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GEOL7333-Seismic Wave and Ray Theory - Day5 09-09-2022
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00:04 Okay so folks uh that was a discussion of the quiz. Today is
00:11 september 9th, we have the rest this afternoon, all day tomorrow and
00:17 more friday together. Uh this is we left off last saturday before I
00:26 there, I want to show you which I should have been doing before
00:34 um I didn't and so now is chance to uh make up for that
00:43 so what I'm going to do is launch uh a different file. That's
01:15 . Okay, so is this show the zoom? Okay, so this
01:22 the uh an Excel spreadsheet and you see down here at the bottom that
01:27 many different um worksheets as part of . And this uh spreadsheet is now
01:36 your blackboard, you can download this later it's replacing a previous version which
01:46 improved uh during the week. And you should uh just delete if you
01:54 the previous version, then delete it uh Download This one. So let's
02:03 first here at this. Oh so we talk at all about this and
02:10 first week of uh um so I that I had this out there.
02:17 I should have talked about this the first lecture. So this is uh
02:23 which will calculate for you all these elastic properties when you enter um velocities
02:31 densities which uh you probably have a feel for. So for example here
02:36 entering that in kilometers per second and reminding over here what it is in
02:41 per second. And then uh it's actually asking you for a p velocity
02:48 also uh V. P two S ratio. Because most of us
02:53 most of us have a better um feel for that ratio than for the
03:00 velocity itself. And then it's giving the sheer velocity, hospital module share
03:05 all these things uh calculated for you ice tropic bodies. And so uh
03:14 think that right lead us all. you. Probably will not help you
03:36 . Okay, next one is the waiver. And so Ricco was a
03:41 justice. He lived in 19 fifties think and he defined this functional form
03:47 here. Um And it looks like when when you plotted out, it's
03:51 a a central peak. And you here this is zero time here and
03:57 it has two characteristic uh has two amplitudes, but only one. Uh
04:08 they combined together to make only one and here's the parameter. And so
04:12 example, here's a 50 hertz ricker and I'm gonna make here 30 hertz
04:18 . Watch the waves change when I enter. So that's lower frequency and
04:23 on. And uh when uh there's of worksheets like this and normally you
04:32 want to look at the tables, just want to look at the figures
04:36 let's see what this lower figure is be like here. I don't know
04:48 this lower figure is. I got thinking think about why that's why I
04:58 that in there. I think I think it shouldn't be there.
05:02 this is uh what this is what mean to show you there.
05:09 let's look at the next one. here's a gathering and you can see
05:15 one of these has got a single and uh uh specified uh this is
05:27 is this is obviously hyperbolic move It's got an RMS velocity here.
05:34 and let's just change that and make faster. And so you see it
05:40 out less because it's faster and let's this to be a shorter wavelength.
05:49 , It looks pretty much the Did you see it wiggle up and
05:52 because it re scale itself. Did . And so this is the curve
05:56 the hyperbole right here. And you it goes right through the middle of
06:00 , it looks like it's not going the middle bit. This point right
06:04 is directly underneath this peak here. , so uh that's a crude as
06:11 crude indication of what data might look . And now this is something that
06:17 didn't talk about very much when you the gather, just like that.
06:23 you see how the far offset has longer um uh period than the short
06:30 that's called an M. O. . Mr. Are you familiar with
06:37 ? Yeah. So the deal is when you when you want to move
06:42 thing up here to have the same instance travel time as this, you
06:47 just move it up by uh sliding uh because you can't do that um
06:57 for every uh every event. Maybe you can imagine doing that for
07:02 simple situation, but the way instead what the way we do it is
07:06 uh we multiply all the times by stretch factor, which stretches this one
07:12 up and as it's as we so not only do we slide it
07:16 , but we uh Lincoln so obviously going to make a problem when we
07:22 these two together. This peak is going to exactly line up with this
07:26 anymore. Even though they start off alike, you can see that these
07:32 all exactly alike. Uh Oops, it looks like there's an Avio effect
07:37 . Yes, there is an Avio . Yeah, so this amplitude is
07:43 than this one. I don't remember I did that this some time ago
07:56 . Uh one of this one is show you the effect of N.
08:00 . O. Strength. And this uh of course for uh simple minded
08:07 , uh simple minded imaging, we the move out and then we stack
08:11 together and call that an image. that was in fact the the standard
08:18 the day for most of the last doing when I came into the business
08:26 very common to do that kind of . So these days we do much
08:30 elaborate imaging. And uh and uh all has the same problem when we
08:39 the arrival times, getting ready to average them together to uh and all
08:48 noise. Um we stretched the far and so that's a problem that you
08:54 to be aware of. We'll talk that in your data processing class.
09:00 , so uh let's see here. yes, I think I showed you
09:06 picture in class uh here. You uh a wave with should be that
09:24 No, sir. So I need uh attend to this worksheet also because
09:35 thing down here is useless and I know what I need to do to
09:39 it more use let's try something here to let's see what happens. Hard
10:05 say. Anyway, so uh don't your time on this. Oh
10:11 this one's interesting. Okay. So talked about um uh abnormal move out
10:19 is adding an additional term to the out equation to correct for a long
10:25 . And this brings us back to to the discussion that we had um
10:33 today. Uh why do we have long since I can assure you that
10:39 I came into the business nobody knew or cared anything about hyperbolic and non
10:45 move out because we designed our our so that we had uh about the
10:52 spread length as we had depth to to the main reflector that we were
10:57 for. And under those conditions you have hyperbolic move out and uh nobody
11:06 or cared much about not having because didn't have for our sets. Then
11:12 the same time I came into people invented what we now call
11:17 V. L. And so somebody oh if we're gonna be studying the
11:23 of offset of the amplitudes, let's more offsets spreads. So when you
11:30 what the hyperbolic equation didn't work So we eventually decided uh the equation
11:37 I showed you earlier for not move . And so um this is an
11:44 of those same equations and it's a complicated. So I'm gonna go through
11:48 here. Um uh we have here cartoon with three layers red, yellow
11:56 green in each layer. We got specifying uh velocity, only velocity.
12:05 so the way we do it is uh with this slide here. So
12:11 gonna grab that slider and slide this curves change. Never mind that.
12:18 uh here it's 24 25 74 m second. That's what I'm reading out
12:25 . Let's make a little bit Watch that. See this as soon
12:29 I let go, this thing Okay now um the thickness here is
12:34 by um uh I uh keyboard. let me change that here uh to
12:54 the thickness watch the grass on the , okay. Oh I had to
13:01 it to 5000. Okay, so so you can change the thickness is
13:11 and and so on and do the thing for formation to information three.
13:16 um let us um look to see we're graphing here. Uh These are
13:29 and or or measuring two kinds of and animal velocity and and general
13:36 So uh those are both the same the upper layer. Ah Can you
13:43 here we have 123 later. They're the same in the apple layer.
13:47 course, since uh since the reflection this boundary right here doesn't know about
13:53 lower layers, that has only one now for the lower layer, uh
13:59 are two velocities. And so uh biggest one uh this is the initial
14:07 and that's exactly the same as you here picked out with this slider and
14:14 the moon move out velocity. Is however, the colors are wrong?
14:22 don't know where the colors are This this should be red but it's
14:29 it's blue, I don't know why be blue. So um uh this
14:37 is obviously an RMS average between this and this one. So it's sort
14:42 halfway in between and down here. slower again. So it's so this
14:48 here uh in red is the the average of all three of so that's
14:56 straightforward Um Uh arithmetic. You can that in your head. Now let's
15:03 over here, eight a star I . I um Give you the
15:09 on the first day, the second or something like that. I give
15:13 a formula for a to start in of these layer properties. And here
15:17 is 123 there this is in the of an exercise. So let's look
15:23 . It says select the move out by adjusting the sliders and or the
15:28 as we did that. Do any the curves above look weird. See
15:33 , uh I want to show you little trick here and go to view
15:40 then split and now I can split scream just like this. And in
15:56 lower screen I'm gonna scroll down and epic skin, I'm gonna scroll up
16:11 of the curves look weird. I say that any of them. So
16:19 one or more of the velocities until weirdness goes away. What have you
16:25 ? You know, it's been a time since I um I wrote
16:30 I forgot what what weirdness I'm looking here. Um So so here it
16:38 that you can reduce velocities until the goes away. So you can find
16:44 by increasing the velocity by golly, gonna do that. Oh, here's
16:55 weirdness. Look at that. so what have you discovered?
17:02 so um ah and what is being what what is being graphed here?
17:23 looks weird. This is offsetting kilometers a reflection time. These things look
18:00 . And these are weird. So now that I know what kind of
18:05 is in there, now, I'm put it back here and these don't
18:14 where you know what, I can't the point that I was making.
18:20 have to refresh myself with the lecture from uh the first uh lecture.
18:29 the second lecture of this fourth. I look here at the notation,
18:37 uh I don't get it. I know what's happening. Uh huh This
18:50 reasonable. Okay, now let's see happens as I make this, making
19:02 thing incrementally faster. I'm getting Okay, so um I tell you
19:18 truth, I don't know what the of that weirdness is because I don't
19:26 this uh notation. It comes from first lecture. Alright, so uh
19:34 put this down as a curiosity and going to now um uh see what
19:42 is in here. Ryker gather animal , interference, abnormal s waves.
19:48 , let's see what we have Oh okay, so this is interesting
19:54 . Uh So here is a proposed structure. And where did that come
20:09 ? Uh by the way it's uh a p wave velocity and a sheer
20:14 loss. Where did that come from came from? This physical characterization which
20:23 is based on laboratory data. And just um change this, this is
20:28 porosity at uh near surface and ferocity infinity. Let's just change this to
20:35 smaller. Change it to 30. watch the curves when I hit enter
20:40 they're fast. And so of course is a water landing. And so
20:46 now um Uh says these values are using exponential decrease of porosity, which
20:54 can adjust by using the slider. , Okay, watch me here,
20:57 gonna change the slider. This changes fast the porosity changes between 30 and
21:05 . Did't change very much did So okay, so uh I should
21:13 you a reference to this, but is all uh implementing a large data
21:22 of measurements on rock in the laboratory by done done by mr uh oh
21:35 know Professor Han. Yeah, you , maybe you don't know. So
21:42 de Haan was uh now retired from department and ran for many years,
21:48 Rock physics lab here in uh of . And he did an outstanding uh
21:54 at stanford many many years ago where measured lots and lots of rocks and
22:02 he has an interesting personal history because old enough. So that when he
22:08 a young man in china, he sort of a victim of the great
22:13 forward, spent a lot of his in reeducation camps in uh in china
22:19 then managed somehow, I don't know to get himself to the United States
22:23 get himself in stanford where he did outstanding work. And then once you
22:28 from stanford, he had a long here in Houston, ending up at
22:33 University of Houston and he the uh caller was there and he uh the
22:46 squares fit of the data to the in the, you know, the
22:52 in the data set and the ferocity the clay content and the data and
22:58 all these things. And so that's implemented here. So this is um
23:05 how you can adjust the velocities. now let's look at here that this
23:11 structure leads to raise here has shown has shown on the right and you
23:17 adjust the offset, it says to right, So let's get a longer
23:25 . Go on. Yeah, that's . Okay. Now, um according
23:38 your calculations, velocity ratio is rather , especially in shallow. How do
23:42 know that? Let's see if we see here anywhere. Yeah, here's
23:52 velocity ratio. So that's not so . 2.5. So let's uh put
23:58 back the way it was before that's 40% ferocity, uh shallow.
