At regional scale, it is common to model groundwater flow as 2-D in the x, y, horizontal plane, by integrating the full 3-D equations over the vertical. Furthermore, adopting the Dupuit assumption results in the local transmissivity T as a formation property, equal to the vertically integrated hydraulic conductivity K. In practice, the related block transmissivity Tb, defined for a volume of area ω (square of side L) in the horizontal plane and height D, is the property of interest. However, most aquifers are of a heterogeneous 3-D structure, and Y = lnK is commonly modeled as a normal and stationary random function which is characterized by the variance σY2, the horizontal I, and vertical Iv integral scales. The Dupuit assumption is generally not obeyed for formations of 3-D spatially variable Y, and transmissivity is no more a meaningful property, independent of flow conditions. Useful generalizations of local and block transmissivity are possible for steady mean uniform flow in the horizontal direction and formations of constant thickness. In that case T and Tb become random stationary variables characterized by their mean, variance, and integral scales. These moments are determined for the first time in an analytical form or by a few quadratures, by adopting a first-order approximation in σY2, and they depend on the ratio D/Iv, e = Iv/I and L/I. The block conductivity expected values are compared with the numerical solutions of Dykaar and Kitanidis (1993), and the agreement is very good for σY2 ≤ 1. The main conclusion of the study is that for this simple flow configuration and for common parameter values, Tb is practically deterministic and equal to Keff(e) D, where Keff is the effective conductivity in uniform flow in an unbounded formation. At regional scale, Tb may be regarded as a local property which changes slowly in the horizontal plane. Analysis of numerous field data shows that this variation is also random and characterized by integral scales Ireg, of the order of kilometers. The separation of scales makes possible to regard the local Tb, as determined along the lines of the present study in a support volume of extent of a few D, as a point value at the regional scale. Practical implications and topics for future investigations are outlined.
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 Transmissivity is one of the properties of widespread use in modeling aquifer flow at regional scale. The availability of advanced codes has made possible the solution of complex problems, for aquifers of different geometries and for various boundary and initial conditions. These codes have become a routine tool, employed for analysis and prediction. A great simplification is achieved at regional scale through replacement of the actual 3-D configuration by a 2-D one, such that the transmissivity T as well as flow variables are functions of the planar coordinates x, y. Solution requires, however, specifying the values of T at the nodes of the numerical grid and a vast body of literature has been devoted to identification of T(x, y) from field data. In these studies and in applications it has generally been taken for granted that T is a medium property, independent of the flow conditions. A close examination (see section 2) reveals that this is based on Dupuit assumption, i.e., negligible vertical flow and constancy of the head H along the z, vertical, direction. It leads to the simple result T = ∫DKdz, where K is the hydraulic conductivity and D is the aquifer thickness. The Dupuit assumption may be not obeyed for aquifers of a 3-D spatially variable conductivity K(x, y, z). Since this is the case for most natural formations, a major question is whether T in its traditional definition is a meaningful property.
 The definition of transmissivity in heterogeneous porous formations and its meaning have been considered by a few articles, but the issue cannot be regarded as settled. In most of the studies dealing with transmissivity in heterogenous formations there is no clear definition of T and it seems that in many cases the one being used is close to the traditional one based on Dupuit's assumption. In the following we briefly examine a few significant papers where the transmissivity and its relation with the underlying three-dimensional, heterogeneous conductivity field are defined.
 Among the first attempts is the definition by Dagan [1989, p. 357], who suggested the expression T = KeffD, where D is the saturated thickness and Keff the effective hydraulic conductivity for mean uniform flow in 3-D heterogeneous media. The definition, of a conjectural nature, is not underlain by Dupuit assumption. It presumes that the support volume of T is much larger than the integral scale of K and that the uniform flow result holds approximately for nonuniform regional flow.
 A few years later, Dykaar and Kitanidis  (hereinafter referred to as D-K) carried out a numerical study in which the transmissivity issue was investigated in a consistent manner. Because of its relevance to the present work (see section 4), we shall recall briefly its content. D-K considered a formation of constant thickness D and of unbounded horizontal extent. The log-conductivity Y = ln K was modeled as a multi-Gaussian stationary random space function characterized by the variance σY2, and the anisotropic autocorrelation ρ, of vertical Iv and horizontal I integral scales, respectively. Assuming mean uniform flow of constant horizontal head gradient −J, in essence T is defined as the ratio between the mean discharge and the mean head gradient, averaged over a square block of dimensions L × L and thickness D. The results are presented for large blocks of L = 30 I, such that Tb (block transmissivity) is practically deterministic. Thus, in the first graph summarizing the main results (D-K, Figure 4) the dimensionless Tb/(KGD), where KG is the conductivity geometric mean, is displayed as function of D/Iv for isotropic ρ and for the three values σY2 = 1,2,3. Subsequently, (D-K, Figure 7), similar graphs are presented for the largest σY2 = 3 and for two anisotropy ratios e = Iv/I = 0.2 and e = 1, respectively. In particular, it was shown that for D/Iv〉〉 1, Tb → KeffD, where Keff is the effective conductivity in mean uniform flow, as proposed by Dagan .
