Type-curve estimation of statistical heterogeneity

Transcription

1 WATER RESOURCES RESEARCH, VOL. 40,, doi: /2003wr002405, 2004 Type-curve estimation of statistical heterogeneity Shlomo P. Neuman Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona, USA Alberto Guadagnini and Monica Riva Dipartimento di Ingegneria Idraulica Ambientale e del Rilevamento, Politecnico di Milano, Milan, Italy Received 18 June 2003; revised 31 January 2004; accepted 16 February 2004; published 3 April [1] The analysis of pumping tests has traditionally relied on analytical solutions of groundwater flow equations in relatively simple domains, consisting of one or at most a few units having uniform hydraulic properties. Recently, attention has been shifting toward methods and solutions that would allow one to characterize subsurface heterogeneities in greater detail. On one hand, geostatistical inverse methods are being used to assess the spatial variability of parameters, such as permeability and porosity, on the basis of multiple cross-hole pressure interference tests. On the other hand, analytical solutions are being developed to describe the mean and variance (first and second statistical moments) of flow to a well in a randomly heterogeneous medium. We explore numerically the feasibility of using a simple graphical approach (without numerical inversion) to estimate the geometric mean, integral scale, and variance of local log transmissivity on the basis of quasi steady state head data when a randomly heterogeneous confined aquifer is pumped at a constant rate. By local log transmissivity we mean a function varying randomly over horizontal distances that are small in comparison with a characteristic spacing between pumping and observation wells during a test. Experimental evidence and hydrogeologic scaling theory suggest that such a function would tend to exhibit an integral scale well below the maximum well spacing. This is in contrast to equivalent transmissivities derived from pumping tests by treating the aquifer as being locally uniform (on the scale of each test), which tend to exhibit regional-scale spatial correlations. We show that whereas the mean and integral scale of local log transmissivity can be estimated reasonably well based on theoretical ensemble mean variations of head and drawdown with radial distance from a pumping well, estimating the log transmissivity variance is more difficult. We obtain reasonable estimates of the latter based on theoretical variation of the standard deviation of circumferentially averaged drawdown about its mean. INDEX TERMS: 1829 Hydrology: Groundwater hydrology; 1869 Hydrology: Stochastic processes; 1894 Hydrology: Instruments and techniques; KEYWORDS: aquifer, heterogeneity, pumping test, spatial variability, statistics, stochastic Citation: Neuman, S. P., A. Guadagnini, and M. Riva (2004), Type-curve estimation of statistical heterogeneity, Water Resour. Res., 40,, doi: /2003wr Introduction [2] Well-testing methods have traditionally relied on analytical solutions of groundwater flow equations in relatively simple domains, consisting of one or at most a few units having uniform hydraulic properties. Among the latter is a transient multiaquifer solution developed by Neuman and Witherspoon [1969], part of which has recently been incorporated into the popular well-testing software package AQTESOLV (available at A method to evaluate aquifer characteristics in the presence of a radial discontinuity around a pumping well was described by Sternberg [1969]. Chu and Grader [1991, 1999] developed a generalized analytical solution for transient pressure interference tests in composite aquifers. Their solution allows the aquifer to contain up to three uniform, isotropic Copyright 2004 by the American Geophysical Union /04/2003WR regions of finite or infinite extent having varied geometries. Active and observation wells may be placed at diverse locations within the composite system. Constant flow rate, pressure, or slug test conditions may be prescribed at active wells having zero or finite radius, the latter including storage and skin. Boundary skins between regions can be used to simulate faults or boundaries between fluid banks. [3] A recent development has been the use of geostatistical inversion to assess the spatial variability of parameters, such as permeability and porosity, on the basis of multiple cross-hole pressure interference tests. The approach yields detailed tomographic estimates of how parameters vary in three-dimensional space, as well as measures of corresponding estimation uncertainty. The approach, originally proposed by Neuman [1987], has been used to interpret cross-hole pneumatic interference tests in unsaturated fractured tuff by Vesselinov et al. [2001a, 2001b]. 1of7

