= _(2,r)af OG(x, a) 0p(a, y da (2) Aj(r) = (2*r)a (Oh(x)y,(y)) A (r) = -(2,r) a Ox Oxj G(x, a)p(a, y) da

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1 WATER RESOURCES RESEARCH, VOL. 35, NO. 7, PAGES , JULY 999 A general method for obtaining analytical expressions for the first-order velocity covariance in heterogeneous porous media Kuo-Chin Hsu Daniel B. Stephens and Associates, Inc., Albuquerque, New Mexico Abstract. Velocity covariance of seepage flow is the essential quantity in understanding the uncertainty of groundwater flow and contaminant transport in heterogeneous porous media. Nevertheless, a closed-form analytical expression is available only for first-order analysis. These available expressions are limited to some specific correlation functions of natural log hydraulic conductivity. The present paper develops a general method for obtaining the first-order velocity covariance for any statistically isotropic log-hydraulic conductivity correlation function. The general analytical expressions are derived for both two and three dimensions. The results are demonstrated for several frequently used correlation functions.. Introduction Velocity covariance of groundwater is important in understanding the uncertainty of flow and transport in heterogeneous porous media. For example, calculating the dispersion tensor requires knowing the velocity covariance [Dagan, 989]. The first-order (in the variance 0 '2 of the naturalog hydraulic conductivity Y) velocity covariance for the exponential correlation Y has been derived in two dimensions by Rubin [990] and Hsu et al. [996] with seemingly different but actually equivalent expressions. For the same correlation function, Rubin and Dagan [992] and Zhang and Neuman [992] have derived the corresponding three-dimensional expressions. As an example of a different correlation function for Y, Hsu and Neuman [997] used a Gaussian model in an attempt to derive the first-order velocity covariance. Hsu et al. [996] and Hsu and Neuman [997] investigated higher-order effects on velocity covariance by using a semianalytical method, while Deng and Cushman [995, 998] used a numerical method to obtain the velocity covariance in the Fourier domain. The closed-form expression of velocity covariance is only available for firstorder analysis. Commonly, in previous research, the analytical solutions for velocity covariance require a specific correlation function for Y. The present study utilizes a Green's function approach to derive the general forms of the first-order velocity covariance. Expressions for both two and three dimensions are derived and are applicable to any statistically isotropic correlation function for Y. 2. First-Order Velocity Covariance Theoretical derivations of velocity covariance usually assume () infinite domain, (2) uniform mean flow, (3) steady state flow, and (4) natural log hydraulic conductivity Y as a stationary homogeneous, multivariate Gaussian random field with constant ensemble mean (Y) and variance 0 '2. With these assumptions, together with the assumption that mean flow takes place parallel to the x direction, we begin with the general expression for the first-order velocity covariance given in (A5) of Hsu et