A stochastic inverse solution for transient groundwater flow: Parameter identification and reliability analysis

In this paper, a stochastic approach is developed for solving the inverse problem of parameter identification for transient groundwater flow. Adjoint state equations are derived for the stochastic partial differential equation (SPDE) relating transient head and log hydraulic conductivity perturbations. The derived equations can be used to calculate the covariance matrix of head observations and the cross‐covariance matrix between head observations and log hydraulic conductivity measurements for each observation time, as well as the covariance and cross‐covariance matrices between different observation times. The number of simulation runs required for obtaining these matrices is equal to the number of head observations. The reliability of model prediction is evaluated through the variance estimate method using adjoint sensitivity analysis and the cokriging estimate. The reliability evaluation of model prediction can be used as an aid in the optimization of experimental design for sampling strategies. A numerical example is given to illustrate the stochastic inverse procedure for a general transient flow problem. The example shows that using head observations of all observation times simultaneously produces much better results than using them sequentially (commonly known as the quasi‐steady approach). The example also shows that it is possible to obtain a reliable stochastic inverse solution for meeting a given prediction objective based on a limited number of observations.