24:06 you see these, these philosophy rations indeed high. Next question is according
24:15 the calculation of ocean bottom seismic receiver have uh these are uh not
24:27 but these are um you see patients the vertical component, The horizontal
24:44 So um I don't know why it p amplitude. It should say P
24:52 here And uh the co signs and signs are like that. So when
24:58 take the co sign of 13.98° you .97 and uh and very similar on
25:11 sheer ways. Yeah. Um Let's here. Uh It says the mismatched
25:25 are small but not negligible. Uh we're gonna neglect this point too
25:31 It's smaller than .97, not that smaller. So it's worthwhile thinking about
25:39 whether or not we should neglect um don't the P. And the X
25:45 emerge at the same point and see don't uh ah We picked the respected
25:51 parameters. Uh We're gonna pick the array parameter for both. Mhm.
26:11 . I have it wrong here. back up here that uh she wave
26:16 .97 on the horizontal component. It's .97 on the vertical component here.
26:26 Those respective angles are here and so comes from this raid calculation here because
26:32 raids are curved and they're curving up more for the share waves than for
26:37 P waves. Uh That's just following law and gets us these these uh
26:45 results. So this is simply confirming intuition about um few ways. Uh
26:56 about convertible. Here's the same thing converter ways. Um I think I'm
27:00 to skip over that and the next is a topic in poor elasticity.
27:05 that's the next topic. So I'm to uh leave that for later.
27:11 I encourage you to download these things puzzle over the questions in the in
27:17 exercises. And uh when I get chance I'll tidy this up. It's
27:22 a while since I looked at this I'll tell you it up. So
27:25 makes more sense and then I'll give a better version later. So I
27:34 I don't need any more of this . So I'm gonna uh not save
27:43 changes. No don't save any Okay now uh stop sharing. And
27:50 we're gonna go back to the lecture pick up where we left. Okay
27:58 remember this, we're in the middle a bunch of complications. And um
28:03 the complication here is concerning resolution. an interpreter always complain to the
28:15 Um processing, complain, give me resolution. You made this too fuzzy
28:25 . So um when we give it them, they don't like because it
28:31 more complexity than they have in the of their mind. But let's talk
28:36 the factors which which effect resolution. now most of our discussion up until
28:45 point in this class has been uh ones with a single frequency and they're
28:52 localized in time. They go on . Yeah we even did that when
28:58 were talking about reflection coefficients. Uh they were independent of frequency, you
29:04 a way of coming in in a of reflected uh same frequency. And
29:10 you can sum up waves like And the summation is the same for
29:15 frequencies. When we've developed reflectivity they were independent or frequency. So
29:22 means all incident waves of whatever frequency in the same way. However,
29:29 wavelet is composed of many such clean and they combined to make it away
29:35 , which is localized and when it , it still has the same
29:41 Since all the uh frequencies reflect Now the reflections from these wavelengths,
29:49 wave localized in time, wavelengths uh nearby interfaces will super pose. And
29:56 a problem. Now, what what does this, what determines the
30:02 of this wave? Well, a of things, including the source
30:07 So if the source is dynamite that sort of an impulsive uh and an
30:15 expansion of the local rocks and of it doesn't remain impulsive as the wave
30:22 out. It spreads out in time of the dispersion effects we were talking
30:29 before. How about an air An air gun um, um,
30:38 not impulsive. When an air gun , it puts a pulse, quick
30:46 , like a dynamite pulse into the . Of air pulse of air into
30:51 water so far? It's pretty much dynamite, but because it's air and
30:56 what it does is it expands and as it expands, it begins to
31:00 up a little bit and then it expands and contracts again. And as
31:05 floats up and back, it's oscillating this. So it has a complicated
31:10 full. We have fiber techniques and here in your processing, of
31:18 , how to uh worked out in computer. So effectively one impulsive.
31:28 can say the same thing about about size Viber size. The truck drives
31:34 to the shot point. Shot point still called the shot point from the
31:38 date. And when I was about age, uh I worked on the
31:44 and using dynamite. And so we uh set out the charges and the
31:51 thing set off. All the last they do is uh connect the charges
31:58 the firing mechanism and then the word go out, Everybody be quiet.
32:05 then uh, tradition would uh, the charges and the dynamite would blow
32:16 Water up into the air 50 And of course, we're all standing
32:20 away and might be standing near a case. We gotta be quiet,
32:27 quiet and stay quiet for about three four seconds after blast. You go
32:33 , come back. And that was way we did it back in those
32:37 . Um we don't do it that anymore on land. Um We got
32:44 case we use air guns uh, land. We don't use dynamite for
32:49 reasons that vibrators are less dangerous. furthermore, there's uh, you have
32:58 in the field, there's always the that somebody would steal it. And
33:03 in the wrong hands is not a thing. So we have,
33:09 vibrators and vibrator, uh, when truck drives up to the shot point
33:15 the pad, jacks the weight of truck up on the pad, so
33:22 uh, bearing down on the and then there's a hydraulic activators on
33:28 truck vibrating pad in a clever it's not just a single frequency,
33:34 a church like that. And or frequencies first, High frequencies
33:45 and the church might last for as as 10 seconds. Then we have
33:50 ways for adding all that into um of an impulsive sort. All of
33:59 techniques had sort of determined shape the signature of the outgoing wave. In
34:10 to those effects, there's raised, don't have just one source at a
34:15 , usually having a race. for example, for the vibrant
34:19 it might be three trucks lined up close together detail, doing the same
34:25 simultaneously. They want to be exactly sync. Exactly that's going to affect
34:35 in the marine environment, you have array of air guns which might be
34:40 10 m long and five m wide have 10 air guns scattered around that
34:47 . And the aerial distribution is designed focus the few ways down, It
34:53 be fired at the same time. they may have inspired different time
34:58 that's all under the control of And affects the shape, the duration
35:07 the shape of the outbound weapon. then as it goes down, it's
35:13 be modified. For example. We about before every single interface that it
35:20 , some of the energy gets reflected and then back down again by the
35:25 above. And we call that a multiple because it's going in the same
35:30 as the primary and with the same polarization. Because I've gone to reflections
35:39 with a little delay. And so little delay means that the shape of
35:44 wave as it goes down changes. finally broadens uh has a longer duration
35:54 the pulse as it has more and of these friendly multiples. Secret
36:03 And also uh here it says it's generation. Of course we're gonna be
36:07 the high frequencies um because of the now. Uh never mind attenuation for
36:16 second. Think of a perfectly elastic . But having these apparent multiples.
36:22 that's going to make it apparent continuation . Uh original pulses like this.
36:29 after it's passed through many uh many multiples, it looks more like
36:35 It gets spread out and that looks attenuation. So before apparently attenuation.
36:41 top of that we have real attenuation we'll talk about both of those
36:51 And then there's reflections from nearby Um Let's see. Well I'm gonna
37:01 you some some examples of that uh for that's what we're going to determine
37:09 the ability to resolve those nearby interfaces that you know that there are two
37:16 down there instead of one that's sort the definition resolution at here, it
37:25 the definition exactly in those terms. resolution is defined as a minimum separation
37:33 in space or in times we have resolution or special resolution. Talk about
37:38 in two different ways between two features uh for which you can constantly
37:46 So that depends upon the uh suppose have two ways separated by this much
37:56 but the way what occupies only this distance. So the way that is
38:02 going to obviously going to be able resolve these two interfaces easily with a
38:08 frequency wave like that, but suppose wave one has a wave and this
38:12 long. Then those two reflections off those two interfaces are gonna superimpose on
38:18 other and be poorly resolved. So . Um So obvious statements about
38:29 And so uh first let's talk about resolution we're gonna use uh here and
38:37 I'm pretty sure that uh the reason put that exercise and the exercise file
38:47 so I can generate figures like the you saw. So it's got these
38:51 times which is the distance between these crossings and the distance between these 20
38:57 and both of those are functions of the maximum frequency. And here,
39:04 don't mean the max, I mean frequency uh it's the maximum frequency and
39:10 spectrum of this. So this is time signature. The spectrum looks like
39:16 . And so the spectrum, the of the uh most energetic part of
39:24 spectrum is called omega max. And here is the MS. Here is
39:38 expression for the uh for the spectrum back up, here's the expression for
39:47 time signature and the expression in this we have closed expression for the uh
39:54 the spectrum and this is the real of it here and there's there's nothing
40:02 complex about this. And here it zero phase. What that means is
40:08 is no imaginary component to the And so Um what that means is
40:21 phase spectrum is zero since the imaginary is zero at all frequencies. And
40:30 here, just to remind you that fun for us to look at these
40:34 wavelengths because they have such simple shapes they are not causative that is to
40:40 uh back up here, this one uh this rival begins at t equals
40:52 infinity. So uh this this is way, way before the peak.
40:59 so uh you have to know when looking at data, what has been
41:08 done to this data to adjust the . And so for many purposes it's
41:16 to look at zero phase data but can be confusing then because when you're
41:22 looking at zero phase data, you that some of it has arrived before
41:28 think it's arrived, I think it's here. Some of it has arrived
41:32 before. Yeah, here's our respect Wayman. So uh for a single
41:45 interface separating two elastic half space is these frequencies that you see here reflect
41:53 independently and reflected wave has the same and so the same time seeing as
42:00 instant way. So in that case didn't have to go through all the
42:06 of all these planets, all we just look at one of them and
42:10 do the summation later to get in the real Earth there's gonna be
42:17 nearby. Perfect. So our friend really established this criterion and semi arbitrary
42:28 um um ah Mhm. It's a definition. So here are two ricker
42:40 separate uh with arrival times here and and here, the the arrival times
42:46 closer together and here they are very together. And so that uh really
42:56 that it changes from resolved unresolved when time separation here is less than a
43:06 of a period. So let's look some wedge models. This used to
43:13 very um this used to be a leading edge, your physics and when
43:19 came into this business there was paper by a friend of my father's and
43:29 received a lot of discussion in the community. And the title of the
43:35 was how thin is a thin And so to answer that question,
43:42 the author uh made this kind of . So let's see what we have
43:49 . Uh these are time normal incidence sections, okay, and here's an
43:55 reflect up here and uh Ryker waving right there now here is a
44:07 you see it's thick part here and thin part of the wedges over here
44:11 now off to the side here is uh the impedance profile, it's implied
44:17 . So in the upper half space have it's called Z zero. And
44:23 the middle part is uh called Z and it's bigger. And then inside
44:28 wedge it's bigger still. And then half spaces bigger still. So it's
44:34 up. So all of these let's have the same polarity. So
44:40 is we're gonna call this positive polarity it reflects off of um an impedance
44:49 , we have another impedance increase another. So so the wages clearly
44:56 you see two peaks. Then somewhere the middle here, the separation here
45:01 so small that we lose the So maybe it's here or maybe it's
45:07 , maybe it's somewhere in here we the resolution. And then the story
45:12 not over because look as you got uh thinner and thinner wedges, the
45:18 gets bigger and that's because down here going from z plus two Z minus
45:25 at once. There's no Z w in between. So you see how
45:30 two uh reinforce each other at the of the of the wedge. So
45:41 thin is a thin bed? you can say that we're going to
45:45 it and you can't argue with the , we're gonna say a thin bed
45:49 thinner than this. What happens if have an embedded increase? Shall we
46:01 from Z. W back to Z minus is the same as E
46:05 . So this part looks the Of course, down here we have
46:09 positive um uh impulse. And here have a negative impulse because it's a
46:16 jump in Z. And so we tell that here, we're not fooled
46:23 by that interior because that's um lower . And we can see my attitude
46:31 . I have to hear uh embedded increased right here. But then right
46:39 here somewhere, maybe here, maybe , maybe here we lose the
46:47 And so uh now look here, one, zero phase. That's this
46:58 does not look at all like this . So this one is symmetrical.