al. [996], i.e., Copyright 999 by the American Geophysical Union. Paper number 999WR /99/999 WR9007 $ /3/v4 (r) + B; (r)] () where ui, is the first-order groundwater velocity covariance, Kg is the geometric mean of hydrauliconductivity K, J is the uniform mean driving force (negative hydraulic gradient), qb is the effective porosity (treated as a constant), /3 is the Kronecker delta, d is the number of space dimensions, and r = x - y is the separation vector of points x and y. Functions.4 and B may be related to the normalized log transmissivity-head covariance and head variogram by comparing () with equation (6) of Rubin [990]. Functions.4 and B can be expressed in terms of Green's function G, via equations (All) and (A2) of Hsu et al. [996] as Aj(r) = (2*r)a (Oh(x)y,(y)) Jo '2 Oxj = _(2,r)af OG(x, a) 0p(a, y da (2) Oxj Oa, Bj (r) = (2 r)d J2o'2 (Oh(x)Oh(y)) OXj Oyk =(2)dff OG(x, Oxj a)og(y, Oyk a')02p(a, Oa Oa dada' (3) Here h is the first-order (in o-) hydraulic head fluctuation, Y' is the random fluctuation of log hydraulic conductivity, p is the log conductivity correlation function, and angle brackets represent the ensemble mean (expected value). Applying integration by parts, the integrated terms vanish as a, a' - +_% and (2) and (3) can thus be written as A (r) = -(2,r) a Ox Oxj G(x, a)p(a, y) da O2(I) (r) = -(2,r) a -- (4) OX 0Xj

2 2274 HSU: TECHNICAL NOTE Table. Expressions for Function Wi Function W Wlo W Expression p(r) - 2T(r)/r p(r)/8 + T(r)/2r- 3U(r)/2 p(r) + T(r)/r p(r) - 3T(r)/r 3p(r) + 3T(r)/2r- 9U(r)/2 p(r) - 9T(r)/2r + 5U(r)/2 p(r) + 5T(r)/2r- 35U(r)/2 p(r) - 3T(r)/2r- 5U(r)/2 p(r) - 5T(r)/2r + 9U(r)/2-3T(r)/2r + 5U(r)/2 p(r) + 3T(r)/2r- 5U(r)/2 For W and W2, T(r) - (l/r) f[ r'p(r') dr' (from (0)), and U(r) = (/r 4) fro r'3p(r') dr'. For W3-W o, T(r) = (/r 2) fro r'2p(r') dr' (from(22)), and e(r) - (/r s) f[ r'4p(r ') dr'. Bjk(r) = (2,r) a 0 4 OxiOylOxjOyk ß ff G(x, a)g(a', y)p(a, a') da da' Expanding the right-hand side of (2) and collecting terms with the same angular dependence, we find that (2) becomes (2,r) 2 A,(r) = 2 [o(r) + W,(r) cos 20] (3) where function W is shown in Table. Similarly, A 2 can be obtained in the form A2(r) = -(2 r) 2 cos ---sin 0 0 0) 0 r r - ß sin0 - + cos0 0 0) (r) (2,tr) 2 2 Wa(r) sin 20 (4) To determine the function B, we begin by writing (9) in polar coordinates and differentiating to obtain T(r) =- d = dr 7 - r - - d d2xi t d3xi t - r d dr 3 (5) where - (2*r)a Oxi Oyl 9xj Oy - G(a', y) (x, a') da' 04xit (r) = (2*r)a 02X0Xj OX k (5) ß (r) = I G(x, a)p(a, y) da (6) ß (r) = f G(a', y)ci>(x, a') da' The functions ß and ß satisfy V2 (r) = --p(r) V2 (r) = - (r) Since p depends merely on distance r = Ir], only the radial component of the Laplacian is required. For two dimensions a first integration of (8) then yields d (r) df F 0 t' (7) (8) (9) r'p(r') dr' = -T(r) (0) and the function T can be related to p through dt T d-7+¾ = () Introducing polar coordinates r - r (cos O, sin O) r and specializing to the case j =, we have 02 (r) Al(r) = --(2'n') 2 -- OXl sin 0 ci) 0 0) 2 = --(2'n') 2 COS 0 r (r) (2) Specializing to the case j = k =, after manipulation by using () and (5), we find that (5) becomes 04xXt (r) B (r) = (2,r) 2-- Oxl ( 0 r 0 r 0) 4 = (2 r) 2 COS