Numerical models that solve governing equations for subsurface fluid flow and transport are commonly applied to analyze quantitatively the effects of heterogeneity. These models require maps of spatially variable hydraulic properties. Because complete three-dimensional information about hydraulic properties is never obtainable, numerous methods have been developed to interpolate between data values and use geologic, hydrogeologic, and geophysical information to create images of aquifer properties. Image creation approaches fall into three general categories: structure-imitating, process-imitating, and descriptive. Structure-imitating methods rely on one or more of the following to constrain the geometry of spatial patterns in geologic media: correlated random fields, probabilistic rules, and deterministic constraints developed from facies relations. Structure-imitating methods include spatial statistical algorithms and geologically based sedimentation pattern-matching approaches. Process-imitating models include aquifer model calibration methods and geologic process models. Aquifer model calibration methods use governing equations for subsurface fluid flow and transport to relate hydraulic properties to heads and solute information through history and steady state data matching. Geologic process models combine fundamental laws of conservation of mass and momentum with sediment transport equations to simulate spatial patterns in grain size distributions. At the sedimentary basin scale, multiprocess models include thermomechanical mechanisms of basin subsidence. Descriptive methods couple geologic observations with facies relations to divide an aquifer into zones of characteristic hydraulic properties. All approaches are capable of reproducing heterogeneity over a range of scales and considering some types of geologic information. Some approaches are strictly spatial while some are linked to the time evolution of sedimentation. Some approaches can be conditioned on measurements. Recent advances aimed at infusing geologic information into images of the subsurface include extracting more information from sedimentological facies models, incorporating qualitative geologic information into random field generators and simulating depositional processes. Classes of research missing from the literature include multiprocess models that incorporate diagenesis and three-dimensional surface water flow, hybrid methods that combine features of existing approaches, and approaches that can make use of all available geologic, geophysical, and hydrologic data.

Heterogeneity in Sedimentary Deposits: A Review of Structure‐Imitating, Process‐Imitating, and Descriptive Approaches

We present an expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order Raviart--Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered finite difference method requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a 9-point stencil in two dimensions and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order approximations in the $L^2$- and $H^{-s}$-norms (and superconvergence is obtained between the $L^2$-projection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If $h$ denotes the maximal mesh spacing, then the optimal rate is $O(h)$. The superconvergence rate $O(h^{2})$ is obtained for the scalar unknown and rate $O(h^{3/2})$ for its gradient and flux in certain discrete norms; moreover, the full $O(h^{2})$ is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.

Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences

The properties and limitations of a number of different dynamic pseudo relative permeability methods are summarized. Severe difficulties common to all methods are discussed: choosing the number and locations of the coarse grid rock types, defining the simulations from which the pseudos are generated, and the dependence of the pseudos on well rates and positions. It is concluded that, in practice, pseudos cannot be reliably used to upscale from a 'fine grid' geological model to a 'coarse grid' fluid flow model, except for cases where capillary or gravity equilibrium can be assumed at the coarse grid block scale. Upscaling from the core scale to the geological model is more likely to be possible because capillary forces are more important at smaller scales. A practical approach to the dynamic upscaling problem is outlined, but one should not expect that the effects of the detail in the geological model will be more than qualitatively captured.

A Critical Review of the Use of Pseudorelative Permeabilities for Upscaling

Immiscible flow behaviour in laminated and cross-bedded sandstones

Abstract In upper-crustal structures, folding by flexural slip leads to the partitioning and ordering of strains, which are commonly expressed by means of faults, fractures and joints. Flexural slip occurs along only some of the numerous layer-to-layer contacts within a sedimentary rock sequence, and these active bedding-plane slip surfaces partition the succession into mechanical units, each of which, although composed of many rock layers, deforms as though it were a single beam. Thus, in upper-crustal flexural-slip folds, the strain domains associated with bending are sharply deliminted by the active slip horizons and also are related to structural position within the beam. Progressive folding leads to the development of additional active slip surfaces and, therefore, to an increased number of mechanical units which each develop bending strains. As a result of this sequence of events, there is a hierarchical overprinting of strains, which leads to considerable complexity in fracture fabrics; nevertheless, order in the fracture patterns is retained. Most petroleum reservoirs appear to be affected by fracture networks created during flexure, and their fluid flow characteristics are probably related to a fracture-dominated, flexural deformation state which is disposed relative to the layering, is inhomogeneous, and is not distributed stochastically throughout the whole rock volume.

Strain partitioning during flexural-slip folding

Any transport property that does not have a molecular analogue must be averaged (pseudoized) or scale-adjusted so that it can be used in making reservoir performance predictions. In modelling flow through permeable media, typical properties so treated are absolute permeability, relative permeability, capillary pressure, dispersion coefficients, fingering factors, and equilibrium constants. These properties are rendered scale-dependent by the average. We review and justify the philosophy behind pseudofunction generation based on the notion of vertical equilibrium, a necessary condition for time-independent pseudofunctions. After discussing the traditional approaches to scale adjustment, we give some interesting special cases and illustrate areas of current research. The central theme, however, is that pseudofunctions represent the only practical method of combining fluid and transport properties with spatially varying reservoir properties.

Pseudofunctions — The Key to Practical Use of Reservoir Description

Analytical method to calculate the effective permeability tensor of a grid block and its application in an outcrop study

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Pseudofunctions — The Key to Practical Use of Reservoir Description

Analytical method to calculate the effective permeability tensor of a grid block and its application in an outcrop study