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GEOL7324 Rock Physics - Day1 08-27-2021-Part1
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00:00 this conference will now be recorded and is the course outline uh This is
00:12 how far we get in this So I'll actually have a pretty long
00:20 trying to set the framework for what doing. Give you some motivation why
00:26 doing it and so forth. Uh the first topic is going to be
00:31 geological, uh looking at poor space , the pore spaces, major factor
00:40 geophysical properties. And so we want talk about the aspects of the poor
00:47 which are important. Uh Then the year physical property of interest is the
00:54 . Of course, that's basic for work, but it's also a necessary
01:02 for seismic work. Uh It's part the seismic impedance and its factors into
01:08 seismic velocities. So we'll talk about , density is nice because the physics
01:16 perfectly in the case of density. We can use theoretical equations with great
01:26 . Next, we're going to talk basic rock mechanics in particular, elastic
01:33 , elastic ma july. Um These necessary building blocks for seismic data,
01:39 there are also an important aspect for work, all kinds of engineering applications
01:46 soil engineering. Civil engineering. Near structures. Um uh Oil and gas
01:55 , drilling wells, depleting reservoirs, mechanics is important in all these different
02:03 cases. Uh So uh I think a pretty important topic. Next,
02:10 fundamental environmental property that controls the geophysical , our pressures and stresses of various
02:21 . And so we'll talk about the kinds of pressure that are relevant for
02:26 now in this course, we're going focus on seismic velocities. Um There
02:33 many other geophysical properties that are of . Electrical properties, magnetic properties,
02:42 properties. Um We're not going to time to cover all of that.
02:48 this is a professional program primarily for applications. And so we're going to
02:55 on seismic velocities. And you could with are the factors controlling compression away
03:05 , But then in Module seven will on to share share wave velocity.
03:10 we have module eight, which is units eight and 9 in your
03:15 And this was talking about fluid properties fluid substitution. And these are very
03:21 for seismic hydrocarbon detection in particular, also reservoir characterization. Now, if
03:30 have time, we'll move on to advanced topics. Generally we don't make
03:35 this far. Uh, Mulele nine composite media. So, when you
03:43 a mineral, logically complex rock, do you deal with that mathematically?
03:49 then module 10 talks about attenuation and . Dispersion being the frequency dependence of
03:58 , which is a necessary consequence if have attenuation. So, um I
04:06 we'll get that far some recommended Uh This is the course will get
04:16 finished awfully fast. It'll be hard you get ahold of these books.
04:20 if you can get them out of library or there's a digital copy floating
04:25 or something online. They're useful Um The one book I will,
04:33 will mention in particular if you're going go on and do advanced work in
04:39 physics, which probably if you're in professional program, you're not looking to
04:44 a career as a research rock But if for some reason you were
04:50 Mapco book is is a great Well, okay, now for applied
05:01 , there are some important papers. mean in rock visits the research literature
05:06 thousands and thousands of pages. Much that work is from a theoretical point
05:13 view and it starts involving very complex . Most of that is of interest
05:23 a scientific standpoint, but not that for applied work. But there are
05:29 papers out there which definitely have an perspective. And so I want you
05:37 read those papers. Gardner, Gardner Gregory talks about the rock physics,
05:44 role of rock physics in strata graphic using seismic data. Gregory's has a
05:53 paper in rock physics. Uh it's it's got some of the same material
05:58 Gardener Gardener and Gregory but it goes that. And it's really a great
06:07 paper, it really sets sets the for everything I'll be talking about.
06:13 course, one of the best papers all time is my own rock physics
06:18 , which was the rock physics. basis for in amplitude versus offset analysis
06:27 my spg book on the same I'm only being facetious by saying it's
06:33 a great paper, but pretty much show you where I'm coming from.
06:40 it's worthwhile for you to read that . A lot of the material in
06:47 paper is covered in this course. there is a very good practical tutorial
06:53 gas men's equations. This is the we do glue it substitution, we
06:58 the fluids in Iraq and we calculate the size properties change as we change
07:03 fluids. So that's the paper by smith in geophysics. I believe these
07:10 all there on blackboard. If they're let me know, I have a
07:15 other miscellaneous papers there on blackboard, should read them all. Um so
07:22 are kind of in order that you want to read these. I would
07:28 early on maybe after tomorrow. Uh start with Gardner, Gardner and Gregory
07:36 then move on to the others. , one of the other things I
07:41 to do in this class is emphasised scientific method and being that this is
07:47 professional class. I think it's it's important, you know, I've worked
07:53 the industry a long time and I technical work being done that doesn't follow
08:01 method and mistakes being made as a . Um and I think one of
08:08 key building blocks of the scientific method the idea of a hypothesis. So
08:16 the geosciences, what do we mean a hypothesis, by the way,
08:22 means something slightly different, but what we mean in the geosciences. So
08:28 like you guys to see if you define this term for me my process
08:37 the foundation falls on the chorus or it is all donations. Perfect.
