Particle velocity interpolation in block-centered finite difference groundwater flow models

A block-centered, finite difference model of two-dimensional groundwater flow yields velocity values at the midpoints of interfaces between adjacent blocks. Method of characteristics, random walk and particle-tracking models of solute transport require velocities at arbitrary particle locations within the finite difference grid. Particle path lines and travel times are sensitive to the spatial interpolation scheme employed, particularly in heterogeneous aquifers. This paper briefly reviews linear and bilinear interpolation of velocity and introduces a new interpolation scheme. Linear interpolation of velocity is consistent with the numerical solution of the flow equation and preserves discontinuities in velocity caused by abrupt (blocky) changes in transmissivity or hydraulic conductivity. However, linear interpolation yields discontinuous and somewhat unrealistic velocities in homogeneous aquifers. Bilinear interpolation of velocity yields continuous and realistic velocities in homogeneous and smoothly heterogeneous aquifers but does not preserve discontinuities in velocity at abrupt transmissivity boundaries. The new scheme uses potentiometric head gradients and offers improved accuracy for nonuniform flow in heterogeneous aquifers with abrupt changes in transmissivity. The new scheme is equivalent to bilinear interpolation in homogeneous media and is equivalent to linear interpolation where gradients are uniform. Selecting the best interpolation scheme depends, in part, on the conceptualization of aquifer heterogeneity, that is, whether changes in transmissivity occur abruptly or smoothly.