# Finite-Difference Interblock Transmissivity for Unconfined Aquifers and for Aquifers having Smoothly Varying Transmissivity

by Daniel J. Goode and Charles A. Appel

U.S. Geological Survey Water-Resources Investigations Report 92-4124, 79 p., 1992.

These alternative interblock transmissivity methods are included in BCF3, part of MODFLOW-96, a three-dimensional finite-difference ground-water flow model. These methods are also included in Modflow-2000 and Modflow-2005.

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**Abstract**

More accurate alternatives to the widely used harmonic mean interblock transmissivity are proposed for block-centered finite-difference models of ground-water flow in unconfined aquifers and in aquifers having smoothly varying transmissivity. The harmonic mean is the exact interblock transmissivity for steady-state one-dimensional flow with no recharge if the transmissivity is assumed to be spatially uniform over each finite-difference block, changing abruptly at the block interface. However, the harmonic mean may be inferior to other means if transmissivity varies in a continuous or smooth manner between nodes. Alternative interblock transmissivity functions are analytically derived for the case of steady-state one-dimensional flow with no recharge. The second author has previously derived the exact interblock transmissivity, the logarithmic mean, for one-dimensional flow when transmissivity is a linear function of distance in the direction of flow. We show that the logarithmic mean transmissivity is also exact for uniform flow parallel to the direction of changing transmissivity in a two- or three-dimensional model, regardless of grid orientation relative to the flow vector. For the case of horizontal flow in a homogeneous unconfined or water-table aquifer with a horizontal bottom and with areally distributed recharge, the exact interblock transmissivity is the unweighted arithmetic mean of transmissivity at the nodes. This mean also exhibits no grid-orientation effect for unidirectional flow in a two-dimensional model. For horizontal flow in an unconfined aquifer with no recharge where hydraulic conductivity is a linear function of distance in the direction of flow the exact interblock transmissivity is the product of the arithmetic mean saturated thickness and the logarithmic mean hydraulic conductivity. For several hypothetical two- and three-dimensional cases with smoothly varying transmissivity or hydraulic conductivity, the harmonic mean is shown to yield the least accurate solution to the flow equation of the alternatives considered. Application of the alternative interblock transmissivities to a regional aquifersystem model indicates that the changes in computed heads and fluxes are typically small, relative to model calibration error. For this example, the useof alternative interblock transmissivities resulted in an increase in computational effort of less than 3 percent. Numerical algorithms to compute alternative interblock transmissivity functions in a modular three-dimensional flow model are presented and documented.