08:47 , that that is the way we hypothesis. An explanation for an
08:54 If you do reading about hypotheses, you'll get different ideas. In fact
09:01 statistics, it means something very Well not very different, something very
09:08 but not necessarily an explanation for But in the in the readings one
09:16 hypotheses they get used in different For example, uh an example of
09:21 hypothesis that was presented by Watson and was that the DNA molecule is a
09:30 from my point of view. As , that's not a hypothesis because it's
09:38 an explanation for anything. It's a of, I think that's the way
09:42 are. Uh and then you we're going to go about ground and
09:47 it. But one could restate that and see and say the x ray
09:53 pattern of a DNA molecule is what is because the DNA molecule is a
10:02 helix. So that would explain the which is the x ray diffraction
10:11 So, you know, one can stay things as in that way and
10:19 find it clarifies the thinking if you so a hypothesis in the geosciences is
10:28 explanation for observations? Actually? I it should be that definition for all
10:33 sciences. Um Now how do we the hypothesis. First of all,
10:40 hypothesis must be testable to be a . It doesn't mean you can achieve
10:45 test. It means you can conceptualize test. And hypothesis testing involves the
10:53 to disprove the hypothesis. It turns that you can never theoretically prove a
11:04 . You can disprove a hypothesis. that's much stronger. You can't prove
11:10 . But you could uh collect data support the hypothesis but that's not as
11:18 a statement as disproving the hypothesis. an analogy is um I'm going to
11:26 my seismic data that I my three . Seismic data set, I'm going
11:30 explain it with an earth model which my interpretation. Now you show me
11:38 interpretation. I can improve your interpretation right. Even if I drill a
11:46 . Well 1st of all the well are never exactly what is predicted anyway
11:52 ? But even once you've drilled well haven't proven the hypothesis. Um Even
11:58 you are relatively close to being correct it could be correct for the wrong
12:03 different reasons right. It could just to you might just happen to be
12:10 depth for example on your prediction. you have compensating errors right? But
12:16 before I drill the well when you your interpretation and I'm listening to your
12:23 . Maybe as an exploration manager I prove it's right but what I can
12:30 is I could prove it's wrong. could find an internal inconsistency in your
12:36 and show your interpretation is inconsistent with . So I can prove it wrong
12:42 I can't prove it right. If failed to prove it wrong and there's
12:46 to support the hypothesis, it will my confidence in it. Uh So
12:53 can never be proved. It can shown to be consistent with the data
12:58 is called confirming the hypothesis or it fail to be falsified, which is
13:05 if you can if you can achieve . Okay, so here's one of
13:10 definitions I got off the internet. hypothesis is a proposed explanation for an
13:18 phenomenon. So if I say I that the wall is green. Is
13:26 a hypothesis? Hold on a Everything's good. I can't project and
13:35 conference at the same time. Full that's not working. So I got
13:40 . So Amir is home. So a video conferencing with him and with
13:47 uh and we could see the screens on the scene. I'll get that
13:54 for you tomorrow. It's good. getting that sorted out. Do you
14:01 to every class of the solution? actually official day that she said I
14:06 enough but like you always have I we're good now we're fine. Okay
14:24 um for it to be a scientific . We have it has to be
14:31 um whether or not you can achieve test right. But you can at
14:38 imagine a test that could be One thing. What's the difference between
14:47 hypothesis and a theory. In there is no precise distinction between the
14:56 . A theory tends to be a and more n compensate encompassing explanation from
15:04 many observations, many different kinds of can be explained. So the difference
15:11 really one of degree. The theory explain many different observations uh and since
15:21 can't prove a hypothesis, um and often we can't come close to proving
15:27 . We can fail to falsify it we can maybe have a little supporting
15:32 but we have to move forward in thinking. So we will carry that
15:38 . We call that a working And so for lack of something
15:44 this is the explanation we're going to with and we're going to see where
15:48 leads. Okay, so which of following our hypotheses, The Earth's diameter
15:56 about 8000 miles. Is that a ? No, no, that's an
16:09 . Okay. Global temperatures will increase one degree in the next 20
16:15 Is that a hypothesis? No, a prediction. It may be a
16:22 . Based on the hypothesis. The being a model to predict past temperatures
16:28 it's not a hypothesis. It's a . Okay, how about the third
16:32 fluctuations in global temperatures over time are caused by natural causes. Could be
16:41 . So we don't know if it's or not. That's not the
16:45 The issue is is it a hypothesis yes, that is. It's an
16:50 for the fluctuations in temperature. So the scientific method is often represented
17:00 something like this wheel and we start the top with an observation. So
17:08 is an observation that needs to be . So you researched the area,
17:14 you read about it, you see other people have done. You derive
17:19 own equations, you conceptualize about it some period of time, then you
17:25 ready to offer a hypothesis to explain observation or observations. Once you have
17:33 hypothesis, then you will design an to falsify the hypothesis and you will
17:42 that hypothesis short of that. If can't design an experiment to falsify
17:49 you may collect data to confirm or confirm the hypothesis. Um you then
17:57 that data and this is where statistical comes in and again in our industry
18:04 pretty cavalier about the way we draw from data. Actually, I would
18:10 to see us looking more at statistical and arab ours and so forth than
18:16 do. Um and then and then report our conclusions in a report or
18:22 or scientific paper and this is cy because the entire way we're getting
18:31 Either from our experiment from our analysis from other people, especially when we're
18:40 and we're making more observations over And so uh you know, we
18:48 start the process over again many times . one of the issues about researching
18:59 topic thoroughly before you formulate a hypothesis is that you're building a bias.
19:09 really there's nothing in the scientific method nothing in your scientific education, they're
19:17 tells you where to get this hypothesis . You know, we don't necessarily
19:24 you two the imaginative and creative and think outside the box. Um come
19:34 with explanations that nobody else has thought , looking at things from their perspective
19:39 nobody else has. And therefore I it helpful when you're looking at an
19:46 why not formulate hypotheses, you could a list of hypotheses. Maybe many
19:53 them are totally naive, Maybe they're wrong, but so what that when
19:59 first faced with explaining an observation, when you're the most unbiased and then
20:09 the area and then decide which hypotheses favor. I think a great example
20:16 this is continental drift. Uh Alfred around the turn of the The 19th
20:24 the 20th century came up with this that south America fit into africa because
20:32 continents had split apart. They were one, they split apart and they
20:36 apart. He came up with the of continental drift and he was laughed
20:41 by geologists. He was a he was not a geologist. So
20:47 didn't have the scientific credibility. He have the gravitas uh to be taken
20:55 , but he had the advantage of being ingrained in the geological theories that
21:01 at the time. So he was to come up with this literally earth
21:06 hypothesis, Which it took over 50 afterwards before serious geologists in a geophysicist
21:18 that, yeah, continental drift does and they were able to explain why
21:23 occurs. Now, as I statistical hypothesis testing is a little bit
21:35 . And if you just google hypothesis , you will get all kinds of
21:41 from statistics, from machine learning, Business administration. All talking about what
21:50 call statistical hypothesis testing and this is different from scientific hypothesis testing. We
21:59 not testing an explanation. We are the observation itself. So basically the
22:10 is, we've got multiple groups of , let's say we have two groups
22:14 data, uh and they're different. , maybe we've got the height of
22:20 and the height of females in Right, there's a difference. So
22:27 have a mean height for males. height for females. Now, is
22:31 difference statistically significant? In other are we confident that it was not
22:40 by random chance? Right. So test this, we test what is
22:46 the null hypothesis, we say, is the probability that this observation could
22:54 resulted by random chance. And we that probability to be very small.