2 NEUMAN ET AL.: ESTIMATION OF STATISTICAL HETEROGENEITY [4] Geostatistical inversion is computationally intensive and requires considerable sophistication. It has been pointed out by Yortsos [2000] that in principle, one should be able to estimate the variogram parameters of a heterogeneous aquifer from the analysis of pressure transients in multiple wells. Copty and Findikakis [2004] used two-dimensional numerical Monte Carlo simulations to investigate the manner in which transient drawdowns due to pumping at a constant rate from a randomly heterogeneous, statistically homogeneous confined aquifer differ from those predicted by the Theis [1935] equation. On the basis of these results, Copty and Findikakis [2003] proposed estimating the mean of log transmissivity from late drawdown data using the Cooper and Jacob [1946] semilogarithmic straight line approach, the integral scale from the time needed for drawdown to approach that of a corresponding uniform aquifer, and the variance through a numerical least squares fit of early drawdown data with ensemble mean drawdown. The authors found their approach to yield reasonable estimates of geometric mean log transmissivity, acceptable estimates of integral scale, and poor estimates of variance. [5] In this paper we explore numerically the feasibility of using a simple graphical approach (without numerical inversion) to estimate the geometric mean, integral scale, and variance of local log transmissivity on the basis of quasi steady state head data when a randomly heterogeneous confined aquifer is pumped at a constant rate. By local log transmissivity we mean a function varying randomly over horizontal distances that are small in comparison with a characteristic spacing between pumping and observation wells during a test. Experimental evidence and hydrogeologic scaling theory suggest that such a function would tend to exhibit an integral scale well below the maximum well spacing [e.g., Neuman and Di Federico, 2003, section 3]. This is in contrast to equivalent transmissivities derived from pumping tests by treating the aquifer as being locally uniform (on the scale of each test), which tend to exhibit regional-scale spatial correlations [e.g., Anderson, 1997, Table 1; Neuman and Di Federico, 2003, Figure 17]. We show that whereas the mean and integral scale of local log transmissivity can be estimated based on theoretical ensemble mean variations of head and drawdown with radial distance from a pumping well, estimating the log transmissivity variance is more difficult. We obtain reasonable estimates of the latter based on theoretical variation of the standard deviation of circumferentially averaged drawdown about its mean. 2. First and Second Moment Type-Curves [6] We consider flow in a confined aquifer of uniform thickness due to a well of relatively small radius that fully penetrates the aquifer and discharges at a constant rate. If the lateral extent of the aquifer is infinite, a steady state flow regime never develops. It is well known, however, that if the aquifer is additionally uniform, a quasi steady state region extends from the well out to a cylindrical surface whose radius expands as the square root of time. On the expanding surface, head is uniform and time-invariant. Inside this surface, head at any time is described by a steady state solution. A rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three-dimensional transient stochastic flow problem in 2of7 an aquifer of infinite lateral extent. In this feasibility study we adopt a simpler approach by considering two-dimensional mean steady state flow in the presence of a circular prescribed head boundary at radial distance R from the well. In analogy to the uniform case, we expect our solution to approximate a mean quasi