Effective pressure law for permeability of E-bei sandstones

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2009jb006373, 2009 Effective pressure law for permeability of E-bei sandstones M. Li, 1 Y. Bernabé, 2 W.-I. Xiao, 1 Z.-Y. Chen, 3 and Z.-Q. Liu 3 Received 10 February 2009; revised 6 May 2009; accepted 13 May 2009; published 24 July [1] Laboratory experiments were conducted to determine the effective pressure law for permeability of tight sandstone rocks from the E-bei gas reservoir, China. The permeability k of five core samples was measured while cycling the confining pressure p c and fluid pressure p f. The permeability data were analyzed using the response-surface method, a statistical model-building approach yielding a representation of k in (p c, p f ) space that can be used to determine the effective pressure law, i.e., p eff = p c kp f. The results show that the coefficient k of the effective pressure law for permeability varies with confining pressure and fluid pressure as well as with the loading or unloading cycles (i.e., hysteresis effect). Moreover, k took very small values in some of the samples, even possibly lower than the value of porosity, in contradiction with a well-accepted theoretical model. We also reanalyzed a previously published permeability data set on fissured crystalline rocks and found again that the k varies with p c but did not observe k values lower than 0.4, a value much larger than porosity. Analysis of the dependence of permeability on effective pressure suggests that the occurrence of low k values may be linked to the high-pressure sensitivity of E-bei sandstones. Citation: Li, M., Y. Bernabé, W.-I. Xiao, Z.-Y. Chen, and Z.-Q. Liu (2009), Effective pressure law for permeability of E-bei sandstones, J. Geophys. Res., 114,, doi: /2009jb Introduction [2] For oil and gas production operations, one of the most important rock properties is permeability. Effective reservoir management not only needs extensive permeability data but also requires understanding of how permeability varies with fluid pressure depletion in rocks subjected to regional stresses [e.g., Vairogs et al., 1971; Schutjens et al., 2004]. For example, a better knowledge of the effective pressure law for permeability may have a significant impact on numerical simulations of fluid flow in the reservoir and, therefore, on setting up the best production strategy. Accordingly, a research project was designed with the purpose of investigating the effective pressure law for permeability in the tight sandstones typical of the E-bei gas reservoir, China. [3] Strictly speaking, rock permeability k is said to follow an effective pressure law if the dependence of k on confining (or overburden) pressure p c and fluid pressure p f reduces to a one-variable function k(p c, p f )=f(p c kp f ), where k is a constant. However, the definition, p eff = p c kp f, can be extended to include cases where k is not a constant, although it may be argued that the effective pressure law looses most of its usefulness if k is a function of p c and p f [Robin, 1973]. 1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, China. 2 Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. 3 E & P Research Institute, Sinopec North China Company, Zheng Zhou, China. Copyright 2009 by the American Geophysical Union /09/2009JB006373$09.00 Whatever the properties of k, the effective pressure concept is so widely used in practice that any discussion of k(p c, p f ) data has to include it. [4] A number of laboratory studies of the effective pressure law for permeability can be found in the literature [e.g., Zoback and Byerlee, 1975; Walls and Nur, 1979; Coyner, 1984; Morrow et al., 1986; Bernabé, 1986, 1987, 1988; Warpinski and Teufel, 1992; Kwon et al., 2001; Ghabezloo et al., 2009]. The main experimental observations can be summarized as follows: [5] 1. Rocks containing a compliant phase within the pore space (most commonly, sandstones with a significant clay content) display values of k greater than unity, which is the theoretical upper limit of k for elastically microhomogeneous porous media [e.g., Berryman, 1992a, 1992b, 1993]. But note that some very heterogeneous rocks do not display k > 1, suggesting that the specific spatial distribution of the phases mentioned above is required to produce this effect [Kwon et al., 2001]. [6] 2. In most other rocks (e.g., crystalline rocks, clean granular rocks, chalks, shales and so forth) values of k close to but lower than 1 are usually found, although values as low as 0.4 have been reported. [7] 3. The coefficient k is not always a true constant. In studies where k is determined piecewise in small p c and p f intervals, it is observed to vary with changing p c and p f [Bernabé, 1986, 1987, 1988; Warpinski and Teufel, 1992; Ghabezloo et al., 2009]. It is then more accurately described as a function k(p c, p f ). When successive loading/unloading cycles are applied to the rock samples, the variations of k(p c, p f ) diminish significantly during the first couple of cycles and tend to stabilize afterward. Thus, it is often 1of13

2 recommended to perform a couple of preliminary, full range, pressure cycles prior to the tests to ensure reproducible results (this practice was called seasoning in early rock mechanics papers; this term has a clear and specific meaning in the context of laboratory testing of rock samples and will be used hereafter). Finally, hysteresis and loading history effects have been observed (e.g., values of k(p c, p f ) and their evolution are different during the loading and unloading stages of a cycle). [8] Early theoretical work was devoted to the effective pressure law for elastic bulk deformation [e.g., Nur and Byerlee, 1971; Robin, 1973]. As pointed out by Robin [1973], this effective pressure law model can be applied piecewise, in sufficiently small p c and p f intervals, when the rock considered is nonlinear elastic. In this case, the coefficient a of the effective pressure law for bulk deformation (not necessarily equal to k) is not constant over a broad range of confining and pore pressures. Berryman [1992a, 1992b, 1993] derived a model of the effective pressure law for permeability assuming that the rock solid framework was homogeneous (implicitly assumed by Nur and Byerlee s model). Berryman s model predicts that k should lie in the interval [f, 1], where f is the porosity, and these limits seem to be consistent with the majority of the experimental results cited above (corresponding to statement 2 in the previous paragraph). As an explanation for the upper bound violations sometimes observed (statement 1 above), Berryman [1992a, 1992b, 1993] also demonstrated that they occur when the solid framework is inhomogeneous and the different phases are arranged in such a way that an equivalent homogeneous medium cannot be defined [see also Al-Wardy and Zimmerman, 