23:00 another example, I'm correlating two I'm correlating rock ferocity to rock velocity
23:10 I see a correlation between the Now, is it possible? And
23:14 are the odds that that correlation happened random chance? Right so we need
23:21 test for that and we need to that the probability of that correlation is
23:25 than some threshold. Usually less than 5% probability. But how low you
23:31 depends on how precise you have to . Okay. So um the smaller
23:41 probability, Well let me put it way, the smaller the greater the
23:50 that is caused by random chance. smaller the probability that it is a
23:55 relationship. The more you need a explanation in other words, the more
24:01 need a scientific hypothesis to accept the . Okay now we're gonna study rough
24:12 . Uh We should we should define terms we're dealing with. So first
24:18 all, what is a rock? Yes we are spending a work naturally
24:28 Yes. Um You know. Yes minerals. Uh No. Right.
24:42 . Um And fluids your fluids and . So the dictionary definition in the
24:51 different dictionary is a naturally occurring aggregate minerals and then I would add other
24:58 . Maybe organic matter. It doesn't as a as a mineral, maybe
25:03 stuff. So but the key terms are natural occurring and aggregate. So
25:10 ceramic teapot is not Iraq because it's naturally occurring even though it's constructed of
25:18 like materials and has a rock like ? Okay, so then what is
25:24 physics mm anything related to the Physical properties of the rock? Like
25:37 texture, the stiffness, the the ? I think it's also it's also
25:47 to the observation we we we we get from from any particular rock.
25:56 , absolutely. You covered it. going to state it a little bit
26:00 succinctly. Yeah. So you could it Number one. It's a relationship
26:07 the rock properties and as you physical properties. But I'm going to
26:12 that to the geological properties of Relationship between that and the geophysical
26:20 Um and that is what you meant measurements. Right? So if we're
26:27 the rock that's geophysical and if it's rock itself, that's geological.
26:33 So it's the relationship between the geological the geophysical properties or it's not,
26:43 example, ballistics of rocks. If take a rock and throw it,
26:47 is its trajectory? Right. That be physics. And it's about
26:52 So it really has to do with internal characteristics of the rock and how
26:57 rock responds to stimuli. The stimulus be a static compression, where it
27:06 be a seismic wave. So um talking about the internal physics of the
27:16 ? So why is rock physics Why are we, why did we
27:22 a course on this in the professional ? Because they tell us about the
27:31 and the contents uh of the So so that because we are in
27:37 for reservoirs and so on. So important for us to use this measurement
27:41 can transform it into, let's say seismic measurement and transform it into uh
27:48 about fluid contents or yeah just in about that. There is a floor
27:55 the nature of the work. Because if we understand the relationship between
28:00 geological properties and the geophysical properties, can invert that process. Right?
28:05 we could infer the geological properties from geophysical observation. That's the inverse
28:12 It's also important in the forward Right? If I'm trying to simulate
28:18 an experiment might look like, like I'm doing a feasibility study. Should
28:22 spend $20 million to acquire a time ? Three D. Data set?
28:29 I want to have some idea of that experiment is going to work
28:34 will the geophysical technique be able to the geological changes that results in geophysical
28:42 ? And then there's my technique have ability to see those geophysical changes.
28:47 a key part of that is rock . So both in the forward direction
28:51 then the inverse direction I rock physics important. And of course it doesn't
28:57 to be in oil and gas. mean it could be environmental it could
29:03 an engineering, right? And it be an academic studies trying to understand
29:10 earth itself? Just for academic Okay, so what factors control the
29:18 properties of rocks? Yeah, I say the physical policy. Okay,
29:30 you mentioned ferocity and the nature of ferocity. You mentioned the matrix.
29:37 the mineral mineralogical composition and the other of Iraq. That's too what
29:45 And okay, but I'm going to that those two things help determine the
29:52 anyway, so something else independent of Russia. Okay, so environmental factors
30:02 pressure, temperature, state of orientation of the experiment. So,
30:09 are external things. Also matter what I'm looking for. One more
30:16 We talked about composition. We talked porosity. What other aspect of the
30:24 will control the geophysical properties uh, your business. Okay, again,
30:32 gonna throw permeability in with the right? It's the poor space and
30:36 gonna talk a lot about the poor and how it affects your physical
30:42 The excuse me, the fluid contract fluid content. Again, I'm incorporating
30:51 in ferocity because the fluids are in pores. But yeah, we can
30:54 that a separate one. That's The fluid properties very important. The
31:00 of the what would affect the resistive , right? Or the fluid
31:04 but also a big influence the presence absence of gas or hydrocarbons affect
31:10 Right. One more thing which may flies under the radar a little bit
31:17 the texture of the rock. Um is the arrangement of grains? What
31:24 the coordination between the grains? The do the contacts between look like?
