| 00:01 | this conference will now this conference will be recorded. Okay you have my |
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| 00:08 | there. I'm glad you reminded me till now. We especially with really |
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| 00:19 | of the equations we've been using uh for some of the multiple regression equations |
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| 00:27 | we had varying composition. But uh V. P. V. S |
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| 00:33 | , we were looking at the VP . Density trans et cetera. Most |
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| 00:40 | these were for pure mythologies. In everything we've been doing with gas mains |
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| 00:48 | as students mono minerality assumes pure In fact it gets a lot more |
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| 00:57 | when you have multiple minerals especially and the minerals have very different properties and |
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| 01:07 | we need to figure out how we're to deal with complex mineralogy is because |
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| 01:14 | of the time we don't have absolutely rocks, we're looking at some mixture |
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| 01:19 | minerals and especially when one of the is much more compressible than the |
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| 01:27 | Uh huh. Yeah. You have problem. That's certainly true in shale |
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| 01:33 | . And we spent left Alaska Mestre a class talking just about the complications |
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| 01:40 | organic material and I barely touched that this class and I don't know if |
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| 01:46 | will but the general idea of having minerals is something we have to deal |
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| 01:55 | and so will deal with it on few different levels. We'll deal with |
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| 02:02 | using bounding equations, we'll deal with using precise numerical models. I like |
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| 02:09 | say precisely wrong because they don't actually real rocks but their precise in giving |
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| 02:18 | an exact answer for a very highly situation. And then we have empirical |
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| 02:27 | ways to combine empirical equations in a that seems to work and has been |
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| 02:34 | by comparing two measurements. Uh So are the three levels we're going to |
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| 02:39 | at. And but I want to in general first about what composite medium |
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| 02:46 | is and that is trying to predict properties of a mixture of constituents. |
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| 02:57 | it's easier to think about when we're with two constituents. But keep in |
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| 03:04 | that we're going to do this in dimensional hyperspace if you prefer that |
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| 03:12 | Um but we're dealing with mixtures. we don't have necessarily rigid rules of |
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| 03:22 | or physics to tell us precisely what's to happen as we mix these things |
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| 03:29 | . Remember what is Iraq? Iraq an aggregate. Uh So the elements |
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| 03:37 | the compounds in that rock are in arrangement, but it's not a crystal |
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| 03:44 | arrangement. And it's it's never, know, like our spear packs. |
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| 03:50 | never a perfect uniform lattice where we actually use physics and predict precisely what |
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| 03:58 | going to have. In fact, are very complex media and to characterize |
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| 04:05 | theoretically in order to make a theoretical , you would have two characteristics at |
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| 04:13 | different scales at the same time, very fine scales where we don't really |
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| 04:21 | know what's going on in the very details at grain contacts in within micro |
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| 04:29 | etcetera. So, uh we could have tremendous computational ability and we could |
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| 04:39 | numerical modeling if we could characteristic no matter how oddly shaped the pores |
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| 04:46 | , no matter how oddly shaped the are. If we could come up |
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| 04:51 | a three dimensional picture of the rock we could characteristics the materials within the |
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| 04:59 | , the constituent properties. Uh then could use a sophisticated finite element program |
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| 05:07 | predict the results. And in I published a paper probably 30 years |
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| 05:16 | where we did just that we did element modeling of rocks. The problem |
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| 05:24 | , it's very difficult to learn from things because rocks are extremely complicated. |
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| 05:34 | how do you verify your prediction? no, there's not a single publication |
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| 05:41 | they were able to characterize a complex , Do the finite element modeling and |
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| 05:49 | the velocities and show that that could done. Reliably, there's been some |
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| 05:54 | doing this with a fluid flow predicting . But that's an easier problem. |
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| 06:02 | more about determining the radius of the throats. Right. So measuring the |
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| 06:09 | space in the rock and how wide is, and then uh numerically passing |
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| 06:16 | through it. But as far as properties. Not very good. So |
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| 06:23 | the ability to do that exists. never seen it used in a practical |
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| 06:28 | . And so I'm not going to any more about it. I will |
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| 06:33 | about theoretical models because we can conceptually from those theoretical models. So, |
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| 06:43 | , we have a mixture. That that I have some property I want |
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| 06:47 | measure. It could be both module it could be a permeability, could |
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| 06:52 | a velocity and attenuation, whatever it , we're measuring a property of the |
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| 07:01 | of the composite medium as we vary concentration of constituents. Right? So |
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| 07:09 | at X. We have the property constituent eh and here at this |
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| 07:16 | We have the property of constituent And then things vary in between. |
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| 07:21 | that's what the composite media model is to tell us how things vary in |
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| 07:35 | . Some reason. I there we . Now, early on, I |
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| 07:42 | about the fact that when we talk heterogeneity and homogeneity, it really depends |
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| 07:52 | the scale of the measurement. For , a sandstone, if you were |
