A Generalized Method for Making Optimal Estimates of Reservoir-Aquifer Parameters
Authors: Dougherty E.L. , Zagalai B. , Khairkhah D. 
Published: December 2007

Abstract
This paper presents a new, faster method for predicting performance of reservoir aquifer systems predicting performance of reservoir aquifer systems and making optimal estimates of aquifer geometry, reservoir size, aquifer size, and reservoir-aquifer fluid conductivity. The method uses existing analytical solutions giving pressure-production performance of a radial or linear aquifer connected performance of a radial or linear aquifer connected to a reservoir with constant compressibility. The novelty of the method is the use of these solutions to predict behavior of a reservoir with compressibility changes. The method's accuracy was verified by comparing results with those calculated using a limite-difference simulator.
The method is built into a computer program that uses an efficient optimization technique to estimate parameters. Results of application to a hypothetical gas reservoir and to an actual oil field are given. In the latter case, a comparison is made with results obtained using the Hurst-van Everdingen water influx calculation method

Introduction
Aquifers, linear or radial, surround many oil and gas reservoirs. Knowledge of the aquifer's strength is essential to predict future performance of a water-drive reservoir. Strength is influenced by geometric configuration, which may be difficult to determine. Often, geological interpretation suggests that the system is neither radial nor linear. Because it is possible that either aquifer shape might approximate the behavior of an actual system, it is frequently desirable to test both in a given problem.
Several methods have been developed for predicting water influx into a reservoir. Of these, predicting water influx into a reservoir. Of these, Hurst's is closest to the method presented here. Hurst used the Laplace transformation to convert standard material balance equations for water-drive reservoirs into a form giving pressure explicitly as a function of other reservoir parameters. To apply his method for predicting reservoir pressure change, it is necessary to approximate all pressure-dependent functions as linear functions of pressure. Hurst gave no indication of the range of applicability of such an approximation and provided no convenient means for updating the approximations.
This study develops an improved method for estimating groups of parameters that control behavior of a reservoir-aquifer system. The system is modeled as a single, homogeneous reservoir block with an active (radial or linear) aquifer. The steps taken to develop the method axe given and then its application to two practical field problems is shown.

DEVELOPMENT OF METHOD
This section presents the steps taken in developing the method. For each step, the main thrust of the work conducted and results obtained are presented; mathematical details appear in appendixes. Step 1 considered the equations describing the behavior of reservoir-aquifer systems. The required solutions gave p vs t in a reservoir with constant compressibility being produced at a constant rate and receiving water influx from either a radial or a linear aquifer. Stop 2 modified the solutions to include changes in production rate and reservoir compressibility during the reservoir's history. A computer program was written to predict reservoir behavior using the method derived in Steps 1 and 2. In Step 3 the method's accuracy was verified by comparing results with those obtained with a more exact tank-type simulator. In Step 4 a computer program was written to estimate values for the reservoir parameters that best matched the given history.

STEP 1: MATHEMATICAL MODEL
This step is concerned with the equations describing behavior of reservoir-aquifer systems. First, a radial aquifer is discussed, and then linear aquifer theory is presented.
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