 Another paper which studies the relation between K and T is by Desbarats and Bachu , who analyzed the spatial variability of hydraulic conductivity K in an experimental basin and the vertical upscaling of K through transmissivity. One of the main findings was that the traditional, Dupuit-based approach (arithmetic vertical average of conductivity) causes an overestimation of transmissivity in two-dimensional flow models; this was found to be consistent with the results of D-K. The authors propose a different, empirically based definition of transmissivity which consists of a geometric spatial average of conductivity. More recently, Tartakovsky et al.  analyzed the problem of upscaling of the local conductivity to grid block transmissivity through stochastic averaging and perturbation analysis. The definition of local transmissivity, which was the starting point of the analysis, is the traditional one, i.e., the vertical integral of K. It was concluded that the solution T = DKeff is not generally valid. The results of the above papers are briefly summarized in the review by Sanchez-Vila et al. [2006, section 2.3].
Neuman and Di Federico  provide a comprehensive review on the scaling of hydrogeological parameters, e.g., conductivity and transmissivity. The latter is defined as the [Neuman and Di Federico, 2003, section 1.3, paragraph 14] “hydraulic conductivity times water-saturated aquifer thickness”; that is, it is strictly related to the traditional definition of T. The review does not consider the process of upscaling from conductivity to transmissivity but rather examines their spatial variations and scaling issues, including the values of transmissivity obtained by pumping tests. The latter problem was also considered by Copty and Findikakis  who outlined a procedure for determining the statistical parameters of transmissivity through the analysis of pumping tests. The authors distinguish between a local and a regional transmissivity, the latter being characterized by a much larger spatial scale of variability. The local transmissivity, which was the object of the study, was defined as the “vertically integrated hydraulic conductivity at the local scale,” i.e., the traditional definition of T.
 A similar analysis and distinction between local and regional transmissivity were discussed by Neuman et al. . The paper has applied the type-curve analysis developed by Neuman et al.  to pumping tests conducted at the Lauswiesen site, near Tubingen, Germany. The pumping test data refer to five wells in which water is pumped alternatively in one well and the head is measured in the remaining four wells. The inferred statistical parameters of transmissivity were compared with the same values obtained independently through 312 flowmeter-based K data in 12 wells. Local transmissivity is calculated by integration of K over the vertical for each of the 12 wells, along the traditional, Dupuit-based approach.
 The aim of the present study is to analyze in a comprehensive manner the concept of transmissivity of natural formations of a 3-D spatially variable conductivity and to examine its validity under general conditions. In particular, the results will be compared with those of D-K for their setting.
 The plan of the paper is as follows. In section 2 the general definition of T is reviewed and in particular its traditional one under Dupuit assumption. Subsequently, local T and block transmissivity Tb and their generalizations as random variables for spatially variable K are formulated. In section 3 transmissivity mean, variance and integral scale are determined for formations of planar boundaries and constant D, by a first-order approximation in σY2, for mean uniform flow. The results are compared in section 4 with those of D-K. In section 5 we review data on transmissivity spatial variability at the regional scale, while section 6 summarizes the paper and presents its conclusions.
2. Definition of Transmissivity
2.1. General Definition
 We consider regional-scale modeling of aquifer flow, pertaining to formations whose horizontal extent Lreg is much larger than the thickness D. The 3-D flow field obeys the equations
where q(qx, qy) is the horizontal component of the specific discharge vector, qz is the vertical one, H is the head, s is a source term associated with elastic storage and ∇ stands for the operator of components ∂/∂x, ∂/∂y. The conductivity K is treated as locally isotropic, while anisotropy is associated with K spatial distribution, as discussed in section 2.4.
 The aim is to reduce the problem to a 2-D one and by following a well established procedure [e.g., Bear, 1972], equations (1) are integrated over z to obtain
where Q = ∫Dqdz is the vertically integrated discharge, the overbar symbol stands for vertical averaging, i.e., = (1/D)∫DHdz, and S is the storativity. In the case of a phreatic aquifer, D is replaced by ζ(x, y, t), the water table elevation, and the r.h.s. of (2) by −n∂ζ/∂t + R, where n is effective porosity and R is the rate of recharge.
 In this set of exact equations, (2) is expressed in terms of 2-D variables whereas the r.h.s. of (3) is of a 3-D nature. At this point it is worthwhile to discuss the traditional approach to simplify (3) by adopting Dupuit assumption, which implies negligible vertical flow such that ∂H/∂z ≃ 0 and H ≃ (x, y). Substitution in (3) yields the classical relationship
where T is coined here as local transmissivity. Equations (2) and (4) form a closed 2-D system for Qx, Qy, , the one usually employed in aquifer flow modeling. The Dupuit assumption is identical to similar ones adopted in various disciplines, e.g., in theory of water waves, where it is known as the shallow water approximation [Stoker, 1958]. Such approximations are underlain by two prerequisites:
 1. The flow domain is shallow; that is, the thickness D is much smaller than the horizontal extent Lreg. This is obviously satisfied by regional-scale modeling.
