So I could say and easier distinction to make. It's
44:26 convenient, but is it more Can I better differentiate uh the stands
44:34 the shell stands from the shelves? I better differentiate that using V.
44:40 . B. S and P Then I can using any of the
44:43 methods keep in mind that as I a coordinate transformation, remember these all
44:57 exactly the same information. They were derived from the impedance and Sharon
45:06 So um essentially lambda Roland euro come PMP Vince and Sharon P even
45:17 And so what that does is it out, it spreads out the
45:26 but the overlap is exactly the same all these cases. Look at these
45:31 that overlap with the other points where with Allah jeez, where there is
45:38 in the law theology, there's the amount of ambiguity in every case.
45:45 so the very important lesson here which people, many practicing geophysicists don't get
45:54 that a coordinate transformation does not improve content. The coordinate transformation never improve
46:05 or signal to noise ratio. If transform the axis, I not only
46:14 out the points, but I spread the arab or is associated with those
46:20 . So I would argue that these all exactly equivalent. So you know
46:31 by swearing and combining p wave impedance shear wave impedance to get land around
46:38 or taking their ratio to get P. B. S. I
46:43 get more sensitivity so that's kind of optical illusion. Okay. Any questions
47:01 then move on to the next Yeah. I'm sorry I need to
47:20 this conference will now be recorded. . Professor. It's not it's not
47:27 . I mean I I see blank slide Dennis. Do you see a
47:33 slide author? Yes sir. Next our cash. I will stop recording
47:43 conference will now be recorded. so I've mentioned this before. What
47:52 pressure? uh 40% well that's stress pressure is a stress but uh when
48:03 I use the term press pressure as to stress? Um It's uh stress
48:13 affected, pressure might not might not um Yeah. Well okay. Yeah
48:20 another way of saying it. So is on on the direction,
48:25 Where stress will depend on direction. do we measure pressure? So essentially
48:48 we use hooks law we can create mechanical devices where we have a known
48:57 . And from the strain of that we could measure the pressure. So
49:03 see how much uh it's been calibrated that we know a certain amount of
49:09 corresponds to a certain amount of So that involves knowing the spring constant
49:17 the, of the pressure gauge, , okay. Um pascal's principle uh
49:27 that you might not have off the of your head. So what is
49:31 principle? Um What pascal's principle says that so the pressure, If I
49:41 pressure in a fluid, it is immediately transmitted to the throughout the fluid
49:52 the edge of the container, we'll back to that in a bit.
49:57 first let's talk about the units of and pressure. Um We talked about
50:05 we had mentioned dimes per centimeter Remember stresses force per unit area pressure
50:13 an omni directional stress as a So it's units will be some
50:20 the unit of force such as a , for example. We prefer to
50:26 Dines in at least I do. prefer to use Dines. Um And
50:32 for uh it could be per square , for example. So pascal's
50:46 If I apply pressure to an enclosed , it is transmitted undiminished to every
50:54 of the fluid and the walls of containing vessel. Um kind of a
51:03 thing to accept. If you think really giant containers, like the atlantic
51:11 , for example, if I apply pressure in new york does, it
51:15 a transmitted undiminished instantaneously to London. right, so we're gonna have to
51:23 you know, Suspend this belief a bit and say, well if the
51:28 is small enough for practical purposes, could assume this to be true.
51:37 Pascal was an interesting guy that made variety of very important contributions and this
51:45 in the 1600s. So this was early on. But uh he had
51:51 mystical experience uh before he died in abandoned his scientific work and uh became
52:02 philosopher and theologian. So uh maybe of us are just have not reached
52:09 point of enlightenment yet or uh we're privileged enough that we can abandon our
52:18 . So now, uh if I a pressure at the surface of a
52:29 , I could calculate the pressure at death in the fluid just based on
52:36 weight of the fluid. So the at any death is equal to the
52:43 pressure. Plus grow jeezy. So is the depth. You aren't a
52:49 . She is the acceleration due to . And row is the density of
52:55 fluid. This is sometimes expressed as G. H. Where h is
53:01 hype of the fluid. Yeah. one way of measuring uh pressure is
53:13 the strain gauge. Uh excuse uh with a tube of mercury.
53:21 if you have a tube of mercury on top of a bath. Uh
53:29 , the atmosphere will push down on mercury and that will push up through
53:34 tube. And so the height in tube uh would be an indication of
53:40 atmospheric pressure and you have to be uh to do that. So the
53:52 static paradox here, we have three with the same fluid level.
54:06 if I go to the base, or Rogue Easy is the same for
54:15 three of these containers. Uh The has the same length for all of
54:21 containers, but you can see the of fluid is different. So this
54:26 the greatest volume. This is the volume. And so this is why
54:31 vessel is the heaviest, which container the highest pressure at the base.
54:42 the answer is they're all exactly the . On the other hand, if
54:51 put if I put these rested these on you, if you were laying
54:57 , I mean if we were truly to face, we could do an
55:02 where I could have tubs of different and I could have a hose and
55:06 them with water and place them on stomach. Which of these is going
55:12 exert the greatest force on your Obviously it's this one. Right,
55:21 , had what is the hydrostatic paradox ? How is it that this one
55:29 exerts the greatest force on you. the pressure at the bottom is exactly
55:34 same in all three. And the is the base of the container makes
55:41 big difference. If you're underneath the you feel the full weight of the
55:48 . If you're above the base. only feel the the pressure of the
55:55 which has to do only with the level, not the amount of
56:00 So you're essentially at the same death you're feeling very different pressures. And
56:06 key is this base is impermeable right , put yourself in the earth.
56:14 I'm the need the base say this an impermeable shell. If I'm beneath
56:24 base, I'm feeling the weight of entire vessel. If I'm above the
56:31 uh the impervious layer, I'm only the column of water above me,
56:43 . So, uh can you imagine situations where this might result in a
56:50 change of pressure beneath the impermeable I'm feeling far more pressure than immediately
57:01 . You could say beneath the impermeable . I'm uh feeling the entire
57:07 a static load, the weight of above me. Whereas above the impermeable
57:14 , I'm only if all my fluids imperfect communication, I would only be
57:21 row G. H. Okay, we're going to assume that the overburden
57:32 is hydrostatic. So the poor is the same pressure in all directions.
57:43 poor pressure is pushing out. And we are in equilibrium then uh
57:51 These forces have to balance if the pressure is bigger than the poor
58:00 that poor is going to want to . So there's a pressure being applied
58:06 that for and how much it closes depend on the material the solid material
58:14 the poor some material will allow that to compress very easily. Other material
58:20 so strong that it won't allow the to compress compress at all. So
58:25 will be some poor volume reduction which going to depend on the module lists
58:30 the solid material outside the poor and module lists of the material inside the
58:38 . Now, if I have a pour, it also depends on the
58:42 shape. So in the case of sphere is the uh is going to
58:47 the compression the most. But if up a flat poor then the fluid
58:53 that platform is definitely going to be and is going to have to help
58:58 the compression. So in that case a flat pour it matters what type
59:03 fluid is in the poor. If have gas in the portal close more
59:09 than if I have water in the that can't get out right. So
59:14 that water is confined I'm gonna as poor compresses is going to try to
59:21 the water. Um So if there's gas in that poor and the poor
59:27 compressing the gas would accommodate most of compression. Okay, So uh we
59:38 the overburden uh with a static load creating an overburden pressure. And we
59:46 call that confining pressure in the laboratory a confining pressure which is usually applied
59:54 placing the sample in an oil bath some kind and the pressure in that
60:01 is increased um Now we calculate the pressure from the weight of the overlying
60:12 fluid column. So that way it depend on the density of the rock
60:19 fluid together. So it depends on total weight above you. And that's
60:25 on the order of £1 per square for foot. And with the conversions
60:31 can convert that to to give the per meter. But it's convenient to
60:39 in terms of one PC for that's an easy thing to remember.
60:44 , it's not exactly one pc per . It depends on the density of
60:49 rock fluid column so that will vary we'll see in an exercise how that
60:55 vary. And again the effective pressure assumed to be hydrostatic. In other
61:02 it's equal in all directions. There also be tectonic stresses supply. So
61:09 it would not be the confining pressure not be equal in all directions.
61:16 That's given a lot of consideration in mechanics. So if you go to
61:21 Mechanics laboratory, they have what are tri axial cells where they could bury
61:27 stress in different directions in rock physics normally dealing with confining pressures. Uh
61:38 pore pressure as we said, is pressure pushing out and if it's normal
61:44 pressure, that is the fluid pressure from the weight of the overlying fluid
61:49 the pore space. So when I row G. H. Is not
61:55 of the rock, it's row of overlying fluid. So the average density
62:01 the overlying fluid. Now, normal pressure Is on the order of .465C
62:11 what it is, depends on the densities, right? So it depends
62:16 uh the salinity of the fluid, depends on temperature, it depends on
62:23 fluid saturation But typically .465 pc So any poor pressure greater than that
62:32 called overpressure or geo pressure. And I said, if you're underneath an
62:39 layer, uh you could have significantly pressure because you're now bearing the weight
62:47 all the material above you. Uh just the weight of the poor
62:54 Now, the effective pressure is a assumed to be equal to the differential
63:04 , which is the difference between the and the fluid pressure. Uh That's
63:11 always the case. Uh Often you see effective effective pressure on a graph
63:19 rock physics measurements when in fact it's differential pressure. And I like to
63:27 the difference between these two because the of pressure is not necessarily equal to
63:33 differential pressure as we'll see. so it's the differential pressure which is
63:44 difference between the overburden and the a pressure. Okay, so now as
63:51 pressure up, Iraq here we have wave velocity here we have shear wave
63:58 as I increase the pressure and that's cut off. Let's see if I
64:04 . Um Yeah, so you can the whole slide down um In this
64:10 the shear wave velocity seems to be flattening out before the p wave
64:17 but both is showing the tendency of increase with pressure, a rapid increase
64:24 low pressure and then leveling off at pressures. If you just plot uh
64:34 vs stress versus the pressure and again is differential pressure. Um You tend
64:42 have a a greater strain for a change in effective pressure. Uh Given
64:53 in differential pressure, uh things strain a higher rate until you level off
64:58 then you have a a slower change strength with increasing pressure. Again,
65:12 first order, assuming effective pressure is to differential pressure. You could look
65:18 effective pressure versus death, assuming it's to the differential pressure. So you
65:24 calculate the overburden pressure versus death from density of the rocks and the poor
65:31 versus depth from the density of the . These would be perfectly straight
65:36 If your density was constant all the to the surface. Uh If there
65:42 the density is changing with death then curve is going to change. It's
65:49 going to be a perfect straight But you have the overburden pressure at
65:54 given depth, you have the four at a given depth and the differential
65:59 is the difference between us now on at the surface you have zero weight
66:14 overlying rock. I mean, you want to include the atmosphere but we
66:18 ignore that way. So at the you have zero pressure. On the
66:25 hand, uh if we're offshore uh the water bottom uh above the water
66:34 , we have no uh rock. we have no contribution from the
66:39 So above the the ocean bottom, we are just the weight of the
66:47 fluids which continues as we go Right? So this is the weight
66:54 the fluids in the four space plus weight of the fluid layer layer on
67:01 . Whereas at the at the ocean , uh we now start adding the
67:07 to brock. So now we the have an increased density above you.
