D= 0/.t(fir +,Bv), (lf) ' fir xx,0<x<l, (la)

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. B1, PAGES , JANUARY 10, 2001 Nonlinear pressure diffusion in a porous medium' Approximate solutions with applications to permeability measurements using transient pulse decay method Yan Liang Department of Geological Sciences, Brown University, Providence, Rhode Island Jonathan D. Price, David A. Wark, and E. Bruce Watson Department of Earth and Environmental Sciences, Rensselaer Polytechnic Institute, Troy, New York Abstract. Transient pulse decay has been widely used to measure permeability of tight rocks and synthetic materials. When the pore fluid is a gas (e.g., dry air, Ar, or N2) as used in a gas permeameter, the pressure diffusion equation governing the pulse decay problem is nonlinear due to a pressure-dependent gas compressibility and molecular slippage effect (also known as the Klinkenberg effect). To simplify data analysis in permeability measurement using a gas permeameter, an approximate solution to the nonlinear diffusion equation was obtained using a regular perturbation method. This solution, which is similar to the original exponential solution of Brace et al. [1968] for a case when the compressibility of the pore fluid is a constant, is valid in the limit when the volume of the interconnected pore fluid is much smaller than the volume of the upstream reservoir. Applications of the approximate solution to laboratory measured pulse decay data show that the estimated sample permeability can be overestimated by as much as a factor of two if the transient gas pressure decay experiment is conducted at low pressures and if molecular slippage is not taken into account. The molecular slippage can be effectively eliminated if the pulse decay measurement is conducted at a mean pressure at least 5 times higher than the Klinkenberg slip factor, which is on the order of 1 bar for texturally equilibrated marble and quartzite used in the permeability study of Wark and Watson [1998]. 1. Introduction Transient pulse decay, a technique pioneered by Brace et al. [1968], has been widely used to measure permeability of tight rocks and synthetic materials. In this method a cylindrical porous sample of porosity q, length L, and cross-sectional area,4 is connected to two fluid reservoirs of volumes V1 and V2, respectively, at its two ends (Figure 1). The pressures in the upstream reservoir, the downstream reservoir, and the porous sample are set to a prescribed value (P20) prior to an experiment. At time t = 0 a small pressure pulse is introduced to the upstream reservoir which now has a pressure of P10 (> P20). The pressure in the upstream reservoir subsequently decays as the fluid in the upstream reservoir flows through the porous sample down to the downstream reservoir. Nonreacting gases (e.g., air, Ar, or N2) or water are typically used. When the sample permeability (k) is small, the rate of this pressure driven flow is much smaller than the rate of heat transfer and the flow is approximately isothermal. Equations governing the evolution of pore pressure (P) in a porous medium can be derived by considering the mass conservation equation and Darcy's law for porous flow [e.g., Brace et al., 1968; Scheidegger, 1974; Lin, 1977; Philips, 1991; Mavko et Copyright 2001 by the American Geophysical Union Paper number 2000JB /01/2000JB al., 1998]. The one-dimensional equation along with the initial and boundary conditions relevant to the transient pulse decay experiment outlined above can be written as ' fir xx,0<x<l, (la) d?2 dt D,4O/ /"2 x...,x=l, (lc) 71(0) :710, (ld)?(x, 0):72 (0): 720, (le) where D is a pressure diffusion coefficient [e.g., Phillips, 1991 ], given by D= 0/.t(fir +,Bv), (lf) # and/5 are the shear viscosity and isothermal compressibility of the fluid, respectively, and is the pore volume compressibility. The nonlinear term on the right-hand side of (la) accounts for fluid compressibility and pressure dependence of permeability. The diffusivity D is a function of pressure if any one of the parameters on the right-hand side of (lf) is pressure-dependent.

