Summary
We propose a novel approach to history matching finite-difference models that combines the advantages of streamline models with the versatility of finite-difference simulation. Current streamline models are limited in their ability to incorporate complex physical processes and cross-streamline mechanisms in a computationally efficient manner. A unique feature of streamline models is their ability to analytically compute the sensitivity of the production data with respect to reservoir parameters using a single flow simulation. These sensitivities define the relationship between changes in production response because of small changes in reservoir parameters and, thus, form the basis for many history-matching algorithms. In our approach, we use the streamline-derived sensitivities to facilitate history matching during finite-difference simulation. First, the velocity field from the finite-difference model is used to compute streamline trajectories, time of flight, and parameter sensitivities. The sensitivities are then used in an inversion algorithm to update the reservoir model during finite-difference simulation.
The use of a finite-difference model allows us to account for detailed process physics and compressibility effects. Although the streamline-derived sensitivities are only approximate, they do not seem to noticeably impact the quality of the match or the efficiency of the approach. For history matching, we use a generalized travel-time inversion (GTTI) that is shown to be robust because of its quasilinear properties and that converges in only a few iterations. The approach is very fast and avoids many of the subjective judgments and time-consuming trial-and-error steps associated with manual history matching. We demonstrate the power and utility of our approach with a synthetic example and two field examples. The first one is from a CO2 pilot area in the Goldsmith San Andreas Unit (GSAU), a dolomite formation in west Texas with more than 20 years of waterflood production history. The second example is from a Middle Eastern reservoir and involves history matching a multimillion-cell geologic model with 16 injectors and 70 producers. The final model preserved all of the prior geologic constraints while matching 30 years of production history.
Introduction
Geological models derived from static data alone often fail to reproduce the field production history. Reconciling geologic models to the dynamic response of the reservoir is critical to building reliable reservoir models. Classical history-matching procedures whereby reservoir parameters are adjusted manually by trial and error can be tedious and often yield a reservoir description that may not be realistic or consistent with the geologic interpretation. In recent years, several techniques have been developed for integrating production data into reservoir models. Integration of dynamic data typically requires a least-squares-based minimization to match the observed and calculated production response. There are several approaches to such minimization, and these can be classified broadly into three categories: gradient-based methods, sensitivity-based methods, and derivative-free methods. The derivative-free approaches, such as simulated annealing or genetic algorithms, require numerous flow simulations and can be computationally prohibitive for field-scale applications. Gradient-based methods have been used widely for automatic history matching, although the convergence rates of these methods are typically slower than the sensitivity-based methods such as the Gauss-Newton or the LSQR method. An integral part of the sensitivity-based methods is the computation of sensitivity coefficients. These sensitivities are simply partial derivatives that define the change in production response because of small changes in reservoir parameters.
There are several approaches to calculating sensitivity coefficients, and these generally fall into one of three categories: perturbation method, direct method, and adjoint-state methods. Conceptually, the perturbation approach is the simplest and requires the fewest changes in an existing code. Sensitivities are estimated simply by perturbing the model parameters one at a time by a small amount and then computing the corresponding production response. This approach requires (N+1) forward simulations, where N is the number of parameters. Obviously, it can be computationally prohibitive for reservoir models with many parameters. In the direct or sensitivity equation method, the flow and transport equations are differentiated to obtain expressions for the sensitivity coefficients. Because there is one equation for each parameter, this approach requires the same amount of work. A variation of this method, called the gradient simulator method, uses the discretized version of the flow equations and takes advantage of the fact that the coefficient matrix remains unchanged for all the parameters and needs to be decomposed only once. Thus, sensitivity computation for each parameter now requires a matrix/vector multiplication. This method can also be computationally expensive for a large number of parameters. Finally, the adjoint-state method requires derivation and solution of adjoint equations that can be quite cumbersome for multiphase-flow applications. Furthermore, the number of adjoint solutions will generally depend on the amount of production data and, thus, the length of the production history.
Optimization Techniques for History Matching and Production Forecasting