On optimization algorithms for the reservoir oil well placement problem

Determining optimal locations and operation parameters for wells in oil and gas reservoirs has a potentially high economic impact. Finding these optima depends on a complex combination of geological, petrophysical, flow regimen, and economical parameters that are hard to grasp intuitively. On the other hand, automatic approaches have in the past been hampered by the overwhelming computational cost of running thousands of potential cases using reservoir simulators, given that each of these runs can take on the order of hours. Therefore, the key issue to such automatic optimization is the development of algorithms that find good solutions with a minimum number of function evaluations. In this work, we compare and analyze the efficiency, effectiveness, and reliability of several optimization algorithms for the well placement problem. In particular, we consider the simultaneous perturbation stochastic approximation (SPSA), finite difference gradient (FDG), and very fast simulated annealing (VFSA) algorithms. None of these algorithms guarantees to find the optimal solution, but we show that both SPSA and VFSA are very efficient in finding nearly optimal solutions with a high probability. We illustrate this with a set of numerical experiments based on real data for single and multiple well placement problems.     

A simultaneous perturbation stochastic approximation algorithm for coupled well placement and control optimization under geologic uncertainty

Development of subsurface energy and environmental resources can be improved by tuning important decision variables such as well locations and operating rates to optimize a desired performance metric. Optimal well locations in a discretized reservoir model are typically identified by solving an integer programming problem while identification of optimal well settings (controls) is formulated as a continuous optimization problem. In general, however, the decision variables in field development optimization can include many design parameters such as the number, type, location, short-term and long-term operational settings (controls), and drilling schedule of the wells. In addition to the large number of decision variables, field optimization problems are further complicated by the existing technical and physical constraints as well as the uncertainty in describing heterogeneous properties of geologic formations. In this paper, we consider simultaneous optimization of well locations and dynamic rate allocations under geologic uncertainty using a variant of the simultaneous perturbation and stochastic approximation (SPSA). In addition, by taking advantage of the robustness of SPSA against errors in calculating the cost function, we develop an efficient field development optimization under geologic uncertainty, where an ensemble of models are used to describe important flow and transport reservoir properties (e.g., permeability and porosity). We use several numerical experiments, including a channel layer of the SPE10 model and the three-dimensional PUNQ-S3 reservoir, to illustrate the performance improvement that can be achieved by solving a combined well placement and control optimization using the SPSA algorithm under known and uncertain reservoir model assumptions.     

Well placement optimization with the covariance matrix adaptation evolution strategy and meta-models

The amount of hydrocarbon recovered can be considerably increased by finding optimal placement of non-conventional wells. For that purpose, the use of optimization algorithms, where the objective function is evaluated using a reservoir simulator, is needed. Furthermore, for complex reservoir geologies with high heterogeneities, the optimization problem requires algorithms able to cope with the non-regularity of the objective function. In this paper, we propose an optimization methodology for determining optimal well locations and trajectories based on the covariance matrix adaptation evolution strategy (CMA-ES) which is recognized as one of the most powerful derivative-free optimizers for continuous optimization. In addition, to improve the optimization procedure, two new techniques are proposed: (a) adaptive penalization with rejection in order to handle well placement constraints and (b) incorporation of a meta-model, based on locally weighted regression, into CMA-ES, using an approximate stochastic ranking procedure, in order to reduce the number of reservoir simulations required to evaluate the objective function. The approach is applied to the PUNQ-S3 case and compared with a genetic algorithm (GA) incorporating the Genocop III technique for handling constraints. To allow a fair comparison, both algorithms are used without parameter tuning on the problem, and standard settings are used for the GA and default settings for CMA-ES. It is shown that our new approach outperforms the genetic algorithm: It leads in general to both a higher net present value and a significant reduction in the number of reservoir simulations needed to reach a good well configuration. Moreover, coupling CMA-ES with a meta-model leads to further improvement, which was around 20% for the synthetic case in this study.     