steady state region whose radius R evolves with time. [7] We treat (natural) log transmissivity Y = ln T as a multivariate Gaussian, statistically homogeneous, and isotropic random field having constant ensemble mean hyi, variance s 2, and integral (autocorrelation) scale l. A fully penetrating well of radius r w l pumps at a constant rate Q, and constant head H R is maintained at radial distance R from the axis of the pumping well. Under mean quasi steady state flow, the mean hydraulic gradient is independent of time so that Q ¼ 2rT a ðþ r dhr h ðþi ; ð1þ dr where r is radial distance from the axis of the well, T a (r) is apparent transmissivity (function of r), h is hydraulic head, and angle brackets denote ensemble mean. Monte Carlo simulations conducted by Neuman and Orr [1993, Figure 13] indicate that when 2l r R (a situation arising at sufficiently large time following the onset of pumping, considering that R expands with time), T a (r) =T G for Y with an exponential covariance and s 2 at least as large as 4, where T G = e hyi is the geometric mean of T. Monte Carlo simulations conducted by Riva et al. [2001] as well as an analytical solution developed by them indicate (their Figure 13) that the same is true when Y has a Gaussian covariance. Whereas both sets of Monte Carlo results correspond to a multivariate Gaussian Y, the analytical solution of Riva et al. is distribution free. Setting T a (r) =T G in equation (1) and integrating from 2l to r yields Dhhr ðþi hhr ðþi hhð2þi ¼ Q ln r 2T G 2 ¼ 2:303Q r log 2T 10 G 2 : [8] Figure 14 of Neuman and Orr [1993] and Figure 14 of Riva et al. [2001] show that for any R/l, T a (r)! T H as r! r w where T H = T G e s2 /2 is the harmonic mean of T (provided the latter is multivariate Gaussian, as we assume here). The results of Neuman and Orr correspond to a finite r w, whereas those of Riva et al. correspond to r w = 0. The same was shown theoretically by Dagan [1989, equation 5.4.3]. The two figures suggest further that for r w r 2l, T a (r)/t G behaves approximately as a cubic polynomial having zero derivatives at r w and 2l. We approximate T a (r) in this range via T a ðþ¼j r H ðþt H þ j G ðþt G ; where a = r/2l, j H (a) =1 3a 2 +2a 3, and j G (a) = 3a 2 2a 3. Since e s2 /2 = T H /T G, this yields ð2þ ð3þ hhð2þi hhr ðþi ¼ Q F ; 2 ; ð4þ 2T G

3 NEUMAN ET AL.: ESTIMATION OF STATISTICAL HETEROGENEITY Figure 1. Dimensionless ensemble mean drawdown W (or, equivalently, its circumferential average) versus log dimensionless radial distance a (solid curves) for various s 2 (0.1, 0.5, 1.0, 1.5, 2.0). Dashed curves represent ±2 standard deviations of corresponding circumferentially averaged dimensionless drawdown about the mean. where Z F ; 2 1 ¼ d ½e s2 =2 ð1 3 2 þ 2 3 Þþð ÞŠ : ð5þ [9] Combining equations (2) and (4) yields where hhðþi ¼ hhð1þiþ Q W ; 2 ; ð6þ 2T G W ; 2 ¼ Hð 1Þ2:303 log10 Hð1 ÞF ; 2 ð7þ and H is the Heaviside function. A semilogarithmic plot of the dimensionless drawdown W versus the dimensionless distance a is shown by solid curves in Figure 1 for various values of the variance s 2. These type curves compare very well with those obtained from 2000 Monte Carlo simulations. We have conducted such simulations by generating random realizations of log transmissivity with the Gaussian sequential simulator SGSIM [Deutsch and Journel, 1998] and solving the corresponding flow problem using Galerkin finite elements with bilinear shape functions on a rectangular grid of nodes, spaced 0.2 arbitrary length units apart. The log transmissivity fields were characterized by Gaussian variograms (as in the analytical solution of Riva et al. [2001]) with unit integral scale. In each simulation, water was withdrawn numerically from the center node at a rate of Q = 100 arbitrary consistent units while maintaining constant head at a near-circular boundary a distance R = = 10 from the center node. [10] Note that we define drawdown as Dh(r, Dq) h(r, q 1 ) h(2l, q 2 ) rather than as h(r, q) H R, where q is angular position and Dq = q 1 q 2. Our unconventional definition of drawdown is motivated by the fact that T a (r)/t G behaves in a fundamentally different manner at 2l r R than at r w r 2l. Whereas our definition of drawdown allows us to estimate l and s 2 without knowledge of R based