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GEOL7324 Rock Physics - Day3 09-03-2021
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00:00 this conference will now be recorded. have a habit of sometimes forgetting.
00:08 if you guys use the recordings do help me remember. All right.
00:13 were looking at this log and we showing how, when I had a
00:17 wide washout and probably this is a log here. And probably the washout
00:25 even bigger than that. The caliper measured with an arm that sticks out
00:30 measures a distance. Well, if bore hole is bigger than the than
00:35 length of the arm, uh then uh it flat lines like that.
00:42 here, you know, the scale the caliper is 6-16". This was
00:47 bit size. So if it was perfectly competent rock and perfect drilling,
00:53 would be at the bit size. in fact it's washed out from the
00:58 size and in this case it's washed so far that the caliper log has
01:06 clipped. As a matter of let me turn on the pointer
01:11 Okay, the caliper log has So actually the whole is much
01:16 And you see the density law goes in there And in fact it goes
01:21 abnormally low density. Now it goes a scale of 2-3, it goes
01:29 And then it wraps around. So has another scale 1-2. So,
01:34 is reading about 1.6 or so, suggests that the logging tool was seen
01:41 drilling fluid at that point. So density here is not indicative of the
01:47 mation density. And here you can it's very low again. And we
01:52 talking about the delta rho, this the degree to which the density log
01:58 compensated for the borehole. Now, delta role is can be an accurate
02:05 if the errors are very small. when the errors get large, delta
02:11 is not accurate at all. And the units of density of grams per
02:18 . Uh delta rho is in units grams per CC also. So this
02:25 a swing of 0.5 g per CC minus 0.25 plus 0.25. So that
02:32 be an enormous porosity change. And you're seeing here where the hole is
02:39 and washed out like this. The Rho correction is large on the order
02:45 .15. In my experience, any rho bigger than 0.5 is usually an
02:54 that the density law reading is I wouldn't, you know, the
02:59 route correction is ineffectual when it has be that large. A small death
03:05 less than .05. And uh there's good chance, it's good, especially
03:10 you have a smooth bore hole. that's what's happening here. Even though
03:14 borehole is washed out, it's smooth enough of an interval that the density
03:20 could be pressed against the borehole wall make a good density reading. So
03:27 see that density laws are problematical. You know, they're designed to give
03:32 porosity in the formation. They're not to give you an accurate density up
03:37 down the bore hole every place. but geophysicists often mistakenly just take the
03:44 log and use that to compute impudence use that to create this synthetic seismic
03:50 . And that's a big mistake. density log has to be corrected.
03:56 you know, we'll talk about how might correct the density log, but
04:00 first order of business is to recognize the density log is bad. So
04:05 look for rough boreholes, wide bore and you look for large density log
04:11 . All of these are indications of bad density log. Okay, so
04:19 does the correction work with the density as we were showing, there are
04:25 detectors. So you have a source emits gamma rays into the formation and
04:31 you have counts at each detector like Geiger counter. And the further away
04:38 get, the more absorption you the more material the gamma rays pass
04:45 , the more of them are And as we mentioned, they are
04:50 by electrons when the gamma ray hits electron that tends to absorb the
04:56 So the long space detective would have accounts than the short space detector.
05:04 if you make a plot of the of counts on the short space detector
05:11 on the long space detector, um should follow a linear relationship between the
05:21 and where you are on that linear and exactly the slope of that line
05:28 depend on the logging tool. And a lot of calibration which is done
05:34 decide what that line should be. you fall on the line, you
05:40 a small delta rho. That's probably good reading. So measurement A.
05:45 on the line. So that was acceptable reading measurement B however was well
05:53 the line. And if I just straight back to the line, drop
05:59 perpendicular to the line, it would reading too low density here. We
06:04 high densities here. We have low , right? So it would be
06:08 too low. And chances are that's of some kind of washout effect.
06:15 so what is done. And this is another logging tool dependent
06:21 It also depends on the formation and fluids in the formation. But what
06:28 done is there's some black magic that the orientation of what is called a
06:35 correction. So that point B is along that slope to intersect the
06:46 So point B. Has been moved point C. And that is the
06:53 density. And like I said, B is not far off the
06:59 that point C could be pretty And and how the further you are
07:07 the line, the larger this Dia's So if I go horizontally back
07:12 the line and I compare the distance that point to the corrected density
07:19 that's distance, the the large India the more likely the correction is to
07:26 inadequate. So everybody, everybody with here. Right, okay, So
07:37 look at another density log. So we have a gamma ray log,
07:44 is the it's the it's the curve a lot of variation in it.
07:52 we have clean rocks. Gamma ray naturally natural radio activity. So here
07:57 have high accounts suggesting more potassium uranium thorium some radioactive minerals. So that
08:06 is indicative of play. Sometimes it be indicative of uranium, for
08:13 sometimes in heavy oils and in black . You can increase uranium. So
08:21 you go. So we have clean , maybe sand stones and we have
08:26 intervals. Maybe shells You can see caliber is not varying very much.
08:32 , a scale of 6-16". It's very in, very much oddly enough
08:39 the sand stones, the caliber reads less than in the shells. If
08:46 look at delta rho in this case rho is usually small. It's usually
08:51 than 0.5. So the for the part, it's a good density law
08:58 . And here are the density Reading itself. The dash line is
09:04 computation which is made from the So don't worry about that off
09:09 But the black line is the bulk and that's indicative of the ferocity.
09:14 the dash line I believe is a ferocity if you had if you were
09:20 limestone. So there's the scale for And really it's a 1-1 relationship,
09:27 ? It's a single grain density, is probably not right. The grain
09:32 is probably varying, but this is they would plot on the log.
09:37 if you are in a sandstone, ferocity would be wrong because it's a
09:41 ferocity. Does anybody want to guess the caliber is slightly small in the
09:49 zones here here, here, here see the well bores is bulging in
09:58 the clean zones that don't have much . Why might that be government having
10:06 your older um No, I think could uh, we could uh not
10:12 about heavy minerals at this point in example. Why would the clean
10:22 Why would the caliber be reading a diameter then? Uh and the shells
10:33 doesn't have to be right. I going to say just just because in
10:41 , dry sand is maybe more rigid wet sand um or sorry, or
10:49 in this case. So if, . So one hypothesis then is that
10:57 wet sand is more plastic and it's deforming, it's less rigid and it
11:03 deformed under pressure and moved into the ? Um that's a valid hypothesis.
11:11 the other hypothesis, how about the tends to be slightly washed out.
11:17 , the delta rho seems to be that's maybe not what's happening. Um
11:27 , I think what's happening here is the clean zones are permissible and they
11:33 developing a mud cake, drilling fluid entering the formation and the drilling mud
11:39 being filtered by the poorest formation and building up on the borehole wall.
11:46 I'm going to suggest that that's maybe mud cake that's occurring there.
11:57 now, here we have a case where uh we have a caliber that
12:02 perfectly engaged in the cleanest zone but in the zones that are a
12:08 bit dirty or a lot dirty, caliber becomes very rough. And I
12:16 what you can see here is that delta rho is zero. Where the
12:23 is very flat, we have neutron and density plotted here at the same
12:31 . So they should overlay if you the right density in it. And
12:35 see the neutron porosity and what you infer the density porosity to be are
12:43 similar in the zone with a good . But where you have a bad
12:49 , the density log is reading much , indicating much lower porosity, then
12:55 new tribal art is suggesting. uh that is another type of quality
13:03 that you can apply by comparing the porosity to the density porosity. If
13:10 density porosity is much larger than the porosity, you should be worried on
13:17 other hand, neutrons ferocity tends to abnormally high when you have place.