Ensemble-based hierarchical multi-objective production optimization of smart wells

In an earlier study, two hierarchical multi-objective methods were suggested to include short-term targets in life-cycle production optimization. However, this earlier study has two limitations: (1) the adjoint formulation is used to obtain gradient information, requiring simulator source code access and an extensive implementation effort, and (2) one of the two proposed methods relies on the Hessian matrix which is obtained by a computationally expensive method. In order to overcome the first of these limitations, we used ensemble-based optimization (EnOpt). EnOpt does not require source code access and is relatively easy to implement. To address the second limitation, we used the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm to obtain an approximation of the Hessian matrix. We performed experiments in which water flood was optimized in a geologically realistic multilayer sector model. The controls were inflow control valve settings at predefined time intervals. Undiscounted net present value (NPV) and highly discounted NPV were the long-term and short-term objective functions used. We obtained an increase of approximately 14 % in the secondary objective for a decrease of only 0.2–0.5 % in the primary objective. The study demonstrates that ensemble-based hierarchical multi-objective optimization can achieve results of practical value in a computationally efficient manner.     

An Improved Gradual Deformation Method for Reconciling Random and Gradient Searches in Stochastic Optimizations

Constraining stochastic models of reservoir properties such as porosity and permeability can be formulated as an optimization problem. While an optimization based on random search methods preserves the spatial variability of the stochastic model, it is prohibitively computer intensive. In contrast, gradient search methods may be very efficient but it does not preserve the spatial variability of the stochastic model. The gradual deformation method allows for modifying a reservoir model (i.e., realization of the stochastic model) from a small number of parameters while preserving its spatial variability. It can be considered as a first step towards the merger of random and gradient search methods. The gradual deformation method yields chains of reservoir models that can be investigated successively to identify an optimal reservoir model. The investigation of each chain is based on gradient computations, but the building of chains of reservoir models is random. In this paper, we propose an algorithm that further improves the efficiency of the gradual deformation method. Contrary to the previous gradual deformation method, we also use gradient information to build chains of reservoir models. The idea is to combine the initial reservoir model or the previously optimized reservoir model with a compound reservoir model. This compound model is a linear combination of a set of independent reservoir models. The combination coefficients are calculated so that the search direction from the initial model is as close as possible to the gradient search direction. This new gradual deformation scheme allows us for reducing the number of optimization parameters while selecting an optimal search direction. The numerical example compares the performance of the new gradual deformation scheme with that of the traditional one.     

Multigrid preconditioned conjugate-gradient solver for mixed finite-element method

The mixed finite-element approximation to a second-order elliptic PDE results in a saddle-point problem and leads to an indefinite linear system of equations. The mixed system of equations can be transformed into coupled symmetric positive-definite matrix equations, or a Schur complement problem, using block Gauss elimination. A preconditioned conjugate-gradient algorithm is used for solving the Schur complement problem. The mixed finite-element method is closely related to the cell-centered finite difference scheme for solving second-order elliptic problems with variable coefficients. For the cell-centered finite difference scheme, a simple multigrid algorithm can be defined and used as a preconditioner. For distorted grids, an additional iteration is needed. Nested iteration with a multigrid preconditioned conjugate gradient inner iteration results in an effective numerical solution technique for the mixed system of linear equations arising from a discretization on distorted grids. Numerical results show that the preconditioned conjugate-gradient inner iteration is robust with respect to grid size and variability in the hydraulic conductivity tensor.     

基于梯度分析法的长江上游水库群供水-发电-环境互馈关系解析

长江上游水资源耦合系统优化调控涉及供水、发电和生态需水等相互竞争、不可公度的调控目标。其供水-发电-环境互馈协变机制难以数学解析和刻画，多维目标效益均衡优化调控难以实施。为此，以多目标优化、运筹学理论及方法为基础，提出了基于梯度分析法的供水-发电-环境两两互馈关系研究方法。通过多目标约束优化方法求解长江上游水库群联合优化调度在供水-发电-环境目标空间的最优解集，并进行插值构造了供水-发电-环境互馈关系多维空间曲面，以此为基础，采用一阶差分近似求解供水对环境和发电对环境偏导函数值，以量化环境对供水变化和发电变化响应的梯度，进而解析环境-供水和环境-发电间互馈协变关系。该方法为水库群优化调度多目标互馈关系研究提供了一种新的思路。     