67:16 so things offshore you would for the debt rock density. You would you
67:21 parallel the on shore line and you deviate from the hydrostatic pressure. But
67:26 at the ocean bottom lip, the and the hydrostatic pressure will be the
67:32 . So you can see that if have an increasing density, this is
67:35 extreme example of it. But compacting will do the same thing. You
67:41 a flattening of the curve here, ? Where you have a greater increase
67:47 pressure with death as you get So we have a concave upwards uh
68:02 . Okay, so here's another example increasing and decreasing pressure. Uh Again
68:12 is um the effective pressure is really reported as effective pressure but it's really
68:21 differential pressure and what it's showing in case I'm pressuring up, I'm pressuring
68:28 and pressure down higher velocities. Um this is by the way the pressuring
68:46 and pressuring down is just by varying poor pressure. Okay so here I
68:54 zero pore pressure. Here I have higher pore pressure and you can see
69:00 it's not exactly um equivalent right? differential pressure is the same in both
69:09 but the effective pressure is different is . This could be a history suspect
69:16 in this case or it could be a deviation from the effective pressure law
69:22 . And from this data alone we know if it's uh due to history
69:29 or it's a due to deviation from stress law. So in fact the
69:37 effective pressure uh we need uh an factor here. So whereas the differential
69:46 is the confining pressure minus the four . The effective pressure is the confining
69:53 minus some constant times the poor Or let's say some scalar, I
70:00 know how constant it is but let's it a scalar value now and is
70:06 close to one but it's not necessarily to one. Uh Just for your
70:18 we have a conversion table for units and this is a good time to
70:25 an exercise. So, uh so fought the overburden pressure versus death and
70:38 Using uh one C per foot and what that and plotted in giga pascal's
70:48 see what that plot looks like. think you can guess just looking at
70:53 question that is going to be a simple plot. But go ahead and
70:57 it and I'll stop recording while you're you're doing that. This conference will
71:03 be recorded. So just to So it's on the recording, the
71:11 to given death is equal to the density down to that death. So
71:16 average of all the dead cities above times the acceleration due to gravity,
71:22 the depth your act. And uh have the units for acceleration due to
71:28 here. Similarly poor pressure is is the weight of the overlying or
71:37 average density above you times acceleration due gravity times the height and this one
71:47 think we can go full screen So, these are some notes from
71:55 Mapco. We just retired from stanford was the head of their Iraq physics
72:02 project there for many years. Um ways that poor pressure impacts velocities.
72:11 I increase the poor pressure, I the compressibility and rigidity of the rock
72:19 by opening up cracks and pushing grains and that will lower the velocities,
72:28 increasing the poor pressure tends to make pore fluids in the rock less
72:39 you know, if I have a and I put that gas under
72:43 I'm forcing the molecules closer together. means they're harder to push together.
72:49 I'm making the port fluid more Excuse me, less compressible as I
72:56 the pressure and that will tend to the philosophy. So, these are
73:02 effects. The first effect is usually than the second effect. Uh
73:09 if I have a fluid mixture, the poor pressure can also change the
73:15 uh as gas can go in and of solution, especially if I have
73:20 gas oil mixture. Uh If I the pressure, I could go above
73:26 bubble point and I could dissolve all gas in the oil and that could
73:31 the velocities to increase also. Um the poor pressure is existing over geological
73:43 , that could result in dire genetic . Um in sand stones that can
73:50 die genesis and preserve ferocity. On other hand, it shells that could
73:57 uh it could result in de uh etcetera. No, I'm
74:03 the high pore pressure, high confining would result in uh the water in
74:09 pore pressure. I would push the apart. Okay, so here we
74:17 a bunch of measurements on Shelly sand that are supposedly dry. Yeah.
74:27 , uh we don't know to what they've been drive. We don't know
74:31 they they've been oven dried or if room dried, but they're not fully
74:38 let's say with water. And uh individual individual laboratory measurements uh you see
74:50 both modules change and a significant ferocity . And so this suite of measurements
74:57 as we're changing the pressure. so at low pressure we would have
75:04 porosity, low bulk modules, increase pressure. We reduce the ferocity.
75:11 in shale, you're going to get porosity reduction than in the sandstone or
75:17 if you have a strong solid uh It's hard to reduce the porosity
75:23 much but in a shell with a compressible matrix and with micro porosity and
75:29 forth, it's easy to reduce the by increasing the pressure. Okay,
75:41 we were looking at a velocity versus , uh the facts um so we
75:53 a high pressure, our philosophies level . Uh we won't get all the
75:59 to the mineral velocity because we have remnant ferocity. We don't close all
76:06 pores. So how close you get the mineral velocity is an indication of
76:14 ferocity. Uh huh. The change velocity uh is an indication of the
76:25 of soft, close herbal ferocity you . So as I'm increasing the differential
76:32 , cracks are closing or pores are , how long it takes you to
76:39 off is an indication of the crack . So the very flat poor is
76:48 close uh with a small pressure improvement as we go to higher and higher
76:56 , what you have remaining are the and rounder pours until you get to
77:02 point where you have only the very pours left now uh fracture pressure is
77:16 engineering concept. It's the it's the way you have to use in order
77:23 fracture the formation around the well So the drilling mud weight is usually
77:31 higher and high enough to counteract the pore pressure. If I drill into
77:40 geo pressure brock, if I were with just water, uh remember the
77:47 pressure in the rock is much higher row G. H. In the
77:51 bore. And so uh the fluids the formation can burst out of the
77:58 and you could have what is called blowout. And these can be very
78:03 if you have hydrocarbons associated, you lots of friction going on blow gas
78:09 of the well bore. And it ignite especially as you get to the
78:16 . Uh So you have to be about this. Um So you have
78:23 have your mud weight fire than the pressure. But if the mud weight
78:27 too high, that makes trillion more . You know, you've got that
78:34 is heavy and it wants to stay it is. You have to move
78:37 with the drill bit so it slows trilling, It also gives you more
78:44 into the formation, which is undesirable you may push hydrocarbons away from the
78:52 and you may never even know there hydrocarbons there. But the other thing
78:57 can happen is you can fracture the and then the formation starts to eat
79:02 drilling fluid and you have a loss fluids in that case, which is
79:08 expensive proposition and causes drilling problems. uh you keep the mud way as
79:16 as possible to avoid blowouts but low such that you don't fracture the
79:24 In fact, when we in hydraulic for intentionally, for stimulation to improve
79:33 permeability of the formation, we do on purpose, but that's under controlled
79:40 . We don't want to do that and down the bore hole. And
79:43 especially don't want to do that uh in in formations above our reservoir.
79:54 uh and important you to application of physics is to try to design the
80:01 program correctly, to be able to the drilling engineers based on what we
80:07 about the rocks, how high should poor pressures be and how high do
80:13 not want them to get? So causes these abnormally high pore pressures?
80:25 , we talked about an impermeable cap pressures from a quick liberating vertically.
80:34 We talked about the weight of the rocks. Remember the hydrostatic paradox
80:40 slightly below the impermeable layer. The can't get out and they are experiencing
80:46 weight of the little static load above . So that causes the poor pressure
80:52 increase another cause of high pore pressures fluids are trapped and can't get
80:58 Is aqua thermal pressuring. So as get deeper you get hotter. And
81:04 the fluids want to expand. Uh if they can't get out they can
81:10 the molecules get pushed closer together and increases the pressure. Uh If I
81:18 trapped fluids that can escape, if have abnormally high tectonic stresses that could
81:25 the fluids and caused the poor pressure increase, I could also add more
81:35 to the poor space by the By taking water out of minerals.
81:42 are two ways that water is and in clay's, it could be trapped
81:48 the crystal lattice non stoking symmetrically. , as I compress that crystal
81:54 I could force the water out of crystal. But there could also be
82:00 transformations where I have bonded by drops . I could force I could have
82:07 diabetic reaction where water is generated. that will also increase the poor pressure
82:16 if the fluids are not free to . And similarly, if I generate
82:22 , so I have solid organic material I start cooking that solid organic
82:28 I will generate hydrocarbons that will be into the pore space. And if
82:33 can't get out they'll create an abnormally pressure as well. Let's say it's
82:44 for a Short break. So let's at a quarter after 10. This
82:56 will now be recorded. Yeah Now mentioned previously that the pressure gradient is
83:05 change in pressure with the change in . Um So if I have a
83:12 versus death curve from a scientific point view it would be the derivative of
83:19 curve. Right? So it would the tangent line at any and any
83:27 . But that's not the way When engineers talk about pressure gradient they
83:34 something else. Uh It's not the of pressure with death. It's the
83:41 divided by the depth. Um Why divided by depth mm Because the engineers
83:54 to know how heavy their mud should and how heavy their mud should be
84:00 equal the formation pressure is the pressure by the death. Remember it's row
84:08 . H. Pressure is equal to G. H. So pressure divided
84:16 h. Is equal to rho Right so the pressure gradient tells them
84:26 what density there mud drilling mud needs be in order to balance the formation
84:37 . Mhm. So if you plot an engineer's point of view uh pressure
84:47 death. Uh huh. What you or pressured radiant versus death as you
84:57 a an increasing pressure gradient gradient with suggesting lower rows lower densities shallow which
85:08 sense. And as you get deeper the pressure gradient uh The pressure berries
85:16 slowly with death. And um older have a higher density. So they
85:28 a higher pressure gradient than younger Yeah. So this increase in pressure
85:36 with death has to do with the increasing density with death. All
85:46 And then just another table relating different . Okay, so now we're going
86:01 do an exercise which is going to some time. So I'm going to
86:07 recording. And what we're gonna do uh we have a table of measurements
86:14 a table of empirical fits density versus trends. And we're going to convert
86:22 into pressure versus death. Um uh we applaud density versus staff average
86:32 versus that. That's the average density all the rocks above you. The
86:39 stress versus death. The true pressure , which is the derivative of the
86:49 the change of stress with the change death and depth divided by I mean
86:56 divided by depth versus death. this is the engineering pressure gradient.
87:01 true pressure gradient. So lots of . The plots hit plot here.
87:08 so this is gonna take a So, I'm going to stop
87:13 And I would suggest that the original for show me for the first
87:23 Each of these plots. And then would be a simple matter to do
87:26 for other curves. Right? as you get results for the first
87:33 , show me, show me the . Uh Show me each of these
87:37 you get them. All right. , I'm going to stop here.
87:41 conference will now be recorded. And see this. I would say in
87:52 measurements. Remember we talked about things off the velocity, leveling off.
87:58 here we have cracks closing and now have leveled up. So you could
88:03 trends to these curves. And uh you could get a relationship between velocity
88:11 pressure, right? And uh and you could for example, with a
88:19 like that where you have the velocity by the high pressure velocity than can
88:27 expressed as a function of pressure. filling an exponential curve given the
88:34 you could calculate the pressure. So the question I have is should core
88:42 be used to determine the instant C velocity versus pressure. In other
88:48 will rocks in the earth. The this expression that you've determined in the
89:02 . Mm. So Exactly right. as I corps, as I drill
89:10 I corps, I'm going to be the rock. That chorus had all
89:15 of torture channel stress on it and worse than that. It's got stress
89:23 . You bring that court to the , you take it out of the
89:26 barrel and you've relieved the pressure on thing. In fact on the drilling
89:32 , you could sometimes see these cores right in front of your eyes.
