Application of PC Spreadsheet Software For Non-linear Parameter Optimization
Authors: Gupta A. 
Published: March 2010

Abstract
Currently available spreadsheet software packages offer an easy to implement and user friendly platform for setting up problems involving non-linear parameter optimization. This paper presents the application of one such software "Solver" embedded in MS-Excel ™ for quantitative interpretation of counter-current imbibition experiments on reservoir core samples. Usually imbibition tests are conducted to obtain qualitative estimates of reservoir wettability and oil recovery. This paper presents a procedure for history matching of counter-current imbibition tests that involves minimization of a user-defined global objective function with respect to various model parameters. Application of this procedure for many counter-current imbibition experiments reveals that a single dimensionless function can represent imbibition recovery history for most rock samples. Also, apparent contact angles for various samples are consistent with their expected wettability.
The parameter optimization procedure presented in this paper offers a powerful and efficient method that can be modified to solve numerous reservoir and production engineering problems.

Introduction
Solution of many common problems encountered in different facets of oil and gas industry requires obtaining values for parameters embedded in highly non-linear mathematical models. Examples of such problems include: determination of petrophysical properties, skin, and drainage radius of a reservoir based on pressure transient data and/or production decline data(1); forward modelling for obtaining petrophysical properties of vertical sequences by matching response of a suite of wire-line logs; and interpretation of laboratory corefloods for determining relative permeability curves(2,3). Solution procedure for these and other similar problems usually requires assignment of initial guesses for the unknown parameters, predicting system response using the available model, and using the quality of match between the actual and predicted system response to correct the value of parameters. This procedure is continued until a satisfactory agreement between system response and model predictions is obtained. For many problems, such strategy is easier to implement than attempting to explicitly obtain various parameters because of the non-linear nature of mathematical formulation obtained in the model.
Approaches suggested in the past for implementing such parameter determination include kriging(1), simulated annealing(2,3), regression(4) and optimal control theory(5). Implementation of any of these parameter determination procedures requires an extensive amount of programming effort or investment in some expensive software which may need to be customized in order to operate compatibly with user developed code. Commercially available spreadsheet software packages offer built-in utility software for determining parameters of a non-linear formulation. This paper describes the implementation of a non-linear model in the spreadsheet software MS-EXCEL ™. The parameters of this model were refined using the built-in utility solver by seeking to minimize a user-defined objective function.
Experiments involving imbibition of water and production of oil are commonly conducted on the core samples from fractured reservoirs in order to characterize mass-transfer of oil from the matrix to the fracture system(4). Such tests are also conducted to understand the wettability of reservoir rocks. Gupta and Civan(6) developed a three exponent transfer function based on mathematical analysis to represent the cumulative production of oil