Stochastic modeling of groundwater flow by unconditional and conditional probabilities: 1. Conditional simulation and the direct problem

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Abstract
Mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated. The formation properties (hydraulic conductivity, transmissivity) as well as flow variables (head, specific discharge, solute concentration) are regarded as random variables subjected to uncertainty. The main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables. The unconditional probability density functions are supposed to be stationary multivariate normal, while conditioning accounts for the measured values at a few points of the formation. Two problems of groundwater flow are discussed in part 1: conditional simulation and the direct problem for steady flow. Analytical results are obtained by using perturbation approximations. Average uniform head gradient flows as well as recharge and flow to wells are discussed. In unconditional modeling the input variable (the conductivity or transmissivity log) is regarded as stationary and is represented by its constant mean and variogram. The ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions. In conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points. Consequently, statistical computations are performed for the subensemble of aquifers which preserve the measured values, and as a result, both input and output variables are nonstationary. The main effect of conditioning is to reduce the variance, i.e., the uncertainty, of these variables. This effect is particularly important in the case of flow toward wells. Application of the modeling method to field problems is outlined.