Hydrologic characterization of heterogeneous geologic media with an inverse method based on iterated function systems

Abstract

One way to estimate the hydrologic properties of heterogeneous geologic media is to invert well test data using multiple observation wells. Pressure transients observed during a well test are compared to the corresponding values obtained by numerically simulating the test using a mathematical model. The parameters of the mathematical model are varied and the simulation repeated until a satisfactory match to the observed pressure transients is obtained, at which point the model parameters are accepted as providing a possible representation of the hydrologic property distribution. Restricting the search to parameters that represent self‐similar (fractal) hydrologic property distributions can improve the inversion process. Far fewer parameters are needed to describe a hierarchical medium, improving the efficiency and robustness of the inversion. Additionally, each parameter set produces a hydrologic property distribution with a hierarchical structure, which mimics the multiple scales of heterogeneity often seen in natural geological media. The parameters varied during the inversion create fractal sets known as attractors, using an iterated function system (IFS). An attractor is mapped to a distribution of transmissivity and storativity in the mathematical model. Thus the IFS inverse method searches for the parameters of the IFS (typically tens of parameters) rather than the values of the hydrologic property distribution directly (typically hundreds to thousands of parameters). Application of the IFS inverse method to synthetic data shows that the method works well for simple heterogeneities. Application to field data from a sand/clay sedimentary sequence and a fractured granite produces reasonable results.