Reservoir Modeling Constrained to Multiple Well-Test Permeabilities
Authors: Campozana F.P. , Lake Larry W. , Sepehrnoori K. 
Published: December 2007

Reservoir Modeling Constrained to Multiple Well-Test Permeabilities F.P. Campozana, SPE, UT, Larry W. Lake, SPE, UT, K. Sepehrnoori, SPE, UT

Abstract
Well-test permeability derived from pressure transient analysis is important information about the interwell permeability distribution of a reservoir. It is available for many active wells in a reservoir. Nevertheless, no conventional geostatistical technique is able to incorporate this information into a reservoir model.
This paper describes an integrated procedure to account for well-test permeabilities obtained from several wells into a geostatistical model of a reservoir permeability distribution. Our procedure is based on a simulated annealing algorithm coupled with a steady-state, single-phase flow simulator that solves an inverse problem. A correlation is developed to relate the effective permeability of each well-test region obtained from the simplified simulation with the true well-test permeability. The problem of nonstationarity between the well-test regions is addressed. The results obtained here show that there is a reduction of reservoir uncertainty when well-test information is added.

Introduction
Most conventional geostatistically generated models use static information only; however, reservoir models generated this way may not match dynamic or field data. History matching is then required. History matching consists of submitting the input parameters of a reservoir model to sometimes arbitrary changes. A flow simulator is then used to observe how these changes impact the production of fluids and the pressure behavior. The model is altered until a satisfactory match between simulated and observed production data is obtained. History matching is cumbersome and time-consuming; it usually does not allow the use of many images of the same reservoir, that is, stochastic modeling is limited. Therefore, it becomes difficult to quantify reservoir uncertainty and its impact on a production forecast.
There are basically three main reasons why well test data should be included when modeling a reservoir:they are available for many active wells,they are an actual response of the reservoir to pressure changes, andtheir support volumes cover large areas around wells,
The purpose of this paper is to extend the previous works by (1) considering the entire reservoir with multiple well tests, (2) allowing for different statistics in each well-test region, and (3) using a new simulated annealing (SA)- based algorithm coupled with a simple simulator.

Background
The effective permeability obtained from classical pressure transient analysis represents an average of the small-scale permeability values within the volume affected by the well test. To consider the inverse problem of obtaining a refined grid of a permeability distribution from the knowledge of its average, it is necessary to define (1) what blocks participate in the average and (2) what type of averaging is relevant.
Van Poolen proposed the following equation to estimate the radius of investigation of a pressure transient well test:
(1)
Equation (1) applies to a reservoir that is homogeneous. However, because of heterogeneity, the actual shape of the drainage area will be irregular, not circular as needed by Eq. (1). If anisotropy is present, the shape of the drainage area is better approximated by an ellipse. In this case, the use of Eq. (1) to determine the small-scale permeability blocks that should participate in the averaging leads to including some blocks that are not participating in the flow and leaving out others that are.
The second problem is to determine the proper averaging to be used. A general procedure commonly used is power averaging.
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