Conservation Equations for Ground-Water Velocity in General Conditions

The conservation equation for the random ground-water flow seepage velocity is derived under general conditions of hydraulic stochastic functions (hydraulic conductivity, storativity, and porosity). The methodology is further extended to obtain two new conservation equations for the covariance of random velocity and the correlation of random velocity with its displacement. All of these conservation equations for the random variables govern the dynamics of the ground-water flow field. In this study, the cumulant expansion method used in the real time-space domain for a single scalar is generalized to cover cases of vector and tensor random variables. The group theoretic methods of chronological exponentials and integrals and the theory of Lie algebra are used extensively in the derivation of the operator form of ensemble average equation based on second-order generalized cumulant expansion. The application of the operator ensemble average equation to the special cases of random velocity and its second-order simple and mixed cumulants resulted in the mean conservation equations of those random functions. The newly obtained conservation equations for the ensemble mean of random velocity and its covariance have convective-dispersive forms. Under general conditions, the ensemble mean and covariance of the random hydraulic field variables dictate the convective and dispersive character of the mean average equations. Stochastic driving forces, hydraulic conductivity, specific storativity, and porosity are functions of time and space, and no a priori assumptions are needed for the statistical characteristics of these random fields. The dynamics and evolution of approximate forms of the conservation equations, where porosity and specific storativity are assumed to be constants, are completely determined by the hydraulic conductivity field. The entire ensemble average conservation equations for the random velocity and its second-order correlations (covariance of velocity and the correlation of velocity with its displacement) have a common mixed Eulerian-Lagrangian feature. This mixed character takes into account not only the local changes but also the interactions of random functions taking place in a period of time and space. The integral of the covariance of random hydraulic field variables determines the magnitude of the diffusive property of the dependent variable.