2 530 LIANG ET AL ß NONLINEAR PRESSURE DIFFUSION IN A POROUS MEDIUM Pl(t), V 1 P1 t P2(t), V2 conditions identical to those of Brace et al. [1968], there exists an exponential solution to the nonlinear pressure diffusion equations (la)-(lf). We will also formally justify the procedure used by Wark and Watson [1998] and compare the permeabilities obtained using the new exponential solution with those reported in their study. As will be shown below, the results from this study can be used to guide experimental design, to simplify the analysis of pressure diffusion data, and to improve the accuracy of permeability measurements using the transient pulse decay method. P2 Figure 1. Schematic diagram showing the geometry of a permeameter. Relevant parameters are A, the cross-sectional area; L, the length of the porous sample; V1, the upstream volume; V2, the downstream volume P, the pressure in the upstream volume; and P2, the pressure in the downstream volume. When the pore fluid is a liquid, it, fif, fly, and k are approximately constant over a small range of pressure. Nonlinear pressure diffusion is important when the dimensionless product (/50-720)/5f > 0.1. When the pore fluid is an ideal gas, its isothermal compressibility fif equals 1/?. The nonlinear term in (la) is important regardless of the initial gas pressures in the two reservoirs. Furthermore, for gas flow in a porous medium the effect of molecular slippage can give rise to a pressure-dependent gas permeability that is higher than the permeability of the sample measured using a liquid [e.g., Klinkenberg, 1941; Aronofsky, 1954; Jones, 1972; Scheidegger, 1974; Freeman and Bush, 1983; Dullien, 1992; Jones, 1997]. This apparent gas permeability ka is related to the true permeability of the sample k via the Klinkenberg relation 2. Analysis of Nonlinear Pressure Diffusion We consider the case of a pressure-driven flow of an ideal gas through a polycrystalline porous rock at a constantemperature. The isothermal compressibility of an ideal gas is fif = 1/P. The pore volume compressibility flv of a crystalline rock is typically much smaller than the fluid compressibility fir, at least at low pressures, and hence will be neglected in the present analysis. To account for the gas slippage effect, the permeability k in (lf) is replaced by the apparent gas permeability ka via the Klinkenberg relationship (2). The resulting nonlinear diffusion equation can be simplified by considering an effective pressure P + b [e.g., AronoJ ky, 1954]. It is convenient to study the nonlinear pressure diffusion equations in their nondimensional forms. The pressure, length (L), and time (to) scales are chosen as follows;?+b=(?lo+b)p',x=lr', t= t0 t' = 0 k. - /t', 1.10, 1'10= 1+, 0Sao where P', x', and t' are dimensionless variables, and fif 0 = 1/P o is the fluid compressibility in the upstream reservoir at t = 0. The nondimensionalized equations, after dropping the primes, are where b is the Klinkenberg coefficient and is a function of the physical properties of the gas and porous sample [e.g., Klink- (3b) j 'x=ø enberg, 1941; Scheidegger, 1974; Dullien, 1992]. The effect of molecular slippage becomesignificant when b/p> 0.1. Although numerical solutions to various nonlinear pressure diffusion problems have been given in a number of stud- =,x=l (3c) ies [e.g., Bruce et al., 1953; Aronofsky, 1954; Jones, 1972; 5(0) = 1, (3d) Yilmaz et al., 1994], exact solutions are in general difficult to obtain, except when the nonlinear effects are negligible. The P(x, 0)=?2(0) = rr, (3e) main focus of this paper is to explore approximate solutions to the nonlinear pressure diffusion equations (la)-(lf) for the where e = (p.4l/i/1 is the ratio of the interconnected pore volcases when the pore fluid is an ideal gas and the Klinkenberg ume within the porous sample to the upstream reservoir volcoefficient b is a constant. A major motivation for the present ume, cr = (P20 + b)/(t5o + b) < 1 is the initial effective study is the recent work of Wark and Watson [1998], who measured permeabilities of texturally equilibrated quartzite and marble using an air permeameter. Wark and Watson [1998] equilibrated their samples with prescribed amount of ratio between the downstream and upstream reservoirs, and v = I/1/I/2. The three nondimensional parameters e, v, and cr completely characterize the nonlinear transient pulse decay problem. To the authors' knowledge, there is no exact solution water at øC and GPa. They calculated sample to the nonlinear pressure diffusion equations (3a)-(3e). In permeabilities by substituting an instantaneous slope obtained many laboratory applications, e<l and (3a)-(3e) can be from the plot of ln(p -P2) versus time at t = 0 and an average solved approximately using a perturbation method. Because e air compressibility into the original exponential solution of is associated with the partial time derivative in (3a), there ex- Brace et al. [1968]. In this paper, we show that under limiting ists an inner solution (a time boundary layer) when the non- (3a) pressure