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| 07:58 | a very microscopic scale could be viewed being heterogeneous. You've got a coarse |
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| 08:04 | , a clay grain, you've gotta filled with fluid, you've got a |
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| 08:08 | bubble in the poor. Right? as you move spatially microscopically around this |
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| 08:17 | , you have very different constituent So you can think of that rock |
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| 08:22 | being in homogeneous. On the other , if statistically the very, the |
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| 08:33 | fractions and the constituent properties etcetera. statistically on the average, they stayed |
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| 08:40 | same, especially uh, and when look at things macroscopic lee at large |
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| 08:47 | . Right? So microscopically are interrogating as a wave length is smaller than |
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| 08:54 | grains smaller than the poorest macroscopic lee device that that's a sonic log maybe |
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| 09:01 | the order of feet. If it's seismic wave on the order of 100 |
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| 09:07 | maybe. Um So that interrogating devices to average the properties over some |
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| 09:18 | And if you measure the properties at very might find microscopic scale and compare |
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| 09:28 | to the properties you measure and a scale they're they're different. Uh You |
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| 09:36 | take the microscopic properties for example, could uh hasta ray through a |
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| 09:43 | An infinite frequency ray. I can up the travel times in every constituent |
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| 09:50 | goes through, take the total travel and from that kid of velocity. |
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| 09:57 | I could treat that medium as a an effective medium where I'm compressing this |
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| 10:05 | volume. This low frequency wave coming with the wavelengths very long, compressing |
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| 10:12 | entire volume and it will generally measure lower lower velocity when you do it |
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| 10:22 | way. When you measure the elastic and then calculate the velocity as a |
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| 10:28 | . And that's what we call an medium. An effective medium is a |
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| 10:33 | wavelength prediction as to what the constituent properties the effective properties of the medium |
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| 10:41 | going to be. And the key here is the wavelength of the measurement |
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| 10:48 | to the diameter of the heterogeneity. by the way, very commonly, |
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| 10:54 | heterogeneity is layering. So this plot be very appropriate for looking at the |
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| 11:02 | of a layered medium. If I'm it at very short wavelengths That are |
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| 11:09 | a 10th the dimension of the then I'm in the realm of ray |
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| 11:19 | . Uh huh. I just basically up to travel times. In that |
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| 11:24 | , if we were actually in that , the time average equation would be |
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| 11:29 | correct. Right? You add up travel times through the constituents and divide |
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| 11:35 | the distance. So that is ray . On the other hand, you |
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| 11:41 | uh two very long wavelength, if more than 10 times or so, |
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| 11:47 | dimensions of the objects, you have effective medium. And if these heterogeneous |
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| 11:53 | occur in layers, then you could the properties of the layered medium from |
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| 11:59 | properties of the layers. Using baptist and we'll look and see what that |
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| 12:05 | uh in a minute. Now, between though between maybe six and 10 |
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| 12:13 | the dimension size. Uh It gets because in that realm you're going to |
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| 12:20 | to move from ray theory to effective , it would be nice if it |
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| 12:26 | a very nice smooth ramp like But in fact they're scattering that goes |
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| 12:33 | and there's interference between reflections from different ease. You know, basically in |
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| 12:41 | layered medium, you'd be talking about effects and so forth. And the |
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| 12:46 | is these interferences give you apparent velocities are, you know, could be |
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| 12:53 | over the place basically. Um I think of these as apparent velocities as |
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| 12:59 | to real velocities right? But uh you have to move from right theory |
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| 13:05 | effective medium theory. It turns out very often this ramp and many, |
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| 13:13 | are many different theoretical theories for All of them have in common that |
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| 13:24 | ramp for a single mechanism, a uh uh mechanism resulting in the different |
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| 13:35 | from ray theory to effective medium one mechanism gives you a very sharp |
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| 13:43 | like that. Now it turns out one can envision different mechanisms at |
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| 13:54 | Uh for example uh layers of different or uh pockets, you know lateral |
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| 14:02 | , ease of different magnitudes, geo and things like that. If you |
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| 14:08 | a number of different mechanisms with different then you might find that this is |
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| 14:16 | more gradual change. Most of the , you'll see have a relatively rapid |
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| 14:24 | from Ray theory to effective medium usually between six and about 10 times |
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| 14:30 | dimension of the object. Okay, I, as I mentioned before, |
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| 14:39 | we're going to deal with composite media various levels. Level one, which |
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| 14:49 | , I would say the least precise using bounding equations. So this tells |
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| 14:57 | what is the range of possible answers we've already looked at the void in |
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| 15:05 | Royce bounds and these are the widest bounce. In fact for layered |
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| 15:13 | they are pretty much the difference between theory and effective medium theory. Um |
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| 15:22 | it's possible to try to tighten those when we start talking about rock fabrics |
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| 15:29 | being uh granular fabrics or uh a with inclusions in it. It may |
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| 15:40 | or it seems to be kind of believed that the machine Strickland bounds are |