2004]. As in the bulk deformation case, Berryman s model can be applied piecewise and thus account for the observed variations of k with p c and p f. From a very different perspective, Bernabé [1986] calculated k for crystalline rocks on the basis of a version of the equivalent channel model [Walsh and Brace, 1984], in which the equivalent channel had an elliptical cross section of aspect ration e (i.e., ratio of the short to the long cross-sectional dimensions). According to this model, it is found that k is a function of the aspect ration e and Poisson s ratio n of the solid phase, varying from 1/2(1 n) fore = 1 (cylindrical channel with circular cross section) to 1 for e = 0 (infinitely thin crack). As a consequence, if e varies during loading of the rock sample, so does the coefficient k of the effective pressure law for permeability. Variations of e during compression could be caused by an increase in contact area during closure of rough microcracks [Bernabé, 1986] or, if a broad distribution of crack aspect ratio exists, by closure of the cracks with the lowest e [Cheng and Toksoz, 1979]. [9] In this paper, we report on a laboratory study aimed at determining the value of k in samples of the tight sandstone formations of the E-bei gas reservoir, China. In particular, one important goal was to delineate the possible variations of k with p c and p f. For that purpose, permeability measurements were performed on rock samples subjected to cycles of the confining and/or the pore pressures and the permeability data were analyzed using Box and Draper s [1987] statistical response-surface approach, which allows k to be defined as a function of p c and p f. The results indeed showed that, along a loading path, the values of k significantly varied with p c and p f, generally approaching very small values at high p c and low p f conditions, possibly violating Berryman s [1992a, 1992b, 1993] suggested lower bound. 2. Materials and Techniques 2.1. Rock Samples [10] The E-bei gas reservoir, China, is a low-permeability and low-pressure sandstone reservoir. Five sandstone samples of the E-bei tight formation were cored from well Da-47 at a depth of 2760 m (they were labeled S3, S4, S8, S9, and S10). At that depth the overburden pressure is estimated to be 63 MPa and the fluid pressure about 24 MPa. The most extreme p c and p f values used in the laboratory tests (i.e., 44 and 6 MPa, respectively) were chosen so as not to exceed the estimated in situ effective pressure (i.e., p c p f ). [11] The original length of the cores was about 10 cm. The cores were cut into three segments. The first one, about 4 cm in length, was used for the permeability tests, the next one, approximately 2.5 cm long, was taken for capillary pressure tests, and the remaining segment was reserved for preparation of thin sections and microstructure observations. Prior to the permeability tests, the porosity of the samples was measured yielding 6.15, 5.48, 2.42, 6.51, and 10.92% for S3, S4, S8, S9, and S10, respectively. [12] The samples are lithic sandstones, composed of quartz (50% 60%), rock debris (30% 40%) and interstitial material (approximately 5%). The major cements are quartz overgrowth and pore filling authigenic clays, mostly illite. Preliminary examination of optical and scanning electron microscope micrographs show that the most common pore types are intergranular pores remaining from the deposition stage of the rock genesis, and, dissolved, intragranular, roughly equidimensional voids (see Figure 1). Some concave/convex and stylolite grain contacts caused by pressure solution compaction can also be observed. Finally, the presence of microcracks can be noted. Most of them are intergranular cracks, along the grain boundaries, while a few intragranular ones developed across grains. Many of these microcracks were possibly closed in situ. Additional information on pore size statistics was gained from mercury porosimetry tests. For example, the left-hand diagram of Figure 2 presents a semilog plot of the mercury capillary pressure versus mercury saturation for sample S4 (the solid dots represent the drainage or mercury injection curve and the open dots the imbibition curve). The mercury capillary pressure data can be recast into pore radius values by means of the Washburn equation. The drainage curve can thus be viewed as the cumulative pore radius distribution, from which the pore radius distribution can be derived (represented in histogram form in the right-hand diagram of Figure 2). Clearly, the pore radius distribution is broad and has a complex shape. Characterizing it quantitatively with a minimum of parameters is a difficult problem. Here, we used the maximum pore radius r Max (corresponding to the threshold mercury pressure) and the median of the distribution of the log of the pore radius r med (the comparison of the two provides a measure of the width of the distribution). The observed values are: r Max = 2.60, 2.60, 2.61, 1.61 and 4.00 mm andr med = 0.068, 0.046, 0.024, 0.030, and mm for S3, S4, S8, S9 and S10, respectively (although the relationship is far from perfect, r Max does correlate roughly with permeability). 2of13

3 Figure 1. Microstructure of rock S4. (a) Image of a fractured section in S4 through the scanning electron microscope in secondary electron mode. (b) Optical microscope image of a thin section of S4 impregnated with dyed epoxy showing the pore space in red Experimental Procedures [13] The permeability measurements were performed using nitrogen gas as pore fluid so that any chemical and capillary effects that might arise from clay/water reactions and imperfect saturation could be minimized. The measurement apparatus used in this study consists of an overburden core holder equipped with input and output pressure transducers and an output soap film flowmeter. The pore and confining pressure systems allows independent control of the confining pressure and the input and output pore pressures (the pore saturating gas is provided by a nitrogen bottle and large dead volume reservoirs were inserted to dampen the pressure spikes produced by the pumps). [14] Permeability measurements were conducted with the steady state method. Pressure drops across the samples varied from 0.16 to 2.82 MPa. The output flow rate was measured with a soap film flowmeter (the fluid volume resolution is 0.02 ml, corresponding to a relative error of 4% or less). Typical values ranged from cm 3 /sec to cm 3 /sec. The viscosities of nitrogen gas at different pressures and temperatures were obtained from the tables of the Beijing Chemical Industrial Company Inc. [1979]. The flow rate measurements were repeated five times and averaged for final determination of permeability. [15] It is well known that using a gas as pore fluid can cause problems at low p f owing to the Klinkenberg effect, i.e., the increase in apparent gas permeability k gas observed when the mean free path of the gas molecules is comparable to the characteristic pore size [e.g., Wu et al., 1998]. Note that the apparent increase in permeability will, therefore, be small either if the mean free path is small (i.e., at high gas pressure) or the pore size is large (i.e., in high permeability rocks). The Klinkenberg effect is expressed as follows: k gas ¼ k 1 þ b ; ð1þ p f 3of13