3 LIANG ET AL.: NONLINEAR PRESSURE DIFFUSION IN A POROUS MEDIUM 531 dimensionl time t <e and an outer solution when t> e. function of distance after a transient period of order e [e.g., Physically, this means that there exists an early transient pe- Brace et al, 1968; Lin, 1977; Hsieh et al., 1981; Trimmer, riod of the order e (or œ2/d10in dimensional time) during 1981], the square of the effective pore fluid pressure, (?+b) 2, which rapid pressure buildup of the porous sample takes decreases linearly within the porous sample in the present case place. The pressure within the porous sample is in quasi [e.g., Freeman and Bush, 1983]. Consequently, the pressure steady state thereafter. This transient period is not crucial in difference normalized by the sum of the (instantaneous) eflaboratory permeability measurement when e < 1. Ignoring the fective pressures in the two reservoirs decays exponentially. transient period, (3a)-(3e) can be solved approximately by ex- There are at least two procedures that can be devised to panding P as a power series of e, viz., measure the sample permeability k, depending on the mean 4 r,/,) =?(0)+ œ?(1)+ œ2?(2)+ 4œ3), (4) gas pressure in the sample and whether the Klinkenberg coefficient b is known prior to a measurement. When where P(") (n = 0, 1, or 2) is the nth-order solution and O (P1 +P2)/2 >> b, the effect of gas slippage is negligibly small. stands for order of magnitude. Substituting (4) into (3a)-(3e) Equation (6b) is reduced to the dimensional form and collecting terms of the same order of e n, one obtains a set of partial differential equations that can be solved by standard methods. In a transient pulse decay experimenthe pressures PI-P2 _]h0-p20 exp - +?I+P P20 /Zl /z2) -. (7) in the upstream and downstream reservoirs are monitored as a The pressure-decay curve can be linearlized via (7) by plotting function of time. The differential pressures between the two the normalized pressure differences (]5-72)/(]5 +?2) as a reservoirs are used to calculate sample permeability [e.g., Brace et al., 1968; Jones, 1972; Lin, 1977; Hsieh et al., 1981; function of time. The exponent obtained from a least squares Trimmer, oe. n..., o,,,,q Walls, oe'>; Chen and exponential fit to the normalizedata is proportional to k. 1984; Haskerr et al., 1988; Wark and Watson, 1998]. In this Similarly, if the Klinkenberg coefficient b is known for the paper, we restrict our analysis in the regime where the sample porou sample and the gas (i.e., from an independent measurepore volume is much smaller than the upstream reservoir volment), one can make use of the implicit solution (6b) by plotume, namely e (( 1. This regime of pressure diffusion is di- ting (Pl-P2)/(?I + P2 + 2b) against time. If, on the other hand, rectly relevant to our laboratory permeability measurements the Klinkenberg coefficient b is not known prior to a meas- [Wark and Watson, 1998] (see section 3). It is sufficient then urement, which is usually the case, one has to estimate b and k to calculate the zeroth order solution, which is given by simultaneously using (Sb). Because there are two unknowns, it is advantageous to estimate b and k using pressure-decay data obtained from at least two experiments conducted at dif- 0)_ 4 0)= (1- or)[(1 + or)(1 + r)- (1- or)(1 - r)] e-( a ferent initial mean reservoir pressures (e.g., different P20 if (5a) //2 >>/q). The sample permeability and the Klinkenberg coefficient can be calculated from a nonlinear least-squares analyor in dimensional form sis of the measured pressure-decay data. Applications of the approximate solutions (Sb),(6b), and (7) to laboratory permeability measurements using a gas permeameter will be dis- (v, (a0- - v,)]r(,) r0r(,) cussed in section Applications to Laboratory