on a single set of type-curves (as described below), the more conventional definition h(r, q) H R would require that R be known and a different set of standard deviation type-curves be developed for each R/l. Since R evolves with time, it is generally unknown and would have to be estimated jointly with l and s 2. This, and the need to develop different type-curves for different values of R/l, would render the analysis impractical. [11] One difficulty arising from our definition of drawdown is that the variance Var(Dh) =s h 2 (r 1 )+s h 2 (r 2 ) 2hh 0 (r 1, q 1 )h 0 (r 2, q 2 )i of a head increment Dh(r 1, q 1, r 2, q 2 ) h(r 1, q 1 ) h(r 2, q 2 ), where s h 2 (r) is the variance of h at r, depends not only on the radii r 1 and r 2 but also on the angular lag Dq = q 1 q 2. (The manner in which hydraulic head covariance C h (r 1, r 2, Dq) =hh 0 (r 1, q 1 )h 0 (r 2, q 2 )i varies with r and Dq is illustrated in Figures 4a and 4b, respectively, of Riva et al. [2001].) Attempts to interpret synthetically generated random drawdown data (corresponding to a single realization of a random log transmissivity field) using type-curves that reflect this angular dependence have caused us to seriously underestimate s 2 in cases where its actual (generated) value exceeded 0.5: The spread of the data about their theoretical mean tended to be much smaller than suggested by ensemble theory. As this was true for several sizeable and mutually independent data sets (similar to those described below), we do not think that the phenomenon is due solely to a sampling error. Instead, we think that it stems in part from lack of ergodicity, which causes the variance of head at a given r (sampled at various angles q)in a single realization to be generally smaller than the ensemble head variance at that same r. This is due to relatively strong angular correlation between heads at a given r, especially at smaller r values, as indicated by Figure 4b of Riva et al. [2001]. The same is suggested by the tendency of head contours in a single realization to be quasi-concentric around the pumping well [Neuman and Orr, 1993, Figures 4 and 5]. [12] To account for this, we filter out the angular dependence by considering mean theoretical increments Dh(r 1, r 2 ) h(r 1 ) h(r 2 ) where hr ðþ¼ 1 2 Z 2 0 hr; ð Þ d ð8þ is average head over the circumference of a circle with center at the pumping well. Since hh(r)i is not a function of q, hh(r)i hh(r)i, explaining why the type curves of dimensionless mean drawdown (relative to mean head at r = 2l or a = 1) in Figure 1 are labeled hdhi/(q/2pt G ) rather than hdhi/(q/2pt G ). The variance of Dh(r 1, r 2 ), D E Var Dh ¼ 2 ð Þþ 2 ð r h 2Þ 2 h 0 ðr 1 Þh 0 ðr 2 Þ h r 1 where s h 2 (r) is the variance of h(r) and h 0 (r) =h(r) hh(r)i, is independent of Dq and vanishes at r = r 1 = r 2. This is reflected in the dashed envelopes in Figure 1, which represent ±2 standard deviations of circumferen- ð9þ 3of7

4 NEUMAN ET AL.: ESTIMATION OF STATISTICAL HETEROGENEITY tially averaged dimensionless drawdown about the mean and vanish at a =1(r =2l). The envelopes are based on Monte Carlo simulations. Kolmogorof and chi-square tests of the null hypothesis that heads generated by the Monte Carlo method are Gaussian were negative at a significance level of 5%. Hence the dashed envelopes in Figure 1 are not strictly proportional to 95% confidence intervals of actual (random) dimensionless (circumferentially averaged) drawdown values. 3. Graphical Method of Test Interpretation [13] The approach we propose applies to situations in which R l, where R is the external radius of a circular mean quasi steady state flow domain. In an aquifer of large lateral extent, R expands with time to eventually satisfy the above requirement, and we limit our analysis to correspondingly late data. Let h i = h(r i, q i )be quasi steady state head values recorded in fully penetrating observation wells at discrete radial and angular locations (r i, q i ). We recall that based on experimental evidence and hydrogeologic scaling theory [e.g., Neuman and Di Federico, 2003, section 3], one may expect the integral scale l of the local log transmissivity field to be considerably smaller than the maximum distance between pumping and observation wells in a test. Hence one