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| 15:49 | and these bounds are much tighter, . I'm not convinced that the machine |
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| 15:56 | bounds are universal, but it's worth at them. You could say these |
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| 16:01 | be tight bounds. These would be bounds. Okay then the next level |
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| 16:07 | could go to infinite precision and we be infinitely wrong but we could be |
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| 16:11 | precise and repeatable by using Theoretical And the most common one is the |
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| 16:22 | tacos theory that we've talked about this where we add inclusions to the |
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| 16:29 | I've shown you some examples using Custer does and we'll look at some |
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| 16:38 | Uh these were actually preceded by Okano which are similar and uh there are |
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| 16:49 | are variations in the theory and there some aspects of the theories I don't |
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| 16:53 | with. Um but you know, Budiansky, there are some claims about |
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| 16:59 | , which I'm not so clear Uh put this one, I'm not |
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| 17:05 | convinced. I believe them so, I prefer custard tuxedos but they're they're |
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| 17:09 | limited in how they're applicable, but looking at because we'll develop conceptual understanding |
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| 17:19 | third level is dealing with real rocks in my experience we have to resort |
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| 17:25 | empiricism. Uh We have global Vp . S friends um Vp ferocity trends |
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| 17:34 | to be locally calibrated. And of , and so if you can locally |
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| 17:38 | the V. P. B. trends, that's even better. Uh |
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| 17:42 | we handle fluids substitution in real rocks video Gasman theory. Video theory for |
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| 17:51 | Gasman for seismic frequencies and for sonic somewhere in between in brian saturated |
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| 18:02 | we assume that gas plants equations are . But if the N. |
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| 18:09 | To Iraq's and hydrocarbons in them, not necessarily convinced that we can use |
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| 18:16 | mains equations. Okay, So we spoke about the boy Royce bounds. |
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| 18:25 | Royce bound is uh the ice so situation where the strain in each layer |
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| 18:32 | different. The soft layers compress more the hard layers and that's your softest |
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| 18:41 | . And the void bound is when in Collins, you can see that |
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| 18:45 | hard layers dominate here, the soft have less impact. And this is |
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| 18:51 | highest possible velocity in most cases uh don't get anywhere near the void bound |
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| 19:01 | log measurements or in laboratory measurements, Royce bound seems to be a practical |
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| 19:07 | limit. When we have poorly lit uh relatively unconsolidated rocks. We were |
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| 19:17 | to the boy band. Uh The limit doesn't seem to get to the |
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| 19:24 | bound and we talked about the critical model and the roemer Hunt Gardner equation |
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| 19:30 | being practical upper limits. Uh It's that the machine Strootman upper bound may |
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| 19:39 | a good upper limit, but I actually actually, I haven't investigated |
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| 19:44 | So that that is something that should done at some point by somebody uh |
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| 19:51 | the machine strict an upper bound to measurements to see if they form some |
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| 19:57 | of envelope or if they are violated the measurements. I don't know. |
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| 20:02 | , what are the machine Strickland Well, the lower bounds has the |
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| 20:08 | material inside the more compressible material. this is the lower bound. And |
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| 20:16 | can see when the more compressible material the fluid that this reduces to a |
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| 20:23 | . And this becomes the Royce Right? They'll be the same. |
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| 20:28 | of course, with machine Strickland, could both be solid. You can |
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| 20:32 | an in compressible followed in a very self. Anyway, this is the |
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| 20:37 | situation because the compressible rock can accommodate of the compression. Whereas the hard |
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| 20:44 | doesn't have to compress very much at . Right. That's soft materials |
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| 20:50 | It will accommodate most of the deformation like in the in the Royce |
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| 20:56 | On the other hand, the machine upper bound is when the soft material |
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| 21:03 | surrounded by hard materials. So that be a poor or it could be |
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| 21:09 | more compressible solid. And you can in this case the way it's |
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| 21:15 | uh we have an arch. So the strongest possible situation you can |
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| 21:24 | here is a spherical poor or historical in general would be the upper |
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| 21:32 | So very literally the machine strictly an bound is a spherical inclusion surrounded by |
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| 21:39 | heart, soft inclusion surrounded by the material. Okay, we have already |
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| 21:46 | that when we have stack layers in of fluids and suspensions, the roi |
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| 21:54 | , we have that simple harmonic average or reciprocal average. Now to take |
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| 22:09 | the fluids into account as we did the last unit. We need to |
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| 22:18 | a fluid module lists. And what said was that we use woods |
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| 22:26 | Right, So that's again a reciprocal of the bulk module i of the |
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| 22:31 | in the solids. Right, So is written with two fluids, |
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| 22:36 | hydrocarbon and solid materials. That one ferocity is the solid volume. Sw |
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| 22:43 | ferocity is brian volume. One minus W times ferocity is the hydrocarbon |
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| 22:49 | And you do the same thing with density, just straight mass balance |
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| 22:55 | Now, we also talked about patchy last time and we'll come back to |
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| 23:01 | when we talk about dispersion, but limit on what the fluid module list |
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| 23:08 | be would be the void average of fluid modules. And what we'll see |
