Experimental design method is an alternative to traditional sensitivity analysis. The basic idea behind this methodology is to vary multiple parameters at the same time so that maximum inference can be attained with minimum cost. Once the appropriate design is established and the corresponding experiments (simulations) are performed, the results can be investigated by fitting them to a response surface. This surface is usually an analytical or a simple numerical function which is cheap to sample. Therefore it can be used as a proxy to reservoir simulation to quantify the uncertainties. Designing an efficient sensitivity study poses two main issues:Designing a parameter space sampling strategy and carrying out experiments.Analyzing the results of the experiments. (Response surface generation) In this paper we investigate these steps by testing various experimental designs and response surface methodologies on synthetic and real reservoir models. We compared conventional designs such as Plackett-Burman, central composite and D-optimal designs and a space filling design technique that aim at optimizing the coverage of the parameter space. We analyzed these experiments using linear, second order polynomials and more complex response surfaces such as kriging, splines and neural networks. We compared these response surfaces in terms of their capability to estimate the statistics of the uncertainty (i.e., P10, P50 and P90 values), their estimation accuracy and their capability to estimate the influential parameters (heavy-hitters). Comparison with our exhaustive simulations showed that experiments generated by the space filling design and analyzed with kriging, splines and quadratic polynomials gave the greatest accuracy while traditional designs and the associated response surfaces performed poorly for some of the cases we studied. We also found good agreement between polynomials and complex response surfaces in terms of estimating the effect of each parameter on the response surface. Introduction Reservoir simulators are capable of integrating detailed static geological information with dynamic engineering data to represent the complex fluid flow in porous media. Therefore they have been used extensively for planning and evaluation of field development projects. Usually economical parameters such as net present value (NPV) or recovery estimates such as cumulative oil production are used to assess the value of different alternatives of a development study. Since most of the inputs to the simulation studies are usually uncertain and uncontrollable (like static reservoir properties), many sensitivity studies have to be performed, which might be prohibitive due to costly simulations. Experimental design methodology offers not only an efficient way of assessing uncertainties by providing inference with minimum number of simulations, but also can identify the key parameters governing uncertainty in economic and production forecast, which might guide the data acquisition strategy during the early phases of a field development project.[1] The commonly used workflow for this purpose is as follows:Define a large set of potential key parameters and their probability distributions.Perform a low level experimental design study, such as Plackett-Burman, which combines the high and low value of the key parameters.Perform simulations corresponding to each of the experiments.Fit the economical or recovery estimates obtained from simulations to a simple response surface, which is usually a line.Using the probability distributions attached to the parameters, perform a Monte Carlo simulation on the response surfaceGenerate a tornado diagram to rank the effect of each parameter on the economical or recovery estimates.Screen the heavy-hitters. From the tornado diagram.Perform a more detailed design such as full/fractional factorial, D-optimal, Box-Behnken, central composite, etc. with the heavy-hitters.Repeat steps 3 and 4.Perform a Monte Carlo simulation on the new response surface to get the probability density function (pdf) of the economical or recovery estimates.
Grid orientation effects can arise in simulations of recovery processes when the mobility ratio of the displacement is unfavorable. These effects result from the application of numerical solution techniques to equations describing physically unstable displacement processes. In theory, the introduction of stabilizing terms (such as physical dispersion in the case of a miscible displacement) in the governing equations acts to minimize these effects. In practice, however, the numerical dispersion resulting from the first order discretization of the convective terms overwhelms the physically dispersive terms, and the stabilizing effects of these terms are not accurately modeled.In this paper, we apply higher order, shock capturing, finite difference methods to the simulation of miscible and immiscible displacement processes. This will be shown to reduce numerical dispersion to a level where the effects of the stabilizing terms can be accurately resolved numerically, resulting in the minimization of grid orientation effects. Two different families of higher order methods are investigated. The first such method, a so-called second order TVD (total variation diminishing) scheme, reduces numerical dispersion somewhat relative to the usual first order scheme. The second method investigated, a third order ENO (essentially non-oscillatory) scheme, reduces numerical dispersion even further, to an extent such that the stabilizing terms can be accurately resolved. This will be seen to nearly eliminate grid orientation effects in some cases in which the usual first order finite difference approach yields highly grid dependent solutions and the second order approach yields slightly grid dependent solutions.
The gridblock permeabilities used in reservoir simulation are commonly determined through the upscaling of a fine scale geostatistical reservoir description. Though it is well established that permeabilities computed in this manner are, in general, full tensor quantities, most finite difference reservoir simulators still treat permeability as a diagonal tensor. In this paper, we implement a capability to handle full tensor permeabilities in a general purpose finite difference simulator and apply this capability to the modeling of several complex geological systems. We formulate a flux continuous approach for the pressure equation by use of a method analogous to that of previous researchers (Edwards and Rogers 1 ; Aavatsmark et al. 2 ), consider methods for upwinding in multiphase flow problems, and additionally discuss some relevant implementation and reservoir characterization issues. The accuracy of the finite difference formulation, assessed through comparisons to an accurate finite element approach, is shown to be generally good, particularly for immiscible displacements in heterogeneous systems. The formulation is then applied to the simulation of upscaled descriptions of several geologically complex reservoirs involving crossbedding and extensive fracturing. The method performs quite well for these systems and is shown to capture the effects of the underlying geology accurately. Finally, the significant errors that can be incurred through inaccurate representation of the full permeability tensor are demonstrated for several cases.
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