may generally expect the radial locations r i to span a range that includes 2l. This should allow one to analyze the head data by (1) plotting h i versus r i on semilogarithmic paper; (2) fitting a straight line to h i data corresponding to the highest range of r i values; (3) determining T G from the slope, 2.303Q/(2pT G ), of the straight line; (4) obtaining an estimate of 2l by equating it to the radial distance at which the experimental points begin to deviate from a straight line; (5) plotting dimensionless head, ~ hi = 2pT G h i /(2.303Q), versus r i on semilogarithmic paper; (6) superimposing the plot of ~ h i versus r i on the semilogarithmic type curves in Figure 1 (note that head differs from drawdown by a constant, equal to drawdown at r = 2l; regardless of whether this constant is known or unknown, such superposition is always possible); (7) verifying that l = r/(2a) based on the corresponding values of r and a; (8) if needed, modifying the estimate of l and the match so as to obtain consistent results in steps 4 and 7; and (9) reading or interpolating the variance s 2 corresponding to the uncertainty envelope (dashed curves) within which about 95% of the data lie, excluding primarily (about 5% of ) data in the vicinity of a = 1 (r = 2l) where these envelops cross the mean typecurves. [14] An alternative approach pursuant to steps 1 4, which we adopt below, consists of (5) obtaining an estimate of hh(2l)i by setting it equal to the head value at which the straight line from step 2 intercepts 2l; (6) using the estimates of T G and l to compute dimensionless drawdowns 2pT G [h i (r i, q i ) hh(2l)i]/q; (7) plotting the latter versus r i /(2l) on semilogarithmic paper; (8) superimposing this plot on the semilogarithmic type curves in Figure 1; (9) verifying that l = r/(2a) based on the corresponding values of r and a; (10) if needed, modifying the estimate of l and the match so as to obtain consistent results in steps 4 and 9; and (11) reading or interpolating the variance s 2 corresponding to the uncertainty 4of7 Figure 2. Positions of wells serving for pumping and observation (1 10) and wells serving solely for observation (11 16). envelope (dashed curves) within which about 95% of the data lie, excluding primarily (about 5% of ) data in the vicinity of a = 1 (r = 2l) where these envelops cross the mean type-curves. [15] Since any given aquifer corresponds to only one random realization of a theoretically infinite ensemble of statistically equivalent but otherwise dissimilar aquifers, one must not expect real data to match an ensemble mean type curve (solid in Figure 1). Instead, one should expect approximately 95% of data to lie within ±2 standard deviations of dimensionless drawdown about the mean. As (1) heads are generally non-gaussian, (2) the dashed envelopes in Figure 1 represent ±2 standard deviations of circumferentially averaged dimensionless drawdown (as opposed to actual dimensionless drawdown), and (3) it is difficult to average sparse head data circumferentially, one should expect some head data to lie outside these envelopes, most notably near a = 1 (r = 2l). Hence there is no avoiding some ambiguity in fitting data to our type-curves for the purpose of estimating s 2. [16] The larger and most widely spread in space is the available data set, the higher is the statistical significance of parameter estimates obtained using the above procedure. This suggests that if there are N wells at a site, one try to analyze jointly data from N pumping tests conducted by pumping one well at a time, while observing drawdown in the remaining N 1 wells. A similar idea was mentioned by Copty and Findikakis [2003]; we illustrate it here by a synthetic example. 4. Feasibility Assessment Using Synthetic Data [17] To explore the feasibility of our proposed approach, we test it against head data corresponding to a series of synthetic pumping tests in aquifers exhibiting various degrees of random heterogeneity. Each such aquifer is associated with a random log transmissivity, characterized by a Gaussian variogram, generated using SGSIM [Deutsch and Journel, 1998] on a rectangular grid of nodes spaced 0.2 m apart. Near the center of the grid, we introduce 10 wells used alternately for pumping and observation and six wells used solely for