47:05 one is anti symmetric. So we that a thin bed uh thin bed
47:14 a thin bed uh interference causes uh the limit. Uh This is the
47:23 of this. Yes. Oh take is a functional time. And from
47:32 time derivative of this, you get that looks like this, so an
47:40 . And then look what happens as wedge gets thinner and that the the
47:48 goes away, it's still anti symmetric goes away. And why is
47:53 Because out here at the end we no uh no wedge in here at
48:00 , we just have this one and one which are the same in this
48:11 now. So back in those uh that guy was a famous uh
48:17 guy worked for Amoco by the uh but they didn't talk about other
48:25 which now we're concerned with, for , that was all normal incidents.
48:35 I don't mean to emphasize here that is oblique incident. I just mean
48:39 uh this is the problem which um was discussed here except I'm spreading it
48:52 here for oblique incident. So you see the different uh ah different ways
48:59 in different ways separately. What about if if it happens again and
49:08 So this one is gonna come up the same characteristics as this one,
49:17 a little bit delayed and and furthermore amplitude because it's got it's got those
49:26 reflections in there. Mhm. And about local murder conversions? So here
49:33 have converted to share and back to that uh reflected his share and then
49:39 to p you know if it doesn't convert back to P here then and
49:50 it goes up this year to the then it's going to be arriving a
49:54 like later. But if only spending time as a share wave in this
50:00 bed, the delay here is not to be much different than the delay
50:04 . So uh that's an effect which ignored in the previous analysis uh For
50:11 reason that was ignored uh because he doing normal incidents and the equations say
50:20 that this conversion should be zero normal and they didn't have very much data
50:26 those days to know whether or not was true as I said today we
50:32 that that not frequently but um well rarely we have normal incidents conversion at
50:48 we have conversion even at normal and reasons are not quite clear. Although
50:55 gave some good examples of good No um you can buy uh a
51:06 analysis packages from companies like Hampson Russell from companies like fruit grow and uh
51:18 those, make sure you understand uh those packages assume about these by the
51:28 , uh Hampson Russell uh interesting I know one of the principles brian
51:38 , I know him well. So the convention just last week is Buddy
51:43 is much less the public figure than , I think Hampson is the president
51:49 Russell is the vice president and I know how they share the money but
51:55 do well uh and they write a service in geophysics for many years.
52:02 only met Hanson maybe twice in my but I see it Russell uh every
52:08 several things anyway they had a successful . They grew it from nothing this
52:15 uh geeky geophysicist. I think they from the same university about the same
52:21 and they said let's start a business help ourselves. Uh And our friends
52:28 this and and the business grew now think they have all over 100 employees
52:35 that is a very good business. got they sold themselves to C.
52:41 . G. 10 years ago I they've been a city area C.
52:49 . G. For about that But recently CG sold them off to
52:55 company called Geo Software. And that's that's a big company that has thousands
53:02 uh the senior to allow Hampson Russell independently, just like they did before
53:13 you know, they're successful. You use their software, make sure you
53:19 uh what their assumptions are now. would think horizontal resolution would be worse
53:31 you have these plane waves coming down they're all uh vertical incidents. They're
53:38 hitting the um surface at the same . And so you think you know
53:44 it's horizontal resolution is gonna be terrible in fact it's not mainly because I
53:51 uh we don't look mainly at the traveling p waves but completely traveling and
54:00 also we don't have to worry so about the finals. Um it turns
54:17 that with modern seismic acquisition we have many rays impacting a given point on
54:25 subsurface resolution turns out to be a better than you would ever imagine.
54:34 when we migrate that we get extraordinary . Uh so there's a number of
54:44 for doing that. What one is uh mean to do that, Hold
54:49 a second, um groups or black um because we do migration, we
55:01 far better resolution than uh mr funnel and I'm not sure that I understand
55:12 reasons for that myself fully, but will talk about more about that in
55:17 imaging course. And then we can have other techniques for for enhancing horizontal
55:26 still further. Uh one combination is coherency processing so that uh you looked
55:36 enhanced to into decrease the coherency, want to increase the incoherence. See
55:45 not a good it's not a good that we call this coherency processing because
55:50 know, if if all the ways coherent, they're all the same,
55:54 don't see any resolution at all. have techniques for enhancing the incoherence.
56:01 the differences, local differences in uh images and some of those are were
56:12 by um I have former professor in department, his name is kurt Mahr
56:18 uh he's now a professor at Oklahoma . And so I'm on zoom session
56:27 yesterday uh and uh other important contributions made by a guy named Mike who
56:37 became a vice president at Apache. he was a colleague of mine at
56:44 and was hired away by Apache did well there vice president and also became
56:51 president of the ScG and I saw again uh two weeks ago at the
56:56 the ScG. Really quite quite a guy. And then there's another set
57:01 ideas called spectral decomposition. So here's idea of spectral decomposition. What you
57:08 is uh to thin beds here and illuminating that with with a broadband web
57:18 is so broad band that you get lost in resolution to reflect it too
57:27 to distinguish. However, suppose you to illuminate it with um a narrow
57:37 of uh air band of frequencies. when you get the top reflection get
57:49 and then it goes on forever since single frequency but also some of it
57:56 transmitted. And then reflected back. suppose you select a frequency such that
58:05 two way round trip within the thin . Exactly makes for one period of
58:13 oscillation of that wave so that it's in phase when it gets back to
58:18 topic back in phase with the primary , All other reflections will not super
58:24 effectively like that. And using that and you find that special frequency by
58:33 with all frequencies and selection the one works. And so uh that technique
58:39 been uh particularly successful. For look at this. So here is
58:47 spectral decomposition image uh computed by my colleague Greg Partyka and you can see
58:56 ancient river delta system here. Uh it looks like an air photo.
59:04 see all these subsidiary channels, You see the sand bars, you can
59:08 so much detail here. But this very 50 10,000 ft of rock.
59:14 just amazing to me that he could out that kind of detail um using
59:22 processing technique. So it was so that he was made to be the
59:28 distinguished lecturer back then. This was of his fault. So here we
59:35 this extraordinary horizontal resolution and mainly it from, well it comes from a
59:42 of things with acquisition and from Yeah. So here's the quiz.
59:54 define something called resolution. And is a good statement of you mean by
60:04 the ability of the data to detect in the substance MS Del Rio,
60:10 you call that? Yeah, I I would agree. That's a good
60:17 for what we said. That's not the words that Lord really is.
60:22 that's probably the way we think Okay. Yeah. In the example
60:29 I showed you some examples where the wavelet. What was that necessary?
60:36 , that was not that was not . It was maybe useful. It's
60:40 a good thing to try and we how to do that by the
60:43 Good. We have um we have processing that techniques which will uh process
60:52 the data so that is locally composed and so maybe that's a good thing
60:57 do and maybe not. Anyway, not necessary to not required.
61:03 okay. In beds are well reserved when the two way travel time between
61:10 and bottom is greater than half of dominant period of women. So that's
61:18 that was what uh that's what really . But using um these ideas of
61:29 decomposition uh well resolved when it's exactly the dominant. Yeah, so uh
61:41 I recall this true in either but because when you do a spectral
61:46 , you don't do it with wave you do it with one frequency at
61:51 time or a narrow band of frequencies the time. That doesn't yield
61:55 but it does yield the spectacular uh city, spectral uh images? Uh
62:06 , I would say that I would that truth. And so so uh
62:11 you have a question like this on quiz for example, and you think
62:17 might be a trick question uh What do is explain your answer. So
62:30 resolution, this is that that was horizontal resolution is limited by the finale
62:38 hence is quite poor, great Uh Well, we just saw an
62:43 uh spectral decomposition that that's not Okay, so let's see here,
62:54 think this is a good uh good to take a break. So let's
62:58 that. Let's break for 15. I'll see you back here at 3
63:03 . Good. So we'll stop the . Okay, coming back from the
63:27 , let's take up the fact that in the subsurface, not all the
63:34 are flat. Not only are they all flat, they're not all um
63:40 . Uh and we have curved reflectors the time. So that's what we're
63:44 to deal with next. So we're put us into presentation mode. And
63:56 when you have current reflectors, they and refocus the energy. This is
64:03 example taken from Sheriff and Girl Dark . This is uh where all these
64:12 are launched from the surface here, to the surface. Uh It says
64:22 normal to the reflector at every And you can see here a lot
64:27 them cross. So we call that very varied focus buried for obvious reasons
64:34 uh they get focused there for obvious . Over here. They get spread
64:39 and here they're not focused, they're . But you see the ones that
64:44 coming up here, see they're coming like this and on the other side
64:50 that they're coming up like. so obviously when you have lots of
64:58 um concentrated around here and the race out over here, that's gonna affect
65:05 received amplitudes and has nothing to do the Avio effect that we were talking
65:12 in lecture six. So that's a example of of how other effects in
65:20 subsurface. Besides reflectivity can call can amplitude variation with Austin. So let's
65:29 some a simple situation. So uh essential idea that we're gonna be talking
65:36 is called the radius of curvature. so here's a defined as a radius
65:42 a circle which locally approximation that Um It's not as straightforward as you
65:50 , because here you have two circles jimmy, we have a blue circle
66:00 and a red circle segment, both which are tangent on this uh elliptical
66:06 . But um uh they're different. so there's a more complicated definition of
66:17 encourager um than what I just local approximation, because you know,
66:23 of these locally approximated. But obviously blue circle approximates the ellipse better than
66:29 red circle. And so uh intuitively what we mean when we say um
66:36 of curvature, which it's expressing how the surface curves and uh from place
66:46 place. So obviously this elliptical service here is gonna have a smaller radius
66:51 courage over here than it does over . Yeah, we're gonna use the
66:59 that the ratings of culture are as for a wavefront curving up like
67:05 So here is a source radiating uh uh um reading it went down and
67:15 a historical wavefront down we're seeing this two dimensions. We're gonna call that
67:19 circle going down. And as it down, the radius of curvature increases
67:25 until it gets down to the And um we call it at that
67:30 , we call the incident greatest of is our survival. Yeah. What's
67:38 shape of this reflector? It has curvature obviously after the reflection and propagating
67:47 up to the surface. Uh this way of propagating back up looks as
67:53 it's propagating homogeneous medium um uh with the source down here we call that
68:04 virtual source point of the mirror source . And so uh this radius of
68:10 is two times the depth and we a minus sign because we have this
68:16 is curving down and the red one curving up. So there's a fundamental
68:24 from geometrical optics optics. People like Newton knew all about this and all
68:30 famous astronomers Galileo and so on. all knew about this. And so
68:34 is the formula which I present without and it says the inverse of the
68:40 wave plus the radius uh etcetera and inverse of the radius of curvature of
68:49 incident wave plus the inverse of the of curvature of the reflected way equals
68:57 the inverse of the race of the surface. Okay, so um let's
69:11 let's uh yeah, see what this . This is our first special
69:19 So there's no source rotating down through uniform overburden to a curved reflector.
69:25 here's a reflector and it has a occurs for the reflector of our sea
69:31 that's here. And we already know radius of curvature for the down going
69:37 is Z. So now we what we have to do is decide
69:41 is the radius of the curvature of reflected wave, which we get from
69:48 equation there. It is very Whatever that is, at the reflecting
70:00 , when it comes back up it an additional value of Z. It's
70:09 off this point. Uh value of radius of curvature increases by an additional
70:18 minus C. Why is it a C? Because it's curved downwards?
