Reservoir simulation

Multilevel Strategies Improve History Matching of Complex Reservoir Models

The complete paper explores the use of multilevel derivative-free optimization for history matching, with model properties described using principal component analysis (PCA) -based parameterization techniques.


The complete paper explores the use of multilevel derivative-free optimization for history matching, with model properties described using principal component analysis (PCA) -based parameterization techniques. The parameterizations applied are optimization-based PCA (O-PCA) and convolutional-neural-network-based PCA (CNN-PCA). Mesh adaptive direct search (MADS), a pattern search method that parallelizes naturally, is used for the optimizations required to generate posterior models. The use of PCA-based parameterization reduces considerably the number of variables that must be determined during history matching, but the optimization problem can still be computationally demanding. The multilevel strategy addresses this issue by reducing the number of simulations that must be performed at each MADS iteration. History-matching results demonstrate that substantial uncertainty reduction is achieved in all cases considered and that the multilevel strategy is effective in reducing the number of simulations required.


Parameterization has two key advantages in history matching:

  • Far fewer parameters must be determined, which simplifies greatly the minimization in some cases.
  • Posterior (history-matched) geomodels will be consistent with the geological scenario or training image for which the parameterization was constructed.

PCA is the foundation for many geological-dimension-reduction parameterization methods. PCA is based on the eigen-decomposition of the prior geomodel covariance matrix and, thus, honors only two-point spatial statistics.
History-matching algorithms largely can be classified as either ­optimization-based or ensemble-based procedures. Optimization-based algorithms generate one history-matched model at a time, while ensemble-based algorithms update a large set of models simultaneously. The resulting ensemble of posterior models is used to quantify uncertainty. In this study, only optimization-based history matching is considered, though the parameterizations applied also can be used with ensemble-based methods.

Optimization-based history-matching algorithms can be classified further into gradient-based and derivative-free optimization (DFO) methods. Gradient-based algorithms often are combined with adjoint procedures that provide the required gradient efficiently. Unlike adjoint-based procedures, DFO algorithms (which are considered in this work) are nonintrusive. They treat the reservoir simulator as a so-called “black box,” and, as a result, they are much easier to implement and can be applied with any simulator.

In this study, MADS, a pattern-search DFO, is applied for history matching. In pattern-search (or stencil-based) algorithms, the number of simulations required at each iteration scales with the number of parameters to be determined. Because the latter typically is much smaller than the number of gridblocks in the model, geological parameterization leads to large computational savings when a pattern search, or other DFO, is used for history matching. The performance of DFO for history matching with non-Gaussian models, a primary topic in the complete paper, has not been studied extensively.

The authors devise a multilevel strategy, applicable for PCA-based parameterization methods, to further reduce the computational cost of history matching using DFO. A multilevel strategy lends itself naturally to PCA-based parameterizations because the PCA representation can be ordered from the most-important to the least-important principal component. Thus, PCA parameters can be determined sequentially, level by level, in groups, rather than all at once.

The complete paper first provides an introduction of the O-PCA and CNN-PCA parameterization procedures. Next, history matching with these parameterizations is discussed and the multilevel strategy is described. Detailed single-level and multilevel history-matching results are presented for several 2D cases, using either O-PCA or CNN-PCA parameterization. Specifically, O-PCA is used for a conditional bimodal channelized system, and CNN-PCA is used for an unconditional binary channelized system and a conditional bimodal deltaic-fan system.

Numerical Results

The multilevel DFO-based history-matching procedure is applied to three cases. All involve 2D systems characterized by multipoint spatial statistics. The first case is a conditional bimodal channelized system, and O-PCA parameterization is used (Fig. 1). In this case, single-level and multilevel MADS results are compared with a gradient-based history-matching algorithm. Case 2 involves a binary (as opposed to bimodal) channelized system, and conditioning data are not included, which makes the problem more challenging. The geomodel in Case 2 is parameterized with CNN-PCA. In Case 3, a bimodal deltaic-fan system that follows a nonstationary geological pattern is considered (Fig. 2). CNN-PCA again is used for parameterization. Previous research has shown that, for the case considered, O-PCA performs adequately in terms of predictions for existing wells, but CNN-PCA performs considerably better for predictions involving new wells because CNN-PCA provides more realistic channel continuity in regions away from existing wells.

Fig. 1—True model and corresponding histogram for Case 1.
Fig. 2—Log permeability of the true model, prior model, and corresponding posterior models from single-level and multilevel MADS for Case 3.


The two methods each have their region of applicability; the much-simpler O-PCA parameterization is acceptable in cases with sufficient hard data, but CNN-PCA is required in cases with little or no hard data (CNN-PCA also appears to give superior performance in cases with very complex channel geometry). The authors aim to demonstrate that each of the PCA-based methods can be used in a multilevel setting. Thus, for each example, they use one or the other (but not both) of the parameterization techniques.


In the complete paper, PCA-based parameterizations for geomodel properties are incorporated into a multilevel derivative-free optimization procedure for history matching. The specific parameterizations considered were O-PCA and CNN-PCA. These parameterizations allowed model properties, such as permeability in every gridblock, to be represented in terms of a relatively small number of coefficients. The derivative-free optimization procedure used in this work was MADS. The multilevel strategy exploits the fact that the PCA representation underlying both O-PCA and CNN-PCA displays a natural ordering. More specifically, the coefficients determined in history matching correspond to basis vectors representing features of decreasing spatial scale. Thus, the PCA coefficients can be determined in groups, using a multilevel strategy. An O-PCA parameterization was used in the first of the three sample cases, where a reasonable amount of conditioning data was available, and CNN-PCA parameterizations were used in the other two cases (in a previous work, the authors found O-PCA to perform satisfactorily in cases with sufficient conditioning data but to potentially underperform with little or no hard data).


Results in all three cases demonstrated that history matching led to posterior uncertainty that was significantly less than prior uncertainty and that the results provided by the multilevel strategy were in general agreement with (reference) single-level MADS results. The multilevel procedure typically requires more iterations to converge than the single-level approach, but each iteration may entail many fewer simulation runs. Thus, significant savings in terms of the total number of simulations can be achieved using the multilevel approach. A number of possible directions exist for future work in this area. The CNN-PCA representation is currently limited to 2D, though work on extending it to 3D systems is under way. Different ­derivative-free optimization algorithms, including stochastic search methods such as particle-swarm optimization and genetic algorithms, should be incorporated and tested within the multilevel history-matching framework. The multilevel strategy itself should also be investigated further. For example, it may be beneficial to use two or more passes through the various levels, with loose tolerances in the first pass and tighter tolerances in the second pass. Other model properties, such as relative permeability or engineering parameters, also could be included in the history matching. Finally, once some of these developments are achieved, application of the overall procedure to realistic cases will be of interest.

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 193895, “Multilevel Strategies and Geological Parameterizations for History Matching Complex Reservoir Models,” by Yimin Liu, SPE, and Louis J. Durlofsky, SPE, Stanford University, prepared for the 2019 SPE Reservoir Simulation Conference, Galveston, Texas, 10–11 April. The paper has not been peer reviewed.