Direct Solution of Reservoir Flow Equations with Uncertain Parameters

This paper presents a direct method to determine the uncertainty in reservoir pressure, and other functions, using the time-dependent one phase 2- and 3-dimensional reservoir flow equations. The uncertainty in the solution is modelled as a probability distribution function. This is derived from probability distribution function for input parameters such as permeability. The method involves a perturbation expansion about a mean of the parameters. Coupled equations for second order approximations to the mean at each point and field covariance of the solution, are developed and solved numerically. This method involves only one (albeit complicated) solution of the equations, and contrasts with the more usual Monte-Carlo approach, where many such solutions are required. The procedure is a development of earlier steady-state two-dimensional analyses arid a transient mass-balance analysis using uncertain parameters. These methods can be used to find the risked value of a field for a given development scenario.