Inversion of Geological Shapes in Reservoir Engineering Using Well-Tests and History Matching of Production Data
Authors: Rahon D. , Edoa P.F. , Masmoudi M. 
Published: December 2007

Abstract
The identification of geological elements (faults, reservoir boundaries, channel positions) using well-tests or history matching of past field performances is a delicate problem in reservoir engineering. Analytical well-tests models can work in the simplest cases but in complex cases the use of numerical simulation models of fluid flow is necessary. With these numerical models, classical methods for calculating well pressure gradients on discrete equations are not readily usable when dealing with parameters that are not found in the model equations.
This article deals with an approach based on calculating the sensitivity on the continuous model - before the spatial discretization - with respect to boundary shifts of different geological objects. This general method will work with geological elements of any shape and in heterogeneous environments. An optimization algorithm was implemented, taking a large number of parameters into account, while assuring the regularity of the solution.
This algorithm allows the geometry of the different geological bodies to be defined using triangulated surfaces. Thus the history matching parameters are the nodes of the triangulation and well pressure gradients are calculated with respect to the displacement of these nodes.
This approach was first applied to monophasic flows and the algorithm was integrated into a numerical well-tests simulator. With this technique, we identified 2D and 3D geometries, especially the shape and thickness of channels and the location of faults by matching well pressure measurements. Several examples, illustrating the efficiency of this method, are presented at the end of this article.

Introduction
In order to characterize reservoirs, automatic history matching techniques have been developed in recent years. Most of them are based on numerical simulation and gradient methods which allow the use of efficient optimization algorithms to minimize an objective function to obtain a history match. In this kind of method the derivative of the simulated production data with respect to petrophysical parameters (permeability or porosity) are calculated by derivation of the state equation 1-5 or resolution of a so-called adjoint-state.
The majority of these automatic procedures only make possible the identification of petrophysical parameters or well parameters like the skin and the well bore storage. The case of geometric parameters such as the position of a fault, reservoir limits, channels thickness, width and position has been treated in recent publications. These works are based on the definition of geological objects using mathematical functions describing the permeability or porosity fields. Thus the classical gradient methods can be used, but the generalization of these solutions for complex geological models remains very difficult.
The method proposed in this paper allows the identification of geological shapes and is derived from shape optimization techniques. It is well-suited for complex geometries and heterogeneous environments. The history matching parameters are the geometric elements describing the geological objects and the difficulty is to calculate gradients of the production data with respect to perturbations of the geometric parameters as they do not appear directly in the state equation.
We present here a description of the method with the calculation of the sensitivities and the implementation of an optimization algorithm adapted to problems dealing with a large number of parameters.
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