Conditioning Stochastic Groundwater Flow Models on Head Data

Abstract

We present a new method for conditioning realizations of the logarithm Y of transmissivity on head data. Each realization is conditioned by subtracting a linear combination of certain basis functions, each corresponding to a measurement, obtained by a process based on co-kriging. The required cross covariance between head and Y can be numerically computed very efficiently using the adjoint method. The coefficients in the linear combination are chosen to try to rninimise the differences between computed and measured heads. The most efficient methods for such non-linear optimisation problems are those that exploit information about the second derivatives of the function being minimised. We show how these second derivatives can be computed very efficiendy using the adjoint method for certain classes of finite-element schemes. We illustrate our method by application to the WIPP-2 test case of the international INTRAVAL project. We show that the match between computed and measured heads is greatly improved. We compare our method with approaches that use pilot points. Our method is exact if the variability of the Y field and the deviations of the measured heads from the values computed for the mean of the Y field are both small, whereas pilot-point methods are only approximate. However, our method may be more expensive computationally. Finally, we discuss the extension of our method to conditioning on transient head data and to dealing with boundary conditions.