SOLUTION OF STOCHASTIC GROUNDWATER-FLOW BY INFINITE SERIES, AND CONVERGENCE OF THE ONE-DIMENSIONAL EXPANSION

This paper investigates analytical solutions of stochastic Darcy flow in randomly heterogeneous porous media. We focus on infinite series so lutions of the steady-state equations in the case of continuous porous media whose saturated log-conductivity (lnK) is a gaussian random fie ld. The standard deviation of lnK is denoted 'sigma'. The solution met hod is based on a Taylor series expansion in terms of parameter sigma, around the value sigma = 0, of the hydraulic head (H) and gradient (J ). The head solution H is expressed, for any spatial dimension, as an infinite hierarchy of Green's function integrals, and the hydraulic gr adient J is given by a linear first-order recursion involving a stocha stic integral operator. The convergence of the 'sigma-expansion' solut ion is not guaranteed a priori. In one dimension, however, we prove co nvergence by solving explicitly the hierarchical sequence of equations to all orders. An 'infinite-order stochastic solution is obtained in the form of a sigma-power series that converges for any finite value o f sigma. It is pointed out that other expansion methods based on K rat her than lnK yield divergent series. The infinite-order solution depen ds on the integration method and the boundary conditions imposed on in dividual order equations. The most flexible and general method is that based on Laplacian Green's functions and boundary integrals. Imposing zero head conditions for all orders greater than one yields meaningfu l far-field gradient conditions. The whole approach can serve as a bas is for treatment of higher-dimensional problems.