Sensitivity Coefficients for History Matching Oil Displacement Processes
Authors: Hirasaki George J. 
Published: December 2007

Abstract
An improved estimate of the reservoir parameters is made during the history-matching phase of a reservoir simulation study by determining the set of parameters that result in the best match of the simulated performance with the observed performance. Often, the process of determining which parameters are to be adjusted is a trial-and-error process. Graphs of the sensitivity coefficients for comparing the cumulative oil recovery with the reservoir parameters are presented to determine the relative significance of parameters are presented to determine the relative significance of the parameters and to provide guidelines for the magnitude of change to the parameters. The sensitivity coefficients are based on a one-dimensional system with dip, incompressible fluids, and polynomial relative-permeability curves. The recovery efficiency can be expressed as a function of the dimensionless cumulative injection with the gravity number (gravity/viscous-forces ratio), mobility ratio, and relative-permeability exponent as parameters. The sensitivity of the cumulative oil recovery (at a given value of cumulative injection) to the movable pore volume, mobility ratio, permeability, and the exponent of the relative-permeability curve permeability, and the exponent of the relative-permeability curve can be calculated from the expression for the recovery efficiency. The graphs of the sensitivity coefficients can be used to determine the relative significance of the parameters, if a unique set of parameters can be determined, and how much they should be adjusted. parameters can be determined, and how much they should be adjusted

Introduction
When the simulated oil-recovery performance differs from the observed performance history, the engineer must determineif the history match is satisfactory, orif not, which reservoir parameters are to be adjusted and how much.
The purpose of this parameters are to be adjusted and how much. The purpose of this discussion is to provide guidelines for the engineer in choosing the parameter(s) to be adjusted and to determine the magnitude and direction of the change. This will be accomplished by first illustrating the sensitivity of water or gas displacement performance to the reservoir parameters so that the critical parameter(s) can be identified, and then graphically presenting the magnitude of the sensitivity coefficients to determine the magnitude of change in the parameter value necessary to achieve a match. The following guidelines will be limited in scope to two-phase displacement processes with negligible interfacial mass transfer (e.g., waterflood, natural water drive, gas injection, or gas-cap expansion). Processes such as solution gas drive or vaporizing gas drive will not be presented. The results will be expressed in terms of the gross fluids produced or injected rather than time. The analysis and results are based on a one-dimensional system. Although the recovery performance of a multidimensional system will be different from that of a one-dimensional system, the relative sensitivity of the recovery performance to the parameters should not differ significantly for most recovery processes. Examples of exceptions that cannot be represented with the one-dimensional system are where well coning is significant or where permeability barriers exist between the injection well and production well. production well. The reservoir parameters that are investigated arethe movable pore volume of the displacement process, SVp;the mobility ratio of the displacing fluid to the displaced fluid, M, where the mobilities are evaluated at the maximum saturation of each phase;the permeability, which is represented as a factor in the gravity number, N(G) if the formation is dipping; andthe shape of the relative permeability curve, expressed in terms of a single parameter, n.

ASSUMPTIONS AND MODEL OF THE DISPLACEMENT PROCESS
The following assumptions are made about the displacement process. process.The saturations, relative permeabilities, porosity, and permeability are averaged over the reservoir thickness. permeability are averaged over the reservoir thickness.The areal displacement is modeled with a linear system.
SPEJ
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