sin 0 (r) (2 r)2i 3 p(r) + W (r) 2 cos 20 + W2(r ) cos ] 4 0 (6) where functions W and W2 are defined in Table. When this procedure is applied successively to the cases (j, k) = (, 2), (2, ), and (2, 2), we find B 2(r) = B2(r) = (2 r)2[ W (r) 4 sin20 + W2(r) sin40] (7) B22(r) = (2'rr)28-p(r)- W2(r) cos 40] (8) Substituting (3), (4), (6), (7), and (8) into (), we obtain the general two-dimensional expressions of first-order velocity covariance for any stationary isotropic function p in the form Ull(r ) = Ky2o-2 2 [3. p(r) - Wi(r) cos 20 + W2(r) cos 40] (9) KY 20'2 W,(r) sin 20 + W2(r) sin 40] (20) t,t 2(r) -- q[2 [-- K2-2 2 gd O' u22(r ) = qb 2 [ -p(r) - W2 cos 40] (2) These expressions are valid for any stationary correlation function of Y for which the resulting integrals converge. For three dimensions, applying spherical coordinates to (8), we have

3 HSU: TECHNICAL NOTE 2275 d (r) df r2 r' 2p(r' ) dr' = - T(r) (22) B3(r) = 32 {sin 2½[6W6(r) cos 0 + 2W7(r) cos 30] dt dr T + 2- = p (23) Introducing the spherical coordinates r = r (sin ½ cos 0, sin ½ sin 0, cos ½)r, so that 0 0 sin O x --= sin ½ cos 0 (24) r Or rsin½00 +/cøs½cøso cos =sin½sino + - cos½sino (25) O y r sin ½ O0 7 - o o o... sin ½ (26) oz-cøs½ r r we find that the functions A become (2 ) 3 A (r) = 4 [W3(r) + W4(r) 0½ ß (-cos 2½ + cos 20 - cos 2½ cos 20)] (27) A2(r) = (2 ) 3 W4(r)(sin 20 - cos 2½ sin 20) (28) where functions W3 and W4 can be found in Table. To derive the function B, we write (5) in spherical coordinates as T(r) = - dti) d = dr d? - d ' (30) u2(r ) = qb 2 sin W 6 ( d ) 2 d2xi 2 r + d3xi dr 3 r r d- r By manipulations similar to those used in two dimensions, the functions B can be obtained in the form Bll(F) = {3Ws(r)- 2W6(r) cos 2½ + 3W7(r) cos 4½ + cos 202W6(r) - 6Wa(r)cos 2½ + 4W7(r)cos 4½] + W7(r) cos 40(3-4 cos 2½ + cos 4½)} (3) B22(r) = (2 ) 3 [Ws(r) - 4W6(r) cos 2½ + W7(r) cos 4½ + W7(r) cos 40( cos 2½- cos 4½)] (32) B33(r ) = {W9(r) + 4W0(r) cos 2½- W?(r) cos 4½ 6 + COS 20[Wll(r) - 4W0(r) cos 2½ - W7(r) cos 4½]} B2(r) = -- {6 W6(r) sin W7(r) sin 40 - cos 2½[8Wa(r) sin W7(r) sin 40] (33) + W7(r) cos 4½(2 sin 20 + sin 40)} (34) B23(r) = - W7(r) sin 4½(3 cos 0 + cos 30)} (35) 32 {2 sin 2½[W6(r) sin 0 + Wv(r) sin 30] - Wv(r) sin 4½(sin 0 + sin 30)} (36) where the functions W (i - 5, 6,.-' ) are defined in Table. Equations (),(27)-(29), and (3)-(36) constitute the threedimensional first-order velocity covariance function with the following expressions, #d O' W 3 3 W 4 3 u(r) = qb2 p- ' -+ - Ws W6 cos 2½ + Wvcos4½+cos20 I( W6 ) + 2 cos 2½ + + cos 40(3 g3j2o '2 cos 4½ ] -4co8 2½ + cos 4½)} (37) u22(r) = qb2 [W5-4W6 cos 2½ + W7 cos 4½ A3(r) = 2 W4(r) sin 2½ cos 0 (29) + W7 cos 40( cos 2½- cos 4½)] (38) g3j2o '2 tt33(r ) = 6qb 2 [W 9 q- 4W0 cos 2½ - W7 cos 4½ + cos 20(Wll - 4W0 cos 2½ - W7 cos 4½)] (39) K2-22{ [ #d O' W cos 2½ + -cos 4½ + - sin 40(3-4 cos 2½ + cos 4½) (40) [( ) #d O' W 4 3 u 3(r) - tk2 cos 0 - -q- - W6 sin 2½ W7 sin 4½ q- cos 30(sin 2½ - sin 4½) (4) g3j2 rr 2 u23(r) = 324)2 [sin 0(2W6 sin 2½- W, sin 4½) q- W7 sin 30(2 sin 2½- sin 4½)] (42) These expressions are valid for any isotropicorrelation function p for which the resulting integrals converge. 3. Applications We apply the results derived above to commonly used isotropic correlation functions, i.e., exponential, Gaussian, spherical, and linear correlation functions IDagan, 989], in both two and three dimensions. These results are readily utilized for the nested correlation function as well. The expressions for isotro-

