First-order variance of solute travel time in non-stationary media

We study solute-flux statistics in heterogeneous porous media exhibiting non-stationary covariance functions of the log-conductivity field. In particular, we investigate the cases of 1) variance scaling, in which the variance undergoes a spatial trend, 2) the blending of multiple covariance functions, including the case of zonal stationarity, 3) non-stationarity because of uncertainty in parameters describing the spatial trend of the mean log-conductivity, and 4)non-stationarity as a result of conditioning. All of these cases can be traced back to stationary covariance functions undergoing particular modifications. We calculate the first-order variance of travel time by quadratic multiplication of the discretized covariance matrix of log-conductivity fluctuations with the sensitivity matrix of the travel-time at the points of interest. The sensitivities are calculated by the adjoint-state method. The matrix-matrix multiplications are performed by Fast Fourier Transformation techniques requiring to embed the stationary counterparts of the non-stationary fields into periodic domains. The method is very efficient with respect to both computational effort and memory requirements. We apply the approach to a two-dimensional field of 1000 x 500 elements in which the correlation length and the variance have a spatial trend, and the parameters of the trend-model for the mean are uncertain. We include the case of conditioning on head and total-flux measurements.