4 Figure 2. Example of a mercury injection test for rock S4. where k denotes the true permeability, p f the mean pore pressure inside the sample and b the so-called Klinkenberg correction coefficient. On the basis of equation (1) and assuming b constant, k can be estimated from a series of k gas measurements performed at increasing gas pressures. However, the constant b assumption breaks down in pressure sensitive rocks [e.g., Warpinski and Teufel, 1992]. The reason is that b is, for a large part, controlled by the characteristic pore size, which, in rocks such as E-bei sandstones, varies strongly with the p c and p f conditions. As a consequence, it is practically very difficult to measure the Klinkenberg correction coefficient and the pressure dependence of permeability simultaneously. The two effects are superimposed in the data and nearly impossible to separate [e.g., Warpinski and Teufel, 1992]. One possible solution relies on the fact that the Klinkenberg correction becomes negligible at sufficiently high gas pressures. In order to verify whether or not p f was high enough in our measurements, we performed additional gas permeability measurements in S8 and S9 samples in p c and p f conditions designed to produce very small changes in effective pressure (p f was kept very low, between 0.2 and 0.8 MPa, while p c ranged from 10 to 40 MPa) and thus allow estimation of b. For each value of p c, we thus estimated b and the true permeability k. The results are represented in a log-log plot in Figure 3. The data show a reasonable correlation of ln(b) and ln(k) and suggest that b follows a power law, b = k 0.43, with b in MPa and k in md (or m 2 ). Similar investigations were conducted by Jones [1972] and Tanikawa and Shimamoto [2006] on a large variety of rocks, resulting in b = k 0.36 and b = k 0.37, respectively (note that, like us, they used nitrogen as pore fluid). These two relationships (light and dark gray thick lines in Figure 3) are very close to ours in the permeability range investigated in Figure 3. Plugging the above relationships in equation (1) and using a recurrence scheme (i.e., use k gas in b to calculate the corrected k 1, then use k 1 in b to calculate k 2, and so on; no more than three steps are needed for convergence to occur in the md range) we can estimate plausible corrections for our permeability data. The largest relative corrections are obtained using Tanikawa and Shimamoto s formula. They are much smaller (i.e., 0.4, 0.3, 0.6, 0.7, and 0.2% for S3, S4, S8, S9 and S10, respectively) than the experimental relative error on permeability (i.e., 5 10%). It is therefore safe to follow the practice of Warpinski and Teufel [1992] and omit the Klinkenberg correction in our data treatment. 4of13

5 Figure 3. Log-log plot of the measured Klinkenberg correction coefficient b versus the true permeability k for S8 (open circles) and S9 (solid circles). The best-fit power law b = k 0.43 is also plotted (thin black line) as well as Jones [1972] and Tanikawa and Shimamoto s [2006] formulas b = k 0.36 and b = k 0.37 (light and dark gray thick lines, respectively). [16] Following the recommendations of Bernabé [1986, 1987, 1988] and Ping [2006], we subjected all samples to at least two seasoning p c cycles, up to the maximum p c value planned for the test. Indeed, history effects (i.e., dependence of permeability, in identical p c and p f conditions, on the previous loading history) make it impossible to interpret the data in terms of an effective pressure law. Fortunately, they become negligible once a few seasoning cycles have been performed. As an example, Figure 4 shows the seasoning cycles performed on sample S10. In this case, history effects have all but vanished after two seasoning cycles. Note that these specific measurements were intended as diagnostic tools rather than to provide accurate permeability values. To save time, we made them at p f values lower than 1.15 MPa and had to apply the Klinkenberg correction. [17] After the seasoning cycles were performed, the samples were loaded to a confining pressure of 44 MPa and left to stabilize overnight. For the tests themselves, we used two different p c and p f cycling procedures. Procedure 1, which was applied only to the S4 sandstone, consisted of a series of loading-unloading p c cycles, each at constant fluid pressure (note that we always made sure that p c > p f ). We started at the highest fluid pressure and scanned the p f range in descending order. For every test point, fluid flow needed 40 min to stabilize during loading and one hour during unloading. Stabilization was achieved as jugged by the flow rate and fluid pressure drop across the sample that became essentially constant after a certain time ellapsed. Procedure 2 consisted of unloading-reloading p f cycles, each at constant confining pressure. We started with a maximum p c of 44 MPa and decreased the confining pressure to 34, 24 and 14 MPa for the S8 and S10 sandstones while40and36mpawereusedfors3ands9.with procedure 2, fluid flow stabilization required 30 min during p f unloading and one hour during reloading Data Analysis: Effective Pressure Law [18] Following Warpinski and Teufel [1992], we applied the response-surface method [Box and Draper, 1987] to represent the permeability data and determine the effective pressure coefficient for permeability k(p c, p f ). The responsesurface method does not presuppose any knowledge about the rock properties. It uses statistical analysis to build an empirical polynomial surface in (p c, p f ) space approximately matching the data. Note that in the vast majority of cases, a preliminary transformation must be applied to the data, otherwise a suitable polynomial response surface cannot be found. [19] Here, we used the Box-Cox transform [Box and Draper, 1987], k (b) =((k/k o ) b 1)/b where k (b) is the transformed permeability and k o is a normalizing constant (e.g., k o = 1 md or m 2 ). The parameter b is normally expected to lie between 3 and +3, with b =0 corresponding to the log transform. For simplicity and Figure 4. Permeability measurements in sample S10 during the seasoning p c cycles. 5of13