Permeability (5b) Measurements where F(t) is given by F(t) = exp - F2 [ œkt ' (5c) As a practical example, we use (5b) to estimate the permeability of a texturally equilibrated calcite aggregate (marble) reported in the recent work of Wark and Watson [1998, their The approximate solution can also be written in an implicit form run P8]. Wark and Watson [1998] measured the sample permeability using an air permeameter with ambient air as the 0)_ 0) _ 1_ cr e_( +,)t 40) +40) - T7-d ß (6a) downstream reservoir and an initial pressure in the upstream reservoir of 5 bars (for details, see Wark and Watson [1998]). From their reported physical parameters (also listed in the caption of Figure 2), we hav e = 10 's and r = 0. Hence (5b) And the corresponding solution in dimensional variables is can be used to estimate their sample permeability. Figure 2 displays their measured pressure differences? -72 (line P1-P2 with solid triangles, in bars), the normalized pressure differ- ]h+p2+2b ences (P1-P2)/(PI +P2) and (Pi-P2)/(Pi +]:'2 + 2b) (solid lines with open circles and open squares, respectively, b = 710 ]50-P b exp{_[ +l/jh0+:/]-4(?20+b),ft} /"2, P20 + L!a '(6b, 1.31 bars, see below) as a function of time (in s) in a semilogarithm plot. For diagram clarity, 10% of their measured This implicit solution is very similar to the exponential solu- data were shown as symbols, and the rest were shown as small tion of Brace et al. [1968] for a case when the compressibility dots joined by solid lines. Clearly, one has to take the nonlinof the pore fluid is a constant and b = 0 (i.e., a liquid). In con- ear gas compressibility into account in this case because the trast to the case of a constant fluid compressibility where the measured pressure difference Pi-P2 deviates significantly pore pressure within the porous sample varies linearly as a from the exponential trend (compare long dashed line in Fig-

4 532 LIANG ET AL.: NONLINEAR PRESSURE DIFFUSION IN A POROUS MEDIUM 1 lo -' P1 -P m ---o-- (P1-P2)/(PI+P2+2b) (P1-P2)/(PI+P2) Time (s) Figure 2. Semilogarithm plot of the measure differential pressures P1-P2 (in bar, solid line with filled triangles, scale given by the ordinate on the right) as a function of time (in s), for a texturally equilibrated calcite aggregate using a transient air permeameter [see Wark and Watson 1998, run P8]. Also plotted are the normalized differential pressures (P1-P2)/(Pi+P2)(solid lines with open circles) and (P]- P2)/(P] + P2 + 2b) (solid lines with open squares, b = 1.31 bars, scale given by the ordinate on the left). For diagram clarity, 10% of their measured data were shown as open or solid symbols and the rest were shown as small dots joined by solid lines. The long and short dashed lines are from the least squares exponential fits to the measured P1-P2 and (P1-P2)/(P] + P2) data, respectively. The exponent from the least squares fit to the latter is x10 -s s -1, while the exponento the exponential fit of (P -P2)/(P + P2 + 2b) versus t(b = 1.31 bars)is x10 -s s -. Relevant physical parameters are from Wark and Watson [1998], St = 1.789x10 -s Pas, qs= 0.03, A = cm 2, L = 0.33 cm, V = 83 cm 3, V2 = P 0 = 5.7 bars, P20 = 1 bar, and a grain diameter of 84 gm. ure 2). An exponential fit (short dashed line in Figure 2) to the normalized pressure difference (?,-72)/(?] +?2) gave a sam- slippage effect. To explore this possibility, we first plot ple permeability of 1.71x10 ']4 m ' (normalized to a grain size (?]-?2)/(?] b) as a function of time for several values of 1 mm for inter-sample comparison). This is in excellent agreement with the value reported by Wark and Watson [1998], who calculated the sample permeability (1.7x1044 m 2) by substituting the slope obtained from the plot of ln(p -P2) versus time at t = 0 and an average air compressibility/3 2 = 2/(P]o+P2o) into the original exponential solution of Brace et al. [1968]. This procedure can be justified by taking the time derivative of (5) while setting b = 0. In dimensional form we have A close inspection of Figure 2 reveals that there is a slight but systematic deviation from the perfect linear relationship in the semilogarithm plot of (?]-?2)/(?]