# Numerical Modelling of Liquid Waste Injection into Porous Media Saturated with Density-Stratified Fluid: A Progress Report

##### Numerical Modelling of Liquid Waste Injection into Porous Media Saturated with Density-Stratified Fluid: A Progress Report

Stephen Wheatcraft
December 1977

ABSTRACT
Waste effluent injected into an aquifer saturated with denser ambient brackish or salt water experiences a buoyant lift. As a result, the effluent migrates both outward from the well and upward in response to the combined effects of injection head and buoyant force. After the injection process has begun, several phenomena can affect the density, shape, and distribution in space and time of the resulting buoyant plume. The most important of these include convection and mechanical dispersion and molecular diffusion. Previous sandbox and Hele-Shaw. Laboratory modeling work have provided a basic qualitative understanding of buoyant plume movement in a porous medium. However, these Laboratory models cannot correctly simulate dispersion phenomena which may have significant effects on buoyant plume movement and distribution. Consequently, it is necessary to mathematically model the problem using coupled sets of partial differential equations which take into account the effects of dispersion and diffusion, as well as convection. For this problem, there are four unknowns (density, concentration, velocity, and pressure), requiring four equations. The four governing equations are: a motion equation (Darcy’s Law), a continuity equation, a dispersion equation, and an equation of state. In addition, boundary and initial conditions must be stipulated. In this study, two sets of boundary conditions are used: the first consists of conditions identical to those in the sandbox model studies, and the second models the geology of a specific prototype area. The resulting governing equations and boundary and initial conditions are numerically solved by both the finite difference and the finite element methods. Finally, the numerical models are calibrated with the results of the sandbox model studies mentioned previously. This report describes in detail formulation of the governing equations and the initial and boundary conditions, and preliminary finite difference modeling work completed to date.