70:22 this is the the radius of the way coming back up to the service
70:28 its measure. So let's think about special cases. This is uh this
70:35 our expression now for the radius of of the reflected ray reflecting off of
70:45 surface reflecting off of a curved whose radius of curvature is this,
70:53 a plain reflector are sebastian is So this term is zero. So
70:58 have uh well, rosie rates the one power minus Z makes a minus
71:06 Z. That's what we we saw . Therefore, Point, this is
71:15 of the one liberty case here for factors that's reflecting off a single point
71:22 occurred to that single point is And so uh this is uh
71:28 divided by zero, divided into is . And then we take that to
71:35 minus one power get goes back to . And so this whole term is
71:40 . So in that case here, uh the radius of character or reflector
71:52 is a pointer founder turns out to minus Z. So we're gonna take
72:01 expression here and divide through by the of the reflected to make it non
72:07 . And then out of that reasonably form, we get this really complicated
72:15 . And so let's see if we figure out uh some special cases that
72:23 already saw before. So the flat , it's located right here. That
72:30 has um radius of curvature is So Z over zero is um zero
72:41 infinity is zero. So it's lying this plane here. And why is
72:48 on here? This is the uh . This is the value of uh
73:26 one half. Is the value of courage or the reflector in this
73:42 Oh, see what I'm missing vertical axis is Z over uh one
73:53 of curvature of the reflected wave. so that's uh minus a half which
74:03 we've seen before. That's for a reflect here's for a point refractor way
74:09 here at minus infinity. And for buried focus, we have this kind
74:15 situation where uh this general area as which are not focused uh completely but
74:25 um substantially. And so uh uh depth of this focus depends upon the
74:34 of this reflected. It's the same that we saw before. And so
74:40 look see what happens. Can you right in here? These uh all
74:47 bow ties and these are times of travel time, equal travel time
74:56 But they get bit into this bow shape because of the shape of this
75:03 . So we have here uh normal , zero offset profiles uh display this
75:13 like a shin uh this bowtie effect , here's the bow tie down
75:17 And uh further up here it's it's a bow tie. And here it's
75:23 it's developing sort of uh quasi bow tie. And so this is a
75:30 sing client and a deep sin You understand what we mean by a
75:34 client means uh reflective shape like Okay, so let's let's look at
75:40 picture as um uh as the source moving in this direction. So we're
75:52 normal incidence reflections. So source here sending our way back here, down
75:58 here, reflect and coming back to same point for each of these red
76:02 . So this one here corresponds to in along this uh line from the
76:16 marching in like, like, so it marches it all the way in
76:26 with the red arrows to this And as it keeps on going to
76:32 to the right, the reflection seems come back to the left because of
76:37 criss crossing of these blue arrows in and then it comes out the other
76:46 . Like so and so uh that's makes this trip lick ation. So
76:58 that make sense? Miss Del Let's go through this again. So
77:06 we marched in from the left and we are marching in from the left
77:12 uh having zero offset profiles, every of these has source and receiver at
77:17 same point, we're just getting to along this branch here. Then when
77:24 get to this point, something new to happen and the rays get crossed
77:29 so because we're at the bottom point as we continue to move to the
77:35 uh um yeah race across each other that. So uh and then as
77:49 continue to move uh to the come out the other side. So
77:55 called a trip, like a Because at times like this, You
77:58 1, 2, 3 arrivals a . Okay, so um uh we
78:11 both curd uh so so let's uh about this quiz. Question curved reflectors
78:20 cause acted anomalies at the recording These amplitude anomalies vary with surface,
78:28 were offset. And so produce an effect. So I'm gonna call that
78:34 , but it's not the one that normally talk about when if we say
78:40 maybe you're effectively if we mean the answer would be false. But
78:46 uh they do because and that happened with offset I call it true.
78:53 this is one of those things you to explain your answer right? This
79:02 is a little bit tricky because uh can have lots of surfaces which are
79:08 lots of lots of circles which are tangent to the reflector. But some
79:13 them uh um better approximation to that than others. So that so this
79:21 is false. Number three for a with a bird focus, a certain
79:31 of source locations. Zero offset ray three arrivals, not one. This
79:36 because of uh look carefully, we all the above and only A.
79:42 . C. So let's check it . This happens because of the propagation
79:47 to the reflection point. What we that was true. Because look back
79:52 here in the shallow doesn't happen. it's the same reflector but shallow,
79:58 doesn't happen. Okay, so that true. Um This happens because of
80:06 velocity distribution in the overburden unless it uniform. So obviously we did uniform
80:14 the example. Actually just to be happens because of the velocity distribution in
80:30 overburden unless it is uniform. So says that um um if it's
80:46 it doesn't affect the trip lick a doesn't matter what value it is,
80:55 uniform, doesn't matter which value it . So uh I would I would
81:02 that one is true. What what would you say? It's a tricky
81:07 . Um I'm going to condense the . That implication happens because of velocity
81:14 in the over word, unless it uniform, if it is uniform that
81:19 I can imagine velocity distributions in the that would cause that. Yes,
81:24 going to say that was true and one is definitely for all of the
81:34 . How about this? We got B and C. Um Oh so
81:57 , what would you say about Miss del rio? Well we didn't
82:11 much about Higgins principle in the, we did talk about it, we
82:15 about planer um um uh playing Uh Yeah, you could imagine that
82:26 that we talked about. Just a case. Um for Higgins pencils.
82:32 pencil might be applicable to occurred reflectors , but you know, um there
82:38 any answer all of the above. we got to pick only one of
82:47 . Yeah, so let's talk our through this certain for these source uh
82:57 where um these source locations, We three arrivals exist, three different rays
83:14 imp ends up on the reflector in places. And so we turned back
83:19 the source point, even if the velocity is nine uniform mm. That's
83:30 a very good question. But uh would I would go with C.
83:39 , you know, we could go and on with complications here. But
83:44 some point we need to uh change focus here. And this is that
83:57 at this point. We're gonna stop about classical waves and raise and talk
84:05 these three. These three effects Real world effects because they affect our
84:13 so strong. And so what I'm do is uh stop sharing here,
84:23 I'm going to um minimize this. . And share my screen.
85:21 so this is the subject now for next lecture of moral elasticity. So
85:32 the end of this lesson, you be able to explain how Hook's law
85:37 modified to apply to real rocks. is good because everything we did so
85:42 is inapplicable to rocks at all. we're gonna uh that's the first thing
85:48 need to deal with, modify the , modify the analysis so we can
85:56 it to real rocks. And then gonna talk uh you're gonna understand that
86:02 concept of effective stress and how the depend upon which fluid is in the
86:08 space. And of course large amounts money turn on that uh ability to
86:14 that question and how the standard equations understanding such fluid dependence are an arab
86:22 how to correct that. So that's brand new stuff. And then uh
86:30 high frequencies a new type of wave in which you haven't discussed at
86:35 And what's the implication of that? , so we're getting on that
86:42 Everything we discussed for seven lectures before has been classic seismology, equally suitable
86:49 either exploring for hydrocarbons or for understanding deep interior of the earth. And
86:56 we know only now, after seven , we know that none of it
86:59 true to truly suitable for exploration, it ignores the effect of process.
87:07 in the deep earth there's no maybe you don't have to worry
87:10 But certainly for the shallow prosecutor, , rocks have all sorts of hydrogen
87:17 . And to tell you the truth law applies has looked um find
87:29 It only applied to homogeneous solids like copper and brass and maybe glass.
87:38 would not have been happy to think how it would be applying to uh
87:44 know, a rock from the lower of the earth. No proxy,
87:48 lots of different minerals. And are have all sorts of heterogeneous. So
87:54 coax law can't be applied to heterogeneous . Are rocks have all sorts of
88:01 has grains, different sizes, shapes minerals. And of course, if
88:06 minerals are all going to be uh tropic, of course. And they're
88:11 gonna be scrambled up. Uh maybe they're scrambled up and maybe
88:15 Uh but since they're anti psychotic, orientation is going to make a
88:21 And we can make an ice a rock out of anisotropy minerals simply by
88:28 them up randomly. And so some are like that and some are
88:32 For example, shales have uh some the minerals are shaped like plates,
88:40 are shape, black plates and those are not um oriented randomly, they
88:47 oriented flat. And so that's a thing that causes shales to be anti
88:54 traffic. So we're gonna we're gonna with anisotropy uh in a couple of
89:02 , but already you can see that in a real rock, that's gonna
89:07 an issue. Then in addition to grains is going to be poor
89:12 which also has many sizes, and also different pore fluids. You
89:17 , the different pore fluids sort of to the different minerals. But here's
89:22 a new a new idea, hydraulic . So, um if the fluid
89:31 squirt around In the four space as wave is going through, that's surely
89:39 to make an effect, is Now? Um that's gonna happen or
89:44 , depending on the frequency of the . So there is immediately, you
89:51 frequency dependence coming from the poor Now, you gotta recognize that some
90:02 are entirely included within the grains. is when that grain was formed.
90:09 of a grain of sand, it have a bubble inside of it,
90:14 is ancient. Air. So that's . And that's not quartz. Uh
90:21 uh ferocity. But we normally don't that. When we talk about
90:30 that kind of we call that included . And we say that it modifies
90:36 properties of the solid instead of being of the process, it's part of
90:43 solid and it modifies the property of solid. So that in the case
90:47 said it's not the uh that grain sand with the bubble of ancient Aaron
90:54 of it wouldn't have the properties of , it would have the properties of
91:00 have module i which are a little different from those reports because of that
91:06 for us. And think about there can be a huge pressure difference
91:16 the grain scale, just millimeters on grain scale. Uh different fluid pressure
91:25 the fluid and in the south. we squeeze this rock and most of
91:30 weight is borne by the south. that uh in in in the simplest
91:36 , you can imagine that the uh total pressure on the rock is due
91:44 the weight of the overlying rock. meanwhile uh right next to the grains
91:51 is um oh, it's lowered which be at a vastly different pressure.
92:01 could uh in the simplest case the in the fluid is due to the
92:08 of the overlying water, not the rock. So it's a lot less
92:15 the than the pressure in the uh the solids and this uh this is
92:23 changes on the millimeter scale. So bound to have an effect, isn't
92:32 ? So when we uh so I said the pressure on the floor is
92:35 be a lot less than the pressure in the grains. Uh And I
92:39 you're thinking of while outs and overpressure so that uh that is happening because
92:51 um the pressure and the fluid is as low as I just said,
93:00 to the weight of the overlying it's higher than that, but it's
93:03 always less than the pressure the grains . So um we'll have a chance
93:12 talk about those sorts of issues Yeah, we're going to handle the
93:24 hydrogenated by simple averaging and it's gonna different for different physical properties. So
93:30 example, for the for the the , this is the density of the
93:35 , not the density of the And uh it's this simply going to
93:40 uh the average of the mineral density you take the sum over all the
93:46 from one to end, however mineral our and uh you specify the volume
93:53 of each mineral and the density of mineral and some of them all
93:58 And uh some of these fractions is to be one of course. And
94:03 that's obviously straight forward. So this just adding up all the mass in
94:08 volume of our straightforward. So so next thing is to um well why
94:18 we doing this? Well because we that uh the velocity depends upon the
94:25 modules and share models and the We know that we're not going to
94:32 add up all the velocity and that's the way we're gonna do it,
94:35 gonna do the different parts separately, density separately and the modular separately.
94:44 so this is the density part very . Now the in compressed guilty of
94:49 boat markets is harder. So the obvious thing to do is to just
94:57 analogous li uh what we did before this was first done by a guy
95:03 Foyt spell pronounced in that way with G. Is silent and I think
95:10 german. And however this idea uh to just adding up all the
95:21 But that that makes no sense. does make sense to add up all
95:25 math, but it doesn't make sense add up all the in compressibility.