89:39 you're changing the core right now you that core, put it in the
89:47 and you pressure it up and you're a lot of the damage to the
89:53 . So here look at that. almost a doubling of velocity from note
90:00 pressure to high pressure. What you've is you remove the cord a lot
90:06 the core damage as well as any fractures in Iraq or low aspect ratio
90:14 in Iraq. But I think given massive velocity difference, I think you
90:20 guess that most of that is core which is being uh fixed. So
90:27 very careful about Using laboratory derived pressure and using those in c.
90:42 Now again this is differential pressure. even my papers, I've been guilty
90:50 using effective pressure instead of differential So this was a paper where we
90:57 suggesting using the PBS ratios to find high pore pressures. Be PBS ratios
91:05 be determined seismically with greater resolution then V. P. Or V.
91:13 . So here was an empirical trend the red in the black curve.
91:19 the red curve was a laboratory laboratory . And uh you can see the
91:27 in V. PBS ratio becomes very . We're seeing BP Ds ratios that
91:33 extremely high compared to liquefied rocks had low and very low pressures. And
91:45 this was the example I showed This is this is the inverted of
91:50 PBS ratio by simultaneously inverting PP data PSV data acquired with ocean bottom
92:02 So we have B. PBS ratios approaching seven. Uh in the very
92:10 rocks as we get deeper we have and lift ification and the B.
92:18 ratios are decreasing. So you see compaction trend in the V.
92:23 V. S. Operations. Uh uh This is a priest like amplitude
92:32 in virgin or it has nothing to with with with with the empathy of
92:36 data. Just purely velocity work. now it's uh it has to do
92:41 the with the velocity velocity in this with the priest. Let me take
92:50 back. This is using um Yeah right. It has nothing to do
92:58 the HBO at this point with inverting p wave impedance and inverting for shear
93:06 impedance and taking that ratio. Okay so this is yes the B.
93:17 . B. S ratio. But actually the ratio of the E.
93:20 and penis to shear wave in And so here we have high
93:25 PBS ratios as we approach the water . Uh But down here we have
93:31 layer with relative to the rocks around . With a very high V.
93:36 . V. S ratio. Normally would expect that to be a geo
93:41 shell. But in fact these are pressured sands. Um It's the stands
93:50 with very high pore pressures get very B. P. V. S
93:54 . Like in this slide here if v. PBS ratios well over six
94:06 into seven right now. So you look at the difference between if you
94:11 the normal compaction trend of the P. V. S ratio and
94:16 subtract that uh from this there you your anomalous li hai the PBS
94:26 You're anomalous li lo be PBS ratios to be sands, you're anomalous lee
94:32 tend to be your shells. And anonymously hy vee PVS ratio is an
94:38 pressured sand. And that was, I mentioned last time, that's where
94:43 had a shallow water flow. But is how we got there.
94:52 that's the end of our pressure Are there any questions on that?
94:57 , I move on to what we've uh building up towards which is seismic
95:14 . There are no questions. I'll on. Oh I should be
95:19 Yeah. The lecture slides for this an open blood body here.
95:25 was it not? Okay, I know for a fact. So what
95:31 do then is we'll take a quick and I'll quickly put them onto
95:38 So let me do that. This conference will now be recorded.
95:50 , You'll remember from Jeff physics one we have two types of velocities,
95:59 shear waves and p waves. Two of body waves for share waves.
96:05 have a transverse wave and the analogy if I have a string attached to
96:14 wall. For example. Uh professor we got the blank screen probably
96:21 Shit, I'll stop the recording This conference will now be recorded.
96:29 , So, uh I don't know you were a kid. Maybe you
96:33 jump rope? Actually. Uh we to do that with the girls in
96:37 neighborhood. We would play jump We attach a rope to the
96:42 I mean to a wall and we flying up and down motion on the
96:49 . And that broke with produce an . All right. So uh the
96:57 displacement of the rope is left right this in this image, but the
97:05 is moving vertically. So the displacement trans verse to the direction of
97:13 On the other hand, if I a spring and I pushed on
97:18 spring on one end, that spring move along the length. Uh and
97:26 compression would then be a longitudinal compression moves through the spring. And that
97:34 would be associated with a rare faction what is stretching only uh other end
97:41 the strengths. As I'm moving in out, I'll have a compression,
97:46 faction, etcetera. They are the of the spring. You know,
97:51 guys have come to have relative to guys. The displacement is longitudinal.
97:58 this is the analogy to compression. waves. And this this is the
98:03 to share waves. So now we're to look at the waves propagating through
98:11 body, a solid body. And going to divide this body up into
98:18 testimonial volume elements. All right, these are these little cubes here.
98:25 before any deformation there were all the size and they're in a regular
98:30 So now if I push this parallel it, if I push it on
98:36 side, that compression will move through rock. So I'm pushing and
98:42 So I have compression moving through the and rare factions uh which we refer
98:49 as debilitation case that we have compressions village stations. Now look carefully at
98:58 volume elements. You notice that the width of the volume elements, whether
99:05 being compressed or stretched the width that that doesn't change right. The width
99:13 doesn't change only the length along the changes. So the mind elements are
99:20 shape and they're changing in body. These compressed funds are smaller volume than
99:32 . The reason they don't the volume don't get fatter trans verse to the
99:38 of propagation is because they have adjacent that are else. Also trying to
99:44 Uh huh better and can't because the are pushing against each other. So
99:52 these volume elements are being constrained trans to the direction of propagation. So
100:01 elastic module is controlling the relationship between and strain in these following elements is
100:08 the plane wave module. Escape was 13. That's why in the p
100:15 velocity equation is square root of Cape four thirds view of arrow square root
100:21 em over row now share waves are different here. The displacement is transferred
100:32 to the direction of propagation and sometimes being displaced upwards, sometimes you're being
100:40 downwards and the result is it change the shape of the volume element.
100:47 changes the shape but not, you , it's strata graphic thickness. So
100:53 fact the volume of these volume elements change only the shape. And that's
101:02 shear waves don't depend on the boat . They don't depend on the ratio
101:08 biometric stress to biometric. Strange, only depend on the ratio of shear
101:15 to shear strength. So uh shear have no volume change associated with.
101:24 can see that if I if I a point in the middle of the
101:29 element as time passes. Sometimes this be a rhombus where the apex on
101:36 right side is up and the left is down. And sometimes as this
101:42 passes it will be the other way love side is the right side of
101:46 camp. So that's why shear waves called rotational waves because it's a rotation
101:53 the shape of that volume element as the way passes. So this is
102:05 the same thing. But at different in time. And now we can
102:12 on one particular volume element, as wave passes. Remember the wave as
102:20 associated with it. And it has factions or debilitation associated with it.
102:27 these compressions and rare fractions are moving the rock as a function of
102:33 you see that. So any one volume element maybe uh stretched at a
102:43 time. So here it's being stretched it could be squeezed here, it's
102:48 squeezed. So that volume element is experiencing a volumetric metric compression and a
103:02 in shape. However, for the wave, the volume of that volume
103:08 is staying the same and all that's is rotation around the share with.
103:14 this difference in behavior greatly affects the fluids in the rock behavior. Now
103:23 a porous rock. And we're going call that a poor oh elastic medium
103:30 the pores are connected and fluids can through the pore space. What we
103:37 with the compression all wave as we while one buying element is being
103:45 another volume element is being compressed. did we say happens when we increase
103:51 pressure on a fluid? It's going want to move to low pressure.
103:55 the poor pressure is high here, low there. That fluid is being
104:01 to move to the to the into the right. You see that
104:08 it's a fluid movement in the direction propagation, right. Either to the
104:15 or to the right, from the pressure volume elements to the low pressure
104:22 elements. On the other hand, at a different time, the volume
104:31 here is now being stretched. So in here and here are one are
104:38 told to move back to that volume . Now, if this is a
104:44 low frequency wave and the rock is , or and or the rock is
104:51 , the fluid will have plenty of to move. So the fluid here
104:57 all the way there and then at time it is time to move all
105:02 way back. But think about if a very um high frequency way or
105:10 a very low permeability such that the rate is very low. In that
105:16 the fluid doesn't have time to make . The fluid here. The fluid
105:22 is being told to move there, before it has a chance to move
105:26 far, it's being told to move to where it was. So it's
105:33 high frequency. That fluid is essentially being kept in place it vibrates,
105:41 it never gets to go very So uh this is uh what we
105:48 call B. O. Flow. this is uh we also refer to
105:53 as sloshing type of flow. It's to the size of of a given
106:00 element. It's a very long distance travel which can be achieved at low
106:07 or very high permeability ease or can't achieved at high frequencies. Uh So
106:15 call that B. O. He was the physicist who was working
106:19 a consultant, Michelle in the 19 , who developed the theory that we
106:25 to this day. On the end other hand, look at the shear
106:32 , the shear waves, there is pressure differential between volume elements. So
106:39 pressure differentials must be very local. , let's talk about local flow as
106:47 deforming the rock, we're changing its . So that means we're going to
106:53 deforming the poor space. The pores changing the shape, their shape and
106:59 pores will be preferentially closed and others be preferentially open depending on their
107:06 So for his closing will squirt fluid , essentially induce a poor a local
107:14 pressure gradient from the closing pour to open pore. So that's what we
107:21 local flow. And both p waves she waves exhibit local 12 P waves
107:33 the longer range flow generally involved more , solid movement or movement between fluids
107:41 solids and therefore more friction between fluids solids. So, p waves
107:47 you have higher absorption, higher frequency continuation. Both are affected by local
107:56 but only p ways are affected by long distance flow. Alright now we
108:08 this idea before that there is a between the physical properties of Iraq,
108:14 as length. Ology ferocity pore, permeability, also the environmental conditions and
108:23 elastic properties of Iraq, which we geophysical properties. Both modules. Sarin
108:29 rigidity and density. Uh These elastic affect the seismic or acoustic properties.
108:39 wave velocity shear wave velocity P wave . This is the quality factor.
108:46 a measure of attenuation and share wave facts. And it's the three dimensional
108:53 of these things which results in the response. And we have a come
109:03 to our velocity equations which I really you to have memorized. And the
109:12 equations show the relationship between the elastic I these are for ice, a
109:18 rocks and the velocities. Now I'm to ask you, what do you
109:25 to happen if I replace the brine Iraq or the water in Iraq if
109:34 replace it with hair or gas, going to happen to V.
109:39 M. V. S and So everything else is the same.