13:24 the neutron porosity itself is not a indicator of ferocity. Okay, so
13:36 is a simple algebra equation exercise from mass balanced equation expressed the prosperity in
13:44 of bulk density, grain density, fluid density and there's the answer right
13:50 . So all you have to do work out the intermediate steps to get
13:56 . But now here's a tough question you're going to have to deal
14:03 Suppose the ferocity log is calculated to correct ferocity in a limestone. How
14:11 this equation be corrected if you're logging a sandstone and in fact, there
14:17 two corrections that need to be First of all, you have to
14:21 the correct grain density. So you to correct the ferocity for the fact
14:27 the wrong brain density was used, also you have to correct the bulk
14:33 here because of the difference in electron between limestone and sandstone. If you
14:40 back here, you remember that uh huh has a small bulk density
14:52 that's needed, whereas limestone is calibrated to need the electron density correction.
15:00 the first thing you have to do you have a log in limestone
15:05 The first thing you have to do correct, correct the bulk density to
15:09 correct in SAm's town and then to the ferocity, you need to use
15:17 correct rain debt city here. So going to ask you to come up
15:24 a correction equation. So I give limestone ferocity, you give me an
15:30 which will convert that to sandstone Okay, again, it's an algebra
15:37 , but it's a little bit tougher . Okay, disadvantages of the density
15:49 . Well, we've seen is very . If you have a small mud
15:55 , it's usually within plus or minus g per CC, But .5 g
16:02 cc is not a negligible amount or is often not a negligible
16:10 Uh, it's particularly susceptible to Do watch the surface tension on the
16:16 , not the surface tension, the tension. Um and that will tell
16:21 if the tool has been hanging If the tool is hanging up,
16:26 can be pulled away from the Warhol , The detectors can be pulled away
16:31 won't be in contact and so often get errors associated with that.
16:38 we've assumed that the borehole is perfectly because if we come back to this
16:49 , okay, this is a cylindrical in a cylindrical bore hole and it's
16:56 , therefore the radius of the tool different from the radius of the
17:01 So that means uh, that there um, the borehole, the detector
17:08 be perfectly flat against the borehole there's, there's an austerity there because
17:15 radius of curvature is tighter than the wall. Now, if you
17:20 uh, the borehole diameter, you correct for that. Uh But at
17:27 borehole in cross section is not perfectly . That could be a pretty tough
17:33 and you may not know the exact of the borehole. So again when
17:39 have a rough irregular borehole that could all kinds of problems. Sorry?
17:48 Another major issue. The penetration of logging tool is only a couple of
17:57 . It doesn't penetrate very far into borehole itself. I mean into the
18:04 itself, that means if you're in permeable formation is probably seeing the invaded
18:12 . So it's you actually have to a correction for invasion on the density
18:20 . Um Otherwise chances are, you for example in the gas reservoir you'll
18:25 measuring too high in density. All now, what happens uh when we
18:36 a bad density log, what do do? Well, you could use
18:41 relations between velocity and density and you get a ballpark correction. So here
18:46 just an equate an example of one relation between velocity and density. There
18:53 a wide variety of these. Here some other relations between velocity and
18:59 Um So these were geographically restricted Each given you their own specific trend
19:08 V. P. Or V. against density. Here's a here's a
19:13 array of measurements uh from the literature a rough relationship there and most importantly
19:25 these are gardeners relationships for common sedimentary and again I want to and these
19:33 just very average values and I do to emphasize that. Uh these tend
19:38 be for uh not entirely liquefied Samson . Uh so uh if you compare
19:48 these fall on the Royce and boy , they're not at the higher end
19:54 , towards the lower end for the and shales. But the nice thing
19:59 these relationships is you have a separate for each mythology, then there is
20:05 average of all of them, which the dash line here. And this
20:09 the famous gardener relationship Which you would read about if you had started reading
20:16 gardener and Gregory in 1974, that's this figure is from most geophysical software
20:25 the understanding that density laws are often available or against the logs are unreliable
20:31 the synthetic seismic grand package. They give you the ability to substitute density
20:38 the garden of density. But keep mind that this can produce large
20:44 Uh if you're not taking into account variations, I mean with ology
20:52 So, for example, when I similar velocities here, I have a
20:58 difference in density between sand stones and for the same velocity. Shells tend
21:07 be higher density than sandstone and shells to be on the high side of
21:13 relation and sand stones tend to be the low side of gardens relations.
21:18 if the velocities happened to be the Gardner equation would predict the same
21:24 So you would see no discrimination, impudence contrast between the sand and shell
21:30 in those cases the density difference, would give you a very significant reflection
21:36 . So using the gardener equation or a constant density where the shells and
21:42 have similar velocities will lead to very miss ties. It'll lead to poor
21:49 ties. A few general rules of , you know, carbonates tend to
21:57 higher velocity, higher density than Uh Rock salt for the velocity has
22:05 abnormally low density. So saul towns even though they're low density and our
22:14 become die appears as a result may relatively high velocities and hide right.
22:23 evaporate happens to have abnormally high So in a carbonate environment, the
22:31 of an of evaporates could be very . Mhm. Okay, so here
22:40 have gardeners relation and this is a challenging algebraic exercise. Usually, uh
22:48 students get this one wrong, so see if you can get this
22:53 But uh here we have density and per cc and velocity in feet per
23:02 . However, if you happen to working in europe or parts of the
23:07 that are using metric units, you have velocities in meters per second or
23:13 per second. The scientific papers, want you to use velocities and kilometers
23:20 second. So in that case re gardeners equation by doing the units conversion
23:29 p wave velocity from feet per second kilometers per second and come up with
23:35 new gardener equation there. It's surprising often students get this one wrong,
23:42 that would be a good exercise for . In fact, why don't
23:54 why don't I make you do that now? So go ahead and uh
24:00 the algebra here? Uh and you use uh well you can just google
24:05 relationship between feet per second and meters second. Right? Yeah. Mm
24:15 I'm going to stop recording at this and give you a chance to work
24:19 this. This conference will now be . Who mentioned in the chat.
24:25 got it from him. Thank Okay, so um yeah, so
24:34 we have gardeners equation and kilometers per now and all the velocity porosity transforms
24:42 will talk about are also velocity density because for a given mythology, if
24:48 know porosity, you know the So here's the widely equation, which
24:55 talked about this is the time average is the heavy solid line and you
25:00 gardens equation is lower velocity at a density or you could say at the
25:06 velocity is a higher density. And because the gardener equation is an average
25:11 all rock types, which includes a of shell. Um there is also
25:18 velocity ferocity transform, we'll talk about reindeer Hunt Gardner equation and for moderate
25:28 , rocks, it tends to give highest velocity, it's kind of an
25:33 limit. I want velocity could you actually can't really extend the roemer
25:39 too far down. They actually you to towards the critical ferocity and they
25:45 change the equation. So ignore this here, but you could look for
25:52 porosity is most of the density rocks that we're dealing with. The Bremer
25:59 is an upper limit. And by way, these curves are drawn for
26:04 . So they converge at the course here and again, gardeners equation is
26:12 all rock types. So it does go through the courts point because it
26:17 carbonates and shells etcetera. Uh some curves on here. Uh The red
26:25 line is an equation that gardener gave sand stones in general. And in
26:34 it veers, it goes wild at velocities, it's actually not well calibrated
26:42 high velocity rocks. So in you can empirically just just distort this
26:49 equation and force it to go through courts courts point. So that's the
26:55 density equation I use for sand sounds what I call the modified Gardner
27:01 And it's the long dash line There's one more curve here which looks
27:07 of like a Royce bound. This called the would like equation. And
27:13 , instead of taking the reciprocal of bulk modules average, it's the reciprocal
27:18 the plane wave modules and we've seen , that is not exactly right.
