Transmissivity of a heterogeneous formation

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Abstract
The objective of this paper is to derive the transmissivity of an equivalent homogeneous medium having the same macroscopic flow behavior as the actual heterogeneous formation. We apply generalized Taylor‐Ans moment analysis in order to determine the phenomenological coefficient of transmissivity for the case of saturated flow confined in a heterogeneous permeable formation with variable thickness. The generalized Taylor‐Aris moment analysis yields a two‐step procedure for determining the constant two‐dimensional transmissivity tensor from the three‐dimensional spatially variable conductivity tensor: solve a flow problem and then perform an integration. For the special case that the conductivity is locally isotropic and the impermeable confining surfaces are parallel, the equations governing transmissivity are discretized using a pseudospectral Fourier‐Galerkin scheme. The resultant system of linear algebraic equations is solved efficiently, using preconditioned conjugate gradients, in order Nt In (Nt) operations, where Nt is the number of spatial discretization points. The numerical method is used in several examples to compute the transmissivity of lognormally distributed hydraulic conductivity, and the results are compared with the transmissivity found using the usual depth‐averaging approach and another method suggested in the literature. The numerical results show that the averaging volume required to obtain an effective value of transmissivity is about 10 horizontal integral scales. When the flow field has a significant three‐dimensional character, the standard method of finding transmissivities by depth averaging can lead to significant errors in the prediction of global scale flows. It is shown that depth averaging consistently overestimates transmissivities. An example illustrates how in the case of tilted strata, anisotropic transmissivities arise and how the degree of transmissivity anisotropy depends on the angle of the dip and the horizontal to vertical integral scale ratio.

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