109:52 have the same rock frame, same . I'm just taking out some of
109:57 water and I'm replacing it with a of some kind. What's going to
110:04 to these different quantities. And so going to happen to DP NBS awake
110:10 you to answer V. S. remain the same. I'm sorry,
110:20 sorry. I'm sorry. I'm The effect of the density will will
110:24 the Yes. Right. So the stays the same. The density drops
110:33 will go up. Yeah. What BP? Um It's a bit more
110:39 because then we have two very key density. Uh huh. I think
110:46 effect of the dynasty at the beginning would be big but then the effect
110:54 the bulk would take over. I looking at that curve from jasmine
111:01 substitution and uh it was showing variation both but each one was limited to
111:07 certain gas situation I guess. if I add just a few bubbles
111:13 gas, am I going to change density very much now? But if
111:20 add a few bubbles of gas from equation, because it's a reciprocal
111:26 I'll reduce the both modules of the . And if the rock frame is
111:32 , the rock frame, uh both will decrease. So the effect of
111:39 , the initial edition of gas is leave rigidity unchanged, reduce the both
111:48 lists and reduce the density but reduce bulk module, it's more so the
111:54 few percent of gas, reduce the . Uh huh. Now, as
112:00 continue to add gas, the bulk is that the fluid won't change very
112:07 the fluid that the gases so that it doesn't matter how much I
112:12 just a few percent of gas and all of the compression. So as
112:18 continue to add gas, the bulk , that the fluid doesn't change very
112:23 . So, the bulk modules of rock doesn't change very much but the
112:28 continues to go down linearly with the saturation. So the effect then after
112:35 initial drop in velocity. When I a little bit of gas, I
112:39 more gas in the p wave velocity up. We're gonna be talking about
112:47 substitution at great length So we're going come back to this idea. Okay
112:58 in the laboratory in the old days start record our acoustic wave forms that
113:07 pass through the rock. We start look at these on in a silla
113:14 and before we digitally recorded them we take pictures or in fact just read
113:20 off the scope. What the arrival of the wave form is. So
113:25 is still a scope is measuring the of these transducers. We have a
113:33 generator. It generates a pulse that trigger the transducer to deform and therefore
113:43 a wave through the sample. We another transducer which will respond to the
113:51 defamation. So it will deform and a voltage or current which will be
113:59 and sent to an a silla At the same time we send a
114:05 directly to the oscilloscope. So we where, you know, we know
114:11 at what time the post was generated that scene on the oscilloscope and then
114:17 could see when the host that has through the sample when that arrives and
114:24 its arrival time and knowing the length the sample, we could measure the
114:30 . You can imagine that there are lot of complications in this process.
114:36 lot of possible sources of error. , this is a dex desktop measurement
114:43 by the way before covid in the learning center we actually had desktop example
114:56 making these measurements, students could actually and do it themselves. It gets
115:02 it's very easy to do on the . Um And of course we don't
115:07 it to sell a scope. We the PC to collect the wave forms
115:12 then we could digitally measure the arrival . Um But so doing this on
115:20 a desktop is fine but that's not of in C2 conditions. So this
115:27 has to be put in a pressure and that's when things start getting very
115:33 and very complicated. We put the in in the vessel, we have
115:39 jacket the sample because the pressure is transmitted through a fluid. So we
115:46 want that fluid at the confining We don't want that fluid getting into
115:50 sample. So the sample is jacketed that starts making uh starts making life
115:57 difficult. That keep in mind the of this must be such that we're
116:05 getting guided waves dominating through the Right? We want to transmit a
116:11 wave through the sample without interference you know, multiples off the sides
116:18 the sample and other things indirect travel . Right? We want so we're
116:24 to want the measurement to be very frequency such that the direct wave can
116:31 cleanly at the transducer on the other of the sample. So these wavelengths
116:38 have to be very short compared to sample. If the wavelength of the
116:45 were very long, then we would a guided way through the sample analogous
116:50 surface wave. Right? So uh we don't want that. And so
116:58 have here is an example of a picture of an acela scope waveform.
117:04 This happens to be a digital oscilloscope the time. And so uh mm
117:11 measuring amplitudes at even time increments. why it started there. But here's
117:19 trigger. And you notice there's a frequency noise associated with the trigger and
117:26 there's high frequency noise before this Have this high frequency noise is uh
117:35 might be, for example using shear transducers. These might be p waves
117:41 got there before the shear wave. so a p wave arriving before the
117:47 way that the p waves are very . They can interfere with your arrival
117:54 . Now think about this. If measuring the velocity of this way,
117:59 just measuring the velocity at one source facing right. It's the velocity at
118:09 spacey for the length of the Um Can you see an inherent area
118:19 ? Where do I pick the onset the way for where where is the
118:26 arrival time at the wave form? it? Uh The first trap here
118:32 the first peak here? Or do have to extrapolate back into the noise
118:39 figure out precisely where the first arrival is. So anyway, there is
118:44 uncertainty in that arrival time. Also I'm pressuring the sample up, I
118:52 be changing its length. So I to be monitoring the length to get
118:58 velocity right yet at every pressure. um so in fact there is error
119:09 with making these measurements and the more the rock is the week or my
119:15 forms are the more error there is with detecting the way for. It
119:21 be nice if we could measure the time difference across two receivers. Those
119:27 generally much more accurate than just measuring arrival time. Okay, so how
119:38 and precise our laboratory measurements and so need to distinguish between what we mean
119:45 accurate and what we mean by precise means how correct is the measurement in
119:55 terms. So I have a true velocity of the rock. What is
120:01 measured velocity? And how different is from the true intrinsic velocity. That's
120:08 accuracy of the measurement precision means, repeatable is the measurement? You could
120:16 precisely wrong but hit the same answer . So uh you know, to
120:23 extent that uh you know, picking first arrival depends on the noise.
120:29 might be an indication of precision. if there is bias in the
120:34 like I'm always picking too late in waning form or something like that,
120:38 affects the accuracy of the measure. The other thing that affects the accuracy
120:44 the measurement. And is the biggest in core measurements is a how representative
120:52 that rock sample after you've acquired it shaped it and put it in your
120:59 apparatus and maybe had variations and stress it before this particular measurement. The
121:10 then of the rock physics measurements is much in question When comparing them to
121:16 c. two measures. Also keep mind that there are sampling differences.
121:21 are resolution differences. Sampling can be by your ability to actually acquire a
121:30 and shape the core into a If the rock is highly fractured you
121:34 lose the core. If it's very , you may lose the core.
121:41 and if you have a mixture of uh rocks that have high integrity versus
121:49 that won't be cord well will be in the court. Then your measurements
121:53 biased towards those samples with high So there are all kinds of reasons
122:02 we we have reason to suspect the of the rock physics measurements. Usually
122:09 are more precise than accurate. If the case then how are they
122:16 Why do we bother to make measurements the laboratory? If they're not
122:24 I'm going to let you answer You don't think they're useful at
123:01 What do we spend the money to these measurements, anybody. Well they
123:10 give us idea of your physical properties we're trying to measure in the
123:16 But the compared to the come the largest experiments are controlled compared to
123:24 in the field where we have the and other factors out there.
123:28 Very good point. Very good. the laboratory. It's a controlled
123:36 So we could vary the parameters. could vary the pore fluids. We
123:42 bury the poor pressure, We can the confining pressure, We can characterize
123:48 sample very well. We could create samples and therefore the laboratory measurements can
123:57 us understand what is happening. The measurements help us predict how velocities will
124:06 with the other factors that we control . And so that's where the laboratory
124:13 are really important. They help us a conceptual understanding. And the empirical
124:21 can often be used because the inaccuracies comparing to a particular insitu rock tend
124:33 cancel that on a trend because we're comparing to a particular rock. We're
124:41 the variation over a group of So there may not be a 1-1
124:48 to what the velocity of that sample be in the earth uh for a
124:54 type of measurement. But the systematics still be applicable. And we found
125:00 to be very true. The empirical we establish in the laboratory have been
125:08 useful when applied to real data. what are these factors, You
125:21 what are these variables that cause velocities bury. So, we'll start with
125:28 wave velocities. We'll find that with exception of pore fluid shear wave velocities
125:34 affected very much in the same way congressional philosophy. First of all,
125:41 have mythology, the composition and We've already seen that effect ferocity,
125:49 amount of ferocity and the type For us now there are a variety
125:55 factors that are depth related pressure uh pressure, confining pressure and deferential or
126:09 pressure. Then there are factors that the degree of lift ification. The
126:18 the rock, the more liquefied it to be, the longer it's been
126:23 at high temperature and high pressure uh liquefied. So also the greater the
126:31 , the greater the degree of lip generally on the average, the degree
126:38 cement, ation and compaction, not as it affects porosity reduction, but
126:47 it affects coordination between graves and so and interlocking of brains. So compaction
126:56 not only decrease ferocity, it could the degree of little indication. Then
127:03 have, as we've seen, we the pore fluids in Iraq affecting primarily
127:07 compression wave velocity and a second order on the sheer weight philosophy, temperature
127:16 affect the velocity through the pore The solid won't be very dependent on
127:21 temperature as we deal with in the few miles. But they will affect
127:26 poor fluids greatly frequency. Especially from to laboratory frequencies. And of course
127:37 Satrapi. So the direction of the counts. So this is just one
127:48 of measurements that was reported about 60 ago. Um These are history,
127:56 of velocities and various rock types and that they didn't separate sandstone and
128:04 But for the most part you had unconsolidated sediments which were lower velocity and
128:12 can see they could be lower than velocity here. So some of these
128:16 above the water table. They're you have sand stones and shells.
128:22 , if dry, they can be than water velocity, which is 5000
128:27 per second, higher velocity or Then you have your igneous metamorphic rocks
128:37 you're in Hyde rights will tend to the highest philosophy salt uh generally around
128:44 , but you're vaporize can be very philosophy Now. Uh in the same
128:54 , they report fiduciary limits. What the fiduciary rule limits? So you
128:59 say Between what range does 80% of data occur. So the, you
129:07 , those that can be represented as bar. And so this is the
129:13 data just represented in terms of The for with 80% of the data occurs
129:20 you see that there's tremendous overlap. fact of these reported values for sandstone
129:26 shale are quite low. I as we drill deeper, we find
129:31 sand stones and shells both could have higher velocities. So I would pull
129:38 stones all the way up to say. And shells not quite so
129:44 . So, velocity alone, it's enough to distinguish uh the mythologies,
129:52 the mythology, the pathology is enough give us an expectation for the relative
129:58 of velocity. Not necessarily right, there is overlap. So it's a
130:04 sand tight sandstone can be faster than porous limestone, for example.
130:16 same concept just showing overlap between different types. Now, we talked previously
130:29 the velocity of the suspension. um this would represent the lowest velocities
130:39 we can have uh in Iraq as loses cohesion. So well, interesting
130:49 here that should say x courts. now I'm just going to ask you
130:57 make the calculation. So, uh the volume of courts from 0 to
131:04 . The other material will be And so you could use a any
131:13 . I and then as you um I could suggest starting module i
131:19 debt cities. Uh for ports use giga pascal's and for water used 2.5
131:28 Pascal's that would be, you either fresh water under some pressure or
131:36 faulty water. So use 2.5 giga for water In a density. You
131:44 use one the water or 1.5 grams cc. It's up to you.
131:52 courts use the density of 2.65. go ahead and create this spreadsheet.
132:01 it's a it's worth doing. But , this is vp computation. I'm
132:22 , I was muted. Yes, Bp. And I'm going to stop
132:26 them. This conference will now be . Which are you seeing this
132:37 Are you seeing fast empirical relationships? , sir. Okay. So the
132:45 thing is showing, Sorry, that cut off and that's something I have
132:49 figure out how to fix. So going to uh escape. And so
132:56 can see the full slide here. in the early 50s when uh sonic
133:05 were first being acquired bounced, collected lot of data and he realized that
133:15 older Iraq was, the higher the and the deeper Iraq was on the
133:22 . The higher the philosophy. And came up with this empirical relationship where
133:31 . Is A. Is a Now, um It's not clear how
133:39 arrived at this form. T to 16 is the to the 16.