6 robustness, we limited our analysis to quadratic response surfaces. The transformed permeability data are, therefore, modeled by km ðbþ ¼ a 1 þ a 2 p c þ a 3 p f þ a 4 p 2 c þ a 5p c p f þ a 6 p 2 f ; ð2þ where the a i coefficients are calculated by least squares regression. The most important problem is to determine the optimal exponent b. Box and Draper [1987] suggested a maximum likelihood technique to optimize the transformation for a given set of data and verify the adequacy of the match using the so-called F test, in which F refers to the ratio of model mean square to the error mean square, F = hk m 2 i/h(k m k) 2 i, where k denotes the observed data, k m = k o (b k m (b) +1) 1/b the back transformed model and the hibrackets the averaging operator. Box and Draper [1987] consider that an F value at least equal to 10 times the F distribution percentage point is needed to ensure that the surface adequately matches the data. Generally, the 95% significant F distribution percentage point is accepted for any data set. Another measure of the fit is the mean square normalized error, E 2 = h((k m k)/k) 2 i (but note that the method described above was not meant to minimize E). Furthermore, in multidimensional regression modeling, the significance of the model does not necessarily imply that the effect of every independent variable on the dependent variable is important. So, it is necessary to test the significance of each regression coefficient. If the significance test values are greater than those from T distribution tables [Wang, 2004], it is inferred that none of the regression coefficients are vanishingly small and, therefore, all independent variables have a significant effect on the dependent variable. [20] Once b is selected and the a i coefficients of equation (2) are calculated, it is easy to obtain an estimate of the coefficient of the effective pressure law, k = (@k/@p f )/ (@k/@p c ) [e.g., Bernabé, 1986], as kðp c ; p m Þ m=@pc ðbþ m =@pf ¼ a 3 þ a 5 p c þ 2a 6 p f ðbþ m =@pc a 2 þ 2a 4 p c þ a 5 p f We see that k(p c, p f ) is a constant if a 4, a 5, and a 6 are negligible compared to a 2 and a 3, or, in other words, if the response surface reduces to a plane (however, remember that the response surface describes k (b) and that k itself may thus not be linear in p c and p f ). As should be expected, k = 1 implies a 2 = a 3, which simply states that p c and p f affect k with equal intensity Data Analysis: Dependence of Permeability on Effective Pressure [21] Once the effective pressure law for permeability is determined we can examine the dependence of k on p eff. Since microcracks appear prominently in the microstructure of E-bei sandstones, we may try to interpret our experimental results in terms of fractured rocks models such as those of Gangi [1978] and Walsh [1981]. Gangi s [1978] model describes the response under normal compression of ð3þ an elastically deforming, rough fracture with a power law distribution of local apertures and is expressed as kp ð eff Þ ¼ k 0 1 p m 3 eff ; ð4þ P 1 where k 0 is a reference permeability corresponding to zero effective pressure, m the inverse of the exponent of the power law distribution and P 1 the effective elastic modulus of the fracture. As shown by Kwon et al. [2001] the constants in equation (4) can be estimated by assuming a reasonable value of k 0 and finding the best linear fit of ln[1 (k/k 0 ) 1/3 ] versus ln[p eff ]. [22] Walsh [1981] also studied the changes in permeability of an elastic rough fracture under normal compression and obtained p ð Þ ¼ k 0 1 ffiffi h 2 ln p 3 eff ; ð5þ a 0 kp eff where p 0 is an arbitrary reference effective pressure, k 0 and a 0 the corresponding reference permeability and fracture half aperture, respectively, and h the standard deviation of the asperities height distribution. A simple method for applying this model consists in determining the constants in (k/k 0 ) 1/3 = A B ln(p eff /p 0 ) (where A should be close to 1) by linear regression. It is convenient to use one of the data points for p 0 and k 0, but note that a 0 and therefore B depend on that choice. [23] It is very important, at this juncture, to emphasize that both models assume purely elastic rock deformation and, therefore, cannot account for any hysteresis and loading history effects that might be observed. Nevertheless, if we assume that the nonelastic processes possibly acting in the compressed rock are not so strong as to destroy the models relevance entirely, we may apply these models to each separate loading and unloading cycle. This approach implies that the constants in equations (4) and (5) may vary during the loading and unloading cycles. Consequently, the curves ln[1 (k/k 0 ) 1/3 ]versusln[p eff ]and(k/k 0 ) 1/3 versus ln(p eff /p 0 ) are not necessarily linear. Linearity or lack thereof cannot be used as a strong argument for assessing Gangi s [1978] and Walsh s [1981] models. Also, a practical consequence for the application of Gangi s model is that linearity cannot serve as a guiding criterion for guessing the most appropriate value of k 0 (in general, increasing k 0 improves the curve linearity and leads to a decrease in m and an increase in P 1 ). Instead, we made preliminary plots of k versus p eff and defined a plausible range of k 0 by visually extrapolating the curves to zero p eff. [24] Finally, for the purpose of characterizing the pressure sensitivity of permeability, it is more convenient to use a simple empirical expression such as k = k 0 Exp( g p eff ) [e.g., David et al., 1994] rather than elaborate models such as Gangi s [1978] or Walsh s [1981]. We emphasize that the goal here is to quantify the pressure sensitivity by a single, easily measured parameter. Of course, the pressure sensitivity coefficient g must be statistically related to the parameters of the Gangi s and Walsh s models (we verified p 0 6of13

7 Figure 5. Three-dimensional diagram of measured permeability superposed on the response surface k(p c, p f ) for sample S4 during (a) p c loading and (b) p c unloading. that g / 1/P 1 and ln(g) / B and that g is also loosely correlated with m) but its usefulness follows from having been recognized in past studies as a good descriptor (e.g., David et al. [1994], Morrow et al. [1994], Evans et al. [1997], and Seront et al. [1998] reported g to range from to MPa 1 in porous sandstones and from to MPa 1 in tight sandstones, and to reach values as high as 0.10 to 0.25 MPa 1 in cores from deep boreholes and fault zones). 3. Results 3.1. Permeability Measurements [25] The permeability data for samples S3, S4, S8, S9, and S10 are given in Tables S1 S5 in the auxiliary material. 