Inverse Stochastic Moment Analysis of Transient Flow in Randomly Heterogeneous Media

Abstract

A geostatistical inverse method of estimating hydraulic parameters of a heterogeneous porous medium at discrete points in space, called pilot points, is presented. In this inverse method the parameter estimation problem is posed as a nonlinear optimization problem with a likelihood based objective function. The likelihood based objective function is expressed in terms of head residuals at head measurement locations in the flow domain, where head residuals are the differences between measured and model-predicted head values. Model predictions of head at each iteration of the optimization problem are obtained by solving a forward problem that is based on nonlocal conditional ensemble mean flow equations. Nonlocal moment equations make possible optimal deterministic predictions of fluid flow in randomly heterogenous porous media as well as assessment of the associated predictive uncertainty. In this work, the nonlocal moment equations are approximated to second order in the standard deviation of log-transformed hydraulic conductivity, and are solved using the finite element method. To enhance computational efficiency, computations are carried out in the complex Laplace-transform space, after which the results are inverted numerically to the real temporal domain for analysis and presentation. Whereas a forward solution can be conditioned on known values of hydraulic parameters, inversion allows further conditioning of the solution on measurements of system state variables, as well as for the estimation of unknown hydraulic parameters. The Levenberg-Marquardt algorithm is used to solve the optimization problem. The inverse method is illustrated through two numerical examples where parameter estimates and the corresponding predictions of system state are conditioned on measurements of head only, and on measurements of head and log-transformed hydraulic conductivity with prior information. An example in which predictions of system state are conditioned only on measurements of log-conductivity is also included for comparison. A fourth example is included in which the estimation of spatially constant specific storage is demonstrated. In all the examples, a superimposed mean uniform and convergent transient flow field through a bounded square domain is used. The examples show that conditioning on measurements of both head and hydraulic parameters with prior information yields more reliable (low uncertainty and good fit) predictions of system state, than when such information is not incorporated into the estimation process.