Effective transmissivity in transient stream depletion

Abstract

Quantifying the timing of streamflow depletion caused by groundwater pumping wells is a central issue in the conjunctive management of groundwater and surface-water resources. It is an important consideration in regions where water supplies and demands are offset annually by season and interannually through variable wet and dry years. Increased water demands, regulatory policy shifts, and aquifer changes have brought scrutiny to this type of stream-aquifer interaction. From this, analytical models of stream depletion have received renewed attention and numerous refinements which have focused primarily on a variety of complex boundary conditions. The question of representing aquifer heterogeneity through simplified input parameters in analytical models has not been directly examined for transient stream depletion. The objective of this research is to identify upscaled transmissivities that effectively model transient stream depletion rates caused by pumping groundwater wells in heterogeneous aquifers. Two-dimensional heterogeneity is considered, with horizontal anisotropy in spatial correlation ranges as a primary independent variable. The subject aquifer is relatively narrow, with an impermeable boundary parallel to a fully connected river boundary. Using numerical flow simulation and the Monte Carlo approach, stream depletion rate curves were computed for transmissivity fields constructed under various geostatistical models of heterogeneity. Effective (expected) and equivalent transmissivities—referring to stochastic ensemble-mean behavior and to individual realizations, respectively—were interpreted from the depletion curves using the Glover analytical solution for stream depletion in a homogeneous, bounded aquifer. The interpreted effective and equivalent transmissivities were related to statistical moments of the heterogeneous fields through power averaging. Effective transmissivity results ranged between the bounding arithmetic and harmonic means, varying with the spatial correlation structure of the transmissivity field, partly as a function of geometric statistical anisotropy. Notably, the shape of that function was similar to what has been derived analytically for steady-state, mean-parallel flow conditions in unbounded domains. Additionally, there was no apparent difference in effective transmissivity results between transient stream depletion conditions and steady-state, mean-parallel flow conditions simulated in the same test domain. Also, unlike some studies on effective permeability under various transient conditions, no time dependency was observed in effective transmissivity for the transient stream depletion case. Results were sensitive to including a nugget effect in the spatial correlation model and to non-stationarity of the transmissivity field. Results were only mildly sensitive to field variance. Ensemble-mean behavior was mildly sensitive to correlation scale, but ensemble variance was strongly sensitive to correlation scale. The latter is to be expected, but is notable for stream depletion considering that transmissivity correlation is often regional in scale and thus often large relative to the scale of the pumping well depletion problem. In such cases, the equivalent transmissivity for a given field and well location was often case-specific and not well-predicted by the expected transmissivity.