133:44 at the same time, theoretical calculations suggesting that sphere packs Should be related
133:52 the 1/6 power of pressure. And that may have steered fast into this
134:01 of relationship. We're not sure what justification was. Uh huh. Now
134:08 is catching the overall tendencies, but doesn't capture the fact that uh more
134:16 rocks of lower velocity than low ferocity . And so he came back and
134:24 tried to incorporate ferocity into the equation including increases Stephanie. So yes,
134:31 L and I think L. That meant to imply with little logic
134:38 And so he has true resistive itty . So so the resisted any of
134:44 rock depends on the ferocity. The mark site, the higher the ferocity
134:50 the lower the resistive Itty, so higher the conductivity. Uh So uh
134:58 he has as the resistive itty the velocity increases. And so another
135:05 relation of a very similar form. gas men about the same time published
135:15 relationship. Now these are not the equation. This is not the famous
135:20 equations that we're going to use for substitution. This was an equation he
135:26 up with for a pack, a packing of spheres. And um he
135:35 various quantities in here But raised to 1 6th power. So Depth to
135:42 16 powers, he's in a homogeneous of spheres. So he knows the
135:50 of this of the sphere pack. he'll know uh you know what depth
135:56 the sphere packed corresponds to what So he's basically got pressure to the
136:03 power. Um Also this factor is to uh the properties of the spheres
136:11 when here he's got the young youngest of the spheres. The acceleration to
136:15 gravity, 1 -2 persons ratio of year's the prosperity of the medium,
136:22 is forgiven spear pac arrangement is And that's the density of the
136:31 And so you see this mimics this a theoretical equation which is very much
136:37 same form as fast equations. So justifies this 16 power relationship. By
136:46 way we use fast equations to this with empirical calibration and modification. But
136:53 use a form like this to this to predict pseudo velocity locks And um
137:05 you plot velocity versus depth trends, see a general tendency for velocity to
137:13 with death. So these are found well fits to the data that Faust
137:26 the rocks of different ages. And he's got velocity versus death. And
137:34 for different ages you have different empirical but you can see uh that the
137:43 the rock is at a given the higher the velocity. So this
137:49 is showing a age dependency which is related to the degree of with ification
137:57 a depth relationship uh which again is to a degree of lift.
138:03 ferocity in semente shin and other So both greater age and greater depth
138:11 a variety of reasons increase the Let's see if I could go back
138:18 full screen. Okay, We already about the time average equation and when
138:24 expressed it, I said the transit is equal to the ferocity times the
138:29 time in the fluid Plus 1 - Times The Transit Time in the
138:36 And instead of transit time here, is expressed in terms of slowness.
138:42 the transit time is equal to one the velocity which is also called the
138:49 . Uh Now it would be good be able to write to extend this
138:59 to multiple constituents. And uh earlier I kind of did that with the
139:10 balance equation where we compute a uh grain density from the constituent densities.
139:21 , well you could do the same with transit time, you can get
139:26 grain transit time or a matrix transit here they say matrix velocity is a
139:35 weighted average of the transit times of solid constituents. Uh So you can
139:42 extend the time average equation to mix ology is that way. Now,
139:50 though this looks like it could be derived, it can't be, in
139:58 , it could be shown to probably theoretically wrong except under certain circumstances.
140:06 it tends to work for poorest and at high pressure and by poorest.
140:11 mean reservoir quality sand stones that are lit defied at high pressure. But
140:20 seen there are many velocity ferocity relationships depending on the local circumstances, the
140:28 degree of consolidation and lift, Uh The pressures and so forth.
140:35 relations will work in different areas. , so uh gardens relation we already
140:43 at and that relates to density to . And because density is directly related
140:53 velocity then it relates ferocity to Uh so let me escape to get
141:03 full figure here. And um Here have some of the data that wally
141:11 and Gardner used this later reprinted in review paper, but I think these
141:16 were made in the in the 50's these were kind of reservoir type sand
141:23 , and that's where the wily time equation was fitting. He also included
141:29 very high porosity rocks, radio larry earth in triple light. Uh presumably
141:38 are related to uh salacious shells. they're the shells tend to be
141:45 So, an aggregation of these not only is it a sphere
141:51 but also the grains themselves have So you can get to some very
141:56 ferocity is here. And Gregory noted they deviate from the time average
142:10 Now I mentioned a few times now rain manhunt Gardner equation and this equation
142:17 from my point of view, I call it an improvement to wireless
142:22 but I do feel that it's a fit to highly liquefied rocks.
142:26 Did you have something? Yes, . I have a question on the
142:30 line. Yeah. So, so though these are beyond 40, we're
142:36 calling them suspended 40% ferocity or they suspended. Well, they may
142:42 I'm not sure. I doubt they're by the time they were in the
142:47 . So the grains are at least against each other. One test would
142:53 to calculate where the Royce bound is . That would be a nice
142:59 Right? Uh Let's see how close are to the void to the Royce
143:04 . I suspect they're not I suspect process, these are so high because
143:09 grades themselves are Horace. Okay, , because they're empty shells basically.
143:18 , so so so porosity can be 40% and we can still have uh
143:27 . Okay. Yes, that's Thank you. And uh so getting
143:36 to this roemer equation basically, he they The equation has two parts really
143:44 parts. So for rocks with he found this relationship which as we'll
143:56 has them very satisfying aspects to But there's a complication, The solid
144:03 is squared. Uh But so the is one minus porosity square times the
144:10 of the matrix because ferocity titus the of the fluid. Um There are
144:18 of this equation that work better than widely equation and we'll come back to
144:24 later in the course. The other he did is for high porosity.
144:29 notice here we're above the critical Um He has something that's like the
144:36 equation except what is roe V. square, that's the plane wave module
144:43 . So here we have the plane module, lists for the composite.
144:48 we have the both module lists of fluid and here we have the same
144:53 modules and solid. So this is exactly a suspension, it's a reciprocal
144:59 weighted average like a suspension, but actually has some rigidity uh because where
145:08 is in the plane wave modules instead bulk modules. And so there you
145:13 , we're not a pure suspension, have some rigidity. I have to
145:19 this is an empirical, I wouldn't call it empirical, I would close
145:24 horrific equation. It's just a way adding some rigidity to the result with
145:32 knowledge that you could have high porosity that are not purely uh suspensions,
145:40 the grains are in some contact with other. So this allows you to
145:45 some rigidity. So it would be slightly higher velocity than uh than the
145:51 equation. And this is what we previously the would like equation. So
145:57 are two branches of the Ramayana Garden fit to the data he had and
146:03 in between and notice he's going higher critical ferocity here. Uh In between
146:10 just Interpol, it's between the So there are three parts, the
146:15 porosity part, the intermediate ferocity which is just an interpretation. And
146:21 the hype ferocity part And notices saying . Well, yeah, we're above
146:27 so called critical porosity of 40%. uh and we still have some
146:34 So this is a little bit at with the critical porosity model. He's
146:39 out to higher porosity is than 40% Sandstone and still having the rock with
146:49 . Okay, okay, so here have the different branches of the Raymond
146:56 equation and we're comparing it to the equation by the way. This is
147:01 the original paper of the Raymond Gardner . And so these guys were slumber
147:09 , they were well log analysts. instead of velocity, they're using sonic
147:13 time. Remember that's the slowness went velocity against ferocity. So they're going
147:19 to very high porosity is here. uh, at the types of ferocity
147:25 in the rocks we're dealing with He has the wildly equation uh,
147:32 uh, the properties of uh, about this 55 would be slower.
147:43 . So this one, this curve using the properties of courts, the
147:48 properties of courts and this other he's or widely curve, he's using
147:55 , is using a slower matrix ferocity course courts only 18,000 ft/s. And
148:02 found is often for clean sands, have to use a lower matrix velocity
148:08 though physically there's no way to explain . And the result of that,
148:15 that basically it's going through the Ramayana equation. All right, So the
148:24 Gardner equation will match the widely equation this matrix velocity here, with this
148:32 velocity there and then back to the matrix velocity there? The high the
148:40 porosity branches is uh, is this line? Over here, Out to
148:50 . And he also to complicate matters more, he gives another empirical relation
148:55 fit the same data. That's pretty to that 1st 1. I tend
149:00 to use this one because it requires information. You need to have the
149:05 density as well. Um and it's little bit more cumbersome. Um Then
149:15 got his high porosity branch. So is the would like equation here and
149:22 in between the two he inter plates between so kind of a very rapid
149:32 where he's losing uh integrity of the . The sample is losing its rock
149:43 characteristics and it's becoming more sediment like you see that's happening around 40%.
149:49 that is uh somewhat in agreement with critical ferocity model. Although you've got
149:56 uh rocks out here which are greater 40%, which have rigidity. And
150:04 this is the would like equation, the wood equations. So there's rigidity
150:08 as well. And just plotting some points and I'm sure these are cherry
150:18 data points. What is cherry Cherry picking means picking only the good
150:23 . So this is an example of data that falls on your trend.
150:29 believe me, there are a lot data points that don't fall on the
150:33 . So we could say these are types of rocks that obey the Raymond
150:38 equation. And these would be the liquefied rocks. And you can see
150:43 this compares to the widely equation. I use the properties of course in
150:49 , the wildly equation seems to be . But here you would need to
150:54 a different wildly equation because of the charity. Whereas the Raymond Gardner equation
151:02 to work. Yeah. And so his would like equation and you notice
151:14 are a lot of points that are than the would like equations. So
151:18 the voice bounds would be further uh the right here. Uh so uh
151:26 are seeing some development opportunity here. , mm now I could change the
151:38 velocity and I could consider different rock . So these are different Raymond and
151:44 curves for different, fully brine saturated . Or I could change the velocity
151:50 the fluid to do a crude fluid and look at the effect of hydrocarbons
151:59 the velocities. Now this is not correct, but it is more correct
152:07 doing the same thing in the widely . So this is one aspect of
152:12 Raymond Gardner equation that is better than wildly equation. You get reasonable answers
152:18 you use the correct fluid velocity. , so, uh it's time for
152:31 exercise. And what I'm going to you to do is compare uh the
152:38 uh mm velocity ferocity hair transforms for sandstone. And so I'm giving you
152:52 courts and the fluid velocities. You use any of these equations. And
153:02 from uh velocity, you could either with velocity and predict ferocity or start
153:09 for us and predict velocity for most these transforms. It's easier to start
153:14 ferocity, plug plug the numbers into equation and get the velocity for the
153:20 of sandstone equation. You'll have to the other way, You'll have to
153:24 with velocities, predict the density and predict the ferocity associated with it.
153:31 you have the gardener sandstone equation in notes. Um and uh then use
153:38 rest of these uh equations to predict velocity ferocity transform. So I'll stop
153:46 here and let me know when you're conference will now be recorded, start
153:53 and I'm going to escape. So can see the full slide and we
153:59 at this previously, this is the relationship. And again, the
154:05 the point to draw from here is course as a density increases, velocity
154:13 , so definitely a strong correlation The gardener relation that we all know
154:19 love and is very famous is a average for all rock types. If
154:25 compare the gardener stand, stone trend the other trends, velocity transforms.