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| 23:14 | that under certain circumstances, at very frequencies it looks like the void average |
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| 23:23 | describes the fluid module as well when have low saturation, um and we'll |
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| 23:31 | come come back to that when we about this version. Yeah. Now |
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| 23:41 | also mentioned that when you we don't what, where we are between the |
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| 23:49 | and the void beyond that, we also use the hill average, Which |
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| 23:53 | just the limit between the two. this is pretty standard operating procedure, |
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| 23:59 | can be very wrong. But when module, I are very similar to |
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| 24:06 | other, if these modules are all similar, then the voice and the |
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| 24:12 | bounds are not that far apart. so if they're close to uh, |
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| 24:17 | begin with, then the hill average going to be a good guess as |
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| 24:21 | what things could be, you're not to be two very wrong. Uh |
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| 24:27 | so if I have a complex mythology courts called dolomite limestone, I can |
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| 24:34 | them uh this way by taking a average and I might predict the results |
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| 24:42 | well. But when one of the , one of the constituents is very |
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| 24:48 | , it could be fluids, it be organic matter. Then what happens |
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| 24:54 | our void Royce bounds become very The boy found is dominated by the |
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| 25:01 | material or the less compressible material, Royce bound would be dominant, being |
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| 25:08 | would be dominated by the most compressible . So they could be far apart |
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| 25:15 | the hill averages, you know, be very uh significantly in error now |
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| 25:26 | layered media when these are elastic So they have rigidity. And the |
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| 25:35 | medium turns out not to be the bound, which would be the volume |
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| 25:42 | reciprocal average of the bulk module. but a bacchus average where it's the |
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| 25:48 | weighted reciprocal average of the p wave is the plane wave modules, K |
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| 25:54 | four thirds mute. And so when have a laminated reservoir and a seismic |
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| 26:00 | is propagating through it, it's not to the average velocity, is responding |
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| 26:07 | the reciprocal average of the plane wave . Um And that would be the |
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| 26:15 | the long wavelength velocity of a highly bedded uh huh, succession. On |
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| 26:23 | other hand, if I had infinite then I would have a time average |
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| 26:29 | of course won over velocity is the . Right? So that you can |
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| 26:33 | this is the time delta T. unit time per unit distance would be |
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| 26:41 | to some of the volume fractions of constituents times the travel time in those |
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| 26:50 | . So the slowness is in those is or one over the velocity. |
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| 26:56 | right. So you could write the average equation this way. And so |
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| 27:02 | results are going to be different. I said, the back is average |
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| 27:06 | to be slower than the time Okay. Which brings us to your |
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| 27:15 | for unit 10 uh huh. Mix sands and shells together. And I |
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| 27:24 | you the velocities of the sand and shell and um net to gross |
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| 27:32 | So these are highly into bedded, thin layers of sand and shale and |
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| 27:38 | with each other. So when I and that's gross. What I mean |
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| 27:43 | 50% is shell right? Um 50% stand if I say net to gross |
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| 27:51 | that means 20% sand, 80% shell cetera. Okay so um so compare |
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| 28:01 | baptist average and the rate theory for case. Also uh use uh the |
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| 28:09 | equation and Gardner little logic trends to the density. You have the |
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| 28:17 | You could estimate the densities and see they compare using the general Gardner equation |
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| 28:26 | using the mythology specific equations and then them back together. Um And tell |
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| 28:33 | if that if that makes a significant . And compare three cases. Stand |
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| 28:41 | of 20%. and 80%. All now. Um for the preceding example |
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| 28:55 | could calculate the vertical and horizontal P. And V. S. |
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| 29:01 | the vertical the the is going to from the back of average. Uh |
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| 29:13 | about the horizontal bp. Well that's to be more of a void type |
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| 29:21 | . So I'm going to ask you do a little bit of research and |
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| 29:24 | out how are you going to calculate horizontal p way Philosophy. So we |
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| 29:30 | a vertical stack. So we have shell, sand shell, sand shell |
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| 29:35 | a vertical sequence. How are you to calculate the horizontal philosophy? Uh |
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| 29:42 | calculate the vertical shear wave velocity and horizontal shear wave velocity, assuming the |
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| 29:51 | wave is vertically polarized. All right to do this, you're gonna need |
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| 29:57 | to know the Balkans share module I use the V. P. |
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| 30:05 | S. Trends and you're not in notes. Um as a rough |
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| 30:13 | you could try using Royce for void and see how that that gets |
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| 30:20 | And then answer the question if the zero and the shear wave velocity is |
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| 30:27 | , what is the P wave? I saw? So these were |
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| 30:33 | what would the P wave on assad be? Mhm. Okay, coming |
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| 30:42 | to the hash in Strickland bounds, I've showed you before was one giant |
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| 30:51 | or one giant inclusion. Right. um how do we justify making the |
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| 30:59 | that way? Well, it turns that's the same as saying I have |
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| 31:04 | infinite sequence of these Eminem's here. . So you have a chocolate filling |