 2. The radius of curvature of streamlines in the vertical plane is much larger than the depth; that is, streamlines are approximately parallel. This is a much more stringent requirement, which is not satisfied for instance in regions like the neighborhood of a partially penetrating well. Far from such regions, the shallow water approximation can be justified if K varies slowly in space.
2.2. Block Transmissivity and Its Identification
 The local T as defined by Q = −T ∇ at a point is of limited usefulness in most applications, for a few reasons: (1) even if (4) holds, measuring the K profile is a difficult task and not a routine procedure; (2) numerical codes discretize the flow domain and the relevant property pertains to a block whose size depends on the particular partition and numerical scheme; and (3) the common methods used in order to identify transmissivity are pumping tests and inverse modeling. In the first case, the identified transmissivity encapsulates the conductive property of a block of the order of the well radius of influence, which is larger than the depth. In inverse modeling the transmissivity values are calibrated such as to match measured heads throughout the aquifer and a vast body of literature is devoted to this topic. The point of principle relevant to the present discussion is that inverse methods rely on numerical codes and again the identified transmissivity values are attached to numerical blocks and not to a point.
 Under these circumstances the interest resides in the block values of the flow variables defined for slowly varying flow by
where ω is an areal element in the plane whose centroid is at x′ = y′ = 0. Assuming that S is constant over ω, it is easy to ascertain that (2) is satisfied by Qb and Hb as well. As for (4), it is replaced by
 If ∇ is regarded as constant over ω, in the spirit of numerical methods, the transition from (4) to (6) renders Tb = (1/ω) ∫ωT(x′ + x, y′ + y) dx′dy′, but this relationship is difficult to apply in view of the discussion above about the identification of T.
 Of particular relevance here is a rectangular or square ω defined by x − L/2 < x′ < x + L/2, y − L/2 < y′ < y + L/2, which leads in (5) to the usual definition of the mean gradient in numerical methods.
2.3. Generalization of Transmissivity Definition for Aquifers of Spatially Variable Conductivity
2.3.1. General Approach
 In natural formations K(x, y, z) varies in the vertical as well as horizontal directions. Because of the seemingly erratic spatial variations revealed by numerous field surveys, it is common to model Y = ln K as a random space function characterized by its statistical moments. This approach is at the core of stochastic modeling of flow and transport, a field which has known a tremendous development in the last three decades (see, for instance, monographs by Dagan , Gelhar , Zhang , and Rubin ). It is common to model Y as stationary and normal, of two point covariance CY(x1, x2) = σY2ρ(x1 − x2, y1 − y2, z1 − z2) and of axisymmetric anisotropy, such that Y is characterized by the four parameters 〈Y〉 = ln KG, σY2, Iv and e = Iv/I, with 〈〉 standing for statistical averaging and Iv and I, for the vertical and horizontal integral scales, respectively. Rubin and Hubbard [2005, Table 21] presents a summary of parameters values identified in a few field investigations.
 However, the main feature of relevance to the present study is that streamlines are of a 3-D nature and wind in space, their radius of curvature being much smaller than the thickness. For instance, fluid particles circumvent blocks of low K, and at the “nose” and “tail” of such elements flow is close to vertical and the radius of curvature is of the order of Iv.
 This flow feature is in variance with the assumptions implicit in the Dupuit approximation, as discussed above, and H cannot be regarded anymore as independent of z. In spite of the essentially 3-D nature of the flow, it is still highly desirable to reduce it to a planar one at the regional scale, in order to capture the distribution of the flow variables with x, y, t after averaging over the vertical. The salient questions is whether toward this aim it is possible to define local and block transmissivities and if the answer is positive, how are they related to the K structure of the aquifer?
 Before attempting to answer these questions in a general manner, we can gain some insight by considering an extreme and simple configuration, which can be solved exactly. Thus, here and in the following we consider a confined aquifer of constant thickness D and unbounded horizontal dimensions under steady flow conditions, such that (2) becomes ∇.Q = 0. For mean uniform flow driven by a constant mean head gradient −J and for the extreme case D/Iv ≫ 1, ergodic arguments can be involved and vertical averages can be exchanged with ensemble averages, i.e., Q/D ≃ 〈q〉, ∇ ≃ ∇〈H〉 and the aquifer may be regarded as vertically as well as horizontally unbounded. As is well known, in this case
where Keff is the effective conductivity for uniform flow. Furthermore, Keff/KG = funct(σY2, e) for a given K structure of axisymmetric anisotropy and for horizontal mean flow. The derivation of Keff is the object of a vast literature and general results were obtained at first order in σY2 as follows:
and in particular Keff/KG = 1 + σY2/6 for e = 1.
 As already mentioned in the Introduction, the expression (7) for T was conjectured by Dagan  as of general validity and was checked numerically by D-K for large blocks.
 Even for this idealized configuration, Keff depends not only on the structure but also on flow conditions. Thus, for another important case, namely, of radial flow toward a fully penetrating well of constant discharge per unit length Qw, Indelman and Abramovich  have shown that 〈q〉 is related to ∇〈H〉 by a convolution and Keff is nonlocal and different from (8).