154:33 find it's between the would like equation the roemer Hunt Gardner equation. So
154:40 not for fully liquefied rocks. Uh we could call these semi liquefied or
154:47 lit defied rocks. It's not for Raymond Gardner relation would be fully liquefied
154:55 , the wood like equation would be virtually unlit ified rocks and the gardener
155:01 would be in between. So this a plot from dr Hahn and again
155:13 showing the wide variation and um velocity transforms. So here is their lower
155:22 , they're using the true lower bounds is the voice equation, which is
155:27 same as the wood equation. The like equation would be slightly faster then
155:34 . So you can see when we're the voice bound, that's probably more
155:39 a more like a pure suspension when slightly above the Royce found the grains
155:46 be in contact. And so you've some rigidity there. You can see
155:51 critical ferocity model, which is this line which kind of acts like a
155:58 upper bound. Uh of course uh know, it does suggest that you
156:06 unlit defied at 40 ferocity. Uh fact maybe uh you're not entirely unlit
156:14 I'd write some of these plots have degree of rigidity. The roemer equation
156:21 similar, gives a similar result to critical ferocity model. So it is
156:27 acting as an empirical upper bound, the critical ferocity model would be a
156:33 upper bound, but they pretty much . Um You see the widely equation
156:39 slightly below that. So the widely not perfectly lit defied rocks, but
156:46 is good for typical reservoir type Uh the gardener relation would fly even
156:55 and so a lot of these you could see if I was fitting
156:58 of these points, that would give something like the gardener relationship there.
157:04 these straight lines represent I think uh the most part where you have many
157:11 forming along the line, that would an individual sample with an individual degree
157:17 consult of lift, ification. Uh the variation along the line would be
157:23 the dependence on pressure. So the dependence is moving you up and down
157:30 that. Okay, so we already about the critical porosity model, but
157:43 at these equations and the critical ferocity , they basically say that the the
157:50 frame properties are essentially linearly related. is uh with as ferocity changes,
157:59 have a reduction in the bulk module of the grain. And so essentially
158:07 have a linear relationship between ferocity and bulk modules of the dry rock,
158:14 a similar relationship for the share Uh What this suggests is that the
158:25 of the dry bulk module lists to dry sheer module list is exactly equal
158:31 the ratio of the mineral bulk modules the minerals share modules and that's not
158:37 far wrong for sand stones, but as a caution, it may be
158:44 wrong for other length. Allah jeez the critical property model is not developed
158:50 theory, it's just a heuristic model was developed based on experience.
158:59 um anyway, that's the answer to question that the frame modules ratio is
159:06 equal to the mineral modules ratio. are some more uh measurements from stanford
159:18 you may remember we looked at a like this previously where we looked at
159:23 was apparent volume, but we could expressed in terms of ferocity. We
159:27 at ferocity versus the mixture of coarse fine particles. So in this case
159:34 coarse particle is courts, the fine is clay and the same thing is
159:41 . There's some minimum value here uh is achieved. So if I have
159:48 framework of course particles, the clays filling the spaces between the courts particles
159:55 the velocity goes down. And similarly I put courts particles into play,
160:04 displacing porous materials so the ferocity goes . Um so there's some some mixture
160:12 the two which gives you minimum And oddly enough, as a,
160:20 happens as the ferocity decreases, the modules increases. So the compression module
160:29 , it's not exactly mimicking this, it's pretty close to be maximum at
160:36 point where this is pretty close to minimum. However, by the state
160:42 token, the sheer modules is relatively . So one question is, why
160:49 the sheer module is not changing too and in fact, this may be
160:56 result of the fact that the court packing has about the same share modules
161:05 um as the pure clay uh So these have similar sheer module is
161:14 varying the amounts doesn't seem to change too much. Now, here's an
161:29 way to prove the effect of micro . Remember we said that if I
161:37 a rapid increase of velocity with I'll be closing micro crafts. Um
161:44 here we have a gap bro, and the measurements of velocity are made
161:50 the dobro and you see a relatively change that's about maybe a 5% change
161:57 velocity. And then this this sample cracked and the way it's cracked,
162:04 heat treated, it's brought to a high temperature and then it's cooled very
162:10 and that induces micro cracks. And see the effect here that it's almost
162:18 change in velocity with pressure. So that is two cracks closing. Now
162:25 never are able to close all the , so we never come back to
162:30 original velocity before micro factory that starts level off. And part of the
162:37 , part of the reason might be this is very specifically an axial pressure
162:43 opposed to a confining pressure. In case, I'm sorry, you can't
162:47 that in this case, it's an pressure. And so what that means
162:54 it's the sample is putting a piston it's compressed from from one end.
163:01 much like our young module. This of experiment, right? It's a
163:06 axial depression. And so if the which were induced by heat seat
163:13 heat treating um are closed. The they're randomly oriented, the horizontal fractures
163:22 close. But the vertical fractures may open up so it may never achieve
163:30 the under the velocity of the Ian rock. Now here we have the
163:39 of a grab it. It's a low velocity, excuse me, Very
163:47 ferocity. And it's left on a and the velocities are measured as a
163:55 of time. So this sample has been jacketed is measured as a function
164:01 time and you see that the velocity down and it's initially a big decrease
164:08 velocity and then it's a slower decrease velocity that continues on as the sample
164:16 left there. And what's happening is are draining out of the sample and
164:22 sample is drying. So even though a low porosity granite, apparently it
164:29 microfractures with fluids in them. As fluids leave those micro fractures, the
164:36 , the rock becomes more compressible and the velocity comes down. So actually
164:43 flu is in the fractures are helping the compression. Uh This is a
164:55 which is uh somewhat complicated. Um there are measurements of VP VS.
165:04 again in a granite and here for line and this line the poor pressure
165:14 equal to the external pressure and again he go so that you can see
165:21 is pressure applied with the poor So this is the external pressure.
165:29 pressure is another word for confining So I'm increasing the combining pressure,
165:35 I'm keeping the poor pressure equal to combining pressure. Uh And you see
165:41 velocity changes very little now if So the differential stresses zero in all
165:48 these cases, if differential stress were to effective stress, then the velocity
165:55 be constant because if I'm holding the stress constant uh and the effective stress
166:02 equal to the differential stress, then effective stresses constant. So what this
166:07 telling me is that the effective stress not equal to the difference of uh
166:13 to the differential pressure. And we find this to be the case in
166:21 impermeable rocks. Now, another aspect these measurements is that If I look
166:31 the measurements at poor pressure equal to the shear wave velocity um for the
166:40 rock is faster than the shear wave . For the saturated rock. Unless
166:46 get to very low, very low . And there may be actually these
166:54 pressures, There may be some chemical happening like repulsion between grains or or
167:02 surface tension effects. Holding things for example. So things might be
167:09 little complicated here, don't worry about . But here we have the saturated
167:15 shear wave velocity being slower than the shear wave velocity that potentially could be
167:23 pressure effect. And you notice that difference gets less and less as we
167:29 that the confining pressure. So presumably got a significant force base enough to
167:39 that velocity difference which has all So maybe this is a highly fractured
167:45 , You have a big increase in with pressure early on. So lots
167:51 these fractures are closing and so the space closes completely by the time you
167:57 to high pressure and there's no difference the dry and saturated rock and you're
168:04 a similar effect here. Uh The rock is actually faster than the dry
168:11 because fluid within the small micro cracks resist the compression as you're squeezing those
168:20 crafts. But uh you uh with pressure, you close them to a
168:26 extent and the difference becomes negligible. , here's an example of the effective
168:39 law working, you know, here have or what is called the external
168:49 . I'm sorry. This is the pressure here and f is the Uh
168:58 . The difference between the external pressure the four pressure. So that's the
169:03 pressure and they're referring to that as skeleton pressure here is equal to the
169:09 , confining pressure minus the poor Or you could say external pressure minus
169:17 fluid pressure. Um So, when is constant, when the differential pressure
169:24 constant, the velocities are constant. the differential pressure is equal to the
169:36 pressure as you increase the external then you increase the differential pressure and
169:42 velocities go up. So in this you can see that what controls the
169:49 is the differential pressure. So that this case is the effective pressure.
169:57 with me on that. Yes Okay. Now these are other sand
170:08 and what you can see is that some cases uh the external, you
170:16 the effective pressure seems to be working the differential pressure being equal to the
170:21 pressure seems to work over a given or given range of pressures. But
170:29 see here the velocity buried. Uh velocity is not constant at a constant
170:37 pressure here. They seem to be of linearly related. So the differential
170:42 to first order you could say is the effective pressure, but not
170:57 Okay, so here we have a versus transit time thought. Um And
171:06 we see are velocities measured over a range. So the velocities here are
171:13 from 1000 ft depth To 13,000 ft death. And here is pure
171:23 And for the deeply buried rocks they to be overbearing, obeying the time
171:28 equation. Of course you need a uh grain velocity there, but more
171:35 less obeyed the time average equation a relationship. But then you go below
171:42 certain death. And you now deviate the time average relations. And the
171:48 you get the further away you deviate the time average equation. So
172:00 why, why is this happening? , these rocks are liquefied to some
172:07 . Can uh beneath, below a debt or above a certain death than
172:12 rocks become more and more poorly lit . So we're seeing a deviation from
172:18 widely equation. Uh here we see same thing. Um so this is
172:28 trend curve fit fit to velocities versus . In fact, these are average
172:36 and uh average velocities for 1000 well or so. And this is in
172:43 gulf coast. And uh so you at a low velocity, uh you
172:51 velocity rapidly with increasing depth and then kind of roll over by the way
172:58 are sandstone. So you reach a where you're fully compacted and you're pretty
173:05 lit defied. And at that point follow more or less the time average
173:12 . Um so uh pretty close to when you're liquefied with ferocity being the
173:20 factor by the way, if you a sand pack and measure its velocity
173:26 pressure and then convert the pressure to equivalent debt death, you get a
173:31 more well behaved Trent right here, uh we're not uh rearranging brains,
173:38 not getting the rapid ferocity reduction that saw from compaction here, the grains
173:44 just getting pushed against each other more more so this is the pressure
173:50 And you can see that at the average equation is the dash line.
173:55 actual data has a slightly different slope the time average equation and you could
174:02 some of that to the changing So there is some pressure dependence
174:08 which the time average equation is ignorant . Remember just pushing the grains against
174:17 other more tightly doesn't change the ferocity much. You're primarily changing the contact
174:25 of the brain context. So that's a big ferocity reduction associated with
174:36 So here's a pretty difficult fly to . Uh, but it's a,
174:42 a really great one. So I'm to take the time to explain
174:47 This was a, an empirical study in europe, uh in the vicinity
174:56 the helps. And what happens here if you're in the basis. So
175:04 you're away from the mountains, you to have a fairly linear variation of
175:11 versus death in shells. You also a fairly linear relationship in limestone
175:19 Of course the lime stones are much than the shells. If you then
175:25 lime stones and shells together, maybe inter bedded. Uh, you can
175:32 interpolate in between. So, if were mostly limestone, uh, maybe
175:39 limestone and 40% shell I would You know, 40% from limestone,
175:46 from shale. So I would have line more or less like that.
