A Comparison of Several Methods for the Solution of the Inverse Problem in Two‐Dimensional Steady State Groundwater Flow Modeling

Abstract

Two geostatistical approaches for the estimation of hydraulic conductivity and hydraulic head from hydraulic conductivity and hydraulic head measurements are developed for two‐dimensional steady flow with sinks. For both approaches the field of the logarithm of hydraulic conductivity (log‐conductivity) is represented as a random field with mean θ1+θ2x+θ3y where x and y denote Cartesian coordinates, variance σ2, and covariance σ2, exp (−αR), σ2, exp (−αR2), or, σ2ΔRr where α, Δ, and r are constants and R is separation distance. The first approach uses linearization of the discretized flow equations to allow the construction of the joint covariance matrix of the hydraulic head and log‐conductivity measurements as functions of the parameters θ, θ2, θ3, σ2, and α or Δ. It then uses maximum likelihood estimation to obtain these parameters and also a parameter associated with log‐conductivity measurement error. Having found values for the parameters, it then uses kriging to form predictors for log‐conductivity and hydraulic head from measured values of hydraulic conductivity and hydraulic head. The second approach uses kriging to form a parameter‐dependent predictor for log‐conductivity from measured hydraulic conductivity, and then uses this predicted log‐conductivity placed into the discretized flow equations to compute hydraulic head. The parameters are determined by the minimization of the sum of the squares of the difference between the measured and computed hydraulic heads. A third approach simply allows the hydraulic conductivity field to be a step function with a different value for hydraulic conductivity assigned to each of several chosen regions in the two‐dimensional aquifer. The assigned hydraulic conductivities are determined by the minimization of the sum of the squares of the difference between the measured and computed hydraulic heads. The three approaches are tested for hydraulic head prediction accuracy on two generated test problems, one of which is statistically generated, and also on a field problem. The third approach, despite its simplicity, performs as well or better than the other approaches.