基于梯度分析法的长江上游水库群供水-发电-环境互馈关系解析

长江上游水资源耦合系统优化调控涉及供水、发电和生态需水等相互竞争、不可公度的调控目标。其供水-发电-环境互馈协变机制难以数学解析和刻画，多维目标效益均衡优化调控难以实施。为此，以多目标优化、运筹学理论及方法为基础，提出了基于梯度分析法的供水-发电-环境两两互馈关系研究方法。通过多目标约束优化方法求解长江上游水库群联合优化调度在供水-发电-环境目标空间的最优解集，并进行插值构造了供水-发电-环境互馈关系多维空间曲面，以此为基础，采用一阶差分近似求解供水对环境和发电对环境偏导函数值，以量化环境对供水变化和发电变化响应的梯度，进而解析环境-供水和环境-发电间互馈协变关系。该方法为水库群优化调度多目标互馈关系研究提供了一种新的思路。     

Reservoir description using dynamic parameterisation selection with a combined stochastic and gradient search

There is a correspondence between flow in a reservoir and large scale permeability trends. This correspondence can be derived by constraining reservoir models using observed production data. One of the challenges in deriving the permeability distribution of a field using production data involves determination of the scale of resolution of the permeability. The Adaptive Multiscale Estimation (AME) seeks to overcome the problems related to choosing the resolution of the permeability field by a dynamic parameterisation selection. The standard AME uses a gradient algorithm in solving several optimisation problems with increasing permeability resolution. This paper presents a hybrid algorithm which combines a gradient search and a stochastic algorithm to improve the robustness of the dynamic parameterisation selection. At low dimension, we use the stochastic algorithm to generate several optimised models. We use information from all these produced models to find new optimal refinements, and start out new optimisations with several unequally suggested parameterisations. At higher dimensions we change to a gradient-type optimiser, where the initial solution is chosen from the ensemble of models suggested by the stochastic algorithm. The selection is based on a predefined criterion. We demonstrate the robustness of the hybrid algorithm on sample synthetic cases, which most of them were considered insolvable using the standard AME algorithm.     

基于粗细网格的有限元并行分析方法

并行计算己成为求解大规模岩土工程问题的一种强大趋势。探讨了粗细网格与预处理共轭梯度法结合的并行有限元算法。从多重网格刚度矩阵推得有效的预处理子。该算法在工作站机群上实现。用地基处理时土体强夯的数值模拟分析进行了数值测试,对其并行性能进行了详细分析。计算结果表明：该算法具有良好的并行加速比和效率,是一种有效的并行算法。     

Hybrid differential evolution and particle swarm optimization for optimal well placement

There is no gainsaying that determining the optimal number, type, and location of hydrocarbon reservoir wells is a very important aspect of field development planning. The reason behind this fact is not farfetched—the objective of any field development exercise is to maximize the total hydrocarbon recovery, which for all intents and purposes, can be measured by an economic criterion such as the net present value of the reservoir during its estimated operational life-cycle. Since the cost of drilling and completion of wells can be significantly high (millions of dollars), there is need for some form of operational and economic justification of potential well configuration, so that the ultimate purpose of maximizing production and asset value is not defeated in the long run. The problem, however, is that well optimization problems are by no means trivial. Inherent drawbacks include the associated computational cost of evaluating the objective function, the high dimensionality of the search space, and the effects of a continuous range of geological uncertainty. In this paper, the differential evolution (DE) and the particle swarm optimization (PSO) algorithms are applied to well placement problems. The results emanating from both algorithms are compared with results obtained by applying a third algorithm called hybrid particle swarm differential evolution (HPSDE)—a product of the hybridization of DE and PSO algorithms. Three cases involving the placement of vertical wells in 2-D and 3-D reservoir models are considered. In two of the three cases, a max-mean objective robust optimization was performed to address geological uncertainty arising from the mismatch between real physical reservoir and the reservoir model. We demonstrate that the performance of DE and PSO algorithms is dependent on the total number of function evaluations performed; importantly, we show that in all cases, HPSDE algorithm outperforms both DE and PSO algorithms. Based on the evidence of these findings, we hold the view that hybridized metaheuristic optimization algorithms (such as HPSDE) are applicable in this problem domain and could be potentially useful in other reservoir engineering problems.     