+?2) versus time (8) (compare(7), b = 0). This deviation may result from the gas of b and found nearly perfect linear relationships in the semilogarithm plot for values of 1 < b < 1.5 bars. A best fit to the measured pressure decay data were then obtained from a nonlinear least squares analysis of the measured decay data via (5b) using the Levenberg-Marquardt method [e.g., Press et al., 1989]. This gave a Klinkenberg coefficient of b = bars and a sample permeability of ( )x10 - s m 2 (normalized to a grain diameter of 1 mm). This slip-corrected permeability is slightly smaller than the (apparent gas) permeability reported by Wark and Watson [1998]. Figure 2 shows the perfect linear relationship in the semilogarithm plot of (?]-?2)/(?] +?2 + 2b) versus t when b = 1.31 bars. Since the permeabilities reported by Wark and Watson [1998] were measured at pressures between 1 and 5 bars, we attempted to recalculate permeability for all of their marble

5 LIANG ET AL ß NONLINEAR PRESSUI DIFFUSION IN A POROUS MEDIUM 533 and quartzite samples to include the Klinkenberg effect using (5b) and the inversion procedure outlined above. Accurate esto solve for the constants n and C (yielding values of 3 and 200, respectively), with d the grain diameter. Of the 13 samtimation of b by this procedure requires a substantial segment ples originally used to solve for n and C, five were among of a complete pressure decay curve, whereas only a small those that were excluded from the k recalculations because portion of that curve is necessary for estimating k without a their pressure decay curves were not sufficiently complete (as slip correction, as in Wark and Watson [1998]. Consequently, a consequence of their low permeabilities), precluding a curve we could recalculate k only for those samples for which a nearly complete decay curve is available, which included only 15 of their reported 31 samples. Of the recalculated permeabilities, all but one is lower (by a factor of 0.43 at most and 0.67 on average) than the published permeability values of Wark and Watson [1998]. For one sample, the new permeability is 5% higher. The measured Klinkenberg coefficient shows a weak inverse correlation with sample permeability, fit from the same previously evaluated sample set. From the remaining data, however, we calculated the same exponent of 3 that had been determined from the original data set for all 13 samples. The denominator (C) in (9) increase slightly, from 200 to 270, for the new results. Although these changes may be important for some applications, it is worth noting that they are relatively minor when compared with the amount that permeability values vary among the synthetic quartzites (sevwith b varying from 0.04 bar to 0.8 bar in high-permeability eral orders of magnitude). Figure 3 compares the recalculated samples (k > 10 ']2 m 2) and 0.6 bar to 2.8 bars in low-perme- permeabilities using (5b) (open circles) with those reported by ability samples (k < 10 -]4 m 2, k normalized to a grain size of 1 Wark and Watson [1998] (open crosses). Also plotted in Figmm). This is consistent with the trend observed in sediment core samples [e.g., Jones, 1972]. Because our permeability calculations utilizing an estimated slip generally produced slightly lower values than those ure 3 for comparison are "adjusted" permeabilities for the five quartzites that were excluded earlier (solid circles). The "adjusted" permeability was obtained by fitting the original pressure decay data using the implicit (6b) with b values ( reported by Wark and Watson [1998], there is a corresponding bars) estimated from the weak inverse correlation between small change in the relationship between porosity and permeability as defined by the earlier evaluation. Wark and Watson log(k) and b defined by the 15 marble and quartzite samples for which we had a reliable b estimate. [1998] fitted their measured permeabilities of quartzites to a power law relation 4. Summary and Discussion k- aesp"(9) Key parameters affecting the nature of transient pulse decay were identified through a dimensional analysis of the E 10'12.: E (D k [Corrected] ß k [Adjusted] ---4>---k [WW O. 1 O Porosity Figure 3. Semilogarithm plot of permeability (normalized to grain diameter of 1 mm) versus fluid fraction for quartzites of Wark and Watson [1998]. Open circles are eight recalculated samples, solid circles are five adjusted samples (see text for details), open crosses are the original values of Wark and Watson [1998]. Solid line is the best fit to the new data set (eight recalculated samples plus five adjusted ones). Dashed line is from the original fit of Wark and Watson [1998].