175:53 you go to the mountains for rocks this ratio of shell to line to
176:01 . And instead of um instead of along this line, they're much faster
176:13 they should be. So what is on here at a given depth in
176:25 mountains? The velocity for the same logic mix is much best is much
176:34 . And the argument that was made this is a velocity versus death
176:43 Suppose the rocks and the mountains were once buried to this death and then
176:51 uplifted to this location and suppose they the velocity, the faster velocity.
176:59 I suppose we have history says buried this step over geologic time, they've
177:05 liquefied to this extent. Then you them to the surface and your velocities
177:12 haven't changed too much. So it tells you how much uplift or it
177:19 you a ballpark figure for how much may have occurred as a result.
177:26 this is an important aspect of understanding . If you're in areas that have
177:35 uh, you know, mountain uh if you had wraps of rapid
177:41 followed by some degree of uplift, the velocities may do things that you're
177:48 expecting them to do because of the now in uh poorly lit defied
178:00 Uh Typically there is a log arrhythmic between shell transit time and death.
178:09 , we're normally pressured. And so an example of a very well defined
178:14 . These points come from, logs. So you go to the
178:18 log you find the pure shells, plot their velocities and they follow depth
178:25 very well. So this is called you fit a curve to that,
178:31 called the normal compaction curve. Uh so here's an example in one well
178:42 we have our normal compaction curve and see the shallow rocks, the points
178:48 following that normal compaction curb quite And then at some point the velocities
178:55 much slower. The explanation is excess pressure. If you have abnormally high
179:03 pressure, you reduce the effect of . And so the velocities are slower
179:09 in fact here were below 10,000 ft the velocities are acting as if you
179:15 at 4000 people. So significantly slower . And from how much slower the
179:23 are, you can estimate the amount geo pressure that has occurred. So
179:32 few more examples of this, here's normal compaction trend and uh you go
179:39 below a certain death and the velocities to be much lower. So uh
179:46 some kind of permeability barrier here preventing escape of fluids and you go into
179:53 high four pressures. Another example we go on and on with these,
180:00 the reason I'm showing this is because reason festivity log is doing the same
180:06 , this is shale conductivity and you it's doing the same thing. So
180:12 in shells the abnormally and by the , density logs, you'll see it
180:17 too. So the abnormally high pore is literally increasing the ferocity of Iraq
180:25 these cases. So you're seeing it velocity, you're seeing it in connectivity
180:31 you're seeing it in density. One more examples that we're on a
180:39 trend. The velocities are slower in this case, the sands happened
180:45 have very similar velocities to the We often assume that the sands are
180:53 affected by pressure than the shells by pressure than the shells. Uh So
181:00 a result, a sand shell combination have a different impedance contrast above pressure
181:08 in in pressure. Uh huh If stands tend to tend to be low
181:15 normally, if we go into pressure , we might find we have very
181:21 impedance. All right. So if sands were lower velocity above pressure,
181:27 shells get slower and then the sands similar velocity below pressure. That will
181:35 the kind of, the magnitude of spots are amplitude anomalies and the kind
181:42 amplitude anomaly that we'll see. these are some actual cases of predicting
181:51 pressure for drilling purposes. Uh These seismic velocities. So they tend to
181:57 inaccurate. Of course you want to able to do this before you've drilled
182:03 well, but here we have seismic velocities and you're following not too bad
182:10 the normal compaction trend and then you to deviate from the normal compassion
182:16 So, from the magnitude of the you predict the poor pressure. Now
182:24 drillers drilling the well need to keep their mud weights which are the thin
182:32 here, need to keep their mud above the poor pressure of the,
182:41 the formation for fluids. And so is an actual Julian result. What
182:46 see here is that the drilling engineers being cautious, they didn't quite
182:52 you know, the the prediction. so they stay up the mud rate
182:56 little bit early And they tried to two or £3 per gallon above the
183:04 predicted ferocity. But you see in couple of cases they've skipped the seismic
183:11 of pore pressure is higher than the mud mud way. Now, if
183:16 happen to have, if that prediction correct and you happen to have a
183:20 zone permissible zone right there, you the potential for a blowout as the
183:28 pressure exceeds the mud way. But fact these are interval velocities. So
183:35 one, they applied to the middle the interval, it doesn't mean that
183:41 uh that velocity is constant throughout that . Right? So, um,
183:47 that may be an artifact of the analysis. So maybe it wasn't a
183:52 for that reason. And then there's the fact that maybe there wasn't permeability
183:58 that precisely at those points where these cross. And we see the same
184:07 again here, the predicted poor pressure in black and the actual mud
184:14 it looks like we're under balanced but in fact, we're in the
184:21 and the seismic velocity analysis is not precise. So, uh maybe the
184:30 were okay, it looks like the were kind of ignoring what the seismic
184:36 was, right. They up the way early on being safe and then
184:41 ignored the seismic prediction here. Uh and or here. So in
184:49 you can see why sometimes the geophysicists are not taken seriously by the drilling
184:57 because this well, had no problems all. Another effective pressure is to
185:10 the anti Satrapi. So, uh this case we're looking at compression.
185:16 waves measured parallel to betting or perpendicular Betty as the pressure is increased.
185:26 for parallel to betting we're going to faster if we're perpendicular to betting,
185:32 the betting this is a shell. betting has facility. There's partying along
185:39 bedding planes. So the bedding planes like fractures. Uh And so if
185:45 perpendicular to betting, we're going to a slow velocity, if we're parallel
185:50 betting we're going to have a high , but apparently that facility is being
185:56 as you're increasing the pressure. So anti Satrapi is decreased similarly with the
186:03 wave uh If we're going across Um So the wave is propagating
186:11 It's abetting um it could be polarized two different directions. It could be
186:19 parallel to betting, or it could polarized perpendicular today? I'm sorry,
186:25 is propagation parallel to Betty, In case if we're moving parallel to
186:31 I could have vertical polarization which would perpendicular or horizontal polarization, which would
186:38 parallel to that. And so the wave is selling that showing the same
186:43 the anti Satrapi is decreasing as you the pressure, suggesting that these bedding
186:50 are being healed. They're being forced . Now, here we have an
187:00 case of fluid effects. I have wave velocity here, P wave
187:09 Here I have a dry rock velocity with pressure, then add kerosene and
187:20 velocity goes down. I add brian the velocity goes down more um likely
187:29 be a density effect. Right? density is uh is uh increasing when
187:38 change the the poor fluid from a to a heavy brine. So the
187:45 wave velocity goes down while the p velocity goes up. But the density
187:53 kerosene is much closer to the density brian. And yet the effect of
188:03 is to slow the velocity disproportionately to density difference. Is everybody with me
188:11 that kerosene being a liquid dry being air kerosene is closer to water in
188:22 than it is to air. um how could you explain what's happening
188:32 ? Give me a hypothesis to explain fact that water reduces gives you an
188:41 reduction in velocity as compared to Do you understand the question? Um
188:56 think he has something to do with bulk models. The compressibility.
189:05 it does, but why? now it's also the rigidity. It
189:14 the both modules. But this is velocity, remember? So we're seeing
189:20 rigidity being decreased because it's bigger than the density effect. So why would
189:30 in water decrease the rigidity of You think it might make the grains
189:44 slippery? I like to call it banana peel effect. Right? And
189:50 , walking around in Houston, you , wet mud is a lot more
189:54 than dry mud. Okay, especially when you have clays in
190:01 the effect of water can chemically interact the place and reduce the rigidity as
190:08 result. And uh as an we could have made these calculations.
190:20 but I'm gonna skip that exercise right . Yeah, you do see kerosene
190:29 a density that's Much closer to water it is to zero. So,
190:36 , this effect is the on the wave velocity is too big to because
190:42 by the density. Yeah, so, I've got different rocks with
190:55 kinds of behavior and again, the pressure here is just the differential
191:01 So, again, mislabeled. Um , I have a VP and uh
191:11 s being measured and I'm comparing brine to dry. So, you can
191:20 the effect on this limestone, The of dryer wet is much bigger on
191:26 p wave velocity than on the shear velocity. Mhm. That could be
191:31 from both modules. Right. Um effect on this granite is much smaller
191:40 that could be explained by the fact the granite may be much lower ferocity
191:44 the limestone. Here, we have dull might also probably low porosity,
191:51 it has this very rapid increase. is likely suggesting microfractures um And
192:02 microfractures or into crystalline force. Remember we re crystallize calcite, we could
192:10 could create low aspect ratio for us the dolomite crystals, oddly enough in
192:18 dole might the saturated rock has the shear wave velocity than the dry
192:25 Uh one way to explain that is that the fracture for the low expectations
192:35 us are such low permeability that the can't get out of the forest.
192:41 actually, as I share the I'm actually trying to compress something for
192:46 . And if the fluids can't get , they will resist the poorest will
192:52 compression caused by the sharing of Um That that means that Uh
193:02 The permeability is low. If the was zero. If they were perfectly
193:08 , you should see you should not that increase as you couldn't get fluids
193:14 those micro fractures at all. Uh yeah, but the permeability is such
193:24 over time, if you give it time to be fully saturated. You
193:28 get fluids into those very flat ports then you have this sullen often limestone
193:36 , which the velocities are essentially in stolen. Often limestone is almost pure
193:44 , right? It's Essentially a ferocity zero and very little change with pressure
193:51 no change with saturation. Right. , we're looking at the effects of
194:10 fluid on p wave velocities. And here we're looking separately at the both
194:16 lists the velocity and the impedance. here you have dry oil, saturated
194:27 saturation and that's pretty typical case the , the oil is usually somewhere in
194:39 the water saturating the result and the results. So, we have some
194:45 variations here beaver sandstone velocity increases with in the sponsor blow sandstone, we
194:54 this rapid increase at very low So maybe that sandstorms fractured. Um
195:03 you look at velocity, it's uh is confounding things a little bit because
195:10 the sheer modules is included in Um And if you look at velocity
195:19 in this sponsored blow sand sound, enough, the dry rock is faster
195:29 the water and oil saturated rock that to be entirely due to density.
195:34 the only way to make the dry faster unless the water is somehow softening
195:39 rock frame, there's a density effect on and it's odd that the density
195:46 becomes larger with increasing pressure. uh, maybe that's not the entire
195:54 , but interestingly, if you look impedance, then uh some of these
196:01 are being complicated in velocity. Uh at in peter's it acts a lot
196:07 like the both modules. So multiplying velocity times density sorts things out of
196:21 . Now, I mentioned that temperature the fluid philosophy. Uh These are
196:30 measurements of uh heavy oil velocities uh a function of temperature and as temperature
196:40 , the velocity goes down, that the bulk module list is going down
196:44 than the density is going down now , the higher pressures, the higher
196:56 is here, sorry, have lower . And so that's a bit
197:04 Uh So what kind of pressure is ? I'm guessing that these are four
197:14 . So I'm sorry, this is rock saturated with oil. So this
197:18 the rock blossom and so the higher pressures you get lower velocity, but
197:24 rock velocity is changing the temperature because both modules of the oil is changing
197:30 temperature. Um Okay, so we temperature here on the horizontal axis,
197:41 have velocity on the vertical axis. is in Maria sandstone, which is
197:46 clean liquefied, since uh people like make laboratory measurements on and you can
197:53 as the temperature increases the p wave decreases and these are different differential
198:01 Um So that makes sense. The expands now for shear wave velocity and
198:10 water becomes more compressible for sure. velocity and high pressure. We're seeing
198:17 change at lower pressure. They're suggesting change. I'm not sure. I
198:24 that if we invoke experimental error, that point out, this is pretty
198:32 . And here we don't have measurements here. Uh So that's pretty
198:37 So maybe that line is overemphasizing the of that gun shear wave velocity.
