Stochastic heterogeneity and dispersion

We study stochastic representation of heterogeneity and the effect of heterogeneity on dispersion for numerical reservoir simulation.;Accurate autocorrelation specification is important in stochastic representation of heterogeneous reservoirs. We study the bias and precision of autocorrelation estimates using both analytical and numerical methods in multiple dimensions. The bias of the sample autocovariance function is shown to be significant at large correlation. The dimensionless correlation length and the number of dimensions are major factors affecting the accuracy of the autocorrelation estimates.;We use the Turning Bands Method in this study to generate stochastic fields to represent spatial heterogeneity. The generation scheme is modified to increase efficiency and accuracy. Extensions include transformation to non-normal distributions, geometric anisotropy in correlation, nested correlation structure, and addition of nonstationarity. The Matrix Decomposition Method is also discussed. We treat random fractals as special correlation models and use them in the same generation methods.;We extend the study of finite difference truncation error to heterogeneous anisotropic media. When approximating the pressure equation, a 9-point FD scheme with heterogeneous media or in the presence of a source term is no more accurate than a 5-point scheme. Discontinuity in heterogeneity changes the truncation error from second-order to first-order with uniform grid and from first-order to zero-order with non-uniform grid. Generally the truncation error in the FD approximation of the convection-dispersion equation is first-order in ;We study the effect of spatial variation of permeability on single-well tracer tests first within a single layer using numerical simulation and then with a layered model analytically. Within a single layer, a deviation from the homogeneous isotropic case includes two opposing effects on the apparent dispersivity. The shifting toward a one-dimensional flow pattern, caused by anisotropy or high permeability channels, reduces the apparent dispersivity. Otherwise heterogeneity increases dispersion. In a layered model, there are also opposing effects. Unequal depths of investigation among the layers reduces the apparent dispersivity. The separation of the tracer fronts when they arrive at the well from different layers, caused by flow irreversibility, increases the apparent dispersivity. Irreversible flow can result from compressibility and permeability variation among the layers.