5 NEUMAN ET AL.: ESTIMATION OF STATISTICAL HETEROGENEITY Table 1. Parameter Estimates Using Data From Individual Pumping Tests and From All Pumping Tests Jointly in a Random Aquifer Realization Drawn From an Ensemble With T G =1m 2 /d, s 2 = 0.5, l =2m Pumping Well T G,m 2 /d l, m s 2 Figure 3. Head versus log radial distance observed while pumping wells 1 10 in random aquifer realization drawn from ensemble with T G =1m 2 /d, s 2 = 0.5, l =2m Average of 10 tests Ten tests analyzed jointly Sample values Ensemble values observation (Figure 2). The well locations were designed to yield numerous head observations sufficiently close to pumping wells so as to allow a good definition of log transmissivity variance and integral scale, and several head observations far enough from pumping wells so as to allow a good definition of geometric mean transmissivity. Initially, head in the aquifer is 100 m, and this value is maintained continuously at a near-circular boundary a distance R = m = 20 m from the center node. Wells 1 10 are pumped in sequence at a rate of 1000 m 3 /d while steady state drawdowns are recorded in all other wells. The drawdowns are computed using the same numerical procedure we described earlier. [18] We start by analyzing observed drawdowns in a randomly heterogeneous log transmissivity field generated by specifying a theoretical geometric mean transmissivity T G =1m 2 /d, integral scale l = 2 m and log transmissivity variance s 2 = 0.5. The generated field has a sample geometric mean transmissivity T G = 1.15 m 2 /d and log transmissivity variance s 2 = 0.46; sample correlation scales l x = 1.81 m and l y = 1.85 m in the x and y directions, respectively; and sample omnidirectional correlation scale l = 1.80 m. Figure 3 is a semilogarithmic plot of head variations with radial distance between observation wells and wells 1 10 when the latter are pumped in sequence. In accord with step 2 of our procedure, a straight line was fitted by regression to head data corresponding to relatively large radial distances from pumping wells. Figure 4 depicts a semilogarithmic plot of dimensionless drawdown versus dimensionless distance, observed while pumping wells 1 10, superimposed on the type curves in Figure 1. Excluding 10 drawdown data from pumping wells (whose computation Figure 4. Dimensionless drawdown versus log dimensionless distance, observed while pumping wells 1 10 in random aquifer realization drawn from ensemble with T G = 1m 2 /d, s 2 = 0.5, l = 2 m, superimposed on best fit (bold, corresponding to s 2 = 0.5, with emphasis on dashed) typecurves in Figure 1. Figure 5. Dimensionless drawdown versus log dimensionless distance, observed while pumping wells 1 10 in random aquifer realization drawn from ensemble with T G = 1m 2 /d, s 2 = 2.0, l = 2 m, superimposed on best fit (bold, corresponding to s 2 = 2.0, with emphasis on dashed) typecurves in Figure 1. 5of7

6 NEUMAN ET AL.: ESTIMATION OF STATISTICAL HETEROGENEITY Table 2. Parameter Estimates Using Data From Individual Pumping Tests and From All Pumping Tests Jointly in a Random Aquifer Realization Drawn From an Ensemble With T G =1m 2 /d, s 2 = 2.0, l =2m Pumping Well T G,m 2 /d l, m s Average of 10 tests Ten tests analyzed jointly Sample values Ensemble values we deem unreliable) yields a total of = 150 data points. Excluding about 5% of the most outlying among these 150 data points in the vicinity of a = 1 leaves 142 data to fit within one of the dashed envelops in Figure 4, suggesting a variance estimate of s 2 = 0.5. [19] Our parameter estimates are compared with sample and ensemble values in Table 1. Analyzing the data from all 10 pumping tests jointly, as we have done in Figures 3 and 4, yields reasonable estimates of T G and s 2 but an elevated estimate of l. Whereas estimates obtained from individual pumping tests vary, their average values are close to their real and ensemble counterparts. [20] Next we analyze drawdowns in a randomly heterogeneous log transmissivity field generated by specifying a theoretical geometric mean transmissivity T G = 1 m 2 /d, integral scale l = 2 m and log transmissivity variance s 2 = 2. The generated field has a sample geometric mean transmissivity T G =0.93m 2 /d; sample log transmissivity variance