27:25 it has the but it has the of adding a little bit of virginity
27:31 the rock. So rather than or the sediment. So rather than pure
27:37 , uh these rocks have some rigidity with them and we'll see that uh
27:45 uh low densities, roemer uses something this. Uh would like equation
27:54 all these curves that go through the point are all designed for sandstone and
28:00 would be those curves would be different different mythologies. So example, for
28:07 , here we have shells and we the rain martin Gardner equation for
28:13 we have the would like equation for . And here is the gardener equation
28:20 a shell. And you see there's lot of scattered because shells have compositional
28:28 and I would argue that some of are more like dirty lime stones than
28:33 . But for the most part you see that the points are contained between
28:39 roemer line and the wood like So the roemer line is kind of
28:44 good trend for the most liquefied rocks have and uh the would like line
28:52 be yeah, poorly consolidated. So closer you are to the would like
28:59 , the less well lit defied. rock is again showing some data points
29:11 sand stones. We see a similar . This was the modified Gardner
29:18 So rough average, the the Remeron equation tends to be a line where
29:25 most liquefied rocks will fall along. as the rocks get less well lit
29:31 , they get closer to the the like equation. So if you are
29:37 unlit defied or virtually almost unlit you would be at the would like
29:44 it's it's slightly faster than the Royce or the wood equation. And this
29:54 just comparing shells and sand stones. a variety of laboratory in log measurements
30:02 for the most parts. And this the overall Gardner equation. The rough
30:07 for all points. And you see kind of by sex. Again,
30:13 you're ignoring calcite Samantha and the shells the most part shells are faster than
30:21 than the gardener equation. Or I say more dance, sorry, slower
30:27 a given density fast, uh more at a given velocity than the sand
30:33 . And the gardener equation is kind divides the the population is there?
30:44 , just for your reference and some the exercises you're going to do,
30:49 are polynomial fits to the gardener So these equations correspond to these lines
31:02 except for the sandstone line which has forced to curve through the courts
31:08 So the sandstone line has been modified And you could uh applaud velocity versus
31:20 . And then compare lines of constant . And again, what you can
31:28 is that if the salmon shell have same impedance. The shell will be
31:34 dense than the sand and the sand be slightly faster than the shell
31:46 Now, if we look at velocity trends at different depths, what we
31:52 that those relations can change. So are shells, this is the gardener
32:00 which remember tends to be an upper for shells. The shales tends to
32:04 more dense than the overall Gardner Uh And so this is where the
32:11 points are plotting, but you can that there's a slightly different trend,
32:16 ? There's a different slope here than . These were from two wells right
32:22 to each other. One was a well, one was a deep well
32:26 the differences these shells are in pressure these are normally pressure. So
32:35 the effect of overpressure is to reduce velocity at a given density. So
32:43 forcing the grains apart. You see the same density. You know,
32:47 we were on the garden line or extrapolating this line straight up, we
32:53 have higher velocity. So this is idea that by increasing the poor
32:59 you will affect the velocity more than density by pushing the grains apart as
33:06 poor fluid is pushing out. It force the grains apart. So here's
33:15 , an example of a shell in ferocity uh that you can see is
33:23 dark ferocity there's also porosity between clay , which you can see. Uh
33:30 , I'm sorry, the black is and I'm sorry the black is
33:35 And so all the ferocity is very , you really can't see it.
33:40 uh you can see that the Karajan mimicking a flat of shape from the
33:47 platelets themselves. Alright, so everything elongate here. Okay, so now
33:59 going to do another exercise and we're to look at the effect of organic
34:07 on the density of a shell. the first thing we want to do
34:15 from this, these are measurements of in shales versus the total organic content
34:23 this is weight percent uh to convert percent organic material to volume fraction.
34:33 need to know the specific characteristics of shell, You need to know that
34:40 shell composition, but as a rough you won't be too far wrong if
34:46 just say the volume of fraction is the weight percent expressed as a
34:52 So toc is 20% That would be volume fraction organic material of .4 or
35:03 . Okay, so the first order business, I'm going to ask,
35:06 , for this suite of shells? I had zero organic content, what
35:14 the shale density be and what would total porosity of that shall be.
35:22 and then I'm gonna ask you to the density of the organic matter.
35:27 the equation you need to use here the mass balance equation And use a
35:34 density of 2.65 g per cc. understand what I want you to
35:56 Yeah. So use the mass balance and a grain done city of 2.65
36:03 per cc. And then look at data points and look at that fit
36:08 the data. Tell me the both of a shell without any TFC.
36:17 at zero TFC, read off that density, convert that to a total
36:23 for the shell from from the mass equation. And then given the that
36:32 relationship between both density and toc calculate density of the organic matter. So
36:44 you had 100 you know, if extrapolated That line to 100% toc Or
36:53 volume fraction of one, what would density P And again, I'm going
37:09 stop recording while you work on This conference will now be recorded.
37:21 uh by definition Karajan is organic matter can't be extracted using organic solvents and
37:37 usually derived from algae and woody plant and the molecular weight is high compared
37:46 bitumen. Bitumen, which is another of organic matter and that's soluble organic
37:55 . There's a fine line between bitumen heavy oil. It often acts like
38:01 thick viscous fluid. Um you can of it as solid petroleum. So
38:12 bituminous rock is a sandstone that has in it. So bitumen will can
38:20 dissolved in organic solvents carriage and There's also a minor constituent graphene which
38:30 a type of single layer carbon. , So similar to graphite, except
38:38 a different with a different crystal Okay, so veteran I is a
38:52 of material found in carriage incense and these are derived from land plants.
39:04 the reason veteran it is important is it has a shiny appearance uh therefore
39:12 vitreous and how reflective it is, on how mature it is. So
39:19 measuring how well veteran it reflects You are in fact measuring the maturity
39:26 the veteran it and by analogy than the organic material around. Quick question
39:35 is there no priests? I laurean . I've so I think the answer
39:55 that there wasn't much in the way land plants back then in the early
40:05 . Okay, so the reflections of veteran. It is a measure of
40:10 percentage of incident light reflected from the of it tonight particles in a sedimentary
40:18 And it's expressed as percent or % Uh And if you have many veteran
40:26 samples, they often report just the of those were many veteran it particles
40:32 a given rock sample, They would the mean are zero. Uh and
40:39 reflections of veteran it changes with And so it's a thermal indicator and
40:46 as a benchmark in maturation studies. this is an example of a publication
41:05 they were getting an organic matter density Pretty High, Right? 1.53 -
41:13 . Usually it's assumed to be on order of 1.1 g per cc.