4 2276 HSU: TECHNICAL NOTE pic exponential, Gaussian, spherical, and linear correlation functions are, respectively, For the two-dimensional isotropic Gaussian correlation p(r) = e -an p(r) = e -br2 p(r) =- 53r +5 ( ) 2 p(r) = 0 >.. p(r) = I <. p(r) = 0 >. < (43) (44) (45) (46) 3.. Two-Dimensional Isotropic Correlation Functions Substituting (56) and (57) into (9)-(2), we recover the expressions in (70), (7), and (72) of Hsu and Neuman [997] In two dimensions, where r 2 = x 2 + x22, the isotropic derived by using Fourier transforms. exponential correlation function (43) yields, from (0), For the two-dimensional isotropic spherical correlation function, we obtain T(r) = a -- 2' [ - ( + ar)e-ar] (47) and, from Table, S ( r ) - - a - r + a- r 63 + a- - r ) e-ar + a r 6 4 (48) Therefore functions W and W2 become, on using the relation a = /X in which X is the integral scale, W = e -r/x (49) r X 2 9A 4 ( X 4A 29A 39 4 W2 = 2r 2 r e -r/x (50) Substituting (49) and (50) into (9)-(2), we obtain the firstorder veloci covarianc expressions. U (r) 3 X r = 2 e-r/x e r/x -2 ß cos20+ ( +-+4 X r ) ß e -r/x cos 40 (5) g 2r2 a 2 { -r/x --[( + -- X ) ß e -r/ cos40 } (52) K2-2 gj 2 { ( X r 2) --2 2] sin20 + [( +-+ r ) e -r/x ] sin40 } (53) u22(r) = 2 e U 2(r) = r -r/x T(r) = 2-- r (- e -br2) (54) r ) S(.) = - 2b2r + 2 e -br2 + 2b2r 4 (55) Substituting (44),(54), and (55) into functions W and W2, and relating b to the integral scale X by b -,r/4x 2, we obtain W -- I q- - F2) 4A2 e -*rr2/4a2,7.f 2 2A 7-/- 2 2A4' 7-2/- 4 ) - 7-? X 2-2 2A 7.2/, (56) W2 = e- 2/4 2 (57) T(r) = r (rlr3) - j + - g r < 2 r T(r) = l Or > 3r r 3 r U(r) = 4 0 l 4 l 3 < 3 4 r U(r) = 40 '4 [ > (s8) (59) Substituting (45), (58), and (59) into function W, W2, and (9)-(2), we then obtain the velocity covariance function. For the two-dimensional isotropic linear correlation function (46), we have T(r) = r 5-5 < 2 r T(r) = 6 r > lr r U(r) = 4 5 l [ < 4 r U(p) = 20 p4 > (60) (6) The velocity covariance function is calculated by using (46) and the above two equations Three-Dimensional Isotropic Correlation Functions For three dimensions, where r 2 = x 2 + x22 + x 23, the isotropic exponential correlation function (43), yields T, from (22), in the form T(r) = a - + a -- 2' + a' 2 e-ar (62) and, from Table, Equations (5)-(53) are given in a different but equivalent 24 form by Rubin [990, equations (9), (0), (2)] and Hsu et al. [996, equations (), (2), (3)]. U(F) a ar a r 2 a- r 3 a a ) = q- q- q- q- e -an (63)