s 2 = 1.89; sample correlation scales l x =1.88m and l y = 2.10 m in the x and y directions, respectively; and sample omnidirectional correlation scale l = 2.01 m. Figure 5 depicts dimensionless drawdown versus dimensionless distance, observed while pumping wells 1 10, superimposed on the Monte Carlo type curves in Figure 1. Like before, we associate the data with those dashed type curves that contain about 95% of them while ignoring the remaining 5% of outlying data near a = 1, suggesting a variance estimate of s 2 =2. [21] Corresponding parameter estimates are compared with sample and ensemble values in Table 2. Whereas T G is overestimated in all cases, the estimates of l and s 2 are again surprisingly close to their actual and ensemble values. 5. Conclusions [22] Our analysis leads to the following major conclusions: [23] 1. It appears feasible to use a simple graphical approach (without numerical inversion) to estimate the geometric mean, integral scale, and variance of local log transmissivity on the basis of quasi steady state head data when a randomly heterogeneous confined aquifer is pumped at a constant rate. By local log transmissivity we mean a function varying randomly over horizontal distances that are small in comparison with a characteristic spacing between pumping and observation wells during a test, which may therefore exhibit integral scales that do not exceed the maximum well spacing. This is in contrast to equivalent transmissivities derived from pumping tests by treating the aquifer as being locally uniform (on the scale of each test), which tend to exhibit regionalscale spatial correlations. [24] 2. Whereas the mean and integral scale of local log transmissivity can be estimated based on ensemble mean variations of head and drawdown with radial distance from a pumping well, estimating the log transmissivity variance requires considering the manner in which the circumferentially averaged ensemble standard deviation of drawdown (about its mean) varies radially. [25] 3. Given N wells at a site, reasonable parameter estimates are obtained by analyzing jointly data from N tests conducted by pumping one well at a time, while observing drawdown in the remaining N 1 wells. [26] Acknowledgments. This work was supported in part by The Institute of Geophysics and Planetary Physics (IGPP) at Los Alamos National Laboratory under contract with the University of Arizona. References Anderson, M. P. (1997), Characterization of geological heterogeneity, in Subsurface Flow and Transport: A Stochastic Approach, edited by G. Dagan and S. P. Neuman, pp , Cambridge Univ. Press, New York. Chu, L., and A. S. Grader (1991), Transient pressure analysis of three wells in a three-composite reservoir, paper presented at the SPE Annual Conference and Exhibition, Soc. of Pet. Eng., Dallas, Tex., 5 9 October. Chu, L., and A. S. Grader (1999), Transient pressure and rate analysis for active and interference wells in composite systems, In Situ, 23(4), Cooper, H. H., Jr., and C. E. 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7 NEUMAN ET AL.: ESTIMATION OF STATISTICAL HETEROGENEITY Theis, C. V. (1935), The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage, Eos Trans. AGU, 16, Vesselinov, V. V., S. P. Neuman, and W. A. Illman (2001a), Three-dimensional numerical inversion of pneumatic cross-hole tests in unsaturated fractured tuff: 1. Methodology and borehole effects, Water Resour. Res., 37(12), Vesselinov, V. V., S. P. Neuman, and W. A. Illman (2001b), Three-dimensional numerical inversion of pneumatic cross-hole tests in unsaturated fractured tuff: 2. Equivalent parameters, high-resolution stochastic imaging, and scale effects, Water Resour. Res., 37(12), Yortsos, Y. C. (2000), Permeability variogram from pressure transients of multiple wells, in Theory, Modeling, and Field Investigation in Hydrogeology: A Special Volume in Honor of Shlomo P. Neuman s 60th Birthday, edited by D. Zhang and C. L. Winter, Spec. Pap. Geol. Soc. Am., 348, A. Guadagnini and M. Riva, Dipartimento di Ingegneria Idraulica Ambientale e del Rilevamento, Politecnico di Milano, Piazza L. Da Vinci 32, I Milan, Italy. S. P. Neuman, Department of Hydrology and Water Resources, University of Arizona, Tucson, AZ 85721, USA. (neuman@hwr.arizona. edu) 7of7