Downloaded 0/9/3 to 99.86.4.8. Redistribution subject to SEG license or coyright; see Terms of Use at htt://library.seg.org/ On the luid Deendence of Rock Comressibility: Biot-Gassmann Refined Leon Thomsen, Delta Geohysics and University of Houston Introduction n imortant issue in using seismic data to hysically characterize crustal rocks is the deendence of seismic rock velocity uon the comressibility of the fluid in the ore sace. t seismic frequencies, the standard theory for understanding this fluid deendence is that of Biot (94 and Gassmann (95. It is well-known that this theory redicts that the rigidity should be indeendent of fluid content, and that the fluid deendence of the incomressibility is given by an exlicit formula (Eqn. (, below. The urose of this note is to oint out an aroximation in the theory that has not been fully areciated. Below it is shown that avoiding this aroximation leads to a refinement of the standard Biot-Gassmann ( B-G formula for fluid substitution in rock incomressibility, with new requirements (comressibility data for imlementation. This revision has been reorted reviously (Brown and orringa, 975 in a limited context (non-uniform mineralogy; here it is shown to be much more general. The refined formula fits exerimental data better than does the classic formula. Exerimental considerations It is sobering to reach the conclusion that a theory that has been well-acceted for over half a century is in fact not correct. In such a case, we should immediately re-examine the exerimental verification of the theoretical redictions. It turns out that the theory has not, in fact, had strong exerimental verification, desite its wide accetance. The ambiguous state of the exerimental test of the B-G conclusions was discussed e.g. by Thomsen (986, Winkler (986, and Z. Wang (000; this work is not reviewed here. rincial difficulty is that most of the revious exerimental aroaches have measured elastic velocities (P and S at ultrasonic frequencies (in gas-saturated and liquid-saturated rock, whereas the theory is exlicitly limited to quasistatic (or low-frequency measurements, with the assumtion that it alies to elastic velocities at seismic frequencies. Of course, the reason for this indirect exerimental strategy is that direct measurements of comressibility under quasi-static conditions are difficult at the low strains (<0-6 beyond which non-linear effects become imortant. The following analysis shows that in fact static comressibility tests are required in order to fully secify the arameters in the (refined theory; recent work by Hart and Wang (00 enables a fresh assessment. Theoretical considerations The roof of the Biot-Gassmann formula is here recaitulated, following the comact and clear argument of Brown and orringa (975 ( B&, but with an imortant change. The argument is conducted in terms of comressibility rather than of incomressibility, but leads eventually to the classic low-frequency B-G formula: fld sat fld ( dry fld ( sol where sat, dry, fld, sol are the incomressibilities of (resectively: the fluid-saturated rock, the rock with comletely emty ore sace, the ore-filling fluid, and the solid grains of the rock. is orosity, and =- dry / sol. Using mainly just calculus, B& derive the difference between the dry comressibility and the saturated 00 SEG SEG Denver 00 nnual eeting 447
comressibility (their Eqn. (3: ( in terms of various measures of comressibility of a mass element ( undrained,,, defined by them. In articular, the unjacketed comressibility and the ore comressibility are: d (3a d (3b where is the volume of the mass element, = is its ore volume, is the fluid ressure in the ore sace, is the total ( confining ressure, and d =- is the differential ressure. Using the Recirocity Theorem (Love, 944, B& obtained =, leading to their rimary scalar result: (4 They remark uon their success in generalizing the B-G result at the cost of only one additional arameter (, but do not emhasize that the aearance in (4 of (instead of the solid comressibility comlicates the imlementation substantially. However, for a rock with homogeneous solid, B& concluded (following arguments discussed below that, = = ( the comressibility of the solid grains, so that Eqn. (4 becomes (5 It is easy to show that Eqn. (5 is exactly the same as the classical B-G result (, written in this notation. Here, we concentrate on B& s conclusion that = = for monomineralic rocks. This conclusion follows from their assertion that equality of stress across all the solid-fluid surfaces of the rock results in a self-similar volume change, with no change in orosity (ore volume/total volume. In fact, it is easy to show that the change in orosity is given by d (6 and that ( ( ( d (7 448 SEG Denver 00 nnual eeting 00 SEG Downloaded 0/9/3 to 99.86.4.8. Redistribution subject to SEG license or coyright; see Terms of Use at htt://library.seg.org/
from which it is easy to see that invariance of leads to the conclusion that = =. Downloaded 0/9/3 to 99.86.4.8. Redistribution subject to SEG license or coyright; see Terms of Use at htt://library.seg.org/ B& simly assert that is invariant, with resect to. However that conclusion is clearly roven by Gurevich (004; the roof allows examination of its assumtions. In the roof, is regarded as an indeendent variable (to be adjusted by an exerimenter, whereas in the resent context, it is a deendent variable, determined by the external ressure, by the rock roerties, and by the assumtion of no fluid flow into or out of the mass element. In this undrained context, is linearly related to the volume (through the fluid comressibility ; this connection means that the deformation cannot be self-similar. In the general case, the distortion of the solid by the alication of uniform throughout the rock is a comlicated function of solid incomressibility, rigidity, and micro-geometry. But, it is easy to imagine an artificial case where none of this matters. Imagine the case where the mass element consists of a cube of homogeneous isotroic solid material with volume, surrounded by a cubic shell of fluid with volume. (If it is desired that the solid should be simly connected, imagine very thin sacers, made