A stochastic Buckley-Leverett model.

Abstract

Even while numerical simulation methods dominate reservoir modeling, the Buckley-Leverett equation provides important insight into the physical processes behind enhanced oil recovery. The interest in a stochastic Buckley-Leverett equation, the subject of this thesis, arises because uncertainty is at the heart of petroleum engineering. Stochastic differential equations, where one modifies a deterministic equation with a stochastic perturbation or where there are stochastic initial conditions, offers one possible way of accounting for this uncertainty. The benefit of examining a stochastic differential equation is that mathematically rigorous results can be obtained concerning the behavior of the solution.

However, the Buckley-Leverett equation belongs to a class of partial differential equations called first order conservation equations. These equations are notoriously difficult to solve because they are non-linear and the solutions frequently involve discontinuities. The fact that the equation is being considered within a stochastic setting adds a further level of complexity. A problem that is already particularly difficult to solve is made even more difficult by introducing a non-deterministic term.

The results of this thesis were obtained by making the fractional flow curve the focus of attention, rather than the relative permeability curves. Reservoir conditions enter the Buckley-Leverett model through the fractional flow function. In order to derive closed form solutions, an analytical expression for fractional flow is required. In this thesis, emphasis in placed on modeling fractional flow in such a way that most experimental curves can readily be approximated in a straightforward manner, while keeping the problem tractable. Taking this approach, a range of distributional results are obtained concerning the shock front saturation and position over time, breakthrough time, and even recovery efficiency.