No abstract

The Bayesian approach allows one to easily quantify uncertainty, at least in theory. In practice, however, MCMC can be computationally expensive, particularly in complicated inverse problems. Here we present methodology for improving the speed and efficiency of an MCMC analysis by combining runs on different scales. By using a coarser scale, the chain can run faster (particularly when there is an external forward simulator involved in the likelihood evaluation) and better explore the posterior, being less likely to become stuck in local maxima. We discuss methods for linking the coarse chain back to the original fine scale chain of interest. The resulting coupled chain can thus be run more efficiently without sacrificing the accuracy achieved at the finer scale.

The problem of mapping reservoir properties, such as porosity and permeability, and of assessing the uncertainty in the mapping has been largely approached probabilistically, i.e. uncertainty is estimated based on the properties of an ensemble of random realizations of the reservoir properties all of which satisfy constraints provided by data and prior geological knowledge. When the constraints include observations of production characteristics, the problem of generating a representative ensemble of realizations can be quite difficult partly because the connection between a measurement of water-cut or GOR at a well and the permeability at some other location is by no means obvious. In this paper, the progress towards incorporation of production data and remaining challenges are reviewed.

This work discusses the development and implementation of a procedure to condition a stochastic channel to well-test pressure data and well observations of the channel thickness and the depth of the top of the channel. The stochastic channel is defined by a set of geometric random variables (referred to as geometric model parameters) that describe the location, size and shape. Channel and nonchannel permeability and porosity are treated as random variables. These four random variables plus the geometric parameters comprise the complete set of model parameters. Multiple conditional realizations of the geometric parameters and rock properties are generated to evaluate the uncertainty in model parameters and the reduction in uncertainty obtained by conditioning to well-test pressure data.

Many enhanced oil recovery processes in reservoir engineeringinvolvelocalized phenomena that could be due to several features, such asinjection fronts, wells or reservoir heterogeneity. In order to reach sufficientaccuracy in field-scale simulation, the localized phenomena need to be resolvedand modeledin appropriate scale-dependent ways. Our approach to treating thelocalized phenomena is to usehigh-resolution discretization methods incombination with dynamicallyadaptive mesh refinement(AMR). The purpose ofadaptive mesh refinement is to concentrate thecomputational work near theregions of interest in the displacement processes, which may evolve constantlyin space. Adaptive mesh refinement requires appropriate techniquesfor datacommunication in a hierarchy of dynamically adaptive mesh. The selection ofappropriate scaling rules as well as computationallyefficient data structuresis essential to the success of the overallmethod. We have exploited the object-oriented features of C++ for the AMR programstructure and data management, while numerically intensive routines areimplemented in FORTRAN. It turned out that adaptive mesh refinement cansignificantlyreduce the computational cost required to obtain a desired levelofaccuracy in the simulation. However, the emphasis here is on the developmentof efficienttechniques for solving linear systems that arise in thenumericaldiscretization of an elliptic equation for the incompressiblepressurefield. We use a conjugate gradient algorithm preconditioned bymultiplicative domain decomposition between refinement levels, in whichadditivedomain decomposition and incomplete Cholesky factorization were employed as"smoothers". In this paper, the combined adaptive mesh refinement technique hasbeenapplied to a single-phase tracer transport model for miscibleflooding. Numerical results demonstrating the effectiveness of the methodarepresented and discussed.