Existing algorithms of geostatistics for stochastic modelling of reservoir parameters require a mapping (the 'uvt-transform') into the parametric space and reconstruction of a stratigraphic co-ordinate system. The parametric space can be considered to represent a pre-deformed and pre-faulted depositional environment. Existing approximations of this mapping in many cases cause significant distortions to the correlation distances. In this work we propose a coordinate free approach for modelling stochastic textures through the application of stochastic partial differential equations. By avoiding the construction of a uvt-transform and stratigraphic coordinates, one can generate realizations directly in the physical space in the presence of deformations and faults. In particular the solution of the modified Helmholtz equation driven by Gaussian white noise is a zero mean Gaussian stationary random field with exponential correlation function (in 3-D). This equation can be used to generate realizations in parametric space. In order to sample in physical space we introduce a stochastic elliptic PDE with tensor coefficients, where the tensor is related to correlation anisotropy and its variation is physical space.
Potsepaev R, Farmer CL (2010) Application of Stochastic Partial Differential Equations to Reservoir Property Modelling. 12th European Conference on the Mathematics of Oil Recovery. Available: http://dx.doi.org/10.3997/2214-4609.20144964.
We would like to thank Ben Hambly and David Allwright for their useful remarks and help. R.V.Potsepaev thanks Schlumberger for support and for permission to contribute to this paper.This publication is based on work by C.L. Farmer, supported by Award Number KUK-C1-013-04, madeby King Abdullah University of Science and Technology (KAUST).