OPERATOR AND INTEGRODIFFERENTIAL REPRESENTATIONS OF CONDITIONAL AND UNCONDITIONAL STOCHASTIC SUBSURFACE FLOW

Operator representations of stochastic subsurface flow equations allow writing their solutions implicitly or explicitly in terms of integro- differential expressions. Most of these representations involve Neuman n series that must be truncated or otherwise approximated to become op erational. It is often claimed that truncated Neumann series allow sol ving groundwater flow problems in the presence of arbitrarily large he terogeneities. Such claims have so far not been backed by convincing c omputational examples, and we present an analysis which suggests that they may not be justified on theoretical grounds. We describe an alter native operator representation due to Neuman and Orr (1993) which avoi ds the use of Neumann series yet accomplishes a similar purpose. It le ads to a compact integro-differential form which provides considerable new insight into the nature of the solution. When written in terms of conditional moments, our new representation contains local and nonloc al effective parameters that depend on scale and information. As such, these parameters are not unique material properties but may change as more is learned about the flow system.