31:30 cemented? Are those grains uh And uh you could kind of talk about
31:38 degree of literacy. Fication, Howlett is the rock. So these things
31:45 separate from composition. You can have same composition and yet very different geophysical
31:52 . You can have the same same ferocity and very different physical properties
31:58 you have a different internal structural Okay, so rock physics is the
32:09 study of the relationships between rock geological and geophysical properties. And we talked
32:19 going into two directions. So let's go in the forward direction here on
32:24 left, we've got mythology which including , um degree of lift,
32:32 etcetera, um uh texture ferocity which including poor space and also including poor
32:42 . We'll talk about four surface area other things. And then there are
32:46 pore fluids, the pore fluids their properties. The mixture of pore
32:51 , the saturation of pore fluids. these are the rock properties. We
32:57 have the environmental properties, stressing temperature and orientation of the experiment.
33:04 as a result of these, we the geophysical properties and I'm going to
33:10 these are velocities. Both p wave shear wave density. Body, let's
33:16 body wave ferocity is there are other of waves, but they can be
33:20 as combinations of body waves. body waves, p wave velocity and
33:25 wave velocity density and also attenuation for waves and S waves. And I've
33:32 that out. That's a complicated And we may not get to that
33:37 this class, but that's what rock does. It allows us to go
33:42 the forward direction from rock and environmental uh to the geophysical properties. Now
33:52 geophysical properties are distributed in space in , Y and Z. So we
33:58 a 3D distribution of these geophysical properties how, what the seismic response that
34:06 from that 3D distribution is done in proper direction by seismic modeling. So
34:14 a separate discipline and that's beyond the of this course. Right? We're
34:19 on the rock physics part. You'll waves and raise or you have done
34:24 and race. So, you have handle on seismic modeling from that
34:31 But that is just the forward What we want to do is we
34:38 to go in the inverse direction, ain't easy, Let me say,
34:43 remember I said, we have thousands pages on rock physics. We also
34:47 thousands of pages in the literature on propagation. So a lot of
34:55 a lot of work. And as result of all the studies we've
35:00 we know at least one thing we that the inverse problem is entirely non
35:08 . A given velocity and density and if you wish, can be produced
35:15 a wide variety of combinations of these factors. Right? So even though
35:20 may be able to go in the direction, in the inverse direction,
35:26 tougher because it's not unique. There thousands and actually theoretically an infinite number
35:32 solutions. And the same thing with modeling of why variety of combinations of
35:40 geophysical properties can produce the seismic So what we want to do is
35:47 backwards and it would be nice if could have a seismic inversion algorithm black
35:54 where you put in the seismic response it gives you the three dimensional distribution
35:58 these things. That would be very . And people seem to think they're
36:02 to be able to do that with learning, right. I beg to
36:06 . We'll be able to do parts it with machine learning. But it
36:10 be as easy as some of the of directors of oil companies seem to
36:15 right, it's a very different problem targeting you with an ad that you
36:20 be interested in it. Right. actually a much tougher problem than that
36:25 much less training. So we're gonna on knowledge, we're going to rely
36:31 our domain knowledge to go from the response back to the things we want
36:37 know. And uh, if we that completely mathematically and scientifically, that's
36:45 seismic inversion. I don't want to that topic. It's one of my
36:50 topics. And I published a number papers on seismic conversion. Uh
36:56 But the whole deal is you can't put the data in, push a
37:03 and get the correct answer out. not that simple. It is an
37:08 problem. And so the way we this in practice is the call of
37:14 art and science of seismic interpretation. , seismic conversion could be a very
37:21 tool. And the more you can that inversion and the more you can
37:26 that inversion smart to look for the that are geologically occurring, the more
37:35 that inversion is going to be rock physics fundamental building blocks to do
37:42 of that. If we needed to forward, we certainly needed to go
37:51 . Okay, so um in that list of rock properties, I talked
37:58 mythology, I didn't break it out I think we need to break it
38:03 in terms of composition and texture. , a a sandstone uh huh may
38:12 very different properties than a silt For example, even if their composition
38:17 exactly the same. So what is texture is uh the way the particles
38:25 arranged and joined in the rock and referred to as a visual or tactile
38:32 characteristics. So the surface looks differently great scales of magnification and it feels
38:41 , feels rough or smooth. That's that's a good description of
38:48 So here we have a picture of happens to be a shell. And
38:52 do you want to say anything about texture of that shell? Anything?
39:07 . Okay. So it's fine Um What else can you say about
39:12 texture? They're kind of the other are created southwest, direct southeast
39:24 More prosperous societies. Yeah. So . So there's definitely an orientation.
39:32 I'm not sure if this is if rock is why it's shown dipping like
39:39 . It may have been part of fold and they actually have been dipping
39:42 it might have been a deposition all . So I'm not sure where up
39:46 is on this figure but you can least see the lamination is here.
