A Semi-Automatic History Matching Technique Applied to Aquifer Gas Storages
Authors: Ory J. , Guedeney K. , Brefort B. , Schirrer L. 
Published: December 2007

Abstract
The scope of this paper is to describe a semi-automatic history matching technique providing valuable information about the complex input-output relationship of reservoir simulators. It is based on the Gauss-Newton local optimization algorithm and on a matrix decomposition theorem called Singular Value Decomposition (SVD). At each Gauss-Newton iteration, the SVD processes the sensitivity matrix and provides parameter combinations having independent effects on the simulated pressure data. The singular values indicate the scale of these effects, while the combinations indicate parameter couplings and can be used to provide an updated model reducing matching errors.
The pressure history of an aquifer gas storage has been successfully matched in very few iterations with this approach, using a two-phase flow, one-dimensional and gridded reservoir simulator meant to perform short-term pressure forecasts. The SVD clearly indicated which model parameters could be reliably estimated.

Introduction
The goal of a reservoir engineering study is to simulate the future performance of a reservoir using reservoir modeling tools. For the forecasts to be valid, the past performance of the reservoir should first be adequately reproduced by a numerical simulation. The task of finding a reservoir description yielding a correct simulation of the past performance of the reservoir is called history matching (HM).
In aquifer gas storages applications, performing a simulation involves compiling in an input file all the available information on the geometry and the petrophysics of the reservoir, on multi-phase flow parameters, on the fluid thermodynamic properties, on well characteristics and on the initial and boundary conditions in the reservoir. The simulation is then run in a discretized reservoir using the well gas flow-rates history and some outputs are computed at each time step, such as gas pressure or saturation. HM then consists in perturbing relevant input parameters until the difference between simulated outputs and measured data, i.e. the data residuals, is acceptably small.
HM means performing at least two loops on the various categories of input parameters. The outer loop is that on the geometrical and petrophysical model. This loop involves defining the geometry of some geological bodies such as faults, the surrounding aquifer, or some homogeneous permeability and porosity zones in the reservoir. Initial permeability or porosity fields may also be generated using geostatistics, either by kriging or by stochastic methods. The inner loop is that on the input parameters attached to these geological or petrophysical bodies: fault transmissivity, reservoir compressibility, permeability, porosity, end-points of relative permeability curves. Needless to say, HM is a complex task. The matching parameters are numerous and the relationship between input and output is non linear and not always predictable, at least in quantitative terms. The simplest HM method is probably the trial-and-error one: reservoir engineers use their experience or their physical intuition to guess which the significant input parameters are, and they make changes by hand in the input file of the simulator. Then, they perform another simulation to check whether the match has improved or not. This method clearly becomes tedious or sometimes overwhelming when the number of parameters gets too large. There has been in the past decade a lot of research aiming to automatize the inner HM loop in oil fields applications. Most techniques consist in computing iteratively a set of matching parameters minimizing a cost function, which is some norm of the data residuals, subject to constraints on the parameter values. A minimization of the cost function using local optimization techniques (see Ref. 1 4.5 for a review of the various algorithms) may not lead to a satisfactory match because of the presence of local minima.
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