1 Although history effects were suppressed by seasoning the samples, hysteresis effects are still significant (i.e., the loading and unloading cycles do not coincide). Therefore, the loading and unloading cycles (p c cycles for S4, p f for all the other samples) were treated separately in the data analysis. For each sample, two response surfaces were thus calculated, one corresponding to the p c or p f loading cycles and the other one to the unloading cycles. In all cases, the back transformed response surfaces fitted the experimental data very well. They all satisfied the F test and, as a confirmation, normalized errors E as low as 3, 12, 12, 4 and 13% were obtained for S3, S4, S8, S9 and S10, respectively. We also verified that the significance test values for each coefficient a i were much larger (by a factor of 10 3 to 10 6 ) than those from T distribution tables [Wang, 2004]. Therefore, all regression coefficients a i are important for modeling permeability. As an illustration, Figures 5a and 5b shows the back transformed loading and unloading response surfaces (three-dimensional set of intersecting constant p c and constant p f lines) superposed on the experimental data points (small solid square symbols) for sample S4 (note that the diagrams include part of the unphysical domain where p f > p c ) Effective Pressure Law [26] We were then able to calculate the effective pressure law coefficient k(p c, p f ) in each case using the corresponding response-surface coefficients and equation (3). We observed that the values obtained varied strongly with p c and p f (see Tables S6 S10 in the auxiliary material). In all cases where the interval of variation of p c and/or p f was about 20 MPa or more, k(p c, p f ) decreased with increasing p c and/or decreasing p f. We also noticed significant differences in k for the loading and unloading cycles. Considering that p c loading cycles are somewhat equivalent to p f unloading ones, the variations of k(p c, p f ) were as follows: for the equivalent p c loading cycles S < k( p c, p f ) < 1.13, S < k( p c, p f ) < 0.85, S < k( p c, p f ) < 1.23, S < k( p c, p f ) < 0.64, S < k( p c, p f ) < 0.77, and for the equivalent p c unloading cycles S < k( p c, p f ) < 1.36, S < k( p c, p f ) < 0.79, S < k( p c, p f ) < 0.83, S < k( p c, p f ) < 1.24, S < k( p c, p f ) < Clearly, some of these values cannot be correct. In particular, we consider the negative k s to be unphysical since we cannot imagine a porous medium (even an heterogeneous one) for which an increase of p f would produce a decrease of permeability. It turns out the most extreme and most unsatisfactory values all occur on the boundaries of the (p c, p f ) domain of definition. For example, Figures 6a and 6b shows three-dimensional plots of k in (p c, p f ) space. We see that the extreme values, k < 0 and k > 1, arise in opposite corners of the (p c, p f ) zone (i.e., high p c -low p f and low p c -high p f, respectively). This could have been anticipated 1 Auxiliary materials are available in the HTML. doi: / 2009JB of13

8 p eff = p c p f. This can be done graphically by comparing plots of k versus p eff for both effective pressure laws and observing which one gives the least data points dispersion. The comparison can be made quantitative by determining a best-fit curve to the k versus p eff data points and calculating s, the root-mean-square of the residuals. Since we noticed that the curves used by Gangi s [1978] and Walsh s [1981] models (see section 2.4) were always smooth and only moderately bent, we decided to use a second degree extension of Walsh s model to represent our permeability data (it did always match with great accuracy the shape of the data point clusters) and calculate s. Note that this function is used for curve-fitting purpose only, so its lack of physical significance is irrelevant. The results of this analysis for the equivalent p c loading cycles are the following: [27] 1. For S3, Terzaghi s [1925] law (i.e., k = 1) gave significantly better results than k(p c, p f ) from equation (3), even though the variations of k are not very large. However, the effective pressure law, p eff = p c hkip f, using the mean value hki = 0.93, produced a lower s ( md) than Terzaghi s law ( md). [28] 2. In the case of S4, k(p c, p f ) of equation (3) was markedly better (s = md) than k = 1 (s = md), provided the negative values were replaced by zero (see Figure 7). Thus, we have very good reason to conclude that, for this particular rock, k(p c, p f ) spanned almost the entire theoretical range from essentially 0 to 0.85, relatively close to 1. [29] 3. For S8, neither k(p c, p f ) of equation (3) nor a constant k equal to hki produced an evident improvement with respect to Terzaghi s [1925] law. However, we obtained better results by restricting k(p c, p f )tok(p c ) measured along the line, p f = 11 MPa, in the middle of the domain of definition (s = md instead of md). This indicates that some amount of variability of k (i.e., 0.20 < k(p c ) < 1.0) was needed to fit the experimental data. [30] 4. For the S9 rock, k(p c, p f ) of equation (3) was satisfactory but we must point out that a constant k equal to hki was remarkably better than both the variable k(p c, p f ) Figure 6. Three-dimensional diagram of the effective pressure law coefficient k(p c, p f ) calculated using the responsesurface method for sample S4 during (a) p c loading and (b) p c unloading. because k(p c, p f ) is defined as the ratio of partial derivatives, (@k/@p f )/(@k/@p c ), and can therefore be rather uncertain when estimated from noisy experimental data [e.g., Bernabé, 1986]. It is apparent, then, that extrapolations outside the domain of definition or even near extrapolations on its boundaries cannot be relied upon. Moreover, we need a way to assess the validity of k(p c, p f ) inside the domain of definition. The most natural method is to compare the effective pressure law determined here, p eff = p c k(p c, p f ) p f (where k(p c, p f ) is computed from equation (3), possibly with some modifications aimed at removing the unphysical or unreliable values) to the ideal Terzaghi s [1925] law, Figure 7. Assessment of the quality of the effective pressure law from the response-surface method applied to sample S4. There is less scatter for k(p eff ) with the responsesurface law p eff = p c k(p c, p f ) p f (solid circles, s = md), than for k(p eff ) with Terzaghi s [1925] law p eff = p c p f (open circles, s = md). 8of13

