We present an analytical and numerical investigation of the finite difference computation of the equivalent conductivity of heterogeneous porous media. The customary harmonic scheme to evaluate finite difference internodal transmissivities produces a systematic bias in the numerical results unless an extremely fine grid is used. In order to quantify such effects, we have developed an analytical approach in the form of a series expansion of the equivalent numerical conductivity in powers of the conductivity variance. This leads to an expression of the numerical answer as a function of the grid block size. The calculation confirms the existence of a strong bias and of a very slow convergence. We propose a simple method to correct it, which is well suited for upscaling. Numerical experiments performed with more contrasting heterogeneous media show similar results. This allows the use of coarser gridding and consequently an appreciable speedup in the numerical simulation approach.
Recent high-resolution tools and novel characterization techniques lead to simulate very detailed geological descriptions of heterogeneous reservoirs. To perform full-field displacement simulations, scaling-up techniques must be used because of the required coarser grids. Pseudoization methods are commonly used to get the relative permeability and capillary pressure curves. But they all consider 2D flow to realize the scaling-up. We propose a three-dimensional approach. When capillary forces are important inside the reservoir, the capillary pressure may be considered as nearly constant inside a coarse grid-block. This assumption enables to compute directly the fine grid-block saturations and the average coarse grid-block saturation for a given capillary pressure. For each fluid phase, we then compute the effective phase permeabilities of the coarse grid-block by means of an absolute permeability scaling-up method applied to phase permeabilities of the fine grid-blocks in 3D. Then renormalizing the resulting phase permeabilities by the average absolute permeability yields the average relative permeabilities curves for the three directions of space. Application of this technique, using an algebraic computation of average permeability adapted to heterogeneous media, will be presented for complex fluviatile cases, characterized by a large amount of shale inclusion at small-scale. Comparison of numerical simulations of oil-water flow on high-resolution descriptions, and on the corresponding scaled-up coarse models, shows the efficiency of this method. We recommend to apply this method to reservoir with low gravity segregation, with rocks exhibiting medium to low permeability and high capillary forces, and moderate injection rates.
Summary The perturbation method provides approximate solutions of the well pressure for arbitrarily heterogeneous media. Although theoretically limited to small permeability variations, this approach has proved to be very useful, providing qualitative understanding and valuable quantitative results for many applications. The well pressure solution using this method is expressed by an integral equation where the permeability variations are weighted by a kernel, the permeability weighting function. As discussed in previous papers, deriving such permeability weighting functions appears to be a complicated calculation, available only for special cases. In this article we present simple and general method to calculate the permeability weighting function. In the Laplace domain, the permeability weighting function is easily related to the pressure solution of the background problem. Since Laplace pressure solutions are known for many situations (various boundary conditions, stratified and composite media etc.), the associated permeability weighting function can be derived immediately. Among other examples, we calculate and discuss the well pressure solution for a horizontal well that is producing from a heterogeneous reservoir. Introduction The trend for reservoir characterization has stimulated the study of well testing in more complex heterogeneous media. Well testing in heterogeneous media has been studied by three approaches: exact analytical solutions, numerical simulations and approximate analytical solutions. Exact analytical solutions exist for a restricted class of problems that involve some simple symmetry: layered reservoirs, single linear discontinuities, radial composite systems etc.1 Rosa and Horne2 computed the exact solution in the case of an infinite homogeneous reservoir containing a single circular permeability discontinuity. Most of these analytical solutions are written in the Laplace domain. Numerical methods can treat much more general situations, but have some disadvantages: their use is cumbersome, investigation is empirical and general insights are difficult to be extracted, results are inaccurate if the time and the spatial discretization were not carefully conducted. Approximate analytical solutions can be a practical way to understand the pressure behavior in geometrically complex heterogeneous media. Kuchuk et al.3 proposed one of these approximate methods. Another popular class of approximate analytical solutions is based on the first-order approximation obtained from perturbation methods. This article is related to these first-order approximate solutions of well pressure in arbitrarily heterogeneous reservoirs. In particular, we propose an easy and general method to calculate the permeability weighting function in various flow geometries. In the next section, we define what the permeability weighting function is and review previous work in the domain. After that, we present our method to calculate the permeability weighting functions. The technique is demonstrated in three situations, including the case of flow through a horizontal well. Permeability Weighting Function The perturbation method is a well known technique by which to solve partial differential equations involving mathematical difficulties, like variable coefficients. According to this technique, we start from an easier problem, the background problem, to modify or perturb it. The full problem is approximated by the first few terms of a perturbation expansion, usually the first two terms. In our context, we start from considering a background medium with permeability k0 and with specified boundary conditions. The k0 may vary in space, i.e., k0(x→D). What is important is that the background problem has a known exact analytical solution, pD0(x→D,tD). The full problem has the same boundary conditions of the background problem but the permeability k(x→D) differs from k0(x→D) in arbitrary regions of space. Strictly speaking, k(x→D)/k0(∙xD) has to be close to 1 in order to obtain sound approximations. In practice, errors tend to be small, say less than 10%, even for relatively greater contrasts up to, say, 10 between these permeabilities, depending on the specific problem. The dimensionless well pressure of the full problem, pwD(tD), is approximated by the sum of two terms: p w D ( t D ) ≅ p w D 0 ( t D ) + p w D 1 ( t D ) , ( 1 ) where pwD0 is the solution of the background problem, which is known, and pwD1 corresponds to the effect of the variation of the permeability. This second term is computed by p w D 1 ( t D ) = ∫ − ∞ + ∞ Δ k D ( x → D ) W ( ∙ x D , t D ) d ∙ x D , ( 2 ) the terms of which will be explained. The dimensionless permeability variation ΔkD may be alternatively defined by Δ k D ( ∙ x D ) = l n ( k ( x → D ) / k 0 ( x → D ) ) , ( 3 a ) Δ k D ( ∙ x D ) = 1 − ( k 0 ( x → D ) / k ( x → D ) ) , ( 3 b ) Δ k D ( ∙ x D ) = ( k ( x → D ) / k 0 ( x → D ) ) − 1 , ( 3 c ) or other equivalent first-order approximations. These three expressions have the same first-order terms of their Taylor series, and produce very close results for k(x→D)/k0(x→D) near 1. However, these definitions are not equally robust for greater permeability contrasts.
∑ = − = Nobs i obs i sim i i d m d w m O 1 2 ) ( ) ( TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractOne of the challenges when making a history match study is to find an adequate parameterization for the reservoir model. The main assumptions of the geological characterization should be respect and the influence of parameters on the fluid flow simulation results should be taken into account. On the other hand, the number of parameters should be kept within reasonable bounds in order to make the process viable. In this work, three examples of novel ways to parameterize the history match problem will be shown. Two of them are real field cases and one is a synthetic case based on outcrop data. Common to all examples is the choice of parameters that are related to the geological model building process, such as the variogram in a geostatiscal modeling or correlations between petrophysical properties (permeability x porosity, for instance). In this context, the use of a versatile history matching tool was essential, allowing for a quantitative evaluation for the quality of the match and for managing a larger number of parameters, when comparing to the traditional trial and error procedure. These examples show how the combination of a suitable parameterization with a versatile assisted history matching tool can improve both the quality and the efficiency of the history matching process.
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