95:29 this is um not an accurate And then furthermore uh to make this
95:36 me boy, assume that all the are all actually topic. That's not
95:41 . So uh this is not a fun. Huh? So we need
95:46 better yeah, with a better you can actually derive that instead of
95:55 it but in order to derive you to assume that that the strain is
96:00 in all the grains. So that's not a good thing that you're gonna
96:07 soft soft minerals and hard minerals, grains and hard grains and the and
96:12 strain is going to be different. each of these when you subject to
96:16 the whole thing to uniform external So the strain is not going to
96:22 informed. So what else? Well not a good assumption. So here's
96:28 bad assumption. But you'll see immediately make it um We're gonna assume the
96:34 uniform everywhere. Now that seems like sort of a better um uh but
96:44 assumption but the more you think about , the more that's not realistic except
96:48 the case, except in the case you're applying these ideas to a mixture
96:53 liquids. So if you have gas brian and oil in the pore space
96:59 rock and you ask yourself what's the of that mixture of fluids? And
97:06 is a good formula. That's what says is actually valid for mixtures of
97:14 . So now uh this was done a guy named Royce. And so
97:22 uh after these guys had made their Royce and fight another guy named Hill
97:32 along. Bill came along and he that if the minerals are uh actually
97:41 , then this one that you see this one that you see why it's
97:48 in this one that you see here a lower limit to the actual after
97:56 the actual uh, in compression And the voice formula, is it
98:04 limb. And so Hill said, , why don't we just take the
98:09 of these? So he didn't give good reason. We could have had
98:18 away there. We know it lies these two, but maybe it's
98:24 we should use a weighted average. , uh, three parts uh,
98:31 a voice divided by four. And , maybe that's a better way.
98:35 is no justification for this form. didn't even try, he said,
98:41 , he said this might be an lesson. However, I can tell
98:45 that very frequently in geophysics, we this seriously and we find the voice
98:51 and the voice average for all the metals and add them up and uh
98:57 the average like that. And then and the hill average and that's called
99:03 VR voice. Voice. Hill So there's no justification for that.
99:10 you need to be alert and somebody laying that on. So you can
99:15 the same thing with the share Uh, that's a white style
99:22 average. And here's the worst style . And Bill also proved that this
99:28 an upper limit. This is the limit. So he suggested you might
99:32 to do this now, because most are more similar to each other than
99:40 are to brian these various differences between various minerals are usually handle in this
99:48 way I said here really no Either experimental or um uh theoretical.
99:59 there weren't celebrities, but we use any anyhow. Since it may not
100:04 too far off, we're gonna show some uh I'm gonna cast some doubts
100:10 your mind about that later. The theory by uh by brightened by
100:23 and by help was all fairly Later. Other guys named Washington Street
100:30 , these are americans derived some uh tight bounds. And those are called
100:37 and lower hs bounds. And here says that the the real in the
100:41 income possibility yes, greater than Lower bound with this formula. And
100:48 less than this upper bound this And you can see that the sheer
100:53 I of the expressions are in there here's the volume fractions of both volume
100:59 here, both volume fractions are Um So these are tighter bounds.
101:06 uh what I've shown you is in case where the subscript one is lower
101:13 subscript two. And you also have same thing. Uh for the sheer
101:18 , if uh if you have in one is less than in compressible
101:28 but sheer models. One is greater cheer models to that's a problem.
101:33 so we should skip over that similar for the share margin line. But
101:41 , I remind you that all these are assuming that the individual minerals are
101:47 should topic and that's never true. uh this is a good PhD thesis
101:53 somebody to huh generalized this theory for for real minerals. But that's not
102:06 focus of this course. But before pass on. So the real issue
102:13 is the force we need to Why are we talking about balance instead
102:19 specific elements, estimate why don't these ? They're so smart. They've done
102:23 this, complicates that. Why didn't give us the exact answer instead of
102:27 and lower balance? And the reason because the average module I depend upon
102:32 micro geometry as well as these So think about that for example suppose
102:39 the softest Merrill is preferentially located at load growing points within the aggregate.
102:47 that's true then the average modules will less than if the softest component is
102:54 located. I think that's pretty obvious the softest mineral is at the key
102:59 within the aggregate, then the whole is gonna depend upon that softness more
103:05 if the soft component is sort of scattered around. And normally we don't
103:13 much about how the uh micro geometry . We can actually take a rock
103:20 rock and study it quite extensively. how we could do it. Here's
103:24 rock, study the outside and we off a very thin slice. Look
103:28 the next layer, slice off another slice. Look for the next
103:31 That's one way to look at Another way is you could shine x
103:36 through it and uh an image with of its micro geometrical complexity with x
103:44 . But once you understand this what does it tell you about this
103:47 for this one? Not much. , I think that's a useless
103:53 It's useless to try to understand the micro geometry of a real rock.
104:02 you can make some idealization that you're and interesting. But I think it's
104:07 waste of time to try to figure what's really there. And here's the
104:13 the most fundamental heterogeneity is the differences the solid and fluid. So all
104:19 solid components are different, but nothing there are a lot more similar to
104:24 other than they are to the So I ask you in situations that
104:31 looked at, where did the ferocity nowhere. Now these elastic properties are
104:41 functions of composition, but now we to consider both properties of the grains
104:47 of the pores, recognizing that on small scale is really competent. So
104:57 another question ignoring this heterogeneity issue. did the pressure appear in this great
105:06 , nowhere Now these things are implicitly upon pressure, whether or not the
105:13 . But we have ignored that up now and now we have to talk
105:21 it because the pressure on the grains the pressure on the force is
105:28 So it's gonna be different in two . One is it's going to be
105:32 because for millions of years these pressures been uh developing and like they say
105:40 major differences on the grain scale between and uh blood pressure and brain
105:46 But also as a wave goes it's gonna put an incremental uh pressure
105:54 the grains and a different incremental pressure the porch because it's a confluence.
106:02 that's gonna turn out to be Now the war's pour fluid is so
106:13 . Same. The greens that we to uh consider that separately. For
106:27 For example, the in compressibility of is about 5% in compressibility of 20
106:35 less. Sure moderates and water is and the share market of the minerals
106:43 something so that ratio is zero and density of the water is about a
106:47 of the density of the minerals. uh these are the biggest differences in
106:57 rock because of crossing we encounter the time we ever need to mention the
107:10 compressed adults. Okay. In wave , the natural stiffness is to discuss
107:16 M launch general models and share We need to, we never need
107:21 mention either K. Orlando in um propagation only M and mute and at
107:32 and and actually topic generalizations. But we have uh waves propagating through
107:40 we need to understand the effects of floods and they depend upon the icy
107:45 application of pressure through uh the That is the the poor fluid is
107:55 gonna have any sheer uh stresses in . If it does it's gonna flow
108:00 readjust those and it's gonna flow. readjust not instantaneously but with little delay
108:10 upon um you know, viscosity and like that. So and and also
108:15 the shape of the pore space. what that means is the high frequency
108:20 are gonna travel uh different velocities then frequency waves because of this uh pressure
108:29 in the flu. Yeah. When did his his analysis, that was
108:38 uniform medium. But now we've got constituents in the simplest case we got
108:44 and pores. Never mind the complicated . But there's the, since we
108:48 two there's a possibility for them to deformed together. So called in
108:54 And this is going to make ordinary like those that we have been
108:58 But with only minor modifications to account the fact that we now have to
109:04 . There's another possible team, they deform independently all that out of face
109:11 which makes a new type of wave is unlike anything we've been discovered,
109:16 been studying completely different. So we um So for, oh an ice
109:29 tropic rock saturated with fluids saturated we're gonna have one kind of uh
109:39 kind of way of actually two cancers . And S waves. Uh Like
109:44 been studying but with minor modifications and other kind of way that you haven't
109:48 about which is unlike anything we've been because of the fact that there's two
109:56 first. Let's do the one that uh ordinary. Okay one question it
110:07 um look at the bottom bottom is of the above. So the theory
110:13 elasticity is not strictly applicable to rocks they're they are and I should tropic
110:22 something high pressure. So um Miss rio. What what would you
110:29 Yeah. Yeah. So uh Yeah. Uh So we're we haven't
110:38 anti centrally yet but the theory of elasticity has an anti psychotropic version which
110:43 gonna discuss uh tomorrow and uh uh tomorrow and the next day and the
110:52 friday and uh high pressure is not . The main thing is they're heterogeneous
111:03 . The density of rock is given this formula. Is that uh is
111:08 correct? Now that's not what I you, I showed you for the
111:13 own and so this is for the . And so would you say that's
111:21 ? I didn't hear you. Yeah that so this is we're just adding
111:25 all the mass in Iraq some of solid uh some of its uh
111:32 Uh and so on and maybe the the porosity has different fluids in
111:37 So maybe we got oil and So that's just included in the
111:43 And so and so maybe the solid has many minerals. That's okay,
111:49 included in the sense. So that true. Now the incomprehensibility of Iraq
111:54 given by this formula here. And um you hear? Okay, so
112:05 got a 1/2 and um uh this looks like avoid some except it goes
112:14 the grains and the ferocity and this like a Royce some. So uh
112:20 talked about the uh the uh so averaging voice type some Royce type some
112:27 we criticized that for Stalin's but I say anything about that for Iraq.
112:34 what would you say? Is this or false? Yeah. So this
112:41 uh really a bad a bad estimate law. Okay, so uh
112:52 I didn't teach you this, but think just with your common sense you
112:56 see that's gonna make a here's one to bring that into focus. Let's
113:05 about this instead of in compressible. think about sheer models. So think
113:10 sheer models here and share models here share models here. Now let's think
113:16 this uh some some of it's gonna of these terms and maybe I equals
113:22 is gonna be the porosity and in is gonna be uh brian and the
113:30 , sure, marvelous. For that inverse is going to be infant.
113:35 that's bad. So that's just uh good example of you're applying your common
113:45 to extend what I taught you express Also he read the the question.
113:58 , Yeah. No, I don't . No, I'm not, I'm
114:12 . Yeah. Yeah, I feel a number please. Okay.
114:22 So I'm not familiar with lots of specialized techniques that companies use. But
114:28 tell you what, if you send uh, description of what it is
114:34 next week, I might be able say something more uh intelligent about
114:44 No. Well, that's that's bad him. Uh You should say,
114:51 , boss, I'm new at Please explain that to me and do
114:55 now because if you wait for six , it's too late, right?
114:59 gotta do it now when he first this on you. And and if
115:03 says read this, that's okay. you read it and also send it
115:07 me and I'd be happy to discuss uh, with you uh and learning
115:14 at the same time, right? time. Yeah. You wanted you
115:22 be able to understand what you're You don't want to function like a
115:26 and he doesn't want you to function a machine. He wants you to
115:30 . And uh you gotta have something think about. So say,
115:36 boss, I'm confused about this. you explain this to me? And
115:40 , he might not understand it He might say, oh, well
115:45 knows this. Say, well, I don't know that. So can
115:49 help me here and he'll eventually give some something written. Yeah, he's
115:54 to. Okay, so, this of effective pressure is um uh working
116:05 us. So, because we have constituents, we're gonna think of this
116:19 enforced to constituents. We're gonna have variables to describe the state of
116:25 The stress is going to be defined the average stress and by the fluid
116:31 . So, this is grant, course, everything. And this is
116:35 pressure only. And the strain is be described by the average strain and
116:42 change in um volume of the The do notation means the fractional change
116:50 volume. So now in the you can adjust the pore pressure independent
116:58 the confining stress by injecting or withdrawing . You know, it's very
117:04 you have some sort of a you a rock, you have it inside
117:07 container and you're going to apply pressure or maybe you got transducers on
117:13 You're gonna send waves to gonna do kind of experiment uh inside this
117:19 let's say, you want to say velocity is a function of pressure.
117:24 would be a good deal velocities as function of pressure. So, it
117:28 you're going to be putting pressure on rock. So, smart thing to
117:32 is to put a jacket around the . Some kind of a membrane around
117:38 rock and then apply pressure with a on the outside of the membrane.
