Spatial averaging of transmissivity in heterogeneous fields with flow toward a well


  • A. J. Desbarats


Steady state nonuniform flow in a heterogeneous medium is investigated using a combined numerical-empirical approach. The objective of the study is to determine the effective transmissivity of a square field with a constant-head external boundary and a constant-rate well at its center. Transmissivity at the point scale is modeled as an isotropic and multivariate lognormal spatial random function. Block transmissivities are defined using an empirical scaling up of point support values within the field. The scaling process is a weighted spatial geometric averaging where log transmissivities are weighted by the inverse square of their distance from the well. Block transmissivities thus obtained and true effective values calculated using a numerical flow model are compared for realizations a finely discretized transmissivity field. The principal finding of this study is that effective transmissivities obtained from the numerical flow model and block values obtained by spatial averaging are in excellent agreement for low to moderate variances of log transmissivity and moderately eccentric field geometries. The geostatistical model for point scale transmissivity and the deterministic spatial averaging law are combined in order to create a geostatistical model for transmissivity at the block scale. Exact expressions are derived for the ensemble moments of block-averaged transmissivities and well drawdowns in both the nonconditional case and the conditional case when transmissivity at the well bore is known. The ensemble mean of block-averaged transmissivity is found to decrease from the ensemble arithmetic mean toward the ensemble geometric mean as the field size becomes large compared to the integral range of spatial correlation. This result contradicts earlier published work. Numerical results presented in this study also disagree with the analytical results of a recent stochastic analysis of specific discharge in radial systems.