 In the opposite case, of lesser interest in applications, of D/Iv < 1, the conductivity K is practically constant over D such that T ≃ K(x, y, 0) D. Although the Dupuit approximation is justified, T is now random and equations (2) and (3) are stochastic. Again for L/I ≫ 1, 〈Q〉/D ≃ 〈q〉 and ∇〈〉 = ∇〈H〉 are related through Keff, which is now the effective conductivity in 2-D flow. It is given exactly by KG for mean uniform flow in isotropic media, but is different for well flow.
 From this simple analysis we arrive at the following conclusions: (1) the traditional definition of T(4) based on Dupuit assumption is not generally valid and flow is essentially 3-D, (2) reduction of the problem to a 2-D one for shallow formations requires defining T as random variable, and (3) T is no more a medium property but depends on the flow pattern and the geometry of the formation.
 Fortunately, for two types of flow conditions prevailing in natural formations it is possible to define useful transmissivities values. The first one is of slowly varying, natural gradient, flow, for which the scale of planar variation is much larger than the thickness. Then, the mean flow can be regarded as locally uniform and the results obtained for random T in unbounded domains of finite thickness can be employed. The second one is of radial mean flow toward a fully penetrating well. The present article is limited to the first configuration, of mean uniform flow, while the discussion of the more complex case of well flow is deferred to future studies.
2.3.2. Definition of Transmissivity of Heterogeneous Aquifers
 The starting point is the general definition of the local T(3), i.e., Q = −T∇, where Q =D = −D and the overbar stands for vertical averaging. It is implicit in this definition that the vectors Q and ∇ are parallel. This is true if Dupuit assumption is adopted or for the extreme cases of thick (D/Iv ≫ 1) or thin (D/Iv ≪ 1) aquifers, for the assumed horizontal isotropy of K. To define T as a scalar we project the vectorial relationship on the direction of ∇ defining T as the ratio between the projection of Q on the head gradient direction and the latter, to arrive at the general definition
 It is reminded that for steady flow −Q.∇ = −∇.(Q) is the local dissipation of the 2-D flow, a positive definite function, ensuring the existence and positiveness of T(9) as well as the preservation of the total dissipation in the aquifer (see a general discussion by Indelman and Dagan ). Obviously, if Dupuit assumption is adopted, then locally = ∇ and (9) leads to the usual relationship (4)T = D.
 In a similar manner, the scalar block transmissivity is defined by
with the areal averages Qb and ∇Hb defined by (5) and it is seen that T is the limit case of Tb for the support ω → 0. In both cases of steady mean uniform flow of constant J = −∇〈H〉 or steady flow to a fully penetrating well of constant discharge/length Qw, the variables Qb, ∇Hb or Q, ∇H are linear functions of J or Qw, respectively. Thus, the latter drop out from the definitions of T(9) or Tb(10), which depend on the structure and geometry only. Hence, for the case of steady uniform mean flow and a square L × L block, the statistical moments of Tb/(KGD) are functions of σY2, D/Iv, e, L/I. These definitions may be assumed to be applicable when mean flow varies slowly in space or time.
2.3.3. First-Order Approximation
Equations (9) and (10) may serve as the starting point for numerical computations and this was essentially the approach of D-K for large square blocks and for a few particular values of the independent parameters. Here we aim at obtaining general and simple results, which are achieved by a first-order expansion in σY2, whose accuracy will be checked in the sequel. Toward this aim we follow a well established procedure [e.g., Dagan, 1989] and expand K = KG exp (Y′) = KG(1 + Y′ + Y′2/2 + …), ∇H = −J + ∇H1 + ∇H2 + … where Y′ = Y − 〈Y〉, J = −∇〈H〉 and H1 = O(σY), H2 = O(σY2),… Substituting in (9) and carrying out the expansion leads after a few calculations to the first-order result in σY2
and similarly for Tb by replacing the overbar by the subscript b.
 Expansion of the exact flow equation (1) for s = 0 leads to the following equation, which is satisfied by the first-order approximation of the 3-D head field
Equations (11) and (12) will serve to effectively derive the statistical moments of T and Tb in the sequel in terms of the parameters characterizing the 3-D structure. It is easy to ascertain that for D/Iv ≫ 1 and invoking ergodicity, i.e., exchange of overbar and 〈〉, leads to → 0, ∇1 → 0 and → σY2. Subsequent substitution in (11) leads to the equation that served [Dagan, 1989, equation 3.4.9] to compute the effective conductivity as given by (8). Similarly, for D/Iv ≪ 1, → Y′, ∇1 → ∇H1 and → Y′2, such that T/(KGD) → 1 + Y′ + Y′2/2, i.e., T/D → K. It is seen that the two limits discussed above are recovered.
3. Effective Computation of Local and Block Transmissivity by First-Order Approximation
3.1. Transmissivity Statistical Moments
 Steady flow of horizontal mean head gradient −J takes place in a confined aquifer of thickness D. Without loss of generality J(J, 0) is taken in the x direction and starting with the expected value of the local transmissivity (11), we get
with similar results for τb = Tb/(KGD), Mb and Nb by replacing vertical averaging by block averaging.