39:51 can see very fine layering there and you also see na jewels of other
39:58 within that layering. Also oriented. , So all of these factors affect
40:05 geophysical properties. The modules affect things . Maybe the dark rocks here are
40:12 clay rich. The light parts are sandwich perhaps. Um What do your
40:20 property do you think would be greatly by this layering velocity? Well,
40:31 , but an aspect of velocity, you see a direction possible directional dependence
40:38 the velocity? You think the velocity change if I'm measuring a wave that's
40:43 perpendicular to the layers as opposed to to the layers. Yeah. And
40:50 we talk about the void Royce you'll see exactly that. There should
40:55 a big difference in the velocities going way across. The betting is usually
41:02 and going this way parallel to betting often usually faster. So what do
41:07 call that when there's a directional dependence velocity? And I was in trouble
41:14 I sought to be. Yes. So um most rocks are anti
41:21 Most minerals are anti psychotropic and uh if I have perfect hysterical courts frames
41:29 are randomly oriented relative to their crystal graphic axes. Uh if they're all
41:37 the same size and they're arranged in things that are regular, those regular
41:43 has become anti psychotropic. So anti is something we're going to have to
41:50 with. Now, I'm not going go through the mathematics of anti
41:56 I think leon Thompson is probably the person in the world to do
42:01 And I believe he teaches waves and your course in your programme. So
42:06 may have done that. Um we deal with an isotopic measurements and at
42:11 conceptually try to understand what's happened And by the way under porosity,
42:17 included permeability and poor structure. one the very sad facts of life is
42:26 seismic waves are insensitive in a direct to permeability. Seismic waves are very
42:36 to porosity but their sensitivity to permeability primarily through the ferocity. There's there's
42:44 first order direct influence of permeability on . It may have an influence on
42:51 attenuation. And at very high frequencies may affect. Well it certainly affects
42:59 at high frequencies. But at low frequencies we don't have a means of
43:06 seismic velocities for permit abilities. In , if you look at our velocity
43:12 , they don't have permeability as as factor in those equations at low
43:19 but we also have high frequency ultrasonic and they are permeability is a uh
43:27 parameter in those high frequency equations. But the sensitivity is very low uh
43:36 frequency. So the way we typically permeability is, we use the seismic
43:41 to estimate the ferocity and then we a correlation between porosity and permeability.
43:47 we infer the permeability. Okay, it's important that we think about the
43:57 types of equations that we have in physics. And there are basically
44:05 Uh well, I'll divide them all three categories and then there are break
44:11 theoretical uh into another category as So what is a theoretical equation?
44:19 one that's derived from physics. So exactly correct. And no matter how
44:27 the rocks are, no matter what arrangement. Uh huh These equations
44:34 And there are very few of these few. The two main ones that
44:39 know definitely work. Our woods which I'll show you here. And
44:45 the mass balance equation, which will a number of times those equations simply
44:53 . The physics is exact and the geological aspects don't interfere with the
45:01 Another theoretical equation which we're going to a lot, his gas men's
45:08 Uh There are other theoretical equations, the p wave velocity being the square
45:14 of K plus four thirds meal of row, shear wave velocity, square
45:18 of you over row. Uh He these equations in physics one their
45:24 I mean geophysics one, their theoretical they are approximately correct actually not exactly
45:32 , but they're close enough to being that we can just accept them as
45:36 correct. And now let's look at equation because this is an important type
45:45 equation is what we call a harmonic . And we'll see this type of
45:51 a number of different times by a equation. We mean a reciprocal,
46:01 this is reciprocal volume weighted average, x here is the volume fraction.
46:09 I have to constituents, I have volume fractional volume of constituent one.
46:15 fractional volume of constituent too. So one plus X two equals one.
46:22 ? Um and then I'm saying that reciprocal of K, which here is
46:27 we call the bulk module lists of effective medium, the bulk module lists
46:34 a mixture of those two constituents is to the volume of constituent one divided
46:41 the both modules of constituent one plus volume of constituent two divided by the
46:47 modules of constituent too. Yeah, it's it's about it volume weighted sum
46:55 the reciprocal zales of the constituents properties the reciprocal of the constituent property.
47:04 what equation. And we'll see this a number of times. By the
47:08 we could have many constituents, we have an infinite number of constituents and
47:13 just handle that with a summation So you have a summation of
47:17 I over K I. For with the equation. Yeah, I'm
47:24 to interrupt process. Uh This is harmonic average. Right? Yes.
47:29 . Yeah, thank you. Which happens to work for gas bubbles and
47:36 ? It works for? Uh I say gas bubbles in a liquid.
47:40 works for a drop of droplets of in a liquid. It also works
47:45 solid grains suspended in a liquid. grains are not in contact with each
47:51 . Uh So uh it's exact, is a low frequency equation. If
47:57 frequencies got high enough you would have scattering and you'd have different things going
48:04 . But if our wavelengths are very long compared to the particles or the
48:11 , then this equation works okay, those are theoretical equations. I wish
48:17 were more of them that were useful rocks are complicated things and there's a
48:23 going on in rocks, their mixtures mixtures are hard to deal with.
48:29 From a theoretical point of view. therefore we usually resort to empirical equations
48:36 these are my favorite type of equations rock physics. These are fits two
48:43 . So, uh for example, least squares fit to a trend line
48:48 multiple regression or or something like And an example is widely is time
48:54 equation. This looks like a theoretical because the coefficients and the values here
49:02 real physical properties at least originally they supposed to be. And what wild
49:08 time average equation is the travel The wave travel time per unit
49:16 delta T through a porous medium is to the fraction of the poorest medium
49:25 is solid. So one minus porosity the solid fraction uh times the transit
49:33 of the solid material plus the which is the volume fraction of void
49:42 uh filled with a fluid times the time per unit distance in the fluid
49:51 T. F. This was used the early days of logging to predict
49:56 is using sonic logs before we had and neutron logs for example. Um
50:04 it looks like it should be derived physics. For example, you could
50:08 , well, my total travel time be the travel time in the
50:12 plus the travel time in the And that might be true at infinite
50:19 . But again, we're talking about that are much longer than the four
50:24 . So we're really looking at an medium. And this equation cannot be
50:30 from physics in fact. Well, was derived by physics, it would
50:35 work within the context of the assumptions derive it. In fact, there
50:41 no physics that predicts this equation. in practice, what they find is
50:49 these physical constants like the Now what the transit time per unit length in
50:56 fluid or in a solid material? the velocity or the reciprocal of the
51:04 . So this could be one over . S. Uh huh has been
51:08 . S. Being the velocity and solid. So one over V.
