Stochastic partial differential equations applied to hydrocarbon reservoir modelling - Aziz, M (2010)
This thesis describe the generation of random fields using elliptic stochastic partial differential equations driven by Gaussian white noise. The intended application is that of sampling from a prior probability density in the solution of an inverse problem. In the first part of thesis, we will study two specific examples of isotropic models, namely the stochastic Helmholtz equation and stochastic biharmonic equation. We will address the analytical properties of a particular solution and the conditions required for the uniqueness of the Green’s function. A useful formula is also derived linking the computation of the covariance function of the solution to that of the Green’s function. Moreover, in each example the correlation function and approximate realizations are computed and simulated. In the second part of the thesis, we will investigate the possibility of using vector fields to guide the random fields. The vector fields will be used to build tensor coefficients for an elliptic stochastic partial differential equation. The model considered is capable of producing non-stationary random fields on curved layers. Approximate realizations of this model will be drawn by using the finite element method, which is applied using the MATLAB PDE toolbox.