 Both M and N, as well as the block values, can be derived with the aid of the generating function
 In order to derive F we have to solve (12), which is carried out in terms of the Green Function G(x − x′, y − y′, z, z′), which satisfies the equation
 With the detailed computations given in Appendix A, the final result for F(14) is as follows
 In (17) and in the sequel, the hat symbol stands for Fourier Transforms in the x, y plane, e.g., (kx, ky, rz) = (1/2π) ∫−∞∞∫−∞∞ρ(rx, ry, rz) exp(ik.x) drxdry. The wave number vector k has components kx, ky and modulus k = ∣k∣. The function , related to the Fourier Transform of G(15) is given explicitly by equation (A3).
 In a similar manner, the variance and the integral scale of τ(13) have the following simple first-order expressions, based on (11)
where τ′ = τ − 〈τ〉 = + … and
 The relationships between F(17) and the M, N functions as defined by (13) are therefore given by
 The expressions of M and N, for a Gaussian log-conductivity covariance are given in terms of a few quadratures in Appendix B by equations (B2) and (B4). Substitution of (20) into (11) renders [〈τ〉 − 1]/σY2 = (1/2) + M − N, as function of D/Iv and e = Iv/I.
 In a similar manner, the expected value of the block transmissivity (11) depends on the functions
 We shall carry out computations for ω a square L × L aligned with the direction of mean flow x. Then, by Cauchy algorithm [e.g., Dagan, 1989, section 1.9] Mb(21) becomes
 The expression of Mb is given in Appendix B (equation (B6)) and substitution in (11) renders the expected value (〈τb〉 − 1)/σY2 = (1/2) + Mb − N as function of D/Iv, e = Iv/I and the additional parameter L/I.
 Finally, the variance and the integral scale of τb are given, similarly to (18) by
which becomes for a square block
 Similarly to (18), the integral scale ITb is given by
 In order to simplify the computations of the moments of τ and τb we adopt an axisymmetric Gaussian log-conductivity two point covariance, i.e.,
where r = ∣r∣ is the modulus of the lag vector in the x, y plane.
3.2. Results and Discussion
 The first-order solution permits us to analyze the general properties of transmissivity in mean uniform flow, as defined above.
 Starting with the local transmissivity τ = T/(KGD), the expected value 〈τ〉 (13) can be determined by two quadratures for the Gaussian Y = ln K covariance (26) using (B2) and (B4). For illustration we represent in Figure 1 the variable (〈τ〉 − 1)/σY2 as function of D/Iv, for a few values of the anisotropy ratio e. Inspection of Figure 1 reveals a few interesting properties:
 1. Since T/D → K(x, y, 0) for D/Iv → 0, 〈T〉/D tends to the arithmetic mean KA = 〈K〉 at this limit. The latter is given exactly by KA/KG = exp (σY2/2) and at first order by KA/KG = 1 + σY2/2, i.e., (〈τ〉 − 1)/σY2 → 0.5.
 2. For D/Iv → ∞ ensemble and space averaging can be exchanged (ergodic behavior); that is, 〈T〉/D ≃ T/D → Keff(8) and its values are represented as asymptotic ones in Figure 1. It seen that 〈T〉/D drops gradually with D/Iv from the upper bound KA to the asymptotic Keff which is not yet attained for D/Iv = 30.
 3. Dupuit assumption and equation (4) imply 〈T〉/D = KA and it overestimates the actual value depending on D/Iv and on e. As is well known, the limit e → 0 pertains to a stratified formation for which 〈T〉/D = KA. Hence, for say e < 0.01 [see Dagan, 1989, Figure 3.4.4] the latter value and Dupuit assumption (4) are a valid approximation.
 Next, we consider the expected value of the block transmissivity 〈τb〉 = 〈Tb〉/(KGD) (13) and (22) in the same dimensionless form (〈τb〉 − 1)/σY2 for a square block of side L. It depends on L/I, the ratio between the block size and the horizontal integral scale, besides D/Iv and e. The equation serving for the calculation of 〈τb〉 results from combining (13), (B2), (B5) and (B6), requiring two quadratures. For illustration, we have represented in Figure 2 its dependence on D/Iv for a few values of the relative block size L/I and for two values e = 1 and e = 0.2. If we regard the heterogeneous structure as made up from elements of different K, then both ratios D/Iv and L/I are indicative of the number of elements within the averaging block. A few features of interest are displayed by Figure 2:
 1. For a given e and for a fixed D/Iv, 〈Tb〉 drops with L/I from the maximal value pertaining to local transmissivity (L/I = 0) to the minimum reached for the largest considered blocks (L/I = 30).
 2. For given e and L/I, 〈Tb〉 tends to the asymptotic value KeffD with increasing D/Iv. However, the limit is approached at smaller D/Iv for large L/I than for small L/I. Even for L/I = 30, D/Iv must exceed the value of 10 for e = 1 and that of around 20 for e = 0.2, in order for 〈Tb〉 to approach the asymptotic limit.
 3. At the limit D/Iv → 0, K does not vary with z and the flow becomes 2-D in the horizontal plane. In this case 〈Tb〉/D varies from the effective value KG for large blocks, say L/I > 8, to the one corresponding to the local 〈T〉/D = KA for L/I = 0.