41:19 you see, we calculated .83 g cc. Right or wrong. Uh
41:33 a plot of another plot of toC both density in a variety of different
41:42 . This is from burning and what arguing is that the very, you
41:53 , sometimes with the same percentage you have a lower uh both density
42:03 what Bernick is arguing here is that toc itself can be porous. So
42:12 suggesting a wide range of both densities the organic matter, depending on the
42:21 of the organic matter itself. well, organic rocks, that's one
42:30 level of complication. Let's go back inorganic rocks. So we're not going
42:37 worry about organic content right now. we're going to look at relationships between
42:44 , density and mythology. So, are equations in the free or
42:51 Uh, not far from here, actually, well, in Elgin
42:57 Uh, and we have p wave is equal to a constant, which
43:04 pretty close to the velocity of courts . The constant times ferocity minus the
43:11 times volume of clay. So what says is as you increase the
43:17 the p wave velocity goes down as increase the volume of clay, mostly
43:24 courts with clay. The p wave goes down, but the sensitivity to
43:32 is greater than the sensitivity to volume clay. As a matter of
43:37 almost four times greater. Similarly for shear wave velocity is equal to a
43:45 , which is similar to the shear velocity of course, minus a constant
43:53 the ferocity minus a constant times the of clay. Now, ideally if
44:03 have separate V. P and S measurements, you could extract,
44:08 could solve two equations, two you could solve for ferocity, you
44:12 solve for volume of clay and it out that that's not a very accurate
44:20 . These coefficients are similar enough to other at least the ratio of them
44:25 similar enough that it's actually not a precise a measure of ferocity or volume
44:33 clay. So, you really need independent measurement in order to extract ferocity
44:42 volume clay by the way, these in brine, saturated rocks, there's
44:47 water saturation in this calculation. so we could go back to our
44:55 balance equation. And so we could that the bulk density is the ferocity
45:01 the density of the water, plus volume of clay, times the density
45:05 clay plus the volume of course, the density of course. And now
45:11 have uh can combine with equation which is usually these measurements are usually
45:17 reliable than the shear wave velocity but we could do it with either
45:22 and that. So equation one in or two and three we'd have two
45:27 . Two unknowns. And what we is um, across plot like
45:34 So p wave velocity versus density we a court's point and you have a
45:41 point. Now, depending on the and depending on whether you're including bound
45:48 in with the clay volume or in the ferocity, that point can be
45:54 different places. So here we're saying density of the plays 2.4. Whereas
46:01 the previous exercise I said use 2.65 would be a dry clay.
46:09 would be a porous clay with some , which with some bound water or
46:14 ferocity associated with the clay. In case then these ferocity has become more
46:20 effective Torosidis. So you have a porosity line, you have a court's
46:26 line. This would be the velocity relationship for courts, you have a
46:32 water line. That would be the density relationship For 100% claret shell.
46:41 then you would have lines of constant here. So from a p wave
46:47 versus density cross block. If the are within this region, then you
46:53 extract the ferocity and the volume with uh buying the play would be where
47:00 are along any one of these lines the process he would be which line
47:07 on. Okay, now we mentioned the density logs are unreliable very often
47:19 we want to estimate the density log one pretty robust way of doing that
47:28 saying, well, my density is to if I have other measures of
47:35 ology like neutron log, like gamma log, uh huh or you
47:42 other log types. Um and I some idea of my volume fractions of
47:49 different minerals. Uh then the density equal to uh Vying fraction of each
47:59 times the density of that component. , these would be porous components.
48:05 this would be density of pure density of pure uh sandstone, density
48:13 pure clay shell for example. And I had divine fractions of each,
48:19 I could calculate the bulk density of rock. Uh huh. Now,
48:27 do I get this density of the mineral? Well, you could use
48:33 ferocity transforms. These could be like gardener equations or the widely time average
48:40 for each Pure mythology. So, have some transform. It doesn't have
48:44 be in this form, but this a typical form that choose some coefficient
48:50 the measure p wave velocity plus a . And these values would be different
48:57 each with ology. Um and uh I I can compute given the measured
49:04 wave velocity, I could compute what density would have been had. I
49:09 a purely mythology and then I could those purely mythology densities. Mix them
49:15 the vying fractions and get the bulk of the rock. And so this
49:24 an example of then a simulated density . So here's my caliper log,
49:31 is my volume clay log that came the gamma ray and using volume play
49:38 courts, just assuming two anthologies. is a plastic section. I can
49:45 the equation the previous equations and I compute the dash line here, which
49:51 the estimated density. And what you is that the estimated density where you
50:00 a nice borehole here, where the is good and flat, it tends
50:05 agree with the measure of density, where the borehole is very rough
50:14 you see the predicted density is larger the metro density measure density being
50:21 And also you're noticing that thin layer , the calipers washed out in that
50:27 formation. And you can see that densities measure density is too low.
50:41 that caliper is really bad. And the way, actually looking at this
50:48 , I've caught this error before, somehow an uncorrected one got through the
50:53 of plays backwards here, clean is to the right. So this should
51:01 1-0. So this would be a formation here. So we reversed one
51:09 the zero there in volume clay but you get the idea if I
51:13 a valid measurement of length. I could correct the density log and
51:21 is just another example of that. cross plotting the predicted density against the
51:27 density. You can see that they're the original law. There's some abnormally
51:32 densities which if you compare the two in zones like this. And in
51:39 you can see a big difference between predicted density and the observed density.
51:49 if we have a neutron log, then that could help me handle the
51:54 complex mythologies. And here's a way predicting ferocity by cross plotting sonic transit
52:03 versus neutron ferocity. You have to corrected both measurements by the way before
52:10 do this. But now we can the ferocity variation in different mythologies.
52:18 if I have a mixture of I could then from the prosperity,
52:23 what the density should be. so some final thoughts on densities.
52:34 sure that the tool has been calibrated the types of rocks. You're in
52:40 for whole washouts. Beware of large corrections. Worry about oddly shaped boreholes
52:52 be sure to take into consideration the density correction. Okay, I think
53:02 is a good time to take a minute break. So I'll stop recording
53:10 . This conference will now be This conference will now be recorded.
53:15 we go. Thank you. what is the vector um uh entity
53:28 has a magnitude and direction. Exactly . Yeah. Who? I don't
53:35 the answer here. Mm. What force? The muscle one here?
53:51 me? Well, that that's what is equal to but what is
54:05 It is a vector quantity that went fly to an object. Could change
54:11 motion of the object. What is ? The difference? Uh distance traveled
54:29 time, yep. And so how's different from speed? Yeah, that's
54:40 right. And we usually ignore the , you know when we talk about
54:45 , right? When we say velocity six km/s, we don't specify in
54:52 direction. Right? But in an psychotropic world you should um what is
55:03 rate of change of velocity? Uh . What is mass? Did they
55:11 define mass in your physics class? . What is math? I can
55:23 you the definition I found is very . It's the quantity of matter.
55:32 find that to be very unsatisfactory. that's what masses. Uh huh.
55:40 are balanced forces, forces leading to ? Yeah. So leading to no
55:54 of motion of an object. What unbalanced forces? Well getting,
56:03 And then we have ethical M A is g gravitational constant? Uh is
56:12 acceleration due to gravity which is not same as capital T. Which is
56:17 gravitational constant. It's the acceleration due gravity and someplace Well, if I
56:26 given it to you. Oh there go. There are the answers.
