History Matching of Three-Phase Flow Production Data
- Ruijian Li (U. of Tulsa) | A.C. Reynolds (U. of Tulsa) | D.S. Oliver (U. of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2003
- Document Type
- Journal Paper
- 328 - 340
- 2003. Society of Petroleum Engineers
- 5.3.2 Multiphase Flow, 4.3.4 Scale, 5.5.8 History Matching, 5.5.2 Construction of Static Models, 5.2.1 Phase Behavior and PVT Measurements, 5.5.7 Streamline Simulation, 4.6 Natural Gas, 5.5 Reservoir Simulation, 5.1 Reservoir Characterisation, 5.6.4 Drillstem/Well Testing, 5.1.5 Geologic Modeling
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Adjoint equations for 3D, three-phase flow problems are developed and implemented to calculate the sensitivity of production data to permeability fields and well skin factors. Typically, the development of adjoint equations is rather tedious, but the formulation given here makes the development and numerical solution of adjoint equations straightforward. The procedure for calculating sensitivity coefficients is coupled with a fully implicit finite-difference simulator to develop an automatic history-matching procedure. Even if one does not wish to fully automate history matching, sensitivity coefficients are useful for understanding the physics that control 3D multiphase flow and the influence of local reservoir properties on production data. Procedures to estimate the value of particular types of data [pressure, producing gas/oil ratio (GOR), and water/oil ratio (WOR)] for reducing the uncertainty in estimates of reservoir properties are applied and discussed.
Automatic history matching is based on minimizing an objective function that includes a sum of production data mismatch terms squared. Minimization using conjugate gradient or variable metric methods requires computation of only the gradient of the objective function, but typically converge fairly slowly (see Makhlouf et al.,1 Yang and Watson,2 and Carrera and Neuman3). Despite their relatively slow convergence, such methods may prove to be the only feasible option for large-scale problems, where the number of data and model parameters are both so large that it is not feasible to compute all sensitivity coefficients. Our focus here, however, is on the Gauss-Newton and Levenberg-Marquardt methods, which typically converge in far fewer iterations. Although the gradient simulator method4 can be used to generate sensitivity coefficients for automatic history matching,5 this is not feasible if the number of model parameters is large. The gradient simulator method requires that we solve a matrix problem with multiple right sides at the end of each reservoir simulator timestep. The number of right sides is equal to the number of model parameters. If one can justify describing the reservoir using only a few model parameters (e.g., some form of zonation5-8 makes sense), then the gradient simulator is feasible. However, if the set of model parameters includes the porosities and permeabilities for every grid cell, computation of sensitivity coefficients with the gradient simulator method is not feasible.
As discussed in Wu et al.,9 the adjoint method3,10-12 also requires solving one matrix problem with multiple right sides at each simulator timestep. But, in the adjoint procedure, the number of right sides is no greater than the number of production data, and is independent of the number of model parameters. Thus, the procedure is applicable when the set of model parameters includes all gridblock permeabilities and porosities. In the examples considered here, the number of conditioning production data is small enough that we can afford to compute the sensitivity of all production data to the model parameters. If the number of data is large, the model can be reparameterized using the subspace method13 and then we can directly compute the sensitivity of each subspace vector to the model parameters.14
If the number of model parameters is very large, there will not be sufficient independent data to resolve all the model parameters. In this case, application of the Gauss-Newton method would fail because the Hessian would be ill conditioned unless some form of regularization is applied. In this work, we use a prior geostatistical model to provide regularization. With this approach, the history-matching problem is equivalent to a Bayesian estimation problem. 9,15-17 We calculate sensitivity coefficients with the adjoint method. The availability of sensitivity coefficients allows the application of the Gauss-Newton and Levenberg-Marquardt algorithms, which exhibit approximate quadratic convergence in a neighborhood of a minimum. Because sensitivity coefficients are calculated, the a posteriori covariance matrix can be generated; this matrix provides a measure of uncertainty in the estimates of reservoir model parameters. The adjoint solution and history-matching procedure presented here are based on a finite-difference formulation of the flow equations. As such, the method can be applied to single-phase oil or gas flow, or to multiphase flow problems; it is not restricted to physical problems where streamline simulators can be used to generate accurate results efficiently.
Model Estimation and Simulation
In this major section, we define the reservoir model parameters and the a posteriori probability density function (pdf) for these parameters. This pdf, which is conditional to production data, defines the the set of plausible reservoir descriptions. We discuss computation of the maximum a posteriori (MAP) estimate. The MAP estimate is the model that maximizes the a posteriori pdf and is thus sometimes referred to as the most probable model. Methods for sampling this pdf to characterize the uncertainty in model parameters and the uncertainty in performance predictions are discussed briefly.
Model Parameters and the Prior PDF.
For simplicity, the reservoir is assumed to be a rectangular parallelepiped that occupies the region
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