 The next important statistical parameter is σTb/〈Tb〉, the block transmissivity coefficient of variation, which is equal to σlnTb at first order and in particular to σT/〈T〉 at the limit L → 0. The equality σTb/〈Tb〉 = στb(24) is obeyed at first order in σY, and the ratio σTb/(〈Tb〉 σY) depends on D/Iv and L/I but not on e as an independent variables. Indeed, this is a direct result of (24) for the assumed separability of the dependence of ρ(26) on r/I and (z1 − z2)/Iv. The ratio σTb/(〈Tb〉 σY) is a measure of the reduction of variability of the transmissivity relative to that of K, and we have represented in Figure 3 its dependence on D/Iv for a few values of L/I (the detailed computations are given in Appendix C). The reduction of the variance with increasing values of these parameters is evident, though the drop is slow for large D/Iv. Of definite interest are the dimensions of the block that render the relative coefficient of variation close to zero so that Tb ≃ 〈Tb〉 can be regarded as deterministic and ergodicity obeyed. Starting with the local transmissivity, a reduction of σlnTb/σY = 0.2 is achieved for D/Iv = 50, so that the formation has to be considerably thicker than the vertical integral scale in order to ensure ergodicity. This important result implies that in many applications T has to be regarded as random, though the variance of its logarithm becomes much lower than σY2 for say D/Iv > 20. In contrast block averaging over a horizontal area has a major impact on variance reduction and Tb ≃ 〈Tb〉 for say L/I > 10. It is seen indeed that the value adopted by D-K, namely, L/I = 30 ensures ergodic behavior for any D/Iv.
 Last, we examine the integral scale ITb(25) of the block transmissivity. For the selected covariance (26) it is seen that (25) becomes
that is, the integral scale depends only on block averaging in the plane. The result of integration over L is given in Appendix C. The integration over r in (24) is computed by a single quadrature and the result is summarized in Figure 4, displaying the ratio ITb/I as function of L/I. Because of the aforementioned separability in (26), the integral scale of the local transmissivity IT is equal to I, the horizontal one of Y. As well known [e.g., Dagan, 1989, section 3.2.4] space averaging increases the correlation scale, which becomes dominated by the block scale. Thus, ITb → L/2 for large L/I, as Tb of neighboring blocks become uncorrelated. The transition to the block dominated scale is evident by the close fit to a line of slope 1/2 (dashed line) for L/I > 3 in Figure 4.
 The setup considered by D-K in order to determine the block transmissivity was described in the Introduction (section 1). By using Fourier transform methods and additional techniques, they were able to derive Tb ≃ 〈Tb〉 numerically, without the limitation of first-order approximation in σY2. Their results were obtained for the specific values L/I = 30, σY2 = 1;2;3, e = 1;0.2 and for an exponential covariance, and it is of interest to compare them with the present ones.
 For the isotropic structure, e = 1, we have reproduced in Figure 5 the dependence of [〈Tb〉/(KGD) − 1]/σY2 based on Figure 4 of D-K and the present first-order solution (equations (13) and (26) and Figure 1). It is seen that the agreement is excellent for σY2 = 1. Furthermore, the results are close to the numerical ones even for the larger values σY2 = 2;3. This is quite encouraging in view of the difference in approach and of the selected covariance. This good agreement may be attributed to the robustness of the first-order approximation Keff/KG = 1 + σY2/6 for isotropic formations.
 In Figure 6 the same comparison is displayed for the other specific results of Figure 7 of D-K, namely, σY2 = 3 and the two values e = 1 and e = 0.2. The isotropic case was already represented in Figure 5, and the additional feature is the anisotropic one for e = 0.2. It is seen that the agreement with the first-order solution is less good for D/Iv > 5, but still the differences are within 20%. The agreement is expected to deteriorate for e ≪ 1, for which Keff/KG → exp (σY2/2). Of course, it is hardly expectable to achieve close agreement for highly heterogeneous formations characterized by σY2 = 3, but D-K did not present results for the smaller σY2.
5. Regional-Scale Variability of Block Transmissivity
 In the preceding sections we have defined local and block transmissivities of aquifers of a 3-D hydraulic conductivity distribution and have determined their dependence on the 3-D structure for conditions of mean uniform flow and formations of constant thickness. In particular we have found that for L/I > 30, Tb becomes practically deterministic, its value being dependent on the structure of K as well as the relative aquifer thickness D/Iv. When both D/Iv and L/I are large, Tb tends to Keff(e) D (Figure 2) for say D/Iv > 20 and L/I > 10. For common values of Iv of the order of tens of centimeters and I of meters, this is ensured for say L > 20–50 m and D larger than a few meters.
 While this result permits one to regard the transmissivity of a block centered at x as constant, the question is what happens at the entire aquifer regional-scale Lreg of the order of tens of kilometers, over which the thickness as well as Keff may change slowly in the plane?
 The analysis of spatial variability of Tb at regional scale has been the object of a few studies in the past. In these studies the values of Tb at different points, as identified primarily by pumping tests, was found to display a random spatial variation which can be characterized again statistically.