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| 31:11 | you have a hard shell around Right? So these are perfectly spherical |
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| 31:17 | and M's with an infinite range of . Such that smaller ones keep filling |
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| 31:24 | . So actually, if I carry to the nth degree, every bit |
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| 31:28 | space, every boy is going to filled. So that's what the rock |
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| 31:33 | made of. It's made of um stacking of concentric spheres that fills up |
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| 31:43 | the void space. But now I have I have no void space. |
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| 31:47 | have constituent A constituent B. If a. is the hard one that |
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| 31:54 | give us the upper bound and if B. It is the hard |
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| 32:00 | Yeah, that gives us the upper . So here we have plotted the |
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| 32:08 | stripped inbounds and uh we're mixing solid liquid. All right. So we |
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| 32:17 | a solid here, Both modules one and a liquid here with both |
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| 32:22 | K. two. And we have upper bound, she's strictly an upper |
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| 32:29 | . And we also have a soft which is acting a lot like our |
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| 32:34 | bounds that we've look at. And uh highway liquified rock would presumably be |
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| 32:45 | the upper bound and a poorly liquified would be near the lower bound. |
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| 32:50 | that's polk modules. That's easy. sheer modules is a problem again because |
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| 32:56 | lower bound is going to be Um If you think about it, |
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| 33:03 | this is a if if the light material here is fluid, then this |
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| 33:10 | just a suspension of the hard material in the fluid. And uh there's |
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| 33:18 | structural framework here, the hard material always completely surrounded by fluid. So |
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| 33:24 | completely floating. It can't rely on from any of the other hard |
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| 33:30 | hard objects. So this just becomes equation for suspension and the suspension Has |
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| 33:37 | share modules. Okay. All well here's the equation. And for |
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| 33:44 | bulk modules is fairly simple And it's same equation for upper and lower bounds |
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| 33:50 | you uh replace the industry. So the c. one Is one material |
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| 33:57 | the c. two is the other . By the way, there are |
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| 34:01 | complicated equations from multiple material, but didn't want to burden you with |
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| 34:06 | Okay, so we're just looking at materials and usually you can get away |
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| 34:12 | that. You could have some average all the hard stuff and then whatever |
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| 34:19 | the unusual soft material embedded in which is the upper and lower |
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| 34:25 | Well, the way they calculate the bounds is two interchange the the |
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| 34:33 | So for one bound is written this for the other band make every one |
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| 34:38 | two and every 21 and absolutely. one gives you the the upper and |
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| 34:44 | bound. I forget. And it on convention what you're calling one and |
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| 34:49 | , but you'll know one is bigger the other one. Right? So |
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| 34:52 | bigger one is the upper bound by way, that's a minus silent sign |
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| 34:56 | , I just noticed. Mhm. of course at this volume fraction, |
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| 35:03 | I like to use X. Uh likes to use f. So this |
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| 35:08 | from his handbook and here's the equation the sheer way. That's a little |
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| 35:15 | more complicated, but still pretty simple . So for two materials then you |
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| 35:24 | calculate the machine stripping bounds quite And this is the kind of results |
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| 35:32 | get and uh mixing a court, mixture and porosity. So this is |
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| 35:39 | constituents. Are actually not using these . There are more complicated equations for |
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| 35:46 | constituents. And we have the upper and we're going we're burying the amount |
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| 35:52 | uh courts That we're going from 0% to 100% courts. So this is |
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| 35:59 | bulk unsure module lists for Uh porous dolomite and porous courts, I'm |
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| 36:09 | , upper and lower bound for for stolen might upper and lower bound. |
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| 36:14 | poorest courts. And you notice the both modules is not changing very much |
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| 36:23 | we vary from dolomite courts for the bound. Whereas the bulk module of |
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| 36:32 | minerals are changed quite a bit, you can see reflected in the upper |
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| 36:37 | there. So you might ask yourself we don't see a big bulk modules |
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| 36:44 | for the lower bound. And think about what the lower bound |
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| 36:49 | The lower bound represents the softest material case and everything else. So, |
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| 36:55 | fact, this is a suspension of and dolomite floating in liquid. And |
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| 37:03 | can see then that uh the compressibility that suspension is going to be only |
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| 37:08 | dependent on the bulk module. I course, and dolomite is going to |
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| 37:14 | more or a great deal on the modules of the fluid. Yeah. |
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| 37:22 | , so that gives us bounds. gonna try to make a precise prediction |
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| 37:33 | to do that. We're going to to make lots of assumptions. So |
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| 37:37 | going to assume an ice a tropic genius aggregate. And we have a |
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| 37:46 | solid. And within that host where putting in a loop seidel |
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| 37:53 | these have various shapes defined by their ratios and they are randomly oriented. |
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| 38:02 | it's again homo genius. Okay, , main exceptions. Uh the medium |
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| 38:11 | I psychotropic because the inclusions are randomly and randomly distributed, the medium has |
|
| 38:20 | same properties in any direction two, assuming it's going to be a perfectly |