of the same solid material, which connect the solid cube to other similar cubes in nearby mass elements, but which do not materially distort the stresses. lication of uniform (at constant d throughout this eculiar mass element results in no shear stress at all, only a uniform comression of both solid and fluid, each according to its own comressibility ( or, resectively. This changes the relative volume fraction occuied by solid and by fluid, hence it changes the fluid-filled orosity, /. Hence, from Eqns. (6, 7 it follows that,, and are all different. Of course this artificial case has no significance in itself, but only illustrates, by examle, the rincile that in the general case,,, and are all different. This argument confirms B& s conclusion that differs from and, and generalizes it to include even the secial case of homogenous solid. Hence, it is certainly necessary to consider the comlications osed by the aearance in Eqn. (4 of, and in even the simlest cases. In terms of incomressibilities, Eqn. (4 becomes (comare with B& (: fld sat fld dry fld fld (8 The difficulty in using this exression for seismic fluid substitution is that (unlike the solid comressibility both and are arameters of the rock, and deend uon comosition, rigidity, and micro-geometry in a comlicated way, not suscetible to theoretical resolution. In recent comressibility exeriments, Hart and Wang (00 indeendently measured all the arameters of Eqn. (4 excet for, in Berea sandstone and Indiana limestone, and found that (esecially in the sandstone that differed from, in a stress-deendent way. (Since this aroach assumes the validity of Eqn. (4, it cannot be used as a check of the same equation. They also measured Skemton s B-coefficient, the ratio of fluid ressure to confining ressure in an undrained exeriment, given in terms of comressibilities by (H.. Wang, 000: B and used this overdetermined dataset to rove their oint, conservatively. However, their data can also serve as a validity check of Eqns. (4, 8. In the course of deriving Eqn. (, B& analyze the variation of the ore volume with alied ressure in an undrained exeriment, finding that (their Eqn. (: ( Using their subsequent result that = -, this is (9 (0 00 SEG SEG Denver 00 nnual eeting 449
B ( ( ( Downloaded 0/9/3 to 99.86.4.8. Redistribution subject to SEG license or coyright; see Terms of Use at htt://library.seg.org/ so that the ore comressibility is given as B ( ( B Using the sandstone data from Hart and Wang (00, Table, ig. confirms their conclusion that (. This conclusion seems to be firm, even with this less conservative aroach (desite the evidence of scatter in the data, esecially at differential ressures greater than 5 Pa. Because of the ressure-deendence of the differences, they cannot be due to the fact that the sandstone has about 0% of non-quartz minerals, but must be due to the microgeometry of the rock, in articular to the closing of microcracks with increasing ressure. igure. Demonstrating that ig.. Neglect of this comlexity causes systematic error. urther, when these measured values are used to calculate the difference, using the refined B-G Eqn. (4, the agreement with exeriment is fairly good (ig.. When the classic B-G formula (5 is used to calculate this difference, the errors are somewhat larger, and biased low, articularly at low differential ressure. Conclusions In the classic exression ( or 5 of Biot-Gassmann for fluid substitution in the rock comressibility, contains an aroximation reviously identified (by B&, in a limited context, whose generality has not been fully areciated. Correction of this aroximation (using Eqn. (4 relacing Eqn. (5 requires data on rock comressibility. With the correction, agreement of theory and the exerimental data of Hart and Wang (00 is fairly good. By contrast, the classic equation gives redictions which are systematically lower. The significance of the correction deends uon the articular circumstances, and is most imortant where the rock contains cracks or fractures. cknowledgments: The author areciates constructive discussion from J. Berryman (Lawrence Berkeley National Laboratory, Boris Gurevich (Curtin University, Gennady Golushubin (University of Houston, and Herb Wang (University of Wisconsin, as well as material suort from LBNL and UH. 00 SEG SEG Denver 00 nnual eeting 450
EDITED REERENCES Note: This reference list is a coy-edited version of the reference list submitted by the author. Reference lists for the 00 SEG Technical Program Exanded bstracts have been coy edited so that references rovided with the online metadata for each aer will achieve a high degree of linking to cited sources that aear on the Web. Downloaded 0/9/3 to 99.86.4.8. Redistribution subject to SEG license or coyright; see Terms of Use at htt://library.seg.org/ REERENCES Biot,.., 94, General Theory of Three-Dimensional Consolidation: Journal of lied Physics,, no., 55 64, doi:0.063/.7886. Brown, R. J. S., and J. orringa, 975, On the deendence of the elastic roerties of a orous rock on the comressibility of the ore fluid: Geohysics, 40, 608 66, doi:0.90/.44055. Gassmann,., 95, Uber die Elastizitat oroser edien, itteiluagen aus dem Inst. fur Geohysik (Zurich, 7, l-3. Translated and rerinted as: On elasticity of orous media, in Classics of Elastic Wave Theory,.. Pelissier, Ed., SEG, 007. Hart, D. J., and H.. Wang, 00, ariation of unjacketed ore comressibility using Gassmann s equation and an overdetermined set of volumetric oroelastic measurements: Geohysics, 75, no., N9 N8, doi:0.90/.377664. Love.. E. H., 944, athematical Treatise on the Theory of Elasticity, Dover Books. Thomsen, L., 985, Biot-consistent elastic moduli of orous rocks: Low-frequency limit: Geohysics, 50, 797 807, doi:0.90/.44900. Wang, Z., 000, The Gassmann equation revisited: Comaring laboratory data with Gassmann s redictions, in Z. Wang, and. Nur, eds., Seismic and acoustic velocities in reservoir rocks, 3: Recent develoments: SEG, 8 3. Wang, H.., 000, Theory of linear oroelasticity: Princeton University Press. Winkler,. W., 986, Estimates of velocity disersion between seismic and ultrasonic frequencies: Geohysics, 5, 83 89, doi:0.90/.4403. 00 SEG SEG Denver 00 nnual eeting 45