The use of Feynman diagrammatic approach for well test analysis in stochastic porous media

Abstract

Transient well test analysis provides valuable information about reservoir characteristics such as permeability and the hydraulic diffusivity coefficient. It is based on the solution of diffusivity equation, which describes mass transfer in porous media. On the whole, analytical solutions are used for interpreting well test data. However, all these solutions were obtained under the condition of reservoir homogeneity. In a heterogeneous reservoir with spatially variable permeability, the exact analytical solutions are not known. The heterogeneous permeability field can be represented as the sum of two terms. The first term is a constant mean permeability value and the second one is a random function with known statistical properties. The second term can be considered a perturbation. The possibility of evaluating geostatistical parameters from well test analysis was considered by various authors and is still a challenging problem. In a randomly heterogeneous reservoir, a flow equation is formulated for the pressure, which is averaged over all the permeability realizations. It can be solved using Green’s function techniques, where the ensemble-averaged pressure is represented as an infinite perturbation series. This series can be represented graphically using Feynman diagrams and its summation can be performed following the rules that are well known in the quantum theory of solid state. For the first time, this framework was introduced to reservoir simulation in King (J. Phys. A: Math. Gen. 20, 3935–3947, 1987), where the stochastic pressure equation was solved for the steady-state case. In this study, we use diagram approach to obtain the solution of the time-dependent stochastic pressure equation which is derived for a lognormal random permeability field under the assumption of the Gaussian correlation function. The expression for transient ensemble averaged pressure is obtained with respect to high-order corrections of permeability variance. In the limit of sufficiently small variance, analytical expressions for the pressure correction are presented. The two limiting cases were considered: (i) the distance between wells is much bigger than the permeability correlation length; (ii) the opposite case where the correlation length is the smallest length parameter. The resulting solution can be used for the analysis of drawdown, build-up, and interference tests in stochastic porous media. The possibility of estimating the parameters of a random permeability field based on well test data is discussed.