A finite-difference model of three-dimensional granular displacement

Recent advances in aquifer mechanics have shown that the hydrodynamic processes associated with land subsidence and earth fissuring due to fluid withdrawal in unconsolidated aquifers are three dimensional in scope. Previous mathematical and numerical models that use hydraulic head or volume strain as the principal unknown variable have traditionally been one dimensional with respect to changes in storage and strain because they assume no horizontal strain. These one-dimensional models can accurately simulate the total vertical compaction of interbeds in a confined aquifer, but they have no way of predicting horizontal changes in strain or granular movement, and hence can not estimate where damaging fissures may occur over time. This report describes a new three-dimensional finite-difference numerical model that has been developed and integrated into the U.S. Geological Survey's modular ground-water flow model. The displacement field of solids is the principal unknown variable within the new model. Because the displacement field of solids is a vector quantity, granular displacement resulting from imposed stresses on an unconsolidated aquifer can be simulated in three dimensions. The new model is not limited to confined or homogeneous aquifers, but can be readily applied to unconfined and heterogeneous aquifers with complex boundaries.; The three-dimensional governing equation is decoupled and each component direction is solved for, first independently, then corporately with the other principal directions. Each of the three decoupled equations is expressed numerically using a Crank-Nicolson scheme. Solution of the set of equations is accomplished with a dual-loop successive overrelaxation technique, while taking advantage of Chebyshev acceleration. Simulation of horizontal displacements compare accurately with analytic solutions for a homogeneous, isotropic confined aquifer. Simulation of vertical displacements of fine-grained interbeds within a confined aquifer compare favorably with results obtained using the one-dimensional interbed storage model. The inclusion of an overlying horizontal barrier to vertical flow results in an increase in calculated subsidence along the edges of the barrier and a decrease in subsidence directly above the pumped well. A vertical barrier to horizontal flow tends to increase subsidence above the pumped well. The horizontal location of the wellbore tends to be drawn toward the barrier resulting in compressional strain between the barrier and the pumped well. Displacement is significant on the side of the barrier opposite the pumped well.