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| 38:26 | material. So again, are stress are small. The resulting strains are |
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| 38:33 | . So we can treat it as elastic medium. And we're going to |
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| 38:38 | interactions between the inclusion. So this that each inclusion is surrounded by a |
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| 38:49 | volume of host material of solid host . So, low concentrations. And |
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| 39:00 | Uh huh. The mathematical limitation is I'm not even sure that that is |
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| 39:07 | sufficient physical limitation. But the mathematical is the amount of uh of inclusions |
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| 39:16 | any given aspect ratio is no more the aspect ratio. So, if |
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| 39:22 | have an aspect ratio of 1, I could have 10% ferocity of an |
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| 39:29 | ratio of 1/10. If I have aspect ratio of 100. Uh then |
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| 39:34 | could have 1% ferocity of that aspect . So it very much limits what |
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| 39:45 | can do. Okay, we've looked these equations previously. Don't need to |
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| 39:54 | the point here. We have the of the inclusions which include terms for |
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| 40:00 | inclusion shape, the inclusion module, hear, and is the host material |
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| 40:07 | module us. And then we have effective medium with the asterisk here. |
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| 40:13 | we have the properties of the host and the effective properties of the medium |
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| 40:19 | uh the characteristics of the inclusions on side. And uh so it one |
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| 40:29 | to solve these equations for the effective . Right. I mean for the |
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| 40:34 | medium properties and by the way, for completeness, these terms which are |
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| 40:43 | shape related are given this way. they have different values for different types |
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| 40:50 | shapes. And uh for our purposes going to deal with what are called |
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| 40:56 | shaped cracks. And I'll define that a minute, basically. They're flat |
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| 41:06 | that are in cross sectional view are . Okay, so just to summarize |
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| 41:18 | inclusions with different module I or with shapes require different terms in the |
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| 41:24 | So inclusions can have different module the K IIs or they could have |
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| 41:30 | shaped terms, the PM E. . And the que mes, the |
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| 41:37 | are distributed and oriented randomly, so they occur and their orientation are completely |
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| 41:47 | . So uh we have a large sample where the statistics works, |
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| 41:54 | And remember we're going to be This a long wavelength approximation. Uh So |
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| 42:02 | a wavelength things statistically average out the are dilute leaning. They're not very |
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| 42:14 | , right? So they're not close to each other where they interact with |
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| 42:18 | other. Ah. A way to about inclusions is uh to think about |
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| 42:27 | . You could have multiple scattering off inclusions and uh the extent to which |
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| 42:34 | closer they are together, the more you have a reinforcing multiple scattering so |
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| 42:41 | far apart. Um Dry pores have module. I it turns out that |
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| 42:54 | is a intelligence a problem using included modeling and fluids in that the inclusions |
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| 43:07 | not connected to each other. And unlike a permeable rock where the poor |
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| 43:13 | are allowed to equip vibrate. For , in gas mints equations, these |
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| 43:19 | are completely disconnected. And so there be no pore pressure equal abrasion, |
|
| 43:27 | we have specific circumstances that allow it happen. For example, if all |
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| 43:32 | pores are spherical, all the pores close to the same degree. |
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| 43:38 | all the fluids in those poorest. , it will have the same poor |
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| 43:44 | . Right? On the other if if I have pores of varying |
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| 43:47 | and varying orientation. Some pores maybe some. Maybe closing some will be |
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| 43:54 | more than others. And so the pressures will be different in every |
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| 44:01 | And so you can't you can't properly for the effect of the poor fluids |
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| 44:11 | way. So what we're going to instead to make things applicable to seismic |
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| 44:19 | is we're going to model the dry . So we're gonna let the pores |
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| 44:25 | dry, have zero modules. And we're going to imagine the pores are |
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| 44:31 | this by the way, the predicting the dry rock properties. That's |
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| 44:36 | the dry frame. Now we could fluids to that dry frame to using |
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| 44:42 | substitution. We could use gas mains and add fluids to the dry |
|
| 44:48 | Now we have something that is applicable seismic frequencies. Uh and of course |
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| 44:55 | fluids we represent the zero share Um Okay, so uh Mapco gives |
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| 45:04 | caveat here in his hand book and says because the inclusions are isolated with |
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| 45:12 | to flow. Uh huh. The toxins model simulates a very high frequency |
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| 45:20 | rock behavior appropriate to ultrasonic laboratory Keep in mind we're still requiring it's |
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| 45:27 | effective medium. So the wavelength is times bigger than the dimension of the |
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| 45:35 | . But it's high enough frequency where fact that the pores are disconnected would |
|
| 45:44 | representative of what's actually happening at high where the poor is, you |
|
| 45:51 | the frequencies were so high that you get pressure calibration between the forest. |
|
| 45:58 | uh these models may be more applicable laboratory measurements. But if we want |
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| 46:05 | emulate what's happening at low frequency we the dry rock and the dry rock |
|
| 46:13 | then saturate with the gasoline equations and another picture of an aqueduct to drive |
|
| 46:24 | . The point that the poor shape a lot to do with how |
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| 46:28 | This is round pores are arches and hard to compress, whereas flat pores |
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| 46:36 | be easily compressible. In fact, term architecture comes from the word |
|
| 46:43 | You see. These tend to be but notice here we have an |