198:43 keep in mind the temperature effect could itself in the fluid expanding and forcing
198:51 grains apart somewhat. Um That would suggest a different pore pressure, but
198:59 supposedly controlling the poor pressure. But can't be sure how well that poor
199:04 is being controlled. So, you , theoretically we should uh we should
199:13 as the share weather as the as go to higher temperature and the density
199:21 the fluid becomes less. We would stuck uh the shear wave velocity to
199:29 , not to decrease. So this kind of going in the wrong
199:34 So, I'm suggesting some kind of error at work. Of course,
199:43 extreme case of temperature has to do you freeze the rock. And this
199:49 a real problem in arctic areas where are times of the year where the
199:54 surface rocks have partially melted and you low velocities. And then in the
200:01 it's all frozen and you have high . In fact, the seismic acquisition
200:07 is in the winter. You don't to be in this range where you've
200:11 pockets of low pockets of five. creates all kinds of near surface
200:17 So, you want to be either or here, but certainly you don't
200:21 your vibrator trucks trying to travel around . All right. So you want
200:28 to be on a hard surface. , this is the acquisition season.
200:41 now in the next section, we're to talk about the factors that control
200:48 bulk modules of the rock and in , but this is correct. Have
200:54 this a few times and sometimes it coming back. So I'm going to
201:00 it now, insert ferocity. Oh, no. Okay, I'm
201:08 gonna do it now. I'm going do it another time. That's not
201:10 question mark. That should be That should save ferocity. So,
201:15 are pore spaces, right? and I think it's funny the way
201:20 attention to Iraq's is cheese. But think of these as fungi
201:25 Okay, so there are areas, module and we need to think about
201:30 order to describe the effects of one is the fluid module list
201:37 Another is the modules of the solid , which is right here. There's
201:43 modules of the rock without the That is called the dry frame module
201:48 . I don't like that term, I'll complain about it repeatedly for the
201:53 of the course. We don't want actual dry frame. We want the
201:59 of the frame independent of the mechanical from the fluids, but in chemical
202:07 with the fluids as the fluids in with the grain, uh with the
202:13 material can actually change the skeleton So we could call what people call
202:19 dry frame modules. I prefer to the frame modules or the skeleton
202:26 Keeping in mind that that's in the of the fluids and that's different from
202:32 saturated modules. That's the modules of rock. If you can find the
202:38 , keep the fluid in the poorest or don't let them escape from the
202:44 and then squeeze the rock and measure modules. That's the saturated both
202:56 And um if you plot uh as result of the bulk module has changed
203:04 fluid properties. If you for a water mixture, if you plot velocity
203:11 water saturation, you typically get a like this for p wave velocity,
203:17 water saturation of 100%. So there's gas in the rock here, you
203:22 a high velocity, I add gasp air and just a little bit of
203:27 . The first five or 10 drops velocity most of the way. And
203:35 as I continue adding gas, the rebounds and comes back up. So
203:42 drop in velocity is caused by the in bulk module lists this increase in
203:50 is caused by the reduction in density the both modular stays relatively constant.
203:57 two effects. Both module is the and the density effect. And you
204:02 there is no bulk module, its on the shear wave velocity. So
204:06 have only the density effect. Uh , uh we're in the next
204:16 we're going to go through some uh complicated equations to be able to predict
204:22 curve like this. So you're going be able to do that on the
204:27 , but there's a simpler way to a rough idea remember? These are
204:32 equations. So they're not going to exactly right. Anyway, so there
204:38 a couple of different empirical relations one use and I'll give this the equation
204:44 this line later. This is if cross bought gas stand velocity versus brian
204:51 velocity. If they were equal, would be on this red line
204:57 So, empirically we find uh that gas and velocity in practice tends to
205:05 slightly low compared to the brian sand . Now, the effect of gas
205:15 a percent change gets bigger and bigger we go to lower velocities.
205:21 when I have a low velocity, means I have a compressible rock frame
205:27 will mean the hydrocarbon effect is big , you know, low velocities near
205:34 water bottom, that effect could you know, pretty enormous,
205:40 100% change here in velocity. Um , um That effect gets as a
205:52 gets smaller and smaller appear at high that might be on the order of
205:57 or two change. So you can that uh gas being in the rock
206:03 cause huge amplitude anomalies deep but can almost negligible or not. Observable in
206:11 little logic variation. I'm sorry, be huge, shallow but negligible
206:22 By the way you can think of stand velocity as a proxy for death
206:27 remember we said velocities increase with So uh you know given a philosophy
206:34 dense depth relationship, you could have depth scale here and so shallow a
206:42 change. Deep small change. And this produces amplitude anomalous. So
206:55 to say an overlying shell which would the square here. So I have
206:59 impedance of the shell density times If my brian stand is low
207:07 what happens when I add gas? lower density. I lower a
207:11 So I make the impedance even So in this case we go from
207:16 negative reflection to a bigger negative And this is what we call a
207:22 spot. Yeah because the polarity issues so forth, europe versus the
207:28 S. What's negative and what's positive people use it in different ways.
207:35 So this I would call a soft and that covers all the possibilities.
207:41 I have a soft reflection due to due to brian, I have a
207:47 , softer reflection due to gas. that's if your sand uh No no
207:54 ready for is low impedance relative to shell. What if your brine,
207:58 rock is high impedance relative to the ? Well, in that case,
208:03 gas reduces the impedance. So it reduce the magnitude of the reflection.
208:11 you go from a very hard reflection a lead guard reflection. That's called
208:16 dirty spot. In fact, sometimes amplitude could go all the way to
208:22 and you don't see a reflection from gas sand at all. So you
208:27 have an invisible reservoir in that Now, when the hydrocarbon effect is
208:33 big or when the brian stand is slightly hard, adding gas can flip
208:41 polarity so you can go from a reflection to a soft reflection. Remember
208:48 is at the top of the at the base of the sand.
208:51 it's the same shell underneath, you see an equal and opposite effect.
208:59 , sometimes it's desirable to have an of what kind of reflection to
209:07 what kind of amplitude should I expect I have hydrocarbons. And one way
209:12 can do that if you have well in the basin is, you can
209:16 depth trend curves. So you could density versus death in jail density versus
209:23 in brian sand and then you could fluid substitution as well due next week
209:29 predict what the gas and uh density be? Well, we could use
209:34 mass balance equation for this. We do that with transit times also and
209:39 would require Gassman situations that we're going talk about next week. And then
209:44 can see what your average reflection coefficients going to be for a gas
209:48 And brian sand. And you can in all of these cases the average
209:53 coefficient is negative. You know, sand is uh or as soft,
210:01 ? The sand is low density, brian stand is low density, low
210:05 . So it's always going to give a lower impedance than the shell.
210:10 always going to give you a negative reflection. Um but you had
210:17 you're even lower impudence in both lower , uh slower transit time. So
210:24 impedance and that gives you a stronger reflection. So these would all be
210:30 bright spots. Now it's not always like that. Sometimes you get variable
210:40 is a function of death. So a shell compaction trend. This is
210:45 versus death. This is a brian compaction trend. So you see the
210:53 Sanders starting out at the very near , higher velocity surface higher velocity than
210:58 shell, but then it becomes lower than Michelle. And then back to
211:04 velocity than the ship. You can substitute oil using gas mints equations.
211:10 that gives you for a he lied , but not a very volatile light
211:18 . Its properties are pretty similar to and down deep. The results is
211:23 indistinguishable from crime or gas, which to be much slower than brian.
211:30 then by comparing if you do the thing for density and construct impedance versus
211:37 trends, you could predict at a death what kind of amplitude anomaly you're
211:43 to have now. Unfortunately these are values. So I'm talking about an
211:55 . Again, actually, these particular came from about 1000 well logs and
212:02 a trend line was spit to average versus death, but there are always
212:10 . And so in fact, at any particular death, you should
212:18 at the history graham for brian for shells for gas stands. And
212:23 you'll find is even though the average may be different, there is potentially
212:30 lot of overlap between this. and so that means, uh,
212:36 know, even though a gas ban this case on the average is din
212:43 I'm sorry, lower impedance than the . Um, you might find particular
212:50 vans, which might be say low which can in fact be higher impedance
212:57 a particular brian stands for example. if you convert that to reflection
213:06 what you'll find, is there a where you can have and you're likely
213:13 could have either gas and their mind . Uh, there's usually some probability
213:20 a reflection coefficient of a certain magnitude be uh, either or okay,
213:27 I have a very large negative reflection this particular case, you have almost
213:34 probability of being brian, but there some finite probability most likely to be
213:41 . But here with still a negative coefficient, It's about a 5050 probability
213:48 been a brian or guests. Uh Hiltermann is has done one better than
214:05 . He looked at thousands of well and he could construct instagram's versus
214:11 Uh and this happens to be for velocity versus death and shale velocity versus
214:19 . And oddly enough, the DP get the more variation you seem to
214:24 . So the average fit that these here represent the average, you can
214:31 the average fit seems to be mon increasing, but you're not always dealing
214:37 the average rock. Right. So , coming back to the average properties
214:50 see this discover the um we could at the average reflection coefficient versus death
214:57 we could look at that for wet and we could look at that this
215:02 gas stamps and there's a crossover point of these curves where the reflection goes
215:10 soft to hard or an american polarities to positive. So here my brian
215:18 are negative. My gas stands are negative. We have bright spots.
215:25 here uh the brian stands are the gas ends are negative, so
215:32 have polarity reversals and here you're brian are positive, your gas fans are
215:39 positive. So you have them Now, if you go into
215:48 you're making your shell impedance is If you go into gear pressure and
215:55 will tend to make the crossover move . So if I were in JIA
216:01 rocks, I uh would would change depth at which you switch from bright
216:11 clarity reversal to dim spot. Now get the idea that all rocks,
216:16 is just a tendency. Don't get idea that all rocks will do this
216:22 any depth. You could have any kind of response. Remember this is
216:28 area dependent curve that's come up that come up with. So uh what
216:39 Idol did is he plotted the crossover at which he moved from uh hard
216:47 soft. He plotted the death at that crossover point occurs and he plotted
216:56 depth for rocks of different ages. . And what he finds is that
217:06 the uh the younger you are, shallower that death occurs. And so
217:14 older rocks that I'm sorry, the that death occurs, I'm sorry,
217:19 is increasing upwards. So older the crossover for shallower younger rocks,
217:28 crossover occurs deeper and that's all I for this section and we've only got
217:40 uh 20 more minutes and I don't to start a new section. I
217:45 to give you time to absorb So actually we moved a little faster
217:49 normal,
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