Well Interference Test Analysis In Stochastic Porous Media

Interference well test analysis provides valuable information about reservoir characteristics such as permeability and hydraulic diffusivity coefficient. Interference well test analysis is based on solution of the diffusivity equation which describes mass transfer in a porous medium. Generally, analytical solutions are used for interpreting interference test data. However all these solutions were obtained under the condition of reservoir homogeneity. In heterogeneous reservoirs with spatially variable permeability, the exact analytical solutions are not known. A heterogeneous permeability field can be represented as the sum of two terms. The first term is the constant mean permeability value and the second one is the random function with known statistical properties. The second term is considered as a perturbation. The possibility to evaluate geostatistical parameters from well test analysis was considered by various authors and it is still a challenging problem. In heterogeneous reservoirs, 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 Green’s function is represented as an infinite perturbation series. This series expansion can be written graphically using Feynman diagrams and its summation can be performed following the rules that are well known in quantum theory. This approach was first introduced to reservoir simulation in [1], where the stochastic pressure equation was solved for the steady-state case.

In this study we use diagrammatic analysis to obtain the solution of the time dependent stochastic pressure equation. This solution was used in interpretation of well test interference data. The calculations were carried out assuming the statistics of the random permeability field are Gaussian and the covariance of the logarithm of the permeability is exponentially decaying. The two limiting cases were considered: (i) the distance between wells is much bigger than the permeability correlation length, (ii) the opposite case when the correlation length is the smallest length parameter. Using different realizations of a synthetic reservoir model with fixed statistical parameters, the ensemble of well interference data was numerically generated. The mean value and correlation length of the permeability distribution were estimated so that the solution for stochastic pressure reproduced the averaged results of numerical simulations.

[1] King P.R. J .Phys. A: Math. Gen 20 p. 3935 – 3947 1987