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| 46:51 | So these would have the Ikhwan An aspect ratio of one ish. |
|
| 46:56 | is this guy is still forming an art. So you can think of |
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| 47:01 | as a poor with lower than one ratio that's vertically oriented and you're compressing |
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| 47:08 | vertically. It's still pretty strong as to but we're on its side. |
|
| 47:13 | . If it was if you are from the side it would be more |
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| 47:18 | from the side than north south. , so we're gonna describe steroids as |
|
| 47:30 | three axes. A. B and . Is the short axis. |
|
| 47:35 | Is intermediate and C. Is the axis. And there are specific cases |
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| 47:43 | we're going to deal with by the ? Yeah, the aspect ratio is |
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| 47:50 | the short axis divided by the long . Uh Then we have oblate spheroid |
|
| 47:58 | the intermediate access coming out of the here is similar to the long |
|
| 48:05 | So in horizontal cross section. That'd pretty circular. All right. So |
|
| 48:11 | close is much smaller. That's an spheroid if you sit on a beach |
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| 48:19 | or you know a pancake or Um By the way the earth is |
|
| 48:28 | the short axis is north. South is actually shorter than the access along |
|
| 48:36 | equator. So uh in a sense an outlet spheroid I wouldn't say is |
|
| 48:42 | smaller than B. But A is than me. Um A pro late |
|
| 48:50 | is like an american football and egg ball uh where A. And |
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| 48:57 | Are closer to each other and they much smaller than see. These are |
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| 49:01 | called needle like chorus. And then course uh piquant steroids or spheres equal |
|
| 49:10 | equal. See they have an aspect one by the way, an oblate |
|
| 49:16 | . And a pro laid spheroid can the same access aspect ratio A divided |
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| 49:22 | C. But have the very different . Okay, so here's an oblate |
|
| 49:32 | . So B and C. Are much the same. A. Is |
|
| 49:38 | . And we also call these penny cracks. So carbonates are famous for |
|
| 49:46 | equant poison and buggy pours molded So here this for Aspect ratio of |
|
| 49:55 | very nice. Um sometimes shell fragments so forth. Or we'll have aspect |
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| 50:02 | on the order of the 10th and granular between crystals or micro factors. |
|
| 50:09 | will be lower. Just as a of terminology, any aspect ratio is |
|
| 50:16 | than a 10 are called piquant. mean what is perfectly equant? Uh |
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| 50:23 | you know we would say these are aspect ratio greater than a 10th. |
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| 50:29 | uh Aspect ratio is less than a of frequency frequently called cracks. It |
|
| 50:36 | out that attempt is kind of a watershed aspect ratio. Kind of separates |
|
| 50:43 | regimes. The equant regime from the regime. Yeah. Now tough, |
|
| 50:53 | talking about lips toys here uh in stones, that's kind of tough |
|
| 51:01 | Right? How do you describe that ? Right. So but you can |
|
| 51:08 | do the exercise and you can calculate short divided by the long axis. |
|
| 51:14 | you might find is that the aspect can be correlated to porosity. For |
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| 51:20 | this low porosity rock tends to have porous whereas the higher porosity rocks tend |
|
| 51:29 | have more piquant. Higher aspect Of course of course there's a correlation |
|
| 51:36 | grain size plays tend to be dominated tend to be more prevalent in the |
|
| 51:43 | grain size, their platelets. So between clay plate that you're going to |
|
| 51:50 | a tendency to have flat forests. Also and the fine grain sizes would |
|
| 51:59 | to be more angular and that could the aspect rations as well. |
|
| 52:07 | we've seen plots like this before, are Custer taxes plus they have uh |
|
| 52:17 | , bulk and shear module versus What do we mean by concentration? |
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| 52:24 | mean the porosity associated with that aspect . So, the concentration of every |
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| 52:31 | ratio sums to form the ferocity. , So the sum of aspect. |
|
| 52:37 | . The film of concentrations is the . Okay, we have broken sheer |
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| 52:43 | in the dash line. Uh is dry rock. The solid line is |
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| 52:49 | water saturated rock. Now on the we have aspect ratios one And on |
|
| 52:56 | right, we have aspect ratios of . And The scale is a little |
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| 53:03 | funny though, that this scale is 0 to .5. So from property |
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| 53:10 | 0 to 50%. This Ale is times 10. So actually this goes |
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| 53:17 | a ferocity of zero to a ferocity .1. All right, so comparing |
|
| 53:24 | plot is a little bit difficult. few points here. Uh Look at |
|
| 53:32 | dry rock bulk module lists can be than the share module lists. So |
|
| 53:39 | high concentrations of low aspect ratio that can happen. Notice here, |
|
| 53:47 | dry p wave velocity is faster, most saturated p wave velocity. This |
|
| 53:54 | to be peculiar to high expectation I've really I've never seen it in |
|
| 54:01 | because even when we have a buggy , we tend to have other low |
|
| 54:07 | for us associated with it. I and you have to have enough for |
|
| 54:11 | city to see that difference, notice bulk modules is less for the dry |
|
| 54:18 | , but the velocity is more. because of the density effect. And |
|
| 54:23 | same thing that the shear wave velocity . Okay, sheer module is |
|
| 54:28 | Pretty much the same for the dry the saturated rock. Remember fluids have |
|
| 54:37 | rigidity so they're not going to contribute much. But again, the density |
|
| 54:45 | is causing the difference now for to these what what might not be obvious |
|
| 54:54 | you uh at these scales is the of the module i on concentration is |
|
| 55:01 | stronger at lower aspect ratios. And because high aspect ratios are not very |
|
| 55:08 | . So I stretched and squeezed these to go from over the same |
|
| 55:17 | Right? So I went from prostitute for high aspect rations. And you |
|
| 55:24 | essentially no no difference between wet and for p waves and share waves. |
|
| 55:33 | For an aspect ratio .1 a big between wet and dry and the bulk |
|
| 55:39 | list, but also the change over same crostini range. The drop in |
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| 55:46 | bulk and share modules is more Then aspect ratios of one. And if |
|
| 55:54 | uh Went further to an aspect ratio .01, you would see that I |
|
| 56:00 | a similar change from 0-1%. So lower the aspect ratio, the greater |
|
| 56:09 | the module lists in velocity reduction associated the boy's face. That goes with |
|
| 56:17 | aspect rations. And so you can that here these are custard. Tox |
|
| 56:23 | uh predictions for a mixture of pure and water. Sorry, pure courts |
|
| 56:31 | water. So on the vertical axis aspect ratio. So one these are |
|
| 56:38 | pours down to very fine cracks Uh huh. This diagonal line is |
|
| 56:46 | porosity equal to the aspect of So you you know this is a |
|
| 56:53 | zone, right? The theory can't after. Okay, so let's |
|
| 56:57 | And these are contours of constant So kind of an interesting way to |
|
| 57:03 | at things. But let's look at single aspect ratio. I mean a |
|
| 57:09 | concentration. So uh so we have ferocity here or for an equant poor |
|
| 57:23 | have a higher velocity, maybe the of course pretty close to it, |
|
| 57:29 | over 19,000ft per second Aspect ratio. Close to 19,000, sorry, But |
|
| 57:39 | would be an 18 person .001 on order of 17,000 down here. So |
|
| 57:47 | a given ferocity, the lower the ratio, the lower the velocity similarly |
|
| 57:55 | a given aspect ratio to hire the , the lower the philosophy. So |
|
| 58:02 | you can see that uh with ferocity or aspect ratio alone, you can't |
|
