Fast History Matching of Finite-Difference Models Using Streamline-Based Sensitivities
- Hao Cheng (Chevron Corp.) | Arun Kharghoria (PetroTel Inc.) | Zhong He (Texas A&M U.) | Akhil Datta-Gupta (Texas A&M U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- October 2005
- Document Type
- Journal Paper
- 426 - 436
- 2005. Society of Petroleum Engineers
- 5.1 Reservoir Characterisation, 5.4.2 Gas Injection Methods, 5.1.2 Faults and Fracture Characterisation, 5.8.6 Naturally Fractured Reservoir, 5.5.8 History Matching, 5.8.7 Carbonate Reservoir, 4.3.4 Scale, 2.2.2 Perforating, 5.6.4 Drillstem/Well Testing, 5.4 Enhanced Recovery, 7.6.2 Data Integration, 5.5.7 Streamline Simulation, 5.4.1 Waterflooding, 5.2.1 Phase Behavior and PVT Measurements, 5.1.5 Geologic Modeling, 5.5 Reservoir Simulation, 5.5.3 Scaling Methods
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We propose a novel approach to history matching finite-difference modelsthat combines the advantages of streamline models with the versatility offinite-difference simulation. Current streamline models are limited in theirability to incorporate complex physical processes and cross-streamlinemechanisms in a computationally efficient manner. A unique feature ofstreamline models is their ability to analytically compute the sensitivity ofthe production data with respect to reservoir parameters using a single flowsimulation. These sensitivities define the relationship between changes inproduction response because of small changes in reservoir parameters and, thus,form the basis for many history-matching algorithms. In our approach, we usethe streamline-derived sensitivities to facilitate history matching duringfinite-difference simulation. First, the velocity field from thefinite-difference model is used to compute streamline trajectories, time offlight, and parameter sensitivities. The sensitivities are then used in aninversion algorithm to update the reservoir model during finite-differencesimulation.
The use of a finite-difference model allows us to account for detailedprocess physics and compressibility effects. Although the streamline-derivedsensitivities are only approximate, they do not seem to noticeably impact thequality 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 robustbecause of its quasilinear properties and that converges in only a fewiterations. The approach is very fast and avoids many of the subjectivejudgments and time-consuming trial-and-error steps associated with manualhistory matching. We demonstrate the power and utility of our approach with asynthetic example and two field examples. The first one is from a CO2 pilotarea in the Goldsmith San Andreas Unit (GSAU), a dolomite formation in westTexas with more than 20 years of waterflood production history. The secondexample is from a Middle Eastern reservoir and involves history matching amultimillion-cell geologic model with 16 injectors and 70 producers. The finalmodel preserved all of the prior geologic constraints while matching 30 yearsof production history.
Geological models derived from static data alone often fail to reproduce thefield production history. Reconciling geologic models to the dynamic responseof the reservoir is critical to building reliable reservoir models. Classicalhistory-matching procedures whereby reservoir parameters are adjusted manuallyby trial and error can be tedious and often yield a reservoir description thatmay not be realistic or consistent with the geologic interpretation. In recentyears, several techniques have been developed for integrating production datainto reservoir models. Integration of dynamic data typically requires aleast-squares-based minimization to match the observed and calculatedproduction response. There are several approaches to such minimization, andthese can be classified broadly into three categories: gradient-based methods,sensitivity-based methods, and derivative-free methods. The derivative-freeapproaches, such as simulated annealing or genetic algorithms, require numerousflow simulations and can be computationally prohibitive for field-scaleapplications. Gradient-based methods have been used widely for automatichistory matching, although the convergence rates of these methods are typicallyslower than the sensitivity-based methods such as the Gauss-Newton or the LSQRmethod. An integral part of the sensitivity-based methods is the computation ofsensitivity coefficients. These sensitivities are simply partial derivativesthat define the change in production response because of small changes inreservoir parameters.
There are several approaches to calculating sensitivity coefficients, andthese generally fall into one of three categories: perturbation method, directmethod, and adjoint-state methods. Conceptually, the perturbation approach isthe simplest and requires the fewest changes in an existing code. Sensitivitiesare estimated simply by perturbing the model parameters one at a time by asmall amount and then computing the corresponding production response. Thisapproach requires (N+1) forward simulations, where N is the number ofparameters. Obviously, it can be computationally prohibitive for reservoirmodels with many parameters. In the direct or sensitivity equation method, theflow and transport equations are differentiated to obtain expressions for thesensitivity 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 flowequations and takes advantage of the fact that the coefficient matrix remainsunchanged for all the parameters and needs to be decomposed only once. Thus,sensitivity computation for each parameter now requires a matrix/vectormultiplication. This method can also be computationally expensive for a largenumber of parameters. Finally, the adjoint-state method requires derivation andsolution of adjoint equations that can be quite cumbersome for multiphase-flowapplications. Furthermore, the number of adjoint solutions will generallydepend on the amount of production data and, thus, the length of the productionhistory.
|File Size||1 MB||Number of Pages||11|
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