Stochastic determination of three‐dimensional capture zones for a fully penetrating well

Abstract

A stochastic methodology is used to investigate transport of a conservative solute by steady flow to a fully penetrating pumping well in a three‐dimensional heterogeneous formation. The spatially variable hydraulic conductivity is modeled as a stationary random function of anisotropic two‐point covariance. Although mean transport is radial, heterogeneity produces three‐dimensional velocity fluctuations which are shown to affect solute recovery and the capture zone distribution. Expressions accurate to the first order in log conductivity variance σY2 are derived for travel time variance, capture zone extent, and solute recovery under the following simplifying assumptions: (1) the vertical integral scale of conductivity is much less than the horizontal integral scale, (2) the depth of the formation is much greater than the vertical integral scale of conductivity, (3) the well bore diameter is much smaller than the horizontal scale of conductivity, (4) longitudinal and lateral local dispersion are negligible while the effect of transverse dispersion is examined. Travel time variance στ2 for transport in convergent flow in a three‐dimensional aquifer is shown to be greater than στ2 in either constant mean flow in a three‐dimensional formation or radially convergent flow in a heterogeneous two‐dimensional medium. Transverse dispersion between layers in a sedimentary formation is found to increase mixing between flow paths leading to a marginal reduction in travel time variance. Because of negative correlation between mass flux and travel time, solute retrieval at the well occurs earlier from a constant flux solute source than from an instantaneous plume. The relationship between initial plume length and relative plume spreading is illustrated for simple plume geometries.