56:35 , 32ft per second squared and you convert that to kilometers per second if
56:42 wish. So what is equilibrium? state of unchanged uh, balance of
56:57 is, yeah, yeah, now this is a physics problem that
57:09 asked many PhD physicists, including professors physics one, I've often had this
57:17 and it amazes me how many graduate , even physics majors Don't get this
57:25 right. So if I drop a on your head from a height,
57:32 ft over your head, there will little effect. Right? A pebble
57:37 on your head from one ft above hurt you. But if I dropped
57:41 same pepel pebble from the top of same mass from the top of a
57:47 building, it will kill you if lands on your head explain west in
58:00 simple course. So the same month changing that generation due to the increasing
58:07 , including the force that conduct except acceleration is the same in both
58:16 Remember the acceleration is g is the due to gravity? It doesn't matter
58:23 far it has fallen. The acceleration to gravity is that was essentially the
58:29 physical? Although the situation is we're that. Okay, but it kills
58:39 because the forces greater does, doesn't ? All right. So how do
58:43 explain the force being greater if the is the same and the acceleration is
58:48 same? Mhm. So, did guys uh, I used to watch
59:10 Gear, you know, Grand Tour Gear? Yes, very nice program
59:19 one of my favorite shows and this from one of the hosts. This
59:23 a show about um what do you them? Not just racing cars,
59:29 high performance vehicles, Right. High cars. Um speed has never killed
59:37 suddenly becoming stationary. That's what gets . Okay. The reason the pebble
59:47 you coming off a tall building because velocity is higher and when it hits
59:53 , you're stopping it, you're decelerating . Right? So because the velocity
59:59 higher, the deceleration is much See my point anyway, this has
60:08 to do with rock. Physics really it is Iraq, right? But
60:13 fact, it's an interesting physics I've always found how many people don't
60:19 that answer. Right, okay, , just reviewing a little bit of
60:26 , one composition of forces. If have to forces acting on an
60:32 it's the same as a result in . Right? And I could I
60:40 uh determine the vector components of that in Cartesian coordinates. If I have
60:48 result in force, I could have force in the X. Direction here
60:54 the force in the right direction. said, I could decompose any result
60:58 force into F of the F. X and F of Y. And
61:04 course, FFC in the 3rd There we go. So any force
61:11 resolved into components in each dimension. . Now we experience different kinds of
61:25 . There is a forest like which is called what what's that force
61:32 ? Extension. A laura tension where have a force like this which is
61:38 what compression good. And here we wrote possibly rotational or share forces.
61:49 that's called a couple. If these are not balanced, this thing will
62:04 . Right? So uh this is if I if I have a moving
62:09 stress, the volume elements of the will be trying to rotate. So
62:14 why shear waves are also called rotational . And of course equilibrium is when
62:22 the forces are balanced and if I a balanced couple again will still be
62:30 equilibrium by the way. What type forces? This the arrows here,
62:45 is called torshavn. And in fact way of uh at one point uh
62:52 we had lots of money to spend the oil industry, we actually had
62:56 anal vibrators and they they were good generating shear waves. Okay, so
63:07 going to be dealing with stress and of pressure as an omni directional
63:15 What is stress? It's force per area? So if you sit on
63:23 nail the forest is your way on now. And the area of that
63:34 nail making contact with the part of that's sitting on the nail. It's
63:39 very small area. So you have high all your forces concentrated in that
63:45 area. You have a high On the other hand, you could
63:49 on a bed of nails and I've that by the way, I don't
63:54 if you ever experienced. It's a bit weird, but if now instead
63:59 one nail you have many nails. the forces divided amongst all those
64:05 And so the stress you experience is and you can actually sleep on a
64:10 of nails. It doesn't hunks for . So that's stress and forces a
64:20 . Then stress is also a vector area is a scalar. Right?
64:25 stress is a vector. And it , like for us it could be
64:29 into components in the X, Y Z directions. So in two
64:36 if I have some arbitrary stress factor our arbitrary orientation and it's acting on
64:43 plane. I could decompose that stress to the plane. I could look
64:49 a component of stress in the and I could look at the component
64:54 stress normal to the plane. And the way, in three dimensions I
64:58 have to tangential components in the plane the inclined plane and one normal.
65:07 an arbitrary stress factor is decomposed into components often called sigma, X,
65:14 , Y and sigma Z. if I apply a stress to
65:23 I will deform the rock. And that stress is small enough and the
65:32 is elastic enough, that relationship between and the resulting defamation is linear.
65:40 we measure the resulting defamation as a change, so change, for
65:47 changed in length over, divided by length or change in volume divided by
65:52 volume. I measure this strain as ratio. So strain is actually uh
66:01 liss. It's a ratio of a to a distance or a volume to
66:06 volume. Right? So strain is liss. And if I'm in the
66:12 field here, I have a linear between stress and strength. That means
66:19 have stress is equal to a constant strength. This constant is called an
66:27 module list. Um If I were with a spring, that would be
66:33 the spring constant. Right? Um if strain is dimension liss, what
66:42 the units of the constant? The as the units of stress. What
66:50 the units of stress force per unit ? So this proportionality constant constant between
67:00 and strain has the same units as force per unit area. By the
67:09 , if I'm in this linear range , and if this relationship holds over
67:17 whole, which range, it doesn't whether I'm increasing the stress or decreasing
67:22 stress, I move up the curve I increase the stress as I increase
67:28 stress, the strain increases. But I decrease distress, the strange decreases
67:36 , that means that the strain is recoverable, that's what elastic needs.
67:43 why we call this elastic field. the strain is entirely recoverable. And
67:51 the way, this idea that stress equal to a constant time strain that's
67:56 Hook's law just being applied to stress of force. Now, at some
68:04 , if the stress is big I'll start to deform the rock in
68:09 way that, you know, and break pieces of the rock. I'm
68:13 permanently change the rock. I'm a , I'm a compact it. So
68:19 the stress gets big enough, the between stress and strain can become non
68:26 . And then when I remove the , not all the strain is
68:32 You can think about squeezing a Can write if you squeeze it a
68:38 bit or even I squeeze this I squeeze it a little bit,
68:42 snaps back to its original shape. if you squeeze it too much,
68:48 will say permanently deformed to some So, um, this field,
68:55 is called plastic behavior when the strain not entirely recoverable. And so that
69:02 called the duct outfield. Now, you'll reach a point where the rock
69:10 . And that could be called the point or the failure point.
69:23 the curve could be a little bit complicated. For example, the yield
69:29 here is where the slope changes the changes. That means for the same
69:39 of stress, I'm getting more You can think about that like
69:47 Just, it reaches a point where just starts to give way right?
69:54 so with increments, a small increment stress, you get a much bigger
70:00 change in strength. So this yield where you have that change in slope
70:05 have to be precisely at the limit Hook's law. Could actually be a
70:10 bit more complicated like this where you a point where you're no longer
70:16 So that might be the limits of law. Even though you're going non
70:21 , you may still find that the recoverable. You just follow the curve
70:26 down. That's called the elastic And the yield point is where the
70:32 flattens out here. Alright. To a few things. 10 south dressed
70:43 a pulling force away from the body stress or a compression is a pushing
70:51 acting into the body. A tensile elongates the body. A compressive
70:59 shortens the body shearing stress, exit angles to the tensile compressive stress and
71:08 , or shears the body. This a change of shape of the
71:18 As I mentioned, units of Our force per unit area. And
71:26 we take Dynes per centimeter squared times to the 10th, we call those
71:32 pascal's. And those are the units usually deal with in rock physics on
71:38 scientific side because if I use giga , my velocities will automatically and densities
71:45 grams per CC. I will automatically out with velocities and kilometers per
71:52 And remember I said this proportionality constant , which are the elastic module i
72:00 the a proportionality constant between stress and is called the elastic module lists.
72:07 has the same units as stress. I could use giga pascal's for elastic
72:15 I as well. Now in the industry, they still like to use
72:19 pounds per square inch. So being to convert here between different units is
72:32 . Okay, so a giga pascal of the 10th, nines per centimeter
72:41 . Okay, quick calculation. I you to make a mineral has a
72:46 module list of 40 giga pascal's and sheer module. List of 40 giga
72:52 and the density of 2.65 grand per calculate VP and Bs In km/s and
73:02 convert that two ft/s. Mhm. these equations here. Oh wait,
73:19 were in the old one. Use the equations for VPN Bs that
73:25 gave you before. I know for video costumes. No, no.