6 534 LIANG ET AL ß NONLINEAR PRESSURE DIFFUSION IN A POROUS MEDIUM nonlinear diffusion equation governing the evolution of pore (P o+p2o)/2 for the two configurations mentioned above. The fluid pressure in an isothermal transient pressure decay ex- calculated permeability is independent of the mean pressure periment (e, v, and or). When the pore fluid is a gas (e.g., dry when the Klinkenberg effect is taken into account (average air, Ar, or N2), the pressure diffusion equation is inherently log(k) = ñ0.015, b = 1.11 ñ 0.31 bars, solid circles in nonlinear and must be treated properly in the permeability Figure 4). The calculated permeability without a slip correcmeasurements using a gas permeameter. When the mean gas tion (via equation (7), open circles in Figure 4)) is slightly pressure is small (e.g., a few bars), the effect of gas slippage higher than the slip-corrected permeability at low pressures can be important, which further complicates the analysis of when the Kiinkenberg effect was neglected. However the esthe transient pressure decay data. To simplify data analysis, timated permeability using (7) approaches the true sample alsproximate solutions to the nonlinear pressure diffusion permeability as the mean sample pressure increases (Figure equation were obtained using a regular perturbation method. 4). These approximate solutions, valid in the limit when e << 1, Hence our overall recommendation for permeability can also be used to guide experimental design and improve measurements using a transient gas permeameter is as follows. the accuracy of permeability measurements using the transient Try to avoid the Klinkenberg correction whenever it is possipulse decay method. For example, one can effectively elimi- ble by conducting transient pulse decay measurements at a nate the molecular slippag effect by running pulse decay ex- high pressure so that (_p] +?2)/2 >>/. Sample permeability can periments at a higher mean sample pressure so that be calculated by plotting the normalized pressure differences ( +?2)/2,,. (?i-_p2)/(_p] +?2) as a function of time. The exponent ob- As a final example, we compare permeability of a syn- tained from a least squares exponential fit to the normalized thetic quartzite (QST2) measured using a modified air per- data is proportional to k. In a case when a high pressure canmeameter that permits investigation at the original pressure not be implemented, equation (5b) can be used to estimate conditions of Wark and Watson [1998] and higher pressures both sample permeability and slip factor. A significant amount (up to 15.5 bars). The sample was fashioned after the quartzite of pressure decay (i.e., > 30%) is needed, however, for a relimaterials of Wark and Watson [1998]: combining vol- able estimate of b. ume fraction of distilled H20 with quartz, and equilibrating these materials 850øC and 1.4 GPa for 4 days. The new Acknowledgments. This paper benefited from an unofficial repermeameter configuration used Ar as the fluid medium, with view of Dr. Joe Walsh, as well as official reviews by Dr. Christian the upstream reservoir volume (V ) of 78.8 mm 3, and the David, an anonymous reviewer, and the Associate Editor. This work downstream reservoir either open to the atmosphere (infinite was supported, in part, by NSF grant EAR and DOE Office of Basic Energy Sciences grant DE-FG02-94ER volume) or closed (fixed volume at mm3). Figure 4 plots the measured log(k) as a function of the mean pressure References ß J, i, i, i, o o o ß ß ß ß o o c o k* (b = O) I- ß k(with est. b) t I, t I I t I, (P1o + P2o )/2(bar) Figure 