51:12 And this would be one over V the fluid. We also call that
51:18 slowness. The reciprocal of velocities house the slowness. Uh huh. That's
51:26 a physical property. Um But uh coefficients aren't always that in particular the
51:38 transit time where the fluid slowness. example, if I have Iraq with
51:43 in it, if I use the slowness for a gas water mixture,
51:50 get the wrong answer. Uh And if you if you look at this
51:56 as if it were exact from a standpoint, you would misuse it.
52:01 I've seen people misuse this equation. seen them put the gas slowness in
52:07 to predict porosity and that's wrong. won't work at all, professor.
52:14 sometimes also there is an additional term I guess consolidated versus unconsolidated.
52:23 absolutely. We'll come to that. the question is, you know,
52:28 , what is the range of of data that this is the danger of
52:36 equations applying the empirical equation um where not applicable. So as you were
52:44 , degree of compaction or actually degree lift, ification Is an important component
52:50 . This equation was derived in the , was derived on a biased sample
52:56 rocks at the time. In the when they came up with this,
53:01 have to be able to cut a of the sample and the sample had
53:04 , had to survive to go into laboratory and make a measurement. So
53:09 were all well liquefied rocks. if you if you take this equation
53:15 apply it to poorly liquefied rocks, will give you the wrong answer.
53:20 one of the great uh pitfalls of empirical equations is using them outside the
53:28 of the experimental data or using them conditions that are different from the observations
53:35 which that the experiment was made. huh. Another thing is who says
53:44 this should be a linear relationship. fact, sometimes the relationship is nonlinear
53:50 when you fit a linear line to , the delta T. Solid that
53:54 get becomes very different from the mineral and cannot be explained by the composition
54:03 very commonly, you know, If I'm in a clean sandstone very
54:09 , I could use 19,000 ft/s for velocity of courts. Um But very
54:17 when the time average equation is they use 18,000 ft/s. And even
54:22 a perfectly clean court sandstone, I , you might try to wave your
54:27 and say there's a little bit of in the rock, therefore the lower
54:30 time. But in fact what we is that lower transit time, uh
54:36 smaller coefficient here actually larger coefficient in time results from non linearity in the
54:46 and you're you're taking a linear fit a nonlinear equations. So the extrapolation
54:53 zero ferocity gets off. So there's why we use time average
55:00 Now, perhaps while this time average may have originated as a heuristic
55:08 we don't know what its origin What is a heuristic equation. This
55:14 the strict definition is a rule based . Really. What it means is
55:22 equation has not been derived just by data and it's not been derived from
55:28 , but it's that it comes from idea that an experienced researcher may have
55:36 a form which later on when he to data uh serves a purpose and
55:43 reasonable. Uh So that's a heuristic . So while this time average equation
55:50 conceptually so nice that it may have as a heuristic equation, but it's
55:56 as a as an empirical equation. example, a good example of a
56:01 equation. In fact, the authors proudly it was a heuristic equation instead
56:07 theoretical or empirical is the critical porosity . This came out of stanford Amos
56:15 and Gary Mapco at stanford. In the germ of this idea came from
56:21 graduate student of there is a fellow the name of Doctor Daiwa Han who
56:26 here at the University of Houston. runs our rock physics laboratory here.
56:31 it was originally his idea that his ran with and turned into a very
56:40 conceptual model here. And what this says is it takes hans idea of
56:49 critical porosity. That is a porosity which the rock loses cohesion essentially.
56:56 do I mean by that rock losing ? I mean when it water if
57:01 water saturated the grains have disconnected from other and we essentially have a
57:10 so it's no longer grain supported, fluid supported, The grains are floating
57:16 . I think sometimes you've seen divers on the ocean bottom where the ocean
57:22 sediments are just sitting there and as work walk, you see this cloud
57:26 sand dust stirring up so the grains are not, are rigidly connected to
57:35 other. Um Okay, so that's critical porosity and for sand stones,
57:43 just empirically that critical porosity turns out be around 40%.4 in fractional terms.
57:54 We'll see later, a simple cubic . So the simplest least dance packing
58:01 grains of the same diameter that you have, That cubic arrangement has a
58:07 of 48%. So any porosity is than that uh in a fluid are
58:15 to be suspended. So in the ferocity model, then what they're saying
58:23 if M is the modular list of mixture, this module lists approaches the
58:33 lists of the solid material as porosity to zero, so as the ferocity
58:39 to zero, this ratio of ferocity critical ferocity goes to zero. So
58:45 equals M of the solid. Um the other hand, uh And this
58:54 by the way for the dry when the porosity becomes the critical
59:00 So as my porosity increases, you decreasing because this number is getting,
59:06 ratio is smaller than one. So term in the parentheses is smaller than
59:13 . So as ferocity decreases, mm relative to the solid. And when
59:21 ferocity becomes equal to the critical M goes to zero, Right?
59:27 this time becomes 1 -1, it to zero. So the rock frame
59:35 cohesion, it contributes nothing essentially to modules of Iraq at this point.