117:43 then the membrane is gonna squeeze the . And it's also going to squeeze
117:49 fluid inside the rock. But no no uh there's gonna be no interchange
117:57 the pressure observed on the outside and fluid on the inside. That saying
118:03 fluid on the inside is gonna have own pressure. It's not the same
118:09 the pressure on the outside. Yeah. So that's called that's a
118:14 test. And the pressure when you the rock in this way uh the
118:21 of the rock gets to be higher . Obviously also the fluid pressure inside
118:27 rock is also higher but by a amount because most of the load is
118:32 by the grains, right? And so the average pressure on the rock
118:37 given by the pressure on the And then you have the fluid pressure
118:42 the inside. You might want to something else might want to put a
118:48 in that um uh in that um and let the fluid pressure drain
118:58 So when you squeeze the rock, fluid from inside rocks squeezes out.
119:04 so that would be called a drained . We allow the rock to
119:09 And so then the fluid pressure inside always zero because uh you allow the
119:17 to drain out. Here's another clever to do. You squeeze it and
119:21 have this hole in the membrane. uh as you squeeze it you allow
119:29 uh the uh flood pressure from the to also get into the inside.
119:36 that means that the flood, the pressure and the inside is the same
119:40 the fluid pressure on the outside. you can imagine variations on this.
119:47 you can imagine if you're rock physics , uh you might want to do
119:53 things experimentally. We observed that when when we increase the average pressure on
120:02 that increases the density of course, when we increase the flood pressure on
120:10 rock, that decreases the density because uh uh the flood pressure is squeezing
120:18 , squeezing the grains apart. So might uh think, okay uh First
120:26 let's do is have a jacketed chest uh the yeah uh the blood pressure
120:39 the inside is less than the than external pressure. And then start pumping
120:44 in with increased fluid pressure and we the density. So first we first
120:50 the density by decreasing the volume and we increase the volume and decrease the
120:57 . So for a certain combination um these two pressures, the density is
121:09 . And we recall that combination the pressure. So increased the average
121:18 Some increase the fluid pressure by some something else. And if the density
121:25 constant we're gonna call that the effective . And to a first approximation that's
121:33 a simple uh difference. First approximation simply the difference between the average and
121:44 portrait. So similarly you can do same thing um uh with regard to
121:54 distances for the velocities. They also on the average pressure and the fluid
122:00 to some sort of combination and to good approach to a certain approximation.
122:07 That would be only the difference. so uh it's it's very common for
122:15 difference to be called the effective but that's really an approximation. Um
122:25 the earth the stress, the stress not a single pressure, you know
122:29 your job that the vertical stress is from the horizontal stress. So uh
122:35 that case we can define an effective uh tensor which is going to be
122:43 equal to the effective to the differential , which is defined in this
122:49 We take the average stress for each here and we subtract off the poor
122:55 only along the diagonal. These shear . Uh We leave alone because there
123:02 no shear stress in the food. it's the uh so the differential stress
123:09 equal to the average stress minus the fluid times. The identity tensor,
123:15 is uh only um which is equal one on the diagonal and zero off
123:22 diagonal. In other words, the wave velocity is approximately given uh has
123:31 pressure depends approximately given by this differential and the same for the V.
123:36 . And it's the same for the . And the same for the uh
123:45 generally we can say that uh effects stress is has a little coefficient in
123:52 , which is an empirical primary And you do experiments to determine that you
124:00 out that an is usually not equal one. So the effects of stress
124:04 usually not quite equal to the differential . And furthermore it's not even a
124:10 but it varies with stress and poor pressure and furthermore different for each different
124:20 . So it makes a complication when include this. So we're gonna talk
124:24 so this is not a course in physics. This is of course in
124:27 propagation. So we're gonna minimize uh issue, we're going to talk about
124:33 the differential stress. Okay, now the sub service or zones where the
124:42 pressure is unknown enormously high. So over pressured zones affect the drilling performance
124:50 the reservoir performance. So uh if reservoir is under high pressure uh it
124:59 to it's easier to produce that reservoir the borehole. But over time the
125:05 you withdraw fluids from the reservoir, pressure in the reservoir drops. And
125:11 uh there comes a time when the stops flowing. And so you might
125:17 to do something if you're the operator say well you know, I know
125:21 a lot more oil down there. got to do something to get it
125:24 . So they have their techniques. And we're not gonna talk about that
125:29 this course, but I'll tell you when he's drilling the well before he
125:33 knows whether it was in the oil there or not. Uh These over
125:37 zones affect the drilling performance. The can tell as he's drilling from the
125:47 the performance of his drill bit, the uh rocks down there 10,000 ft
125:54 our overpressure or not, let's see what he can control is uh the
126:01 of the mud in the borough and can control the weight on the drill
126:05 and he can control the rate of of the drill bit and all these
126:11 go into uh his understanding so that um when he when he encounters a
126:19 of over pressured rock, he knows from the from the drilling performance of
126:27 drill bit and he usually doesn't want wait until until um uh water starts
126:37 out of the top of his uh considered bad luck when uh when it
126:45 out like that. And so immediately soon as he detects the the um
126:51 performance of his drill bit, he starts putting more mud into the
126:56 in other words, holes already filled mud, but he starts in putting
127:00 heavier mud to keep that overpressure down . So it doesn't blow out,
127:12 ? But I don't consolidation your Well now it's on the same rock
127:29 . Right? Okay, Okay. uh so that that rock sample might
127:36 consolidator might be unconsolidated depending on where was collected from. And now I'm
127:41 to think what could those two terms . Um Are the pressure conditions different
127:47 the two conditions? I'm thinking so both cases are drained, right?
127:55 both cases are under. Okay. So why what could it mean
128:04 I'm thinking that um Well I can a number of different things that might
128:11 you need to understand for yourself exactly those mean. So you know,
128:17 gonna be like somewhere around the there's gonna be a manual which was
128:23 years ago uh to explain what they by these things. And you need
128:27 get your hands on that main. , but he needs to make time
128:35 and you need to make him make . Uh because if you ask him
128:40 questions uh these are ignorant questions but okay. You're supposed to be
128:45 You're supposed to be smart but ignorant this point. So six months from
128:49 ? You're not you're not supposed to ignorant. You got to get these
128:52 answered now. And so uh maybe boss is the wrong person. Maybe
129:00 where is the previous guy who did uh this job? Yeah. Oh
129:11 boss. Okay. Well so if nobody else to ask. You gotta
129:19 him by the collar and say explain we mean by these tests.
129:25 And I think he'll appreciate it. , if he wanted you to be
129:31 , he probably would. He shouldn't hired you. Oh, well,
129:39 should be glad. Is that the I ask these questions is so that
129:43 can do a better job for Yeah. Okay. Now,
129:50 uh, so in the earth, , the occurrence of over pressured zones
129:56 hard to predict in advance, but one of the things that they asked
130:01 us to do these days. They for us as us is as they
130:04 , okay, you're so smart. got this all this education from the
130:08 of Houston. Tell us where in subsurface we're going to find oil.
130:14 good. And also the ask us uh, you're gonna do this seismic
130:20 . We're gonna do this, this sign, the seismic acquisition.
130:24 going to do all this sergeant We want more than just an image
130:29 the surface. Something. We want to tell us what's down there.
130:33 much you can, for example, you tell us what kind of
130:37 What are the mythologies down there? you tell us what are the fluids
130:40 there? Is it brian or its and gas? All these things were
130:44 , also we want to know is over pressure or not? And the
130:48 is because we hate it when we a blowout. You know,
130:53 guys quit, uh, billions of get lost when there's uh we know
131:01 to handle high pressure, especially if expecting it. We have a well
131:08 and we we drill the well and we go, we have the capability
131:15 deal with overpressure when we find But if we find unexpected oil
131:20 that's a problem. So can you , please tell us what or what
131:27 should we expect down there? um uh the general idea is that
131:38 say, well sir, we we our acquisition, we did our
131:42 we did our processing, we did analysis, that we did our velocity
131:47 . And uh we noticed that at certain depth at 8000 ft, the
131:53 are low abnormally love. So you that means high, high pressure.
132:01 if the pressure down there are high pressure, the floods are squeezing the
132:05 apart which are lowering the blossoms. that's a good idea. Except that
132:13 things can also cause lower velocity. maybe it comes from uh mythology with
132:21 of play in there. Maybe it's porosity. Maybe it's a zone of
132:27 lee, high porosity. Or maybe gas in the pore space. All
132:30 things are going to lower the So uh we can't be too simple
132:37 about with our predictions. And in inexact science. Yeah. Well,
132:54 we're not measuring density from the right? We're measuring seismic arrivals.
133:01 from that. We can deduce And uh so uh Mhm. Um
133:11 know what you're thinking, you're thinking can take these reflectivity equations and uh
133:18 the jumps in velocity and the jumps density and the jumps in sheer
133:23 And we talked about that um uh than six. But if you go
133:29 there, you realize you'll remember that said we're not very good at um
133:38 those things. Um And so uh here, I'll remind you of the
133:51 there uh in the first place, reflectivity is gonna be telling us about
134:02 jumps and uh in elasticity, the Vp and the jump in V.
134:08 jump intensive three jumps that we're looking . And we have three things to
134:13 . We've got uh intercept gradient and solved problem except that I talked about
134:20 today. We're not going to be to determine the curvature very well.
134:25 we have three we um properties that want and only two data. So
134:33 not gonna solve that problem. There's problem, which is that what we
134:40 if we could solve that problem, we have is the velocity jumps and
134:45 density jump, we don't have the itself. And so the way you
134:50 reduce the density itself is you oh, well, I know the
134:54 up here and I know there's a here. I know the jumping density
134:59 that interface. So the density and lower layer is simply the density in
135:03 upper layer plus that jump work our down. So that's called uh seismic
135:10 really all it is is integration but call it seismic conversion. So um
135:20 if we could get the density jobs we could do that but we don't
135:25 the density jobs so we can't do . And then we have the
135:31 It's gonna give us the porosity. It's a pretty straightforward uh calculation from
135:38 trip ferocity but still it doesn't tell the pressure. So you see it's
135:45 X. We don't have a really idea for that, how to predict
135:54 in the subsurface. And so that's we still have mistakes from time to
136:00 . And there's one uh Has put on my heart and my brains which
136:07 the so called oil spill in the of Mexico that was done by
136:13 P. in the year 2010. um that was not an off
136:23 that was a blowout and it was off not caused by uh the drawing
136:29 drill uh successfully and just and made discovery very nice discovery. In fact
136:38 discovery was done by one of my . We're all proud of ourselves at
136:43 point. And that the boy out at a later point when they were
136:47 a so called so called completion of well to get it ready for
136:52 And and the completions, engineer made grievous mistake which caused the boil,
136:59 17 lives and $50 billion. One less. So it was not our
137:07 as your physicists, but we we some of the blanks. And the
137:12 we bear the blame is because we not make an accurate prediction in
137:18 So as they were drilling down thousands feet for the reservoir, as they
137:22 drilling down, they encountered pressure which higher than they expected and it took
137:29 time and money to solve that And so when they finally got
137:33 when they finally solved all those problems and got down to the reservoir,
137:39 were behind schedule and over budget. that put pressure on the management decisions
137:46 to the to the error by uh , if we could do a better
137:54 of predicting overpressure, if this that would be a great wow contribution
138:00 me. Well, um huh I a big step in that direction about
138:10 25 years ago with a colleague and called the of the scott Thompson pressure
138:17 prediction algorithm hardly ever used. Uh reasons which um I won't go into
138:30 uh there's uh there's hope for doing better job in the future than we
138:35 today. So let's have a request of the following is the best complete
138:44 of this statement. So the statement the mechanical properties of rocks, including
138:51 velocities are determined by a. Or C. Which is the difference
138:59 D. Which is a combination. didn't hear you, I didn't hear
139:06 . D as in David. D as in David. That's what
139:09 looking for. Okay, now. with that preparation. Now let's talk
139:17 body waves in poor oral elastic And our principal aim here is to
139:24 the effects of fluids on sizing philosophy uh this will be important for amplitude
139:30 as in lecture. See we'll revisit topic and for four D.