74:02 If you use giga pascal's and grams CC, the velocities will come out
74:07 kilometers per sec. That's just a . That's why I like to use
74:14 pascal's VP is 5.9, you want per second, yep. And 19,000
75:23 476 correct. The v. s 3.9. Yeah. Uh and we
75:32 12746. 50%. What was Uh 12? Uh 746. And
75:42 those numbers are very close to those the eastward equations that we talked about
75:48 the last unit. Mm. Uh . Mhm. So here we have
76:02 arbitrary stress factor acting on a face a parallel pipe head here and this
76:15 a slate. I got out of Mechanics textbook and it says the stress
76:24 which is also called attraction, is on an infinite testicle area. Ds
76:31 this little black square over here because an infant intestinal area. So these
76:41 differentials here and that area's dx dy this terminology a stress vector acting on
77:02 face. The the face it's acting is the face that's normal to the
77:12 axis. It's in the it's parallel the XZ plane but it's normal to
77:19 y axis. So that's why this called sigma. Y. It's the
77:26 acting on the wide face, which the face normal to the y
77:32 So rock mechanics, you have to very close attention to this notation,
77:38 the terminology here. So there's an on this flight, can you see
77:51 ? There are always going in the , it's I said rather than what
77:59 go to a girl. Okay, the stress factor by by the
78:04 You mean this? Okay, so notation says nothing about the direction of
78:13 vector of the stress factor. What saying is it's acting on the y
78:19 . That's what sigma's that why needs the Y face is the face normal
78:26 the y axis. So this notation correct. I'll give you a hint
78:36 errors in the caption. Mhm I there is more than one ever.
78:52 , I don't know why the numbers like Roman and also English like 5:00
79:02 . And yeah you mean italic? like Ivy and then three.
79:14 Oh here. Yeah. The figure . Yeah, that's just the Chapter
79:20 Figure 3. I mean that's a way of saying things there. No
79:25 a technical there's a technical error on slide and it isn't the caption mm
79:33 white. I'm not sure what iso to the X. Y axis.
79:38 , again that's just terminology. They from the origin out in the Y
79:44 . That's what they mean by the axis. So this would be their
79:47 . Y axis. Okay, these are my two observation. That's
79:52 not sure what Okay, so where acting on an infinite intestinal area?
80:00 is the area of that square? that rectangle? What is the area
80:08 that? That this infinite decimal That the stresses acting on what is
80:19 Y? That's a distance in the direction on that plane which is normal
80:27 the y axis, D Y is . The area. This is actually
80:34 DZ Yeah. So it gave me delight finding this error in Iraq mechanics
80:42 . Okay. okay, so reminder some. Why is that vector is
80:55 parallel to the Y axis? That is acting on a plane normal to
81:02 Y axis. Now I could then this vector and I could look I
81:08 do a decomposition into the force in Y direction, in the Z.
81:14 and to the X direction. So of forces that give me that resulted
81:21 . All right. So then I have a force in the in the
81:25 . Force in the UAE and of in the Z direction. So that
81:30 me three different stresses. All of stresses act on the Y plane but
81:39 in the X direction, the wind in the Z direction. You see
81:45 Y direction is the normal force when plane and the direction of the forests
81:51 the same designation. That's a normal . Right? That would be normal
81:59 that plane along the Y axis. that So sigma, why why is
82:06 normal sigma? Y X is on Y plane acting in the X
82:15 That's a sheer force and sigma, . Z. Is in the UAE
82:20 acting in the Z direction. That's the sheer component. So one general
82:28 with General Orientation has three components acting that wide plane. The two sheer
82:35 , X, Y. Z and normal component. Why? Why follow
82:44 that's just this space. But I have the X face normal to the
82:50 axis and I have the z face to the Z axis. So each
82:56 those I have sigma's X X six X. Y. Sigma X.
83:04 . And I have cigna, Sigma , X, Z, Y.
83:09 Z. Z. Right so the normal components are xx sigma,
83:15 X. Sigma, Y. And sigma's easy. Is everybody following
83:21 on this? So one so acting a parallel pipe, the pipe head
83:28 this distress applied to that entire body applied to all the faces then?
83:41 many components of stress act on that pipe? Ed. Uh huh.
83:49 period? Well on all the So I have three equivalent faces.
83:55 we're assuming a uniform stress field. the stress on the top face is
84:01 same as the stress on the bottom . So we only need to consider
84:05 stresses on three faces. The opposite have the same stresses. Yeah.
84:12 huh. Right. No we have acting on each face and there are
84:19 three faces that are independent. So . And again I've assumed the uniform
84:25 field that means as I move around stress hasn't changed. Okay so believe
84:36 or not you weren't bargaining for this we're talking about a tensor now the
84:44 tensor remember uh stress has magnitude and and There are nine components of stress
84:57 on an object on this face on face. And on that face on
85:03 infinite decimal volume so I can express stress as the tensor. It's just
85:13 matrix With all nine stress components in . That's the stress tensor.
85:25 by the way this object isn't we're going to require that this object
85:30 equilibrium. So if I have reversed they must be equal. Otherwise the
85:37 going to move. So sigma X. Which is on the y
85:46 acting in the Okay, let's get axes right, where is the white
85:50 ? This is the white plain. it's that plane acting in the X
85:57 is this guy? This guy must must equal sigma X. Y.
86:04 guy if it doesn't, the cuba going to start rotating. Now this
86:18 the more complicated situation where you have non uniform stress field so the stress
86:24 varying. So you have to take account the rate of change with the
86:30 with position along each axis. you know people engineers in rock mechanics
86:39 need to worry about this for our . We're going to assume a uniform
86:45 field so we're not going to worry this kind of situation. So what
86:55 defamation? Here's an under formed solid here is a deformed solid. The
87:05 can be expressed as the changes in of the size of an inscribed
87:19 Now suppose the angles angle angle. an angle be suppose these angles off
87:26 the same. I could have a stress where the sides become shorter but
87:32 angle stay the same. So I've the volume but I haven't changed the
87:38 . I haven't changed the shape of rock. So a volumetric strain will
87:48 change these angles, but a distortion I change the shape will change those
88:02 . So strain is the deformation caused stress stress. Dilation is a changing
88:11 distortion is a change in shape. , so coming back, let's define
88:23 terms again. This should be reviewed now. What does elastic mean?
88:34 strain is recoverable mostly? Yes, right. You're both right. The
88:39 is reversible. What is the elastic ? Uh is the yield point?
88:50 , okay, so in the simplest the elastic limit and the yield point
88:56 the same. But literally the elastic is the stress beyond which the strain
89:03 no longer entirely recoverable. So some all the defamation, but some of
89:11 deformation is irreversible. What is hooks a distress? The spiritual proportional to
89:20 strength. Well, both. And strain is proportional to stress. Um
89:29 does plastic mean? You can Right. Yeah. So it's at
89:38 partially reversible visco elastic is kind of tough one. Uh We use that
89:45 a lot in uh in geophysics and has specific uh implications about how
89:55 Uh how the rock continue weights, would say just very loosely. It's
90:02 elastic and plastic deformation rupture is where material breaks. Now here's an interesting
90:14 brittle. We haven't talked about what means yet. What is a brittle
90:26 bill across? Yeah. But what what does the term brittle mean?
90:33 the definition of brittle? Uh Is erupt that does not go through a
90:42 ? Uh It moves from elastic to to eat. Exactly. So coming
90:49 the way back here. That's very . Most people don't realize this if
90:55 is a lot of defamation before That is called plastic. If there's
91:03 little depth defamation before fracture. So put it this way there's very little
91:13 plastic deformation before fracture. That is a lasting. So if I break
91:19 after the U. Point here that's . If I have a lot of
91:26 deformation before fracturing that's called Doctor. people think brittle means it's easier to
91:38 . That's not what it means. it means is when it breaks it
91:43 right away it doesn't perform very Okay so now let's uh start to
91:56 some cylinders. And again we're looking this in two dimensions but I have
92:06 cylinder and on one end it's attached a perfectly rigid anchor there. So
92:15 diagonal lines means that that thing cannot . It's perfectly richard and I have
92:24 material here and I compress it. is a uni axial compression. It's
92:31 compression in one direction and it's a compression. So it's equal force at
92:40 point along here. What will happen when you squeeze that rock with a
92:47 axial compression, you will shorten it you have a change in length.
