Permeability Averaging for Well Tests in 3D Stochastic Reservoir Models
Authors: Noetinger Benoit , Haas Andre 
Published: December 2007

Abstract
Geostatistical models are useful to generate equiprobable realizations of oil reservoirs that allow the study uncertainties in the production forecasts. A key problem is the constraint of these models according to known production data, to reduce uncertainties. Due to the relative simplicity of well tests, numerous authors attempted to include thems in the geostatistical models, especially in a 2D context.
The goal of this work is to propose a simple analytical formula which relates the apparent permeability given by a well test interpretation and the original permeability map around the well for 3D heterogeneous reservoirs. The pressure derivative is used to interpret drawdowns. Assuming that this permeability is related to the original permeability map in the drainage volume by a simple power averaging formula, the best value of the averaging exponent is computed using perturbation methods. It depends on time, and on the vertical to horizontal permeability ratio kv/kh,, and geostatistical anisotropy. For sufficiently long times, when the apparent permeability is stabilized, we obtain an apparent permeability equal to the well-known steady-state equivalent permeability of the medium. This regime occurs when the investigation radius of the well test is larger than the correlation length. Numerical simulations of well tests performed in this regime are in good agreement with the calculation and demonstrate the validity and robustness of the proposed formula.

Introduction
Progress in the power of computers and in reservoir characterization now makes it possible to represent detailled distributions of oil reservoir heterogeneities which leads to more realistic predictions of flow. Well data are generally scarce and must be complemented by indirect measurements such as seismic (references 1 and 2) and dynamic data. In particular, well-test data are important in the validation of stochastic models around wells. Ideally, starting from an initial geostatistical model, one should be able to generate reservoir realizations respecting all known data, particularly well tests constraints. Here, we suppose that petrophysical attributes of the different faciec are known, so we cannot adjust them to get a correct match (ref 3). In the present problem, the variability originates from the random character of the geostatistical images. The number of degree of freedom is on the order of magnitude of the number of grid blocks. This number is large such that "gradient methods" (see ref 3) cannot be used easily to get a practical solution. All proposed solutions (see ref 4 through 7) require a solution of the "direct problem" i.e. the well test pressure variations for a given permeability map.
This particular problem has already been investigated by many authors (references 3 and 4), mainly in 2 dimensions. Since no analytical expression exists for well tests in heterogeneous reservoirs, the first idea is to perform numerical simulations using a gridded reservoir simulator. As most methods to constrain geostatistical images to dynamic data use a great number of calls to the direct problem solver, simplifications are needed to get exploitable results. In this spirit, several empirical averaging formulas have been proposed and checked for their match against apparent permeabilities (references 3 through 5). These formulae are accurate only if the well test is stabilized, i.e. a single large scale permeability value K can be defined by a standard interpretation of the test. On the other hand, theoretical developments have been limited in the case of permeability heterogeneities of limited contrast, and no explicit solution is known to 3D cases.
Our present goal is to generalize these formulae for three dimensional reservoirs which advantages will be to mix the robustness of empirical formula with the limited permeability contrast results giving a predictive formula. In particular, we will check that the solution depends only on the reservoir petrophysical and geometrical anisotropies through the so called global anisotropy ratio: where is the vertical to horizontal permeability ratio, and Lh and Lv denote the horizontal and vertical correlation lengthscales.
The proposed formula has the structure of a power averaging of the small scale permeabilities over the investigation radius of the test:
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