9 Figure 8. Assessment of the quality of the effective pressure law from the response-surface method applied to sample S9. There is less scatter for k(p eff ) with the responsesurface law p eff = p c hkip f (solid circles, s = md), than for k(p eff ) with Terzaghi s [1925] law p eff = p c p f (open circles, s = md). and Terzaghi s [1925] law (s = md instead of md; see Figure 8). The implications are important because hki = 0.54 produces an effective pressure law, p eff = p c hkip f, which, when applied to the field, will lead to very different estimations of in situ permeability than Terzaghi s law, p eff = p c p f. [31] 5. For S10, we found that k(p c, p f ) (with 0.04 < k(p c, p f ) < 0.77) was better than k =1(s = md instead of md). [32] Since the equivalent p c unloading cycles results are quite similar to the ones listed above, we will simplify our presentation by simply reporting the most satisfactory effective pressure law coefficient in each case (but note that extreme k values are more common than before and that we always replaced negative k s by zero; also, the gains in s were not as substantial as before) S3 p eff = p c hki p f (with hki = 0.86), S4 p eff = p c k(p c, p f ) p f (with 0 < k(p c, p f ) < 0.79), S8 p eff = p c k(p c, p f ) p f (with 0 < k(p c, p f ) < 0.83), S9 p eff = p c hkip f (with hki = 0.46), S10 p eff = p c k(p c, p f ) p f (with 0.06 < k(p c, p f ) < 0.50). Thus, the k values observed for the p c unloading cycles (or p f loading) tend to be lower than for the p c loading cycles (or p f unloading) by 15% on average Effective Pressure Dependence of Permeability [33] We applied Gangi s [1978] and Walsh s [1981] models to each sample and each equivalent p c loading and unloading cycle. We used either the effective pressure laws reported in section 3.2 or Terzaghi s [1925] law. Examining the results listed in Tables 1 and 2, the following three general comments can be made: (1) the curves ln[1 (k/k 0 ) 1/3 ] versus ln[p eff ] and (k/k 0 ) 1/3 versus ln(p eff /p 0 ) were always fairly smooth although relatively nonlinear (this is consistent with the generally nonelastic behavior of E-bei sandstones); (2) we observed marginally significant variations (comparable or slightly greater than the uncertainties) of the constants P 1, m, and B with the loading and unloading Table 1. Values of the Best-Fit Constant for Gangi s [1978] Model a P b 1 (MPa) m b P c 1 (MPa) m c k 0 (md) S3 L 67 ± ± ± ± ± 30 U 89 ± ± ± ± ± 30 S4 L 56 ± ± ± ± ± 70 U 9 ± ± ± ± ± 100 S8 L 57 ± ± ± ± ± 100 U 52 ± ± ± ± ± 70 S9 L 45 ± ± ± ± ± 120 U 48 ± ± ± ± ± 100 S10 L 80 ± ± ± ± ± 300 U 70 ± ± ± ± ± 300 a Here L and U refer to equivalent p c loading and unloading cycles, respectively. b Effective pressure laws of section 3.2. c Effective pressure laws of Terzaghi s [1925] law. cycles; and (3) using Terzaghi s law yielded substantially different values of P 1, m, and B. [34] More specifically, in the case of Gangi s [1978] model we almost always found m on the order of 0.5, an intermediate value between the allowed bounds (i.e., 1 and 0, characterizing polished and rough fracture surfaces, respectively). The only exception is S9 with m = 0.75 indicating a rather small degree of roughness. In general, the effective modulus P 1 ranged between 45 and 89 MPa, slightly greater than the value measured in shales by Kwon et al. [2001] but much smaller than that for the single fracture data used by Gangi [1978] (note that the relative uncertainty on P 1 is always rather small, while, for m, it is comparable to that on k 0 ). No clear trend could be identified in the variations of m and P 1 with loading and unloading cycles or with assuming different effective pressure laws. The largest variations were observed for S9. In this case, using Table 2. Values of the Best-Fit Constant B for Walsh s [1981] Model a B b k 0 (md) B c k c 0 (md) S3 L 0.63 ± ± U 0.54 ± ± S4 L 0.72 ± ± U 0.69 ± ± S8 L 0.74 ± ± U 0.88 ± ± S9 L 1.18 ± ± U 0.95 ± ± S10 L 0.64 ± ± U 0.69 ± ± a The values of A were all relatively close to 1 (in the average, A =0.97± 0.13) and p 0 was always chosen as close as possible to 23 MPa (±1.5 MPa). Here L and U refer to equivalent p c loading and unloading cycles, respectively. b Effective pressure laws of section 3.2. c Effective pressure laws of Terzaghi s [1925] law. 9of13

10 [36] As to the empirical pressure sensitivity coefficient g, we observed: (1) 0.076, 0.097, 0.11, 0.17 and MPa 1 for the equivalent p c loading cycles of S3, S4, S8, S9 and S10, respectively, and (2) 0.064, 0.085, 0.10, 0.16 and MPa 1 for the equivalent p c unloading cycles. These values tend to be high (see section 2.4) and indicate that permeability may be controlled by highly compliant microcracks in the E-bei rocks. Figure 9. Assessment of the quality of the effective pressure law from the response-surface method applied to (a) Chelmsford granite and (b) Pottsville sandstone. There is less scatter for k(p eff ) with the response-surface law p eff = p c k(p c ) p f (solid circles, s = nd for both rocks), in which p f was set to the middle value 20 MPa, than for k(p eff ) with Terzaghi s [1925] law p eff = p c p f (open circles, s = nd for Chelmsford granite and s = nd for Pottsville sandstone). Terzaghi s [1925] law leads to a very strong decrease of the inferred m (from 0.75 to 0.3) and an increase in P 1.It is not easy to understand the reason for these variations (For example, why does S9 appear rough when Terzaghi s law is used but not with k(p c, p f ) from section 3.2?). We can nevertheless conclude that it is very important to determine the proper effective pressure law in order to interpret the data accurately. [35] The analysis in terms of Walsh s [1981] model yielded values of A fairly close to unity (in the average we found A = 0.97 ± 0.13) and B between 0.54 and 1.18 with no discernable correlation with loading and unloading cycles and with differences in effective pressure law (note that the analysis was applied with a nearly constant p 0 of 23 ± 1.5 MPa to allow comparison of the estimated values of B). S9 has the largest B s but this does not necessarily indicate highest roughness. It may also indicate that the microcrack aperture is smaller in S9 than in the other rocks (this is consistent with the mercury injection data). 4. Discussion [37] The response-surface method allowed us to determine an effective pressure law, p eff = p c k(p c, p f ) p f, more pertinent in some of our samples (i.e., S4, S8, and S10) than Terzaghi s [1925] law, p eff = p c p f. In these samples, we found that k(p c, p f ) was a strongly varying function of the confining and fluid pressures. In particular, k(p c, p f ) decreased to essentially zero in high p c -low p f conditions. However, the method was not as successful in other samples (i.e., S3 and S9). It seems that the main reason for the failures is the narrow (p c, p f ) domain experimentally investigated for these samples (p c ranged between 36 and 44 MPa, an interval of only 8 MPa). The effective pressure coefficient k(p c, p f ) defined as a ratio of partial derivatives, can be very uncertain near the boundaries of the domain of definition, affecting all apparent values of k(p c, p f ) in small domains. We also tried to interpret our k(p eff ) data in terms of Gangi s [1978] and Walsh s [1981] models, but no clear-cut conclusion can be reached. It may even be that the E-bei sandstones are too nonelastic to be properly represented by these models. [38] In order to assess the response-surface method more comprehensively, we applied it to the permeability data reported by Bernabé [1986, 1987, 1988]. In these papers, the permeability to water was measured in four New England granites (Barre, Chelmsford, Pigeon Cove and Westerly) and one low-permeability sandstone (Pottsville). The (p c, p f ) cycling procedures involved a broad range of