92:55 you will also it will also bulge because the material that you're pushing into
93:02 shorter distance to conserve the volume, has to bulge app. So you'll
93:10 a change in length and a change with. So this uni axial
93:19 So compression in one direction has strained two directions. It's got a longitudinal
93:26 delta L. Over L. In trans verse strain. Delta with over
93:37 in a perfectly elastic rock or situation whether Iraq behaves elastically depends on the
93:47 of the stress and the time over that stress is applied. So let's
93:52 an instantaneous stress. And it's small small enough magnitude that I'm still obeying
93:59 law. Then it doesn't matter if compressing or I'm pulling pulling will give
94:06 the same situation. It will uh L. Over L. Or delta
94:13 . Over W. So as I on, it gets thinner.
94:17 if I'm applying a sheer force for most part I keep the volume the
94:24 . And what changes is the angle . So I have a change of
94:30 . So a sheer force involves only change in shape. In the case
94:35 solids. When I have a unique compression, I'll have a change in
94:40 and a change in volume in a , I won't be able to change
94:45 volume, I'll maintain the volume constant that's what we mean by being in
94:52 . This is an unconfined experiment. it if it were confined, I
94:57 change the volume of that fluid by it. But unconfined like this um
95:03 can't change the volume of the A solid will change its volume
95:15 I thought this was an interesting quote Nigel Anstey. Um if you ever
95:20 um very clear, concise introduction to , his course notes are fantastic.
95:29 , they're expensive to get usually um h R D C International Human
95:40 I think Development Corporation sells his notes they're quite expensive, but if you
95:47 get your hands on them, they're . And this is a quote from
95:53 . Yet, despite all the problems velocity, we are reluctant to give
95:57 the hope that somewhere, if only could find it, there must be
96:01 master variable connecting velocity and geology, master variable meaningful to a geologist and
96:09 leading directly to the physical property of and uh to the state, we
96:16 have a master variable really. Um same factors that affect elastic module I
96:24 velocities. So, you know, the entire science of rock. Physics
96:29 , connects geology to velocity. Uh a few words on tensors and if
96:40 course we're being taught by dr you would be doing a lot of
96:46 mathematics and we're not going to in class, but you need to be
96:52 of what they are. So, zero order tensor is a scalar and
96:58 can think of that as a a first order tensor is a
97:03 so you can think of that in computer as an array, And the
97:08 order tensor is a two dimensional Now there's a strain tensor and a
97:20 tensor. So, I have a , remember the designations of stress on
97:28 face? A law indicate direction. . So along the K access on
97:37 I face. Right. And we we have nine of these guys.
97:45 . Xx xy xz y x. ? Why? Why is the uh
97:52 x z, y, z Right, so we have nine of
97:57 and each one of these produces strains all nine directions. So, I
98:05 um I have a strange tents are again epsilon for strain, same nine
98:17 , So I. K stresses resulting LM strength, is that clear?
98:26 , each stress has a proportionality constant with each strain. So there are
98:35 of these And there are nine of . Right, So the like a
98:41 produces the LM strength. So how proportionality constants do I have here?
98:55 yep, I have nine of these to nine of these. Each one
99:02 these has nine of these and there nine of them. So 81.
99:08 is called the stiffness tensor. So one of these is the elastic module
99:14 connecting a stress component with a strained . Got it. Now I could
99:26 this differently, I could talk about compliance tensor where I have strained is
99:33 to the compliance tensor times the Just that I'm just offering that for
99:44 . We're not going to talk about sensors any further. We're going to
99:49 entirely with stiffness tenses. Now, , They're these 81 module. I
100:01 not independent of each other. If have an icy tropic rock, There
100:12 only two module. I buck module sincere modules. I don't have 81
100:19 I. So depending on the symmetry the anti Satrapi, you have a
100:31 number of independent components. And the complicated case is try clinic symmetry where
100:38 have 21 independent components. That would a layered medium with fractures crossing the
100:48 . That would be a kind of clinic symmetry. The ones that are
100:53 important to us, Our idea tropic two independent stiffness coefficients and hexagonal
101:01 which is the same as trans I Satrapi. This is a layered
101:07 or it could be a set of fractures in a homogeneous medium. This
101:12 trans verse and ice octopi And that five independent components. Now there's further
101:26 to worry about. Uh this I is cumbersome to carry. So there
101:34 a short hands where Ikea is represented M. And L. M.
101:43 represented by N. And so I . Of 11 gets an M.
101:52 of 22 gets an M. Up 33. So these are the normal
101:57 And instead of six trans verse Now you only have to worry about
102:03 because remember that if I alternate the those have to be equal. So
102:12 becomes four etcetera. So now I write the stiffness tense tenser in this
102:21 convenient fashion And many of these values going to be zero along the
102:35 I have the normal components here and off diagonal terms. In some cases
102:43 are new values, but in other they can be expressed in terms of
102:49 other ones, so they're not So uh this is hexagonal symmetry,
102:57 have 1234, five independent components. huh. Now, this was a
103:15 of a confusing statement from Sean. So let's see, let's see if
103:22 can understand what he means by He says to observations are important for
103:27 description of the stress strain behavior. number one, the elastic module I
103:34 on stress. Therefore the stress strain are non linear. Wow, that's
103:44 a little bit something to chew Let me say that we don't worry
103:53 this, we don't worry about the module, I being stressed dependent,
104:00 look at one curve. Iraq. , We have one stress strain
104:09 Let's say that one. Let's make simple as I increase the stress.
104:15 stays linear. We have not changed stress strain relationship. And that's because
104:27 wave propagation we have very small stresses applied. So we're not thinking about
104:35 linear, nonlinear charity over the strange change and stresses. In
104:43 uh over geologic time in the earth I bury Iraq, we know the
104:49 becomes harder. And so the elastic list is going to get larger and
104:54 . Right? This slope is going increase. You're going to have less
104:58 less as the rock gets buried deeper deeper is under more confining pressure.
105:05 really under higher stress. Uh You'll less. Strange. So in
105:12 if we're talking about the ambient the stress under which the rock exists
105:19 the subsurface. That is a non relationship between ambient stress and how much
105:26 rock has strained over geologic time to where it is. Right. That's
105:32 non linear relationship. But we deal the Viet oric stress. We bury
105:38 stress by passing away through the rock we very distress in the laboratory.
105:46 stress strain relationship will be more And we think about that as a
105:54 . V. A. Taurus a deviation from the ambiance stress.
