|Publisher||Society of Petroleum Engineers||Language||English|
|Content Type||Journal Paper|
|Title||Gradient-Based History Matching With a Global Optimization Method|
|Authors||Susana Gomez, Inst. for Applied Mathematics and Systems, Natl. U. of Mexico; Olivier Gosselin, John W. Barker, TotalFinaElf Exploration U.K.|
|Volume||Volume 6, Number 2||Pages||200-208|
|Copyright||2001. Society of Petroleum Engineers|
We investigate the application of a global optimization algorithm called the tunneling method to the problem of history matching of petroleum reservoirs.
Results are presented for two test cases. The first is a small synthetic case in which the global minimum is known. The second is a real field example. In both cases, a series of minima was found. The computational cost of each tunneling phase is found to be comparable with the cost of each local minimization.
It is concluded that the tunneling method may have a practical application in history matching as an alternative to immediate reformulation of the problem if the first minimum found does not represent an acceptable match.
Gradient-based history-matching techniques1 are becoming more widely used as appropriate software becomes commercially available.2-4 In these techniques, optimization methods such as Gauss-Newton, Levenberg-Marquardt, steepest descent, and others are used to minimize an objective function that measures the difference between the simulator response and the observed field data.
Such techniques are usually quite efficient in converging to a local minimum in the objective function. However, there is no guarantee that this will be the global minimum. In general, the objective function is likely to have many minima. Only those minima that give a sufficiently good match to the data are of interest.
The aim of global optimization methods is to track down the global minimum of the objective function. However, well-known stochastic global methods such as simulated annealing and genetic algorithms usually require a very large number of iterations to converge (see for instance Ref. 5). This makes them too expensive for practical history-matching applications because each iteration requires a reservoir simulation to be run, and each simulation may take up to several hours of computing time, depending on the size of the reservoir model.
In this work we test the benefits of using a global optimization method called the tunneling method for history matching of oil reservoirs.6-8 This is an essentially deterministic method that makes use of gradient information. It seeks to find a series of minima, each one with a lower objective value than the previous one. The last minimum in the series will be the global minimum, although in practice one is unlikely to push the process to the end. The attraction of the method is that, having found a first minimum with a conventional method, one or more better minima may be found using tunneling at an affordable computational cost.
Definition of the Problem
The objective function that we seek to minimize is
subject to . Here Yobs=the measured values of observable quantities such as bottomhole pressure (BHP), water cut (WCT) or gas/oil ratio (GOR) and Y (x) are their calculated values from a reservoir simulation with a certain set of values of the parameters x. These parameters characterize the reservoir (e.g., porosity or permeability cell values) and have to be found so that the simulator approximates the data closely. The index k=the type of observable data (BHP, WCT, GOR), the index j runs over the number of wells, i=the index of the measurement dates for each well, and σ=a normalization factor or weight.
The Tunneling Method
The method that we use has been designed to find the global optimum of a general nonconvex smooth function such as the objective function of Eq. 1. The code that we are using in this work has recently been developed and incorporates the numerical experience obtained in recent years by the authors. 6,9,10 It has already proved to be very efficient in solving difficult problems, both in academic and real settings.7,8,11
The basic idea of the method is to tunnel from one valley of the objective function to another. This means to find a sequence of local minima with decreasing function values,
where xG=the global minimum of f(x). An important feature of the method is that it is able to ignore all the local minima with larger objective function values than the ones already found. This characteristic makes the algorithm faster and more efficient than other general purpose methods like simulated annealing, random search, clustering, and genetic algorithms.
Once a local minimum has been found, tunneling amounts to finding parameter values x*m that solve the inequality problem f(x* m)−f(xm) 0 subject to x*m ∈B.
Starting from an initial point x0, the tunneling method thus has two phases that are repeated alternately until convergence is achieved (see Fig. 1 ):
In this phase, the problem is to find a local minimum of the objective function defined by Eq. 1. Any algorithm designed to solve local optimization problems with bounds on the variables can be used. In this work, we use a limited-memory quasi-Newton method.12
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