139:36 So uh and for the sizing is you do is you make an image
139:41 the reservoir and then you do some and then you make another image of
139:45 reservoir and you see exchange. And you want to interpret those changes.
139:50 the changes will be changes in fluid pressure in the restaurant and changes
139:59 fluid um content in the reservoir. mean if you're pumping out oil that's
140:06 replaced by brian uh from deeper in section and maybe a change in
140:14 And so uh you need to do post seismic interpretation of the 40 differences
140:21 the civic data which is seen. so all that is depending upon these
140:26 which we're going to discuss next. gonna analyze separately the density and the
140:32 marshall. Okay. Start by assuming a psychotropic grates and by recognizing that
140:43 pressure in the four flu is different the stress in the solid and the
140:48 in the porcelain is different than the in the salad. And this is
140:52 both with respect to the rock before wave gets there. And the very
140:58 due to the passage of the way consider um we consider a mass
141:07 not a volume element but a mass large enough to contain many grains.
141:13 reason we call it a mass element because um uh we're going to consider
141:24 untrained wave propagation and there's no opportunity the fluid to drain out. So
141:32 when they talk about the volume they say well, so the fluid
141:36 gonna squeeze in and out. But volume of the rock is gonna be
141:40 arm of the grains is gonna be same. So that's the volume.
141:43 I'm going to emphasize that we have a mass element here uh call them
141:50 , It comes from the word you know, know what a pixel
141:54 and it contains many grains but but small enough to the average stress and
141:59 average strain does not very much across volume. And after thinking about
142:05 we conclude that the point to point on micro scale within the solid part
142:11 much less than the difference in these the solid and the food. So
142:16 you're looking at point to point inside grains, that's uh differences but not
142:21 much. The big difference is the between solids and should. And we're
142:26 going to assume that the poor fluid uniform within this maxim. So that
142:33 that when the wave is going through gonna squeeze the rock and it's gonna
142:39 some parts of the pore space are to be compliant and others are going
142:42 be not so compliant fluid is gonna around inside the pore space of the
142:48 that's going to affect the stiffness of rock. And so we want to
142:55 that um we want to consider that have um the frequency of the seismic
143:04 is so low. Yeah, that happens quickly while the wave is
143:13 you know, passing through. And can assume that the fluid pressure is
143:19 everywhere in this sample of rock. , you know, it means long
143:25 low frequencies. And so that means we're gonna expect velocities to be different
143:36 sonic logs or an ultrasonic laboratory experiments expect them to be different because they
143:45 have necessarily this condition that the four pressure is uniform within each maximum.
143:59 . Um The father of poor Destiny, there was a guy named
144:06 . O. Belgian fellow and from speaking Belgium. And so that word
144:14 pronounced B. O. In the way. And he proved that for
144:18 sound ways the velocities are just like , except that what we deal with
144:24 the undrained uh marvelous and the undrained marvelous instead of before when we talked
144:33 the corresponding expressions that we talked about had no sub scripts and now we
144:40 studies indicating that uh as the wave no food interest or leaves the maximum
144:48 dream. So this is really good . We can just use all the
144:54 that we did in the first seven . And now all we have to
144:58 is recognize that the parameters depend upon composition, solid composition and the fluid
145:06 and the pressure and the fluid pressure . Now that's all just contained in
145:10 . So that's really good news. did not waste your time with the
145:14 seven lectures talking about elasticity because we're use all those results right here and
145:23 . Uh So uh that's very good and we owe it all to be
145:31 now. Uh let's think about the of these things on these things.
145:54 Are you telling me that this slide not in one of the other files
145:58 gave you. Okay? So that be I will uh I will send
146:06 an updated file. I'm surprised by , but I'll send you the updated
146:17 send me email tonight to remind So what I do is I uh
146:28 before I come in here and give lecture, I go through the the
146:34 file and I make uh improvements uh think to myself I can do this
146:42 than I did before. Uh, I make improvements, but usually the
146:46 are so minor that I don't bother point them out to you. But
146:50 you bring it up, um, remind me and I'll send you the
146:56 file tonight to the blackboard, you , Notify years what would happen?
147:12 there. Okay, so you write me, telling me which, which
147:18 we're talking about? Okay, now density is easy, density is always
147:27 . We say that the density of rock is equal to the density of
147:31 solid times the solid fraction plus the of the fluid times the fluid fraction
147:38 we call porosity. And uh, of course the fluid might be composed
147:45 grind oil and gas. And uh, we can uh say that
147:54 uh, the density of the fluid given by this and these saturation parameters
147:59 add up to one, just like . These are the volume fractions
148:05 they obviously add up to one. I think all of that notation is
148:19 . Yeah. The effects of fluid of on um share models and and
148:27 modules are commonly understood understood using uh paper written by gas mark in
148:36 And this is what he said, said, the sheer models of the
148:42 rock is the same as the sheer of the empty frame. In other
148:50 , sheer models does not depend upon fluid content. Doesn't even depend upon
148:56 there's any fluid in there at And of course the reason is because
149:01 fluid does not resist the sheer. as you share a rock the third
149:07 isn't going to be resistant this So that's why he said this.
149:16 , right here, it says And so we specify undrained because during
149:21 propagation, the fluid masses council. course. Now there's a corresponding formula
149:27 for the boat models or in other , in compressibility. And here is
149:34 notation. Same notation uh for the and for the empty frame and now
149:40 the fluid and for the solid. of course you see here the porosity
149:45 here. And so this is uh formula is a bit complicated but uh
149:51 very familiar to uh most exploration And before we pass on, I
150:00 you to note here that we don't either one of these only the
150:05 We don't predict either one of Only the difference. I mean when
150:08 say we Gas monkey didn't predict either of them. Only the difference.
150:13 you see the difference depends upon only frame. The frame modules uh appears
150:19 also. You can rewrite this by way, it's not commonly done,
150:23 you can rewrite this so that there's only the upgrade modules over here.
150:31 so um as it's written, this a good way for predicting the unknown
150:38 the frame because you just move this the other side and then you have
150:43 you want to predict on the left and on the right side only the
150:46 models and you're going to use that predict the undrained market. You can
150:50 it the other way as well. is the form we normally think of
150:57 business. And here is the This is uh the title page of
151:05 paper. And if you read german can uh you can recognize this,
151:09 the guy's name and this fame papers really one of the most famous and
151:14 geophysics. Incited many times a day over the world Mostly in 40 size
151:23 . It's so important that the scg for it in 2005 to be uh
151:30 . So this is uh period in in this book which you can buy
151:37 the S. E. G. think about 30 bucks. And uh
151:43 of us don't read german but um all read english because english is the
151:50 language of science and business. And mr wu who uh grew up not
151:59 English speaks english well now because he to be a scientist. And so
152:04 of the scientific skills that he has acquired is english. Well since 2005
152:11 had no excuse for not reading it I can tell you, I can
152:15 you that most of us have not it. Why should we read
152:19 Because it helps us understand the food of the rocks that are waves are
152:26 now. Uh This is not um quietly now don't say it's fairly wisely
152:38 , widely known experimental support for I mean everybody believes, I would
152:44 99% of all believe this, but shouldn't be believing it because the experimental
152:54 is very thin. See what I by that here we see a pretty
153:00 confirmation Concerning share modules. This is bunch of of observed share model light
153:08 with the predicted modules in the prediction this case is trivial. We say
153:14 prediction equals um uh observation. So is this line is at 45° And
153:23 published this confirmation this compilation in maybe before some of you were
153:31 Um based on data which I did , I did not acquire but my
153:38 friends at Amoco acquired that data and uh pretty good confirmation. There's some
153:45 but you know, uh the noise uh the errors are pretty much um
153:53 equal positive negative errors. But on other hand, for the boat models
153:58 a difference. And so you can that all the observations why um uh
154:08 higher than the prediction. And you see here some of them are lime
154:17 . That's the excess. And that's line. And some of them are
154:22 . But really there's there's a there's bias data. Do not confirm the
154:34 . This observation was predicted to be . I have what 9% was predicted
154:41 the center as a function of Uh This is typical data and this
154:48 Andrea sandstone, a well known uh sandstone. And here we have the
154:54 between untrained and frame. I should this um this is untrained and frame
155:01 as a function of confining pressure at pore pressure I think. And look
155:06 this uh Mr do you know this Arthur Ching? He has a different
155:15 name, I forgot what his chinese is. He's in Hong kong.
155:19 uh he's notable because he is the president elect of the scg from
155:28 So he was elected just last And uh so I knew it was
155:33 happen sooner or later. And uh now it's happened and he is the
155:39 elect. He will automatically become president year at this time. And then
155:45 that a year later he'll be automatically president emeritus, former president. And
155:52 he'll he'll automatically have about five years declining responsibility as in chairing of important
156:00 and so on. So he's signed for seven years of service for the
156:05 . E. G. And he's that young. He's about my age
156:09 little bit younger than me. Oh luck Arthur In 2006. So let's
156:23 at what we have here. We the data is higher than the
156:29 And furthermore, you see the the discrepancy uh decreases as the pressure
156:37 . So you should have enough physical about rocks already. You were so
156:42 but you should have the physical understand as we apply the pressure here,
156:47 squeezing cracks out of the rock. the rock has a very complicated for
156:54 and some of its thin and And we're gonna call that cracks and
156:58 of its uh sort of equal in dimensions. And we're gonna call that
157:03 cracks. And so uh as you out as you apply pressure, you
157:08 close the cracks. And so uh what's happening here as we close as
157:15 increase the pressure. But in the by gas, not, let's
157:23 Here's the theory. You see there's cracks in here. There might be
157:28 , implicit notion that there are no here in the in the south and
157:32 are no cracks here in the And the cracks here in the
157:37 hardly any if you have 20% no more than 1% um is due
157:43 cracks and probably less than that. you don't see cracks anywhere in
157:48 Although there might be cracks in here implicitly inside here. But you can
157:55 that the error is changing. Uh we as we uh increased pressure.
158:05 frame models, which is the dry is changing and the saturated one is
158:11 . But the discrepancy coming from gas is uh the decreases that the discrepancy
158:23 decreasing. So the theory is always . But it's wrong uh uh differently
158:32 pressures. And that's puzzling because Gasman no explicit assumptions concerning poor mike
158:40 He's assuming that all the effects of the cracks are included in here,
158:46 I call earlier, the frame Um That's obviously not true because if
158:58 were true, we'd have the same . Everyone where we have zero
159:04 So, here's the resolution. This been known for years and years.