| 58:09 | the ferocity right? There's there's a family of solutions, right? That |
|
| 58:15 | give you the same velocity and you start making this real and find it |
|
| 58:24 | geology. This was a study in rocks that I participated in many years |
|
| 58:31 | . And we were looking at the P. V. S ratio. |
|
| 58:35 | we had modeled the back greek sand it was showing as for us that |
|
| 58:40 | the VPs goes up acting a lot a classic sand stones do the reef |
|
| 58:49 | which had higher aspect ratio for is know this this is more like a |
|
| 58:54 | type of texture. The high aspect poorest showed an actual decrease of the |
|
| 59:01 | with increasing ferocity. Um I'm sorry . And in both cases this is |
|
| 59:08 | brian saturated or oil saturated rocks. you add fractures, you increase the |
|
| 59:15 | . P. S ratio. So B. P. S ratio should |
|
| 59:19 | higher when you have more highly Unfortunately, this won't be the case |
|
| 59:27 | you're a gas that trade. But oil reservoirs you can spot the abnormally |
|
| 59:35 | zones by having higher V. PBS . Okay, so remember I said |
|
| 59:46 | this 0.1 aspect ratio kind of separates regimes. The Ikhwan poor is and |
|
| 59:53 | cracks. So we could take the porosity and we could divide it into |
|
| 60:00 | ferocity is the Ikhwan ferocity. And crack ferocity. And uh we're also |
|
| 60:07 | this equal ferocity plus this funny term , four thirds pi alpha epsilon alpa |
|
| 60:15 | the aspect ratio epsilon is a term the crack density. And the cramp |
|
| 60:23 | is defined this way it's basically has do with the uh huh the number |
|
| 60:33 | crafts of certain dimensions. Yeah, ci is the spheroid access, |
|
| 60:46 | So every see there are a certain of those cracks. Right? So |
|
| 60:52 | add these all up and we get crap density and from relations. I'm |
|
| 61:02 | show you soon as a homework exercise that this thing is equal to the |
|
| 61:08 | ferocity. And I don't mean use equation. Right. It's obvious there |
|
| 61:13 | , cracked ferocity equals this. I you to go ahead and do some |
|
| 61:18 | for and derive this from equations which show you now rocks don't have one |
|
| 61:30 | ratio. Right? They have many with many different aspect rations. And |
|
| 61:36 | far as crack stone, this smoke very well for equant ferocity, but |
|
| 61:42 | crap ferocity it turns out that Iraq an effective aspect ratio given by this |
|
| 61:58 | . We'll have a velocity similar to you had used, if you had |
|
| 62:05 | rock were composed of a single aspect . So the effective aspect ratio should |
|
| 62:15 | you the same velocity as a rock a concentration of that aspect rations. |
|
| 62:24 | to get the effective aspect ratio, sum up the concentrations of all the |
|
| 62:31 | aspect ratios. Well that's the total . So that's ferocity there. And |
|
| 62:37 | you divide by the ratio of the to the aspect ratio. So the |
|
| 62:43 | of that for every different aspect And it turns out that if you |
|
| 62:51 | you do this and you stick this the customer talks those equations, uh |
|
| 62:57 | get a velocity very close to what could have gotten if you had just |
|
| 63:03 | a single aspect rations. So it's way of describing a spectrum of aspect |
|
| 63:10 | by a single number. Um, yeah, some of all the concentrations |
|
| 63:17 | the total porosity and this guy here related to the crack density. This |
|
| 63:26 | O'connell and Budiansky ease, crack which they defined on the previous page |
|
| 63:35 | . Okay, so uh this is is nice because there's a mathematical trick |
|
| 63:41 | could employ here and that is suppose want to get a concentration of a |
|
| 63:51 | aspect ratio but I want to go the mathematical limit of the custard toxins |
|
| 63:58 | . Well I could keep adding finer finer cracks uh in order to accomplish |
|
| 64:05 | aspect ratio. So what I'm saying you could accomplish and a forbidden concentration |
|
| 64:14 | an aspect ratio by summing cracks that that effective aspect ratio and um show |
|
| 64:27 | the effective aspect ratio is related to and Budiansky. He's cracked density |
|
| 64:38 | All right now, some practical consequences all of this. Remember I mentioned |
|
| 64:44 | long time ago at the widely time equation acts Like uh your granule of |
|
| 64:53 | stones have an aspect rations on the of .1. And I drew that |
|
| 64:59 | from this slide here, where were for courts transit time. Now at |
|
| 65:09 | time we were looking at logging transit time versus porosity and this is |
|
| 65:15 | wildly time average equation And here's an ratio of the 10th. And you |
|
| 65:24 | the wildly time average equation is kind spanning aspect ratios from .062 .13 or |
|
| 65:34 | like that. I mean you wouldn't to go out to those five ferocity |
|
| 65:38 | anyway, So in the vicinity of 10th for the wildly time average equation |
|
| 65:48 | you could do the same thing for shear wave time average equation. And |
|
| 65:51 | draw a similar conclusion On the order aspect ratio, not too far different |
|
| 65:59 | a 10th. So somehow the point and granule Iraq's act like an effective |
|
| 66:06 | ratio of attack going in the wrong . Okay, so let's try to |
|
| 66:18 | this theory to sonic log measurements. so we have with ology and porosity |
|
| 66:28 | we have a measured VP here and the blue curve is measured VP and |
|
| 66:39 | could then take the prostate and say if all my Horace were equal? |
|
| 66:44 | if they were all spherical. What Custer Tacos model predict? And that |
|
| 66:50 | be the red curve here. And the way this is the V |
|
| 66:53 | V. S ratio. I'm sorry sounds ratio From 0 to .5. |
|
| 66:59 | uh blue is measured. Red is you had a piquant force and so |
|
| 67:07 | can see it doesn't match very well this is work from ted smith. |
|
| 67:14 | he went ahead and said, what if I assumed an effective aspect |
|
| 67:17 | of .1 get it to advance. , there you go. It's much |
|
| 67:26 | similar. Not exactly right. There's over swings, but you know, |
|
| 67:32 | least for the higher velocity rocks. aspect ratio .1 seems to predict things |
|
| 67:41 | well, but then you can turn problem around and you can say, |
|
| 67:47 | , given the velocity and the ferocity the composition, what aspect ratio is |
|
| 67:55 | . And that's this curve here. it would have been nice that had |
|
| 67:59 | to spread over a larger range. , So This is .1 here. |
|
| 68:07 | um anyway, interesting exercise there. of course the ultimate goal was then |
|
| 68:15 | compare the effective aspect ratio to the permeability or the porosity, permeability |
|
| 68:24 | presumably at a given ferocity, the effective aspect ratios would have higher |
|
| 68:31 | but uh don't have that kind of to work with and um that would |
|
| 68:38 | to happen inside an oil company, it's an interesting idea. Okay, |
|
| 68:44 | , I might as well stop There. Are there any questions Quick |
|
| 68:52 | this 1? Yeah. Excuse Yeah. When do we want to |
|
| 68:58 | check with what like our faces are stuff like that to make sure that |
|
| 69:01 | when we're solving for the aspect like it actually makes like geologic sense |
|
| 69:05 | we're not just essentially getting numbers to . Okay, so yeah, I |
|
| 69:11 | the theory allows you without knowing anything the geology except you have to know |
|
| 69:17 | composition. It allows you to compute aspect ratio as if the theory were |
|
| 69:23 | . Right? We don't think it's valid. So you can think of |
|
| 69:27 | as just a phenomenal logical number, ? It's an apparent aspect ratio, |
|
| 69:33 | not real, but then doesn't correlate anything and that you might want to |
|
| 69:39 | within different faces to look to see you have relationships between this derived aspect |
|
| 69:46 | and something else. Unfortunately, I saw that done, but it would |
|
| 69:51 | interesting. I think the best correlation can see here of all these laws |
|
| 69:58 | the effective aspect ratio seems to correlate the ferocity, but maybe there's a |
|
| 70:04 | for that. Mhm. Okay. , I just got curious about |
|
| 70:08 | like it was done. But does actually, what does it mean? |
|
| 70:16 | , exactly. But it's an observation you now have an opportunity to correlate |
|
| 70:23 | geology. Only that bit was never . The guys that that did |
|
| 70:29 | Well uh we're out of the University florida they do and so many and |
|
| 70:35 | associates uh did a lot of this carbonate sediments and that's probably the most |
|
| 70:45 | work I've seen published trying to do like this. Okay guys, we'll |
|
| 70:50 | you on thursday. Yeah, |
|
