A derivative-free methodology with local and global search for the constrained joint optimization of well locations and controls

In oil field development, the optimal location for a new well depends on how it is to be operated. Thus, it is generally suboptimal to treat the well location and well control optimization problems separately. Rather, they should be considered simultaneously as a joint problem. In this work, we present noninvasive, derivative-free, easily parallelizable procedures to solve this joint optimization problem. Specifically, we consider Particle Swarm Optimization (PSO), a global stochastic search algorithm; Mesh Adaptive Direct Search (MADS), a local search procedure; and a hybrid PSO–MADS technique that combines the advantages of both methods. Nonlinear constraints are handled through use of filter-based treatments that seek to minimize both the objective function and constraint violation. We also introduce a formulation to determine the optimal number of wells, in addition to their locations and controls, by associating a binary variable (drill/do not drill) with each well. Example cases of varying complexity, which include bound constraints, nonlinear constraints, and the determination of the number of wells, are presented. The PSO–MADS hybrid procedure is shown to consistently outperform both stand-alone PSO and MADS when solving the joint problem. The joint approach is also observed to provide superior performance relative to a sequential procedure.     

Accounting for Estimation Optimality Criteria in Simulated Annealing

This paper presents both estimation and simulation as optimization problems that differ in the optimization criteria, minimization of a local expected loss for estimation and reproduction of global statistics (semivariogram, histogram) for simulation. An intermediate approach is proposed whereby an initial random image is gradually modified using simulated annealing so as to better match both local and global constraints. The relative weights of the different constraints in the objective function allow the user to strike a balance between smoothness of the estimated map and reproduction of spatial variability by simulated maps. The procedure is illustrated using a synthetic dataset. The proposed approach is shown to enhance the influence of observations on neighboring simulated values, hence the final realizations appear to be better conditioned to the sample information. It also produces maps that are more accurate (smaller prediction error) than stochastic simulation ignoring local constraints, but not as accurate as E-type estimation. Flow simulation results show that accounting for local constraints yields, on average, smaller errors in production forecast than a smooth estimated map or a simulated map that reproduces only the histogram and semivariogram. The approach thus reduces the risk associated with the use of a single realization for forecasting and planning.     

A Blocking Markov Chain Monte Carlo Method for Inverse Stochastic Hydrogeological Modeling

An adequate representation of the detailed spatial variation of subsurface parameters for underground flow and mass transport simulation entails heterogeneous models. Uncertainty characterization generally calls for a Monte Carlo analysis of many equally likely realizations that honor both direct information (e.g., conductivity data) and information about the state of the system (e.g., piezometric head or concentration data). Thus, the problems faced is how to generate multiple realizations conditioned to parameter data, and inverse-conditioned to dependent state data. We propose using Markov chain Monte Carlo approach (MCMC) with block updating and combined with upscaling to achieve this purpose. Our proposal presents an alternative block updating scheme that permits the application of MCMC to inverse stochastic simulation of heterogeneous fields and incorporates upscaling in a multi-grid approach to speed up the generation of the realizations. The main advantage of MCMC, compared to other methods capable of generating inverse-conditioned realizations (such as the self-calibrating or the pilot point methods), is that it does not require the solution of a complex optimization inverse problem, although it requires the solution of the direct problem many times.     

Automatic Histogram and Variogram Reproduction in Simulated Annealing Simulation

Simulated annealing simulation (SAS) is a flexible tool for generating stochastic simulations conditioned to data at various scales and precision. However, a number of important drawbacks exists: (1) SAS requires considerable CPU time, (2) it requires experience in setting the so-called cooling schedule to obtain convergence, and (3) its space of uncertainty is not well understood. In this paper, I propose a new simulated annealing method that guarantees histogram and variogram reproduction through the implementation of a novel perturbation mechanism. The novel perturbation method is based on the Metropolis–Hastings sampler for a Markov-type random field. This new annealing method allows removal of the histogram and variogram from the objective function in the simulated annealing procedure. Furthermore, for the proposed method one finds that (1) the uncertainty space can be quantified, (2) the convergence properties can be linked to convergence of stationary Markov chains, and (3) the convergence speed is improved over traditional simulated annealing.     

Performance studies on six heuristic global optimization methods in the location of critical slip surface

Recently, many heuristic global optimization algorithms have evolved with success for treating various types of problems. Majority of these algorithms have not been applied to slope stability problem for which the presence of soft band and convergence problem (discontinuity of the objective function) may create difficulties in the minimization process. In this paper, six heuristic optimization algorithms are applied to some simple and complicated slopes. The effectiveness and efficiency of these algorithms under different cases are evaluated, and it is found that no single method can outperform all the other methods under all cases, as different method has different behavior in different types of problems. For normal cases, the particle swarm method appears to be effective and efficient over various conditions, and this method is recommended to be used. For special cases where the objective function is highly discontinuous, the simulated annealing method appears to be a more stable solution.     