159:10 here's the discrepancy. These ultrasonic easy to do in the laboratory violate
159:15 low frequency assumption in Gaston theory. , since the test, don't uh
159:24 say, okay, we're not going pay attention to this because this is
159:27 frequency. Our data are seismic data low frequency. So, we're going
159:31 simply assume that gas mon is The low frequency. Even though we
159:36 assume we can see that it's not that high frequency. So that's just
159:42 assumption. That's not good science. we've done that for a long,
159:47 time. So, people who know rock physics say, okay, when
159:57 we wanna know seismic band frequencies in saturated rocks. And we want to
160:05 that in the laboratory, it's very to look to do in the
160:09 a low frequency measurement. Um because you have a low frequency measurement,
160:16 wavelength is a lot bigger than the , Right? And so uh a
160:27 easier to do. High frequency uh . You just put a, have
160:31 sample and maybe you have it under and put a transducer on it and
160:39 energize the transition and it sends a frequency wave through the, through the
160:44 sample. And you have another transition the other side which uh detects
160:49 And uh he said, hey, we do, I think that's a
161:00 . I'm not gonna answer that. know that these high frequency waves are
161:11 to be traveling faster than the seismic because of the frequencies work. Because
161:19 frequency effect, if you call fluid high frequencies we have for instance,
161:24 we're going to have that in our sample. But we don't want that
161:28 to compare with saving data. So what we're gonna do. We're gonna
161:32 the uh, yeah, losses on rocks, whether ultrasonic equipment. And
161:41 we're gonna make a uh, we're to compute the saturated velocities using the
161:46 , monetary that of your show. is very common Francis, but it's
161:52 valid because of the arguments which follow . And so what I'm about to
161:56 you is uh um leading edge not classical additions. Leading edge
162:06 I gave this same lecture earlier this on Tuesday in china. Were you
162:12 ? No you were not. Um Our our chairman uh Professor joe arranged
162:21 me to give this lecture via zoom Tuesday evening our time. It was
162:27 morning china time And there were 2000 listening. Amazing. You get the
162:35 lecture 1 to 1. Now I'm not gonna get they were interested very
162:40 in this topic. So I talked them for an hour and a half
162:44 that topic. And so uh since is a topic since this course is
162:50 wave propagation, not about what I'm gonna minimize it here. But
162:55 you're gonna be exposed to this so times in your studies um and everybody
163:01 who who teaches you this, we'll teaching you these old ideas from
163:06 And so I'm gonna teach you then why? Uh it's more complicated than
163:15 . Yeah. As preparation Gassman assume solid is micro homogeneous. That is
163:22 say one mineral. Yeah, like said, almost all rocks have
163:27 several minerals. But since most minerals similar compared to brain, this has
163:34 a minor issue. So people have all along that. He was making
163:39 assumption and they said, never we're gonna ignore that complications. But
163:46 something else to think about. All are anti psychotropic. And when they
163:51 randomly oriented, you know, to an isil tropic rock in some
163:55 You've got stiff axis sticking up against faces but not in other places.
164:00 we can call this orientation heterogeneity orientation homogeneity leading to uh in homogeneity of
164:11 even though the composition is uniform in uh in this circumstance. So this
164:19 orientation in homogeneity is always ignored without . Maybe it's important. Maybe
164:25 We frankly don't know yet. Now is the formula that I just showed
164:30 in terms of of compress abilities. so it's gonna be easier for us
164:36 talk about this in terms of compress . Kappa is one over in compressibility
164:43 you see the formula looks very So this is what this is what
164:49 going to. Okay. Now earlier mentioned B0, here's a picture of
164:57 . And he was the actual guy actually invented for elasticity uh 10 years
165:08 . So he was the first guy say sheer Modelers did not depend upon
165:15 mon uh copied that from him without . Maybe he developed that idea himself
165:23 or not. We'll never know because these guys are now dead. Now
165:28 regards to the in compression bills, is what Hook said for the income
165:34 that lawyer dependence is given by um one over K. Or equivalently by
165:44 . That's the relation between volume change pressure. That's for homogeneous bodies.
165:52 for in homogeneous bodies, all we was make averages big deal, but
165:58 poor for porous rocks. B. . Said we need something very
166:06 I need something very different in the place. We need to have a
166:12 here which describes the framework. And the second place we have a term
166:18 describes the fluid pressure separately with a physical parameter which he called H.
166:26 is why um for elasticity is so more complicated than cocaine elasticity because we
166:32 a separate dependence on the average pressure understood pressure. Now those focus was
166:41 consolidation. And by the way he uh in 1941, remember that was
166:48 before the war started, he was at Columbia University, it said.
166:54 then after that he went to work shell oil. And I should tell
166:58 that he's a very famous guy outside rock physics. He's uh famous in
167:05 different fields of science, including you like aeronautics, aeronautics. So there
167:11 B. O. Effects in the as well. Look him up in
167:17 . So while he was at shell focus was on consolidation. Uh I
167:23 he was interested in how the soil the underneath uh an oral storage tanker
167:33 and squeezes the water out. So had this process where the water is
167:39 squeezed out and he had to end of the process the beginning and the
167:44 end at the beginning. He called instantaneous compressibility after the instant of application
167:52 the load. But he is going assume that the uh the pressure,
167:59 fluid pressure is uniform. So uh gotta be after the relaxation of local
168:06 in Portland. So we now call to be the undrained compressibility and it's
168:12 kappa that you see. And then the end of the process uh where
168:18 all the water has been squeezed the load is supported only by the
168:22 . We call that the frame compressibility in your in your work um Stephanie
168:29 might call it the grain compressibility. same thing. Now back in 1941
168:39 found the correct expression for fluid dependence in the case of uh one
168:47 compassion with an in compressible fluid. so here I'm showing you his uh
168:54 his results really for three D. . And it looks like this here
168:58 this instantaneous compressibility. Now we call undrained compressibility. Here is his final
169:04 . We call that the frame compressibility it differs from the uh the undrained
169:11 differs from the frame compressibility by this here where this is a second
169:19 a second physical parameter which uh vo which I'll show you later. So
169:28 is such a minor change one day three D. That that I uh
169:33 this result to be all himself, to me. He considered only the
169:39 of in compressible fluid. And when fluid is compressible then the change of
169:46 due to compression is not zero. we assume uh in this case,
169:51 it depends upon the compressibility of um of the food and the pressure in
170:00 fluid. And then the previous three . result, it's generalized to
170:08 So I want you to compare these expressions. This one is from Gasman
170:13 this one is from D. And you can see they kind of
170:16 a lot a lot of luck, they? Here's the correspondence in
170:23 This difference here is what do you H. Inverse. And and then
170:29 inverse is given down here. And can see that gas mod has replaced
170:35 parameters from B. O. With O. Called H. And
170:38 With one which is the um of compressibility of the fluid which appears here
170:47 here. So if you want to about uh the difference between these two
170:58 gas man, all you need to is uh solid compressibility. But for
171:05 . O. You need to know two other quantities H. And
171:08 Which we don't know what they are . But they're uh two different things
171:12 they're not the same. Which is um uh which is quite gas
171:20 So we immediately suspect logical error. guess. So we ask ourselves the
171:32 , could he be wrong? Could be wrong after being accepted by everybody
171:38 all these years? And the answer yes. He improperly applied a theorem
171:47 to love back in 1927 years this theorem assumes a hydraulically open system
171:54 he applied what gas one did as applied this theorem to wave propagation in
171:59 closed system. This there is strictly for uh for gas man's case micro
172:09 , but it's also approximately valid for heterogeneous seidel like we have in rocks
172:16 it can't be applied to the untrained . So he did make a
172:28 Yeah back here, here's boko has here. The the frame compressibility
172:39 The frame compressibility be measured by just out the system. So if you
172:51 out the system, that means that the load is being supported by the
172:56 only. So that's why we call the framework here. But that's obviously
173:00 open um an open system. So we all have also made a
173:09 The EOS duration also be invalid To that question. We turn to another
173:15 paper published um 24 years after gas . And among their advances, they
173:26 . Mhm. Hetero Geneti of the . So here's their title and I
173:41 tell you I actually met these It was let's see um 19,
173:50 was not in 1975. Uh It about in this in that same time
174:01 time period. So Brown was working Chevron. Uh And Karenga was a
174:09 at the University of State Ohio State and he's actually quite a famous guy
174:14 solid state physics and he was coming there every summer to chevron in California
174:21 working as a consultant with Brown. when I met them I was pretty
174:26 , I was fresh PhD and did know what were the important questions to
174:35 and I did not know how to the important questions. So I just
174:39 them and I was polite, polite we shared some coffee and so on
174:43 the Chevron Research Center in California. I didn't, I went away uh
174:49 understanding how important this work is, later I found out and here is
174:56 result which they derived in 1975 about a year or two after I met
175:05 . And they probably already knew that published it in service and they didn't
175:10 the same logical error that Gasman did is the result. But look at
175:15 result, all kinds of strange parameters you have no clue what they
175:21 And that's a big problem with This notation has confused generations of
175:29 No, I'm just going to point that that Brown and Karenga is exactly
175:36 to B. O. Only with notation. You can see here this
175:40 and notation, this correspondence and Uh and you can see instead of
175:46 and R. You got campus of and campus of fire here. There's
175:53 there's the correspondence and notation. Uh we don't have too much time
176:02 But we have enough time I think describe the uh why why did he
176:12 capper frame? I would say that pretty obvious why we have the subscript
176:19 here for frame. Why did he a subscript capital A. Instead of
176:25 ? Well, let's consider the asymptotic case. Uh What we have.
176:44 This is not the formula that I'm for. But read what it says
176:49 . It says that the writing of on the dependence of the elastic properties
176:54 the porous rock on the compressibility of poor fluid. So this is the
177:01 is the elastic properties of the forest . And he's gonna find the functional
177:07 of this on the poor foot right . Okay, so here's the expression
177:22 we have had before. And now showing here I'm looking for the functional
177:28 and we just consider the case of very compressible fluid. A compressible
177:34 So we put in here infinite right . This whole term goes away and
177:39 left with this term here. So , I'm gonna interpret campus of
177:45 They didn't explain what it meant but interpreting it as the asymptotic compressibility in
177:51 limit of highly compressible fluid in this limit, the load of support only
178:00 the frame. So we call it a frame. Although it can be
178:06 as with a drained experiment. It's by the functional dependence of kappa undrained
178:16 this is drained. Kappa undrained upon . The asymptotic compressibility and they also
178:25 a clue to the meaning of campus . I remember there was another point
178:31 they called kappa phi and they actually a name for that in english.
178:36 call that the poor compressibility. And you can interpret that as the compressibility
178:43 the pore space of the rock. makes sense. And then they show
178:47 formula on page 6 14 that the cap M. Is given by this
178:57 of cap S for the solid and phi for the horse race. In
179:03 words, kappa M. Is the weighted average of these 24 compressed
179:11 Now I think you're thinking that this here. The pore space is gonna
179:15 a lot more compressible than the compressibility the solid. But that's uh that's
179:22 really true because um when you squeeze the water is not gonna be draining
179:31 . So the pore space is not change that much differently than the some
179:37 differently. But it's not going to uh completely different. Uh In general
179:43 least all three things are gonna be uh different from each other but maybe
179:48 very different. We have to do experiments to find out. So we're
179:53 interpret campus of them as the mean , not the middle compressibility. Many
180:00 have taken seen this. So, that means the middle capable mental
180:06 And let me let me just back here. You see uh this is
180:11 Brown and Pringles formula. You don't anywhere in here. Capra some
180:17 And so if you're thinking, if sloppy, you might think, oh
180:21 we go. That's kappa sub. that's the purpose of mineral same as
180:29 and solid. I understand that. the only thing I have to do
180:32 worry about this. No, that's true, that neither one of those
180:36 the solid comparison instead is mean Okay, so now when you compare
180:45 expressions now they now this is brown karina. And this is an early
180:52 by Gas Monkey. And you see look a lot alike. And
180:56 you can see that if these three are all the same, then um
181:03 their identity. So in fact brown chris decided that they argued that if
181:11 solid is micro homogeneous, then these things are the same. And so
181:16 that case there result uh reduces to mark. But you remember as they
181:22 it, I said it, you see it as they derived this.
181:25 did not assume micro homogeneity. They multi mineralogy, multi minerals micro
181:33 And so they concluded that the reason have two parameters instead of one is
181:45 they didn't restrict themselves to micro homogeneous the way gas Mark did. How
181:53 you? This particular argument made the mistake as Gasman did applying loves theorem
182:02 UN for uh, for open systems the untrained case. And so the
182:12 , the difference you see here doesn't from the issue of solid home ingenuity
182:17 not, comes from um, gas error. So I think this is
182:24 good place to stop. Yes, is a good place to stop
182:27 and we'll uh we'll take this up morning, 8 30 online. And
182:36 way we won't get tangled up in football game. Of course, we're
182:41 be cheering for the home team, we're not gonna be inconvenienced by
182:45 So let's uh, stop the
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