5 HSU: TECHNICAL NOTE 2277 Substituting (43), (62), and (63) into (27)-(29) and (3)-(36), we obtain the first-order velocity covarianc expressions, which are given in a different but equivalent form by Rubin and Dagan [992, equations (A2)-(A8)] and Zhang and Neuman [992, equations ()-(4)]. For the three dimensional isotropic Gaussian correlation x/- r] ) r(r) = - 2b e-br2 q- 4b3/2r 2 erf [ ( ( 2 + 4b2r 34 ) e -br2 + 8bS/2r 3x/ 5 err [ x/- r] ( 65) U(r) = r where erf is the error function. Substituting (43) and ()-(65) into (27)-(29) and (3)-(36), we obtain the expressions for the isotropic Gaussian model. For 0-0 ø, ½ = 90 ø, and relating b to the integral scale X by b =,r/4x 2, we recover the results of (6)-(63) of Hsu and Neuman [997] derived by using the Fourier transform. For the three-dimensional isotropic spherical correlation r(r)=r (3rlr3) 5- [+ - x r [< l 3 r T(r) = 24 r 2 > lr r 3 r U(r) = - [ +63 < l 5 r U(r) = 80 r s [ > (66) (67) Equations (45), (66), and (67) are now used for the calculation of functions Wi, and the velocity covariance function (37)- (42). For the three-dimensional isotropic linear correlation function, we have r(r) = r ( 5- ) r [ < 3 r T(r) = 2 r 2 > (68) The corresponding velocity covariance function is readily calculated by using (46), (68), and (69). 4. Summary We have utilized the general expressions of the first-order velocity covariance given by Hsu et al. [996] to derive the ready-to-use expressions in both two and three dimensions. These expressions can be applied to any stationary isotropic correlation function of natural log-hydraulic conductivity for which the resulting integrals converge. We have applied the results to exponential, Gaussian, spherical, and linear functions in two and three dimensions. Some of these expressions are presented for the first time; others appear in the same or different forms given in previous research. The first-order velocity covariance expressions derived here offer a general method for obtaining analytical expressions for first-order seepage velocity covariance in heterogeneous porous media. Acknowledgments. The author thanks George L. Lamb Jr. for helpful discussion in developing the methodology. Comments from Dongxiao Zhang and an anonymous reviewer are appreciated. References Dagan, G., Flow and Transport in Porous Formations, 465 pp., Springer-Verlag, New York, 989. Deng, F.-W., and J. H. Cushman, On higher-order corrections to the flow velocity covariance tensor, Water Resour. Res., 3(7), , 995. Deng, F.-W., and J. H. Cushman, Higher-order correction to the flow velocity covariance tensor, revisited, Water Resour. Res., 34(), 03-06, 998. Hsu, K.-C., and S. P. Neuman, Second-order expressions for velocity moments in two- and three-dimensional statistically anisotropic media, Water Resour. Res., 33(4), , 997. Hsu, K.-C., D. Zhang, and S. P. Neuman, Higher-order effects on flow and transport in randomly heterogeneous media, Water Resour. Res., 32(3), , 996. Rubin, Y., Stochastic modeling of macrodispersion in heterogeneous porous media, Water Resour. Res., 26(), 33-4, 990. Rubin, Y., and G. Dagan, A note on head and velocity covariances in three-dimensional flow through heterogeneous anisotropic porous media, Water Resour. Res., 28(5), , 992. Zhang, D., and S. P. Neuman, Comment on "A note on head and velocity covariances in three-dimensional flow through heterogeneous anisotropic porous media" by Y. Rubin and G. Dagan, Water Resour. Res., 28(2), , 992. lr r U(r) =5 6 l [ < i l 5 r U(r) = 30 r s J > (69) K.-C. Hsu, Daniel B. Stephens and Associates, 6020 Academy NE, Suite 00, Albuquerque, NM (khsu@dbstephens.com) (Received July 4, 998; revised March 29, 999; accepted April, 999.)