59:43 and the rock becomes a when suspended fluid, that rock becomes a
59:54 Okay, by the way, there some important, very important equations related
60:04 the relationship between velocity and density and they're all empirical equations and they have
60:15 common theme of having uh someone with surname Gardner working on the equation or
60:24 was involved in publishing the paper where equation was derived. So the first
60:32 is widely Gregory and Gardner LW Gardener was anyone, we'll come back to
60:44 . But anyway, as we they came up with the wildly time
60:48 equation. A few years later, Gardner, LW Gardner and Gregory published
60:58 very famous paper in geophysics and it'll the first paper you read for this
61:04 . And and uh in that paper presented gardeners equation. This was a
61:12 , a very general relationship, empirical between velocity and density and therefore velocity
61:19 ferocity, if you know the density you know the composition, you know
61:22 ferocity from the mass balance equation. , um jerry Gardner, uh I
61:34 to be lucky enough to briefly work him, uh probably the most brilliant
61:41 I've ever encountered for many years, thought uh he was the son of
61:47 Garden. In fact, these researchers at gulf research gulf was an oil
61:55 in the old days before it was by Chevron and they had a research
61:59 in Pittsburgh pennsylvania. And these guys all there. So jerry Gardner became
62:05 of that team and I always thought was LW gardener's son. In
62:10 there is no relation between the L W Gardner is the Texan jerry
62:17 was a native Irishman who immigrated Um and anyway, they're interesting stories
62:25 jerry Garner. He was very politically and the FBI had a file on
62:31 , was investigating him and so Um One of the smartest guys I
62:35 met and he was a professor here the University of Houston. In
62:41 he started the Ally Geophysical Laboratory here the way. Another fellow part of
62:49 group was Ray Gregory. When I a graduate student at the University of
62:55 , we used to see this old dressed guy warm during around in the
63:00 . I thought he was the And then one day in class we
63:04 him giving a presentation on the time equation. Um, and uh,
63:11 of my classmates raised his hand and said, you know, there are
63:15 with the time average equation and his was yes, I know. Well
63:19 was Ray Gregory. Okay, um, later on, uh,
63:27 fellows at slumber, J. Roemer and Gardner came out, came out
63:31 the reindeer Hunt Gardner equation, which do consider an improvement of the widely
63:37 average equation, um, John Gardner no relation to the others as far
63:43 I know. Um, so if was a regular class, I'm not
63:51 make you do, it has an , but I'm going to ask,
63:57 certainly, I'm going to ask you read the wily Gardner in the famous
64:03 that's also on blackboard that talks about critical ferocity model. So I'd like
64:09 to start with Gardner, Gardner and , but then go ahead and read
64:13 other papers as you find time to that. And uh maybe next week
64:19 might tell me if you were estimating from velocity, which approach would you
64:26 to use and when and why? be talking about velocities not this
64:37 but we'll be focusing on velocities and week. So we'll come back to
64:42 and we'll have this discussion. Then did want to talk some more about
64:48 critical ferocity model. This is from and the leading edge which you have
64:55 blackboard and he plotted uh elastic ma of rocks versus ferocity in an interesting
65:08 , he, you know, this the critical ferocity equation. He normalizes
65:13 measured elastic modules of the rock by mineral module list. So that scale
65:19 0-1 and he normalized the ferocity by critical porosity and by the way,
65:26 little ology has its own critical Usually around 40%. Uh so,
65:35 if I have a prostitute, then I'd be here one on this
65:40 and went across it is zero. here and you see a pretty good
65:45 there, right. Uh if the porosity model were exactly correct. The
65:53 would be precisely on the diagonal. let me draw the diagonal here and
66:01 think maybe this gives you a more view. I mean if you think
66:06 it compared to the value on the , some of these points are way
66:15 ? Um By the way, I'll you in a second what you think
66:20 saturating fluid is? But let's just back to here. And so if
66:26 were using the elastic module list or derived from the elastic modules like velocity
66:33 predict the porosity, right? So have this module list. My my
66:40 would tell me my Karasu cities here in fact the points are lower.
66:46 fact this critical porosity model predicts a velocity. You see it works as
66:54 bounding equation. There are very few on the other side of the line
66:59 not very far, probably close enough experimental error. Right? So this
67:06 the fourth type of equation. I to talk about buying abounding equation.
67:11 is in fact a heuristic bounding equation seems to work when compared to
67:18 So as a uh as a bounding , I'm going to say this is
67:25 moved because of the agreement with it's moved from the category of purely
67:30 to now be an empirical, because satisfies data. Uh Now, given
67:39 results, can you guess what the fluid is here and why? So
68:15 give you a hint. Uh When I have uh when I'm at
68:21 critical ferocity. Now I'm being dominated the fluid modules. Uh Do you
68:31 when I'm at the critical process, do you see what the module
68:35 Is um you zero if I if were on the line. Right.
68:50 um yeah, what, what If modules was 0? What fluid would
68:57 be? Uh water? Well, is actually a little bit higher.
69:04 has a finite module list, which have to consider when we're when we're
69:13 module I and velocities, we'll see we look at gas mains equations will
69:17 that water significantly increases the modules of over the dry frame. So,
69:28 would have a non0 answer if the fluid were water. What else?
69:35 ? These are experimental measurements, What could the saturating fluid? B
69:44 Well, brian is even stiffer than is. So it's going to have
69:48 similar result spring, by the What's the definition of a fluid?
70:02 Yeah, it can be, it be either gas or liquid.
70:09 So if it's if it's not brine the laboratory and it's not water,
70:15 would would what is it likely to in the four space? They're most
70:23 . And at surface, you uh at surplus pressures anyway, The
70:30 of gas is essentially zero and air going to be zero. Uh So
70:38 fact that this goes to zero suggests the saturating fluid here is air.
70:45 And if that's important to note for for this relationship to be working,
70:51 this is only applicable to what we the dry rock. Right. Uh
70:58 this relationship would not work if I fluid in the rock. So really
71:04 critical ferocity model is addressing the properties the rock frame and not taken into
71:12 the contribution of the water. But there are very useful theoretical bounding
71:23 and some two very simple ones. first ones derived the Royce and boy
71:32 . And these are the widest possible . In fact, if we were
71:39 go into our composite medium unit, would talk about tighter bounds like the
71:45 stricken bounds. Uh But for our in this class, we're going to
71:52 with the Royce and Boy bounds. useful is bound in equations. It's
71:59 four types of Ukraine vertical. You wish to the morning resting and
72:08 critical. Okay. So yeah, had theoretical, empirical and heuristic.