confining pressures (from 40 to 200 MPa) while a shorter range of fluid pressures (from 10 to 30 MPa) was used. This procedure was designed primarily to investigate the effect of p c. Note also that one of Bernabé s goals was to study the effect of repeated p c cycles and, therefore, the rock samples were not subjected to seasoning cycles prior to the measurements. As in the analysis of the E-bei data, we treated the p c loading and unloading cycles separately. We found that, for each rock and each p c cycle, the back transformed response surface matched Bernabé s permeability data within a few percent accuracy. We then calculated the corresponding effective pressure coefficient, k(p c, p f ), from equation (3) and observed an unmistakable gain with respect to Terzaghi s [1925] effective pressure law when k(p c, p f ) was restricted to k(p c ) measured along the central line, p f = 20 MPa (s was up to 4 times lower in this case than for Terzaghi s law, see Figures 9a and 9b). Thus, we were able to resolve the p c dependence of k. The p f dependence could not be determined with the same level of confidence although the p f range investigated was not overly small (between 10 and 30 MPa). The most probable reason for failure was that too few values of p f were used by Bernabé s studies (10, 20, and 30 MPa [Bernabé, 1986] and 10 and 30 MPa [Bernabé, 1987, 1988]). For the data sets covering only two fluid pressure values, 10 of 13

11 Figure 10. Correlation between the range of variation of k and the pressure sensitivity coefficient g for S3, S4, S8, S9, S10 (the corresponding symbols are indicated), and the five rocks studied by Bernabé [1986, 1987, 1988] (gray dots). the response-surface method yielded an unphysical p f dependence of k, as should be expected since the variations f cannot be measured in this case. From the above considerations, we conclude that the response-surface method is capable of delineating the p c and/or p f dependence of k, provided that sufficiently large p c and p f ranges are explored using as many p c and p f intermediate values as possible (we recommend at least five). [39] The present analysis confirmed the results reported by Bernabé [1986, 1987, 1988], i.e., the coefficient k of the effective pressure law for permeability varies significantly with p c and is also strongly sensitive to loading-unloading hysteresis and to loading history (i.e., k changed with the number of cycles applied and depended on the order in which p c and p f increments were applied). The complete picture is quite complicated and, since a detailed description is given by Bernabé [1986, 1987, 1988], we will only report here the results for the first p c loading cycle of each rock. One general observation is that, in all rocks, k decreased with increasing p c. The k and p c ranges, respectively, were ( ) and ( MPa) for Barre granite, ( ) and ( MPa) for Chelmsford granite, ( ) and ( MPa) for Pigeon Cove granite, ( ) and ( MPa) for Westerly granite, and ( ) and ( MPa) for Pottsville sandstone. The low end of these values is significantly higher than corresponding values for the E-bei sandstones. Pottsville sandstone, the rock most similar to the E-bei sandstones, does not have k values lower than 0.42 (i.e., much higher than its porosity, 7%). [40] The main question is: What causes the occurrence of unusually low k in the E-bei sandstones S4, S8 and S10? Unfortunately there is no evident answer to this question. We tentatively looked at the pressure sensitivity of permeability in these rocks. As reported in section 3.3, the pressure sensitivity coefficient g tends to take relatively high values (in the average, g is on the order of 0.1 MPa 1 for E-bei sandstones compared to 0.05 MPa 1 for other tight sandstones, see references in section 2.4). We also measured g for Bernabé s [1986, 1987, 1988] data and found 0.021, 0.020, 0.014, 0.036, and MPa 1 for Barre granite, Chelmsford granite, Pigeon Cove granite, Westerly granite and Pottsville sandstone, respectively. When plotted against each other for all these rocks and the E-bei sandstones, g and the domain of variation of k show a relatively good correlation (see Figure 10; note that there are outliers corresponding to the p f unloading cycles of S3 and S9, but the response-surface method failed to delineate the pressure dependence of k for these rocks). This correlation suggests that very low k values occur principally in rocks with a high-pressure sensitivity. [41] We also applied Gangi s [1978] and Walsh s [1981] models to Bernabé s [1986, 1987, 1988] data. In the average, the inferred values were the following, P 1 = 280, 330, 470, 160 and 370 MPa, m = 0.36, 0.36, 0.32, 0.43 and 0.37, and B = 0.41, , 0.55 and 0.36 for Barre granite, Chelmsford granite, Pigeon Cove granite, Westerly granite and Pottsville sandstone, respectively. Moreover, we observed hysteresis and loading history variations of P 1, m, and B, but no general trend could be identified. The values mentioned above are consistent with a lower pressure sensitivity of Bernabé s rocks than for the E-bei sandstones. As a possible explanation of the enhanced pressure sensitivity of E-bei sandstones, we speculate that the clays present in these rocks (and absent from Bernabé s rocks) may increase the roughness of microcrack surfaces on which they are attached (see Figure 1), without forming a compliant phase within the pore space that would lead to k >1.Coyner [1984] pointed out that individual clay particles are just as rigid as other silicate crystals. In order to behave as a compliant phase and produce k > 1, clay clumps within the pore space must be effectively impermeable, for example, by being saturated with organic matter preventing penetration by the pore fluid at nominal pore pressure. [42] Permeability is not the only rock property dependent on p c and p f. For example, porous rocks are known to be nonlinear elastic. One very important observation is that the bulk modulus K of the vast majority of rocks increases with increasing p c [e.g., Coyner, 1984; Zimmerman, 1991]. As already mentioned in section 1, this is often explained by the closing of low aspect ratio pores or cracks [e.g., Cheng and Toksoz, 1979; Zimmerman, 1991]. A consequence of the increase of K is that the coefficient a of effective pressure law for bulk volumetric deformation should decrease with p c. Theoretical analysis shows that, if the rock solid frame can be considered as made of a single homogeneous and isotropic constituent (this assumption is often called the Gassmann limit), we must have a =1 K/K S, where K S is the effective bulk modulus of rock solid frame [Nur and Byerlee, 1971; Robin, 1973; Zimmerman, 1991; Berryman, 1992a, 1992b]. Likewise, Bernabé s [1986] model suggests that the coefficient k should decrease with increasing pore aspect ratio and, therefore, with increasing confining pressure. However, Bernabé s [1986] model predicts a lower bound for k of 0.5 (i.e., for e = 1 and n = 0), a value much 11 of 13

