Fault models are often based on interpretations of seismic data that are constrained by observations of faults and associated strata in wells. Because of uncertainties in depth migration, seismic interpretations and well data, there often is significant uncertainty in the geometry and position of the faults. Fault uncertainty impacts determinations of reservoir volume, flow properties and well planning. Stochastic simulation of the faults is important for quantifying the uncertainties and minimizing the impacts. In this paper, a framework for representing and modeling uncertainty in fault location and geometry is presented. This framework can be used for prediction and stochastic simulation of fault surfaces, visualization of fault location uncertainty, and assessments of the sensitivity of fault location on reservoir performance. The uncertainty in fault location is represented by a fault uncertainty envelope and a marginal probability distribution. To be able to use standard geostatistical methods, quantile mapping is employed to construct a transformation from the fault surface domain to a transformed domain. Well conditioning is undertaken in the transformed domain using kriging or conditional simulations. The final fault surface is obtained by transforming back to the fault surface domain. Fault location uncertainty can be visualized by transforming the surfaces associated with a given quantile back to the fault surface domain.

An algorithm for the computation of the solution to Laplace's equation in a 2-dimensional region is given in terms of equivalent sources on the boundary. The region may be of an arbitrary shape, and the boundary conditions may be an arbitrary combination of Dirichlet, Neumann and impedance types. The solution is obtained by a moment method, using either a step approximation to the source, or a piecewise-linear approximation. Point matching is used for testing the boundary conditions. Computer programs are available for the general problem, and some electromagnetic-field applications are discussed.List of symbols a, j8, y = arbitrary functions of c = solution to Laplace's equation V F, = potential from strip of uniform source on AC, A,-= length of element AC,-of curve C c = length variable along curve C g = conductivity k = constant Iji = elements of matrix [/] n = outward unit vector normal to curve C u = unit vector tangential to curve C w = complex function x, y -co-ordinates of point in space z -x + Jy = complex variable C = capacitance D = electric-flux density E = electric-field intensity H = magnetic-field intensity / = current J = current density P = power • P-, = pulse function Q -electric charge R = resistance V = constant potential Z o = characteristic impedance of a transmission line * denotes a complex conjugate

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A multigrid Markov mesh model for geological facies is formulated by defining a hierarchy of nested grids and defining a Markov mesh model for each of these grids. The facies probabilities in the Markov mesh models are formulated as generalized linear models that combine functions of the grid values in a sequential neighborhood. The parameters in the generalized linear model for each grid are estimated from the training image. During simulation, the coarse patterns are first laid out, and by simulating increasingly finer grids we are able to recreate patterns at different scales. The method is applied to several tests cases and results are compared to the training image and the results of a commercially available snesim algorithm. In each test case, simulation results are compared qualitatively by visual inspection, and quantitatively by using volume fractions, and an upscaled permeability tensor. When compared to the training image, the method produces results that only have a few percent deviation from the values of the training image. When compared with the snesim algorithm the results in general have the same quality. The largest computational cost in the multigrid Markov mesh is the estimation of model parameters from the training image. This is of comparable CPU time to that of creating one snesim realization. The simulation of one realization is typically ten times faster than the estimation.