Stochastic Hydrology and Hydraulics

, Volume 3, Issue 2, pp 135–153

# Stochastic partial differential equations in groundwater hydrology

Part I: Theory
• T. E. Unny
Originals

DOI: 10.1007/BF01544077

Unny, T.E. Stochastic Hydrol Hydraul (1989) 3: 135. doi:10.1007/BF01544077

## Abstract

Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.

Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.

There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.

Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.

This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.

### Key words

Stochastic partial differential equations maximum likelihood estimation parameter estimation moment equations stodhastic contaminant transport