 Thus, Hoeksema and Kitanidis  have analyzed data from different publications and reports, for 30 formations in the USA. Although the relationship between Tb as identified by pumping tests and the one applying to mean uniform or slowly varying flows is still a matter of debate, the results are still indicative of the variability of Tb at the regional scale. By using direct measurements of transmissivity (with no inverse procedure which takes advantage of head data), Hoeksema and Kitanidis  arrived at estimates of Ireg, the integral scale of the stationary log-transmissivity field. For 19 aquifers made of unconsolidated material the median value of the identified Ireg was around 1.4 km whereas for 10 consolidated ones it was around 15 km. The identification of Ireg was quite imprecise, but the values are indicative of its order of magnitude. Similar values were found by Delhomme  for a few aquifers in France.
 Recently, Jankovic et al.  have used separately conductivity data and head measurements to identify the parameters characterizing the unconsolidated aquifer of the Eagle Valley basin in Nevada [Arteaga, 1982], under conditions of natural gradient flow. Direct identification was based on transmissivity data obtained primarily from specific capacity tests made in 65 wells and the value was Ireg ≃ 470 m. By using 184 head measurements and a flow model in an inverse mode, the result was Ireg ≃ 1400 m. Although there are a few possible explanations for this discrepancy, one of them being the limited size of the block surrounding the wells that contribute to the specific capacity test, whereas heads data reflect the impact of conductive properties of larger blocks. In any case it is seen that Ireg was of the order of magnitude of the other unconsolidated aquifers considered by Hoeksema and Kitanidis .
 Summarizing, the log-transmissivity integral scales have been estimated to lie between hundreds to thousands of meters, which are considerably larger than the scales of variability of T or Tb resulting from the spatial 3-D variability of K, which are analyzed here.
6. Summary, Conclusions, and Practical Implications
 Natural formations (aquifers, reservoirs) are characterized by horizontal length scale Lreg = O (104m) and thickness D = O (101 − 102m). For such a shallow system (D/Lreg ≪ 1) it is common to simplify the complex 3-D flow equations to 2-D ones in the horizontal x, y plane, by integration the equations over the vertical and by adopting the Dupuit assumption. Subsequently, the local transmissivity, defined as the coefficient of proportionality between the vertically integrated specific discharge Q and the mean head gradient −∇, is given by T(x, y) = ∫DK(x, y, z) dz. The more useful block transmissivity Tb is defined in a similar manner for a support element ω in the horizontal plane, e.g., a square of side L = O(101D). Since L ≪ Lreg it is common to regard Tb in numerical models of regional flow as a point function of x, y, attached to the centroid of ω.
 In most aquifers, K(x, y, z) varies in a complex manner in space and it is commonly modeled as a stationary random space function within regions of the aquifer of extent of a few D. For axisymmetric anisotropy of Y = ln K, the statistics of the normal Y are captured by the 3 parameters σY2, I and e = Iv/I, for a given shape of the two-point autocorrelation ρ. For such 3-D heterogeneous structures the Dupuit assumption is generally not obeyed and the question is whether the traditional definitions of T and Tb can be generalized in an appropriate manner, so as to model regional flow as 2-D. By using the basic definition of T as relating linearly Q and −∇, where = (1/D) ∫DHdz, it is shown that generally: (1) T(x, y) is random; (2) it depends on the mean flow pattern; and (3) for a given pattern, the statistical moments of T depend on the geometry of the aquifer (and of the block in case of Tb) and on the statistics of Y. This is in variance with the traditional definition of T as an aquifer property solely.
 In spite of these limitations, useful definitions of T and Tb can be forwarded for two particular cases of flow conditions, of wide application: mean uniform and radial well flows. The present study addresses the first case only, and for steady flow in an aquifer of constant thickness, the statistical moments of the transmissivity are determined effectively for the first time by using a first-order approximation in σY2. With τ = T/(KGD), the expected value (〈τ〉 − 1)/σY2 is a function of D/Iv and e solely, and the results of the computations for a Gaussian covariance are represented in Figure 1. It is seen that for D/Iv ≫ 1, 〈T〉 → KeffD, where Keff(e) is the effective conductivity for horizontal flow, a result conjectured by Dagan . In contrast, for the less interesting limit D/Iv < 1, 〈T〉 → KAD which is also the value which results from the traditional (Dupuit) definition. Hence, the latter overestimates 〈T〉 considerably, depending on the values of σY2, D/Iv and e. The block transmissivity 〈Tb〉 depends on the additional parameter L/I (Figure 2) and it is seen that for L/I > 10 it grows from KGD (for D/Iv ≪ 1) to KeffD (for D/Iv > 15).
 Comparison of the present first-order solution for 〈T〉 and the numerical one of Dykaar and Kitanidis  in a few particular cases (Figures 4 and 5) shows very good agreement for σY2 ⩽ 1 and differences less than 20% even for σY2 = 3.