72:15 then the 4th type is bounding And the bounding equations could be
72:23 theoretical or empirical. Right? But a different type of equation. In
72:28 words, a bonding equation is not to predict precisely what the value is
72:36 the mixture from the from the properties the constituents. It tells you how
72:41 or how small, what's the maximum of properties of the mixture? All
72:48 . So, the Royce Boy bounds derived theoretically. They're very useful and
72:55 they actually correspond to are either alternating or alternating layers. Right? Uh
73:06 again we were talking about is our compression for example, perpendicular to to
73:13 layers, or is it parallel to layers? Um I like to think
73:20 the void situation as columns because conceptually helps you understand what's going on.
73:28 so the dark grey are the columns the white is the air for
73:34 between the columns. So the gray the hard, the white is the
73:40 if I and the reason columns are in architecture is because it's the hard
73:46 . It's the the in compressible columns control the compressibility of the whole
73:54 Right? So if I'm trying to this guy, so I have a
73:58 compression. Imagine a piston here with plate and I'm compressing that plate.
74:06 so I'm applying uh the same force all of the elements here. Uh
74:15 , even though the air in the is not doing much to resist that
74:21 , the columns are the ones doing of the work. And so what
74:26 find is if you work through the , the module lists. Uh This
74:33 be the compressibility, it could be share module is, it could be
74:38 the plane wave module list. That is whatever it is, is a
74:44 weighted average of the module at the . Now I stole these these equations
74:49 stanford. So instead of using I. They use they like to
74:54 F I For decimal fraction. So that's a number of small between
75:01 and 1. So it's the volume of the constituents? And you can
75:08 , and so we have an infinite and is not infinite, we have
75:12 sum over and the number of And so the module lists of the
75:20 is a volume weighted average of the I of the constituents. And you
75:25 see in this case of columns and air is going to have a module
75:29 near zero, columns are going to a large module list. So the
75:35 is going to be dominated by the lists of the columns. The more
75:41 I have, I'll have a linear in that module says F I for
75:46 air increases, F I for the will decrease. Right? So that
75:53 sense that if I only had one slender column, the thing would be
75:58 compressible. Right? So it's a volume weighted average. That's the maximum
76:07 lists you could get for any Um Similarly, if I have parallel
76:14 like this, that's the minimum possible you could have and once again,
76:20 have our harmonic some here, the of the modules of the mixture is
76:27 to the sum of the volume weighted of the reciprocal of the module I
76:33 the constituents. And this makes sense , because if I have hard material
76:38 soft material, imagine planks of wood foam rubber in between. So,
76:43 I have that stank, what that . Wood, foam rubber,
76:48 foam rubber and I sit on what's going to squeeze? It's going
76:52 be the foam rubber is going to . That would is not going to
76:55 very much at all. So the rubber is going to dominate. So
77:01 a reciprocal average, the smaller the modules is what dominates. Now,
77:10 you seen this form before? Have seen a reciprocal harmonic, volume weighted
77:21 before? Yes, Woods from the looks different, but conceptually is the
77:30 thing. You see it's a some the recipient volume weighted sum of the
77:37 is the reciprocal of the whole Right? So that's what we have
77:43 in this equation. It's just a way to write the same thing.
77:50 , now, also, when if had a we would call this a
77:58 compression, right? I have a where I'm applying that compression. That
78:06 effective medium will strain the same amount every constituent, right? Because the
78:17 are stronger than the air, it compress more than the columns. Can
78:24 see that? So, if you about the deformation of each of those
78:31 , it's exactly the same degree of image. Like if I if I
78:38 this and I shorten it all of columns, all of those constituents will
78:43 short and the same amount. So what we call is a strain.
78:52 , in the case of the Royce , you see that the softer material
78:59 going to compress more than the hard . So the strain is different in
79:05 constituent. That's what we call the stress found. The stress is the
79:11 in every layer. And as a they strain differently because as we'll
79:15 stress is proportional to strain. Um uh so if if the strain is
79:24 same, the stress has to be . If the strain is different,
79:28 stress will be the same to have correct proportionality constant constants, which we
79:34 elastic module i in each of these constituents. Okay, So we have
79:46 second exercise for you and this one we will do and we'll do that
79:55 with the break. Uh And so going to calculate Royce boy bounds.
80:02 , so, I've got to I've got courts and it's module
80:10 It's a both module lists his 38 pascal's. I've got water with both
80:20 of 2.5 Gigaal's and I've got sheer of 40 giga pascal's for the courts
80:28 zero for water. Just hypothetical Remember fluids have no rigidity. So
80:35 zero. So I want you to the plane wave module. Life for
80:42 and water. K plus four thirds . I want you to calculate the
80:47 bounds and the Royce found for the way of modules, the plane wave
80:54 is m is equal to K plus thirds mu So we're following these equations
81:02 now. Alternatively, what if I at the Royce and void bounds separately
81:08 the bulk module lists and share So I'm going to ask you to
81:13 those also. Uh and then I'm to ask you to recombine them.
81:19 I take the Roy spoke about the roi sheer modules that gives me
81:23 roi sam and I'm going to do void both modules and the void share
81:29 . That gives me the boy All right. So the way you're
81:34 to do this, you're going to this in Excel on the spreadsheet,
81:38 going to vary the ferocity from 0-1 small increments. So you'll make a
81:46 which is the ferocity And you might it, you know, maybe 100
81:52 every .01. So 101 values, go from uh ferocity of zero to
81:59 ferocity of one and then you'll compute both module I and share module i
82:07 these equations for every porosity. And I want you to plot those and
82:16 one thing I want you to do I want you to compare your
82:20 If you computed uh the results using you computed the plane wave module is
82:31 from these equations. That's one The second result is computing them independently
82:37 K. And you and then recombining to him. And I would like
82:41 to compare those two results, and will tell us what is the right
82:46 to compute the Royce and boy So uh go ahead and get started
82:54 that. And also take a Now, if you need it,
82:58 going to stop recording at this Do remind me to make sure I'm
83:03 when we pick it up
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