106:01 let's clear up that confusion. But uh he goes further. He says
106:11 are not ideally elastic materials. Their to stress depends also on the velocity
106:18 defamation and the history of defamation. deviations from hook floor result in one
106:27 phenomenon of energy absorption. And that's that's why I said, you know
106:35 we talk about visco elastic, absorption can be described mathematically, would
106:42 visco elastic model and that involves uh plastic deformation. And he also talks
106:51 the discrepancy between statically and dynamically determined . I A static module is you
106:59 the rock, you measure its length its volume, you put it on
107:05 , er pressure, hold that pressure and then re measure its dimensions,
107:13 a static measurement. Those are very stresses which are applied. A dynamic
107:21 is you pass away through the you measure the velocities and you back
107:25 the elastic module knowing the density, are dynamic module. So, dynamic
107:33 I number one are dealing with much , much more other stresses. Devia
107:42 stresses as opposed to large ambience Right? So smaller strain amplitudes for
107:50 dynamic measurement, also a dynamic measurement a frequency of oscillation. It could
108:00 uh kilohertz or megahertz. And what find is that if Iraq is
108:12 it's also going to be disperse That means the velocity varies with
108:16 That means the elastic modules varies with . So dynamic measurements have dispersion and
108:26 have a small strain amplitude relative to measurements which are essentially a zero frequency
108:40 . So now the minerals we deal our anti psychotropic, certainly think about
108:46 , they're going to be an icy . But even quartz calcite,
108:51 these are the stiffness coefficients. And see there are different, in
108:58 Uh We're beyond uh huh. Uh beyond hexagonal symmetry, even though that's
109:10 we up in the soup. Um how can we talk about rocks being
109:20 tropic If the minerals composing the our anti psychotropic, uh it's a
109:34 of skill. It is a matter scale. Right? And so and
109:40 the tropic minerals, if they're randomly , can produce a nice the tropic
109:54 . Okay, so let's get to elastic module I that are important to
109:59 . Number one is the bulk module also called the in compressibility and that's
110:06 resistance to volumetric compression. It's the stress divided by the volumetric strain.
110:16 sheer module is also called the rigidity the resistance to share deformation and that's
110:23 to the shear stress divided by the strength. The shear strain measured by
110:28 change in angle of Iraq, by way, both of these cannot be
110:36 . These particular elastic module, I be zero or greater. Where would
110:43 have zero share module list? Some . No, well uh the the
111:02 modular says, if I have stress is how much strain I'm going to
111:07 . Right. Um So the sheer is is what it is for the
111:14 under the conditions it has. So the stress doesn't change the share
111:20 So if I I have a rock a certain share modules, I can
111:23 it under zero stress and it would zero strain, but it has some
111:31 share modules. What would have zero modules? Exactly. Right. So
111:47 and liquids have zero shear modules. again, remembering the stress is equal
111:59 a constant time. Strange if I a shear stress, the shear stress
112:13 is equal to some constant times the strength which is measured by a change
112:24 the angle, right relative to the angle. So if it was originally
112:31 , these were right angles and then share the rock, I produce a
112:38 angle there. And so uh shear is related to the change in the
112:47 . Uh if I apply a hydrostatic , so the same stress in all
112:55 , I will shrink the volume of and the ratio of the stress is
113:01 to the bulb module is times the strain here. Now, engineers like
113:08 deal with a concept called Young's we don't use it very much in
113:15 , but the engineers like it and go back to our cylinder. And
113:20 we uni actually compress this, this an unconfined cylinder. So I can
113:28 it or I lengthen it, it matter if I compress it, I
113:33 it better if I lengthen it. make it a slimmer. But the
113:38 in length divided by the original length is called Young's modules. Now if
113:49 take this cylinder it's unconfined, I a uni axial stress to it.
113:59 will strain longitudinal li it will also trans firstly. And the ratio of
114:07 verse strain to longitudinal strain is called . The ratio of verticals or uh
114:24 stress to or longitudinal stress to longitudinal . These young's modules. So we
114:42 say that the Parsons ratio here, using sigma for persons ratio. That's
114:49 because we've been using sigma for uh . But oddly enough in geophysics we
114:56 to use sigma group with songs So it's the fractional change in with
115:03 by the fractional change of length and depending on how you choose your
115:10 You might have you might see this with the minus sign in front of
115:14 . I tend to ignore the minus . Is this this author did.
115:19 the practical limits of lessons ratio are zero and .5. In fact zero
115:29 not the theoretical limit. You could have a negative plaisance ratio. That
115:35 as I squeeze the rock, it thinner and they say that good french
115:42 Kirk corks have negative response ratio. never seen a rock with a negative
115:49 ratio. So for for all practical the smallest lessons ratio can be zero
115:58 the way. What does persons .5 correspond to? Uh huh.
116:14 uh fluids. Yes, that's And he could derive that actually if
116:23 take a fluid cylinder and you change length, keep the volume the same
116:33 see how much it's with its with . So I'm not going to give
116:39 to you as an exercise. But very doable. And what you'll find
116:47 that you'll approach a maximum value of . So that would be a
116:58 So by the way, uh in book they said persons ratio is usually
117:03 .25 for most rock materials. Well doesn't include the fluids but these are
117:11 minerals. And you can see that a variation but there is one very
117:18 significant anomalous mineral here. Oh, you see it? Yeah.
117:33 courts of all the minerals we tend encounter court says the lowest Hassan's
117:42 And so it's perhaps not surprising that rich rocks tend to have low Hassan's
117:59 . Okay, now we're going to a little bit different experiments. Instead
118:09 having an unconfined cylinder, we're going put a rigid constraints around it in
118:15 laboratory that might be a thick aluminum or something. Um So it at
118:24 theoretically. Now we're not going to it strange trans firstly. Right,
118:29 it can't change its with and we're to longitude only compress that rock.
118:36 the transfer strain is zero. Now measure uh the stress divided by the
118:45 strain and that is called the plane module is hem also called the constraint
118:53 . Us this guy is very, important to us because when I pass
119:01 plane compression all way through the any volumetric infant intestinal volumetric elements,
119:13 may want to strain trans firstly, remember I have a plane wave.
119:19 if I have an infinite intestinal element , that wants to strain trans
119:25 right next to it. I have international tell element which also wants to
119:31 trans firstly. And those lateral stresses cancel each other out. You see
119:41 So passing a plane wave through Iraq I'm in the middle of the plane
119:47 , it's like I have a rigid . So compression all waves involved.
119:58 plane wave module is this constrained modules by the way, happens to be
120:05 Plus 4/3 view. That's why that's module lists in the equation for p
120:11 philosophy. Is this also the youngest for young mothers do not have the
120:20 is unconstrained. That will that if unconstrained, that will give you a
120:28 larger change in length because it's freer change length by fattening. But now
120:35 can can fatten. So it's going resist your change in length more.
120:43 the plane wave modules should be bigger young's module. Now it turns out
121:00 if I'm in an icy tropic I can express all of the
121:06 all this stiffness coefficients or all the module. I I could think of
121:12 any kind of stress in any kind strength. I could express any of
121:17 elastic module i in terms of any other elastic module. So in this
121:26 lama is constant and the rigidity rigidity is uh uh an overturned,
121:33 there, I forget what that's Um but lambda and sometimes we use
121:40 in geophysics because velocity is K plus that's two lambda. Um here it's
121:50 in terms of Hassan's ratio and Young's . E often e issues for Young's
121:58 . Um but I can express any elastic module. Is the land that
122:04 be expressed in terms of K and for example. Um Any modules can
122:10 expressed in terms of the other, terms of two other elastic module.
122:20 this is one thing I'm not gonna you to memorize for the test.
122:28 These are equations expressing the different elastic . I and here the V
122:36 I forget what that's called in Uh The V this is Poseidon's ratio
122:42 . So, new rigidity am plane module lists. Hassan's ratio here,
122:52 . Young's module is bulk module So you can express any one of
122:58 in terms of two of the And here they are using you and
123:02 if you express some of the Yeah, I've seen, I don't
123:11 you uh, in terms of VPs well, which makes sense, we'll
123:16 talking about that. But, so, uh, that's as far
123:22 I want to go today. We'll it up Bright and early 8:30 AM
123:26 morning and we'll continue talking about elastic . I I'll stop
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