4. Plot of calculated permeability versus mean initial pressure of measurement for a synthetic quartzite QST2. Twelve pressure-loss curves were collected using a modified gas permeameter of Wark and Watson [1998] with Ar as the fluid medium (see text for details). Sample permeability was calculated using either the original method of Wark and Watson [1998] which is equivalento the case when b = 0 (open circles) or the full inversion method outlined in the text (solid circles). The estimated Klinkenberg coefficient from an average of the 12 measurements is 1.11 ñ 0.31 bars. i Aronofsky, J. S., Effect of gas slip on unsteady flow of gas through porous media, d. Appl. Phys., 25, 48-53, Bourbie, T., and J. Walls, Pulse decay permeability: Analytical solution and experimental test, Soc. Pet. Eng. d., 22, , Brace, W. F., J. B. Walsh, and W. T. Frangos, Permeability of granite under high pressure, d. Geophys. Res., 73, , Bruce, G. H., D. W. Peaceman, H. H. Rachford Jr., and J. D. Rice, Calculations of unsteady-state gas flow through porous media, Trans. Am. Inst. Min. Metall. Pet., 198, 79-92, Chen, T., and P. W. Stagg, Semilog analysis of the pulse-decay technique of permeability measurement, Soc. Pet. Eng. d. 24, , Dullien, F. A. L., Porous Media: Fluid Transport and Pore Structure, 574 pp., Academic, San Diego, Calif., Freeman, D. L., and D.C. Bush, Low-permeability laboratory measurements by nonsteady-state and conventional methods, Soc. Pet. Eng. d., 23, , 1983 Haskett, S. E., G. M. Narahara, and S. A. Holditch, A method for simultaneous determination of permeability and porosity in lowpermeability cores, SPE Form. EvaL, 3, , Hsieh, P. A., J. V. Tracy, C. E. Neuzil, J. D. Bredehoefi, and S. E. S illiman, A transient laboratory method for determining the hydraulic properties of 'tight' rocks - I. Theory, Int. d. Rock Mech. Min. Sci, Geomech. Abstr., 18, , Jones, S.C., A rapid accurate unsteady-state Klinkenberg permeameter, Soc. Pet. Eng. d., 12, , Jones, S.C., A technique for faster pules-decay permeability measurements in tight rocks, SPE Form. EvaL, 12, 19-25, Klinkenberg, L. J., The permeability of porous media to liquids and gases, Drill. Prod. Prac., , Lin, W., Compressible fluid flow through rocks of variable permeability, Rep. UCRL-52304, 15 pp., Lawrence Livermore Natl. Lab., Livermore, Calif., 1977.

7 LIANG ET AL.: NONLINEAR PRESSURE DIFFUSION IN A POROUS MEDIUM 535 Mavko, G., T. Mukerji, and J. Dvorkin, The Rock Physics Handbook: Yilmaz, 0., R. C. Nolen-Hoeksema, and A. Nur, Pore pressure pro- Tools for Seismic Analysis in Porous Media, 329, pp., Cambridge files in fractured and compliant rocks, Geophys. Prospect., 42, Univ. Press, New York, , 1994 Phillips, O. W., Flow and Reactions in Permeable Rocks, 285 pp., Cambridge Univ, Press, New York, Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge Y. Liang, Department of Geological Sciences, Brown University, University Press, New York, Providence, R (Yan_Liang brown. edu) Scheidegger, A. E., The Physics of Flow Through Porous Media. 353 J. D. Price, D. A. Wark, and E. B. Watson, Department of Earth pp., University of Toronto Press, Toronto, Ont., and Environmental Sciences, Rensselaer Polytechnic Institute, Troy, Trimmer, D. A., Design criteria for laboratory measurements of low NY (pricej rpi.edu; warkd rpi.edu; watsoe rpi.edu) permeability rocks, Geophys. Res. Lett., 8, , Wark,D. A., and E. B. Watson, Grain-scale permeability of texturally equilibrated, monomineralic rocks, Earth Planet. Sci. Letters, (Received November 9, 1999; revised August 4, 2000; 164, , accepted September )