12 larger than the lowest values observed for S4, S8, S10, and even Pottsville sandstone. Bernabé s [1986] model is a very simple conceptual model and is not meant to make accurate quantitative prediction but rather to provide a plausible explanation for the trend. Berryman [1992a, 1992b, 1993] derived a lower bound for k much more consistent with our experimental results. First, he noted that 1 a f (a consequence of the Reuss-Voigt bounds [e.g., Zimmerman, 1991]). Then, he argued that permeability can be modeled as k = HL 2, where H is a scale-invariant function of the pore geometry (i.e., invariant under an arbitrary uniform dilation of the pore space) and L is a characteristic fluid flow length scale. On the basis of the fact that, in the Gassmann limit, any scale-invariant rock property (including H) must obey Terzaghi s [1925] effective pressure law, p eff = p c p f, and that L scales as the bulk volume to the power 1/3, Berryman [1992a, 1992b, 1993] demonstrated that the coefficient of the effective pressure law for permeability must verify the following relation: 2f 1 a k ¼ 1 ð Þ 3nða fþþ2f ; ð6þ where n is the exponent of the power law, H / f n (it is customary to assume that scale-invariant rock properties are related to each other through power laws). It is easy to see that equation (6) implies that k has the same properties as a, i.e., k should decrease with increasing p c and verify 1 k f. Our experimental results are consistent with this prediction, except that the lower bound for k appears to be violated in S4, S8 and S10. We do not want to overemphasize this observation since it could be caused by experimental errors or by uncertainties of the responsesurface method (e.g., owing to insufficient pressure ranges or insufficient data sampling). Nevertheless, the possibility of a violation of Berryman s lower bound for k exists and it is interesting to examine the validity of Berryman s model assumptions. One important assumption is that the solid frame of the rock is made of a single, effectively homogeneous and isotropic constituent (i.e., the Gassmann limit; its validity is analyzed by Berryman s papers and an alternative model allowing k > 1 is proposed). Although we know that the E-bei sandstones are mineralogically heterogeneous, our results cannot be used to invalidate the Gassmann limit hypothesis conclusively. On the other hand, Bernabé [1988] simultaneously measured permeability k and formation factor F (i.e., the ratio of the rock electrical resistivity to that of the saturating fluid) in Chelmsford granite samples. Since F is a scale-invariant quantity, it should verify Terzaghi s effective pressure law. We reanalyzed Bernabé s F data using the response-surface method and found that the coefficient of the corresponding effective pressure law did not take values significantly lower than 1 (e.g., 0.8 and 1.1 during the first p c loading and unloading cycles, respectively). Therefore, we can conclude that Chelmsford granite is in reasonable agreement with the Gassmann limit. In fact, the Gassmann limit is considered a robust assumption in many rocks [Zimmerman, 1991] and it seems reasonable to extend this conclusion to the E-bei sandstones, although they are probably more heterogeneous than Chelmsford granite. Another assumption of Berryman s model is that the rock should be elastic (possibly nonlinear elastic [see Robin, 1973]). The strong hysteresis and loading history dependence observed by Bernabé [1986, 1987, 1988] suggests that the rocks investigated were not exactly elastic (i.e., reversible). Here, the seasoning cycles removed the loading history dependence for the most part but a significant hysteresis effect remained. Thus, the explanation of the ultralow values of k in E-bei perhaps lies in a combination of inelasticity and strong pressure sensitivity. 5. Conclusion [43] Our conclusions are as follows: [44] 1. The response-surface method is capable of identifying and delineating the dependence of k on p c and p f, provided the pressure ranges investigated are sufficiently extended and sufficient pressure increments are used. It is also important to pay close attention to the p c and p f cycling procedures in order to accurately assess hysteresis and loading history effects. [45] 2. In E-bei sandstones, the coefficient k is strongly sensitive to on p c and p f. It generally tends to decrease with increasing p c, eventually reaching ultralow values possibly lower than the theoretical lower bound f. Significant hysteresis effects were also observed, indicating that these rocks are not perfectly elastic. [46] 3. The most likely explanation for this behavior is a combination of inelasticity (as evidenced by the hysteresis effects) and strong pressure sensitivity (as suggested by a qualitative correlation of ultralow k and high pressure sensitivity coefficient g). [47] Acknowledgments. We would like to express our appreciation to J. G. Berryman and Han Zhao Hui for many informative discussions. This work was supported by a grant from the National High Technology Research and Development Program of China (Program 863, 2007AA06Z209) and by the Open Fund PLN0802 of the State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University). Y.B. s work was partially funded by the U.S. Department of Energy under grant DE-FG01-97ER References Al-Wardy, W., and R. W. Zimmerman (2004), Effective stress law for the permeability of clay-rich sandstones, J. Geophys. Res., 109, B04203, doi: /2003jb Beijing Chemical Industrial Company Inc. (1979), Cryogenic Handbook, pp , Beijing. Bernabé, Y. (1986), The effective pressure law for permeability in Chelmsford granite and Barre granite, Int. J. Mech. Min. Sci. Geomech. Abstr., 23(3), , doi: / (86) Bernabé, Y. (1987), The effective pressure law for permeability during pore pressure and confining pressure cycling of several crystalline rocks, J. Geophys. Res., 92, , doi: /jb092ib01p Bernabé, Y. (1988), Comparison of the effective pressure law for permeability and resistivity formation factor in Chelmsford granite, Pure Appl. Geophys., 127, , doi: /bf Berryman, J. G. (1992a), Effective stress for transport properties of inhomogeneous porous rock, J. Geophys. Res., 97, 17,409 17,424, doi: / 92JB Berryman, J. G. (1992b), Exact effective-stress rules in rock mechanics, Phys. Rev. A, 46(6), , doi: /physreva Berryman, J. G. (1993), Effective-stress rules for pore-fluid transport in rocks containing two minerals, Int. J. Mech. Min. Sci. Geomech. Abstr., 30, , doi: / (93)90087-t. Box, G. P., and N. R. Draper (1987), Empirical Model-Building and Response Surfaces, John Wiley, New York. Cheng, C. H., and N. Toksoz (1979), Inversion of seismic velocities for the pore aspect ratio spectrum of a rock, J. Geophys. Res., 84, , doi: /ja084ia05p of 13