Monte Carlo simulation of permeability fields and reservoir performance predictions with SVD parameterization in RML compared with EnKF

In a previous paper, we developed a theoretical basis for parameterization of reservoir model parameters based on truncated singular value decomposition (SVD) of the dimensionless sensitivity matrix. Two gradient-based algorithms based on truncated SVD were developed for history matching. In general, the best of these “SVD” algorithms requires on the order of 1/2 the number of equivalent reservoir simulation runs that are required by the limited memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm. In this work, we show that when combining SVD parameterization with the randomized maximum likelihood method, we can achieve significant additional computational savings by history matching all models simultaneously using a SVD parameterization based on a particular sensitivity matrix at each iteration. We present two new algorithms based on this idea, one which relies only on updating the SVD parameterization at each iteration and one which combines an inner iteration based on an adjoint gradient where during the inner iteration the truncated SVD parameterization does not vary. Results generated with our algorithms are compared with results obtained from the ensemble Kalman filter (EnKF). Finally, we show that by combining EnKF with the SVD-algorithm, we can improve the reliability of EnKF estimates.     

多点地质统计学建模的发展趋势

从算法研究、训练图像处理和实际应用三个方面详细解剖了国内外多点地质统计学的发展历程,在此基础上,分析了多点地质统计学主流的几种算法的核心原理、适用范围及优缺点,以此来对储层建模的发展趋势作出展望。目前,多点地质统计学虽是随机建模的一种前沿研究热点,但由于其尚未成熟,仍需对建模算法进行研究。为此,在前人研究的基础上,重点分析了多点地质统计学的发展趋势：合理处理训练图像;合理利用软信息;选择合适的相似性方法;选择合适的标准化方法;合理利用平稳性;算法间的耦合;选择合适的过滤器;拓展缝洞型碳酸盐岩模拟。最后,提出多点地质统计学在储层建模方面,应从增加储层的模拟区域、提高模拟精度、扩大储层相的模拟范围和提高计算机模拟效率等方面进行改进。     

History matching time-lapse seismic data using the ensemble Kalman filter with multiple data assimilations

The ensemble Kalman filter (EnKF) has become a popular method for history matching production and seismic data in petroleum reservoir models. However, it is known that EnKF may fail to give acceptable data matches especially for highly nonlinear problems. In this paper, we introduce a procedure to improve EnKF data matches based on assimilating the same data multiple times with the covariance matrix of the measurement errors multiplied by the number of data assimilations. We prove the equivalence between single and multiple data assimilations for the linear-Gaussian case and present computational evidence that multiple data assimilations can improve EnKF estimates for the nonlinear case. The proposed procedure was tested by assimilating time-lapse seismic data in two synthetic reservoir problems, and the results show significant improvements compared to the standard EnKF. In addition, we review the inversion schemes used in the EnKF analysis and present a rescaling procedure to avoid loss of information during the truncation of small singular values.     

混合加速遗传算法在流域模型参数优化中的应用

在实编码遗传算法中加入单纯形搜索算子和加速搜索算子,提出了混合加速遗传算法.通过实例对该法与其它一些遗传算法进行了比较.并在大坳流域模型的参数优选中得到成功的应用.结果表明,混合加速遗传算法具有直观、简便、快速及适用性强等特点,是一种既可以较大概率搜索全局最优解,又能进行局部细致搜索的优秀非线性优化方法.     

基于并行预处理算法的三维重力快速反演

随着地球物理设备和探测技术的不断发展,快速处理大规模地球物理数据的需求也随之增长。为了解决三维重力数据密度反演的耗时问题,提出一种并行的预处理共轭梯度算法来提高计算效率。本文分别采用两种不同的预处理算子通过组合模型数据反演进行测试比较,并利用迭代残差和计算用时共同评价其加速效果。结果表明：对称逐次超松弛预处理方法比对角预处理方法反演计算速度快,密度结果更贴近实际模型;与传统串行的共轭梯度算法相比,本文并行预处理快速算法可以获得近19倍的加速比。将该算法应用于美国Vinton盐丘的实测重力数据中,反演结果能够很好地圈定出岩体的位置,验证了本文并行预处理共轭梯度法在三维重力数据快速反演中的高效性和可行性。     

基于RAGA的供水库群水资源配置模型研究

针对动态规划在供水库群优化调度中存在"维数灾",且难以获得真正最优解这一缺点,将改进的加速遗传算法应用到供水库群的水资源优化配置之中。在加速遗传算法中,嵌入局部搜索,以加强算法的全局寻优能力。以库群间的水利联系为线索,设计了相应的算法框架,并提出了供水策略区间的概念。为了体现该算法的优越性,以二个串联供水水库的水资源优化配置为例,选用该方法与基于动态规划的轮库迭代法进行比较研究。结果表明该方法合理可行、收敛速度快,有一定的实用性。