 The upscaling process leading from K to Tb results in the diminishing of the variance σlnTb2 relative to σY2. We have determined the ratio σlnTb/σY ≃ σTb/(〈τb〉σY) by the same first-order approximation and the results are displayed in Figure 3. It is seen that the local T becomes practically deterministic for say D/Iv > 50, and the same is true for Tb for any D/Iv if L/I > 10. Similarly, the integral scale of T is equal to I (Figure 4) and increases with L/I for Tb, tending to L/2 for L/I > 5.
 It can be concluded that for the common cases of Iv = O(10−1m), I = O(101m), D/Iv = O(10) and L/I = O(10), the block transmissivity in mean uniform flow can be modeled as deterministic and equal to Keff(e) D. The same conclusion is valid for the local transmissivity T provided that D/Iv 50; otherwise T has to be modeled as random and to be characterized statistically.
 Because of the separation between the scale L of the block transmissivity and the aquifer-scale Lreg, it is possible to regard Tb as a point value attached to the centroid of the block. Then, D and the statistical parameters of Y, that contribute to the definition of Tb, are related to a volume of horizontal extent O(101D). At the regional scale, Tb changes slowly in the plane and its seemingly erratic spatial variation is again modeled as random. Analysis of data of numerous aquifers indicate that the integral-scale Ireg = O(103m), justifying again the separation of scales between the locally defined Tb and the regional scale. Numerical codes that use elements of the order of Ireg or larger require upscaling of regional-scale Tb and this issue was discussed in two recent articles [Dagan and Lessoff, 2007a, 2007b].
 A salient question is what are the implications of the present study to current codes employed in aquifer flow modeling. The answer depends to a large extent on the method used in order to infer the values of the block transmissivities of numerical elements and on flow conditions.
 Thus, if Tb is to be determined from measurements of K values in space (a procedure which is not yet of wide use, but may become so with advances in field tests techniques), the present study presents a methodology to determine Tb for slowly varying natural gradient flow in terms of the statistical moments of K. Although the validity of results for unsteady or nonuniform mean flow (e.g., in presence of recharge, leakage, variable thickness) has yet to be assessed, the extreme and common case which calls for further investigations is the one of well flow. Nevertheless, in the particular case of highly anisotropic formations (say e < 0.01), the aquifer behaves like a stratified one and the common Dupuit assumption may be adopted.
 If Tb is determined by pumping tests, its applicability to zones in which natural gradient flow occurs is a topic of recent research, and the matters cannot be considered as settled. We propose to investigate it in the future along the lines of the present study.
 Lastly, Tb is determined quite often by an inverse method. The problems encountered when using such procedure is the subject of a vast literature. Inasmuch as the results are reliable, they already incorporate the effect of local heterogeneity. A salient question is how to apply the identified values to different flow regimes, e.g., pumping by future wells in a zone of slowly varying flow.
 Concluding, transmissivity is a parameter whose identification and application need further elucidation. We hope that this study has contributed to its setting in a rational framework.
Appendix A:: Computation of the Generating Function F
 The Green's function G (x − x′, y − y′, z − z′) is defined by (15). To derive it we take a Fourier transform in two dimensions (represented by adding a hat to the symbol) and require that = (1/2π) ∫−∞∞∫−∞∞G(x − x′, y − y′, z − z′) exp(ik.x) dxdy be continuous. By (15) satisfies
 The solution of the linear equation (A1) with boundary conditions (A2) is given by = exp (ik · x′) where = + for (z > z′) and = − for (z < z′) and
Appendix B:: Computation of Flow Statistics for a Gaussian Covariance
 For the a Gaussian log-conductivity covariance F(r, z1, z2) is obtained by combining (17), (26) and (A3) as follows
 Substituting (B1) into the function N(20) and integrating analytically over β and z′ results in
and the integrations over k and z must be computed by quadratures. Such quadratures were carried out using standard mathematical software in the computations for Figures 1, 2, 5, and 6.
M is obtained by combining (17), (20) and (26) in a similar manner we get
which can be integrated analytically over z′, z1, z2 and k with the final result
Note that in this case M is independent of e and is a function of D/Iv only.
 For a block defined by a square of side length L in the x, y plane and height of D, flow is stationary in the x, y plane therefore the block averaged Nb = N is given by (B2) whereas Mb(21) is computed as follows by combining (17), (22), (A3) and (B1)
the integrations over z1, z2 and z′ can all be carried out analytically as above (for M). The integrals over r1 and r2 can be performed analytically as
whereas the integrals over k and β in (B5) were computed by quadratures, leading to the results illustrated in Figures 2, 5, and 6.
Appendix C:: Variance and Integral Scale of Block Transmissivity
 Computing 〈τ2〉 − 〈τ〉2 from (11) and keeping terms only to the second order in Y′ results in (18). For a Gaussian log-conductivity, the local transmissivity variance can be solved analytically as
Similarly for a block of height D and length and width L having a Gaussian log-transmissivity covariance aligned with the axes [which can be expressed as ρ(x, y, z) = ρ(x) ρ(y) ρ(z)] (23) can be resolved into the analytical expression
 This research was supported by a grant from the Ministry of Science, Culture and Sport of the State of Israel and FZK Forschungszentrum Karlsruhe, Germany.