地震反演与非线性随机优化方法

地震反演问题常常需要一些非线性全局最优化的方法。近年来，基于生物学、物理学、人工智能和一些非线性科学而发展了一些具有全局优化性能且通用性强的随机搜索算法，如：遗传算法、模拟退火、禁忌搜索和混沌搜索等。作者主要针对地震勘探反演问题的应用综述了这四种非线性随机优化方法的原理和特点，并对它们的性能作了一定的分析和比较。     

Geostatistical Seismic Inversion with Direct Sequential Simulation and Co-simulation with Multi-local Distribution Functions

Stochastic sequential simulation is a common modelling technique used in Earth sciences and an integral part of iterative geostatistical seismic inversion methodologies. Traditional stochastic sequential simulation techniques based on bi-point statistics assume, for the entire study area, stationarity of the spatial continuity pattern and a single probability distribution function, as revealed by a single variogram model and inferred from the available experimental data, respectively. In this paper, the traditional direct sequential simulation algorithm is extended to handle non-stationary natural phenomena. The proposed stochastic sequential simulation algorithm can take into consideration multiple regionalized spatial continuity patterns and probability distribution functions, depending on the spatial location of the grid node to be simulated. This work shows the application and discusses the benefits of the proposed stochastic sequential simulation as part of an iterative geostatistical seismic inversion methodology in two distinct geological environments in which non-stationarity behaviour can be assessed by the simultaneous interpretation of the available well-log and seismic reflection data. The results show that the elastic models generated by the proposed stochastic sequential simulation are able to reproduce simultaneously the regional and global variogram models and target distribution functions relative to the average volume of each sub-region. When used as part of a geostatistical seismic inversion procedure, the retrieved inverse models are more geologically realistic, since they incorporate the knowledge of the subsurface geology as provided, for example, by seismic and well-log data interpretation.     

改进的模拟退火-变尺度优化方法在测井解释中的应用

实际中的测井参数反演是一个多参数、非线性优化问题,所采用的目标函数,即度量由参数化的理论模型得出的预测值与观测值的吻合程度,往往存在多解的现象。针对这种状况我们提出了模拟退火与变尺度综合反演方法用于参数计算,经过实际试算证明该方法效果很好。     

A comparative performance of least-square method and very fast simulated annealing global optimization method for interpretation of self-potential anomaly over 2-D inclined sheet type structure

Inversion of self-potential anomaly for 2-D inclined sheets of infinite horizontal extent has been studied. Least-square inversion and very fast simulated annealing global optimization has been used to model the five parameters of self potential anomaly. The method of least square and very fast simulated annealing global optimization method is compared and analyzed. Very fast simulated annealing can model the noisy and field data of self potential anomaly very precisely than linear inversion technique. However, time taken by very fast simulated annealing inversion is larger than linearized inversion. The comparative analysis has been done on synthetic data (noise free and noisy) and two field data from Bavarian woods anomaly, Germany and Surda anomaly, India to show the efficacy of both the methods. The estimated parameters were compared with those from previous studies using various global optimization algorithms, mainly neural network, genetic algorithm and particle swarm optimization on the same field data sets. It can be concluded that the global optimization algorithms considered in this study were able to yield compatible solutions with those from least-square methods. The present global optimization method is in good agreement with the other global optimization methods in terms of results and computation time.     

Optimal water and waste load allocation in reservoir–river systems: a case study

In this paper, a new methodology is developed for optimization of water and waste load allocation in reservoir–river systems considering the existing uncertainties in reservoir inflow, waste loads and water demands. A stochastic dynamic programming (SDP) model is used to optimize reservoir operation considering the inflow uncertainty, and another model called PSO-SA is developed and linked with the SDP model for optimizing water and waste load allocation in downstream river. In the PSO-SA model, a particle swarm optimization technique with a dynamic penalty function for handling the constraints is used to optimize water and waste load allocation policies. Also, a simulated annealing technique is utilized for determining the upper and lower bounds of constraints and objective function considering the existing uncertainties. As the proposed water and waste load allocation model has a considerable run-time, some powerful soft computing techniques, namely, Regression tree Induction (named M5P), fuzzy K-nearest neighbor, Bayesian network, support vector regression and an adaptive neuro-fuzzy inference system, are trained and validated using the results of the proposed methodology to develop real-time water and waste load allocation rules. To examine the efficiency and applicability of the methodology, it is applied to the Dez reservoir–river system in the south-western part of Iran.     

An interactive global optimization algorithm for geological problems

A number of problems in geology can be formulated so that they consist of optimizing a real-valued function (termed the objective function) on some interval or over some region. Many methods are available for solution if the function is unimodal within the domain of interest. Direct methods, involving only function evaluations, are particularly useful in geological problems where the objective function may be strongly nonlinear and constructed from sampled data. In practical problems, the objective function often is not unimodal. Standard optimization routines are not capable of distinguishing between local extrema or of locating the global extremum, which is the point of interest in most cases. The usual approach—trying several different starting points in the hope that the best local extremum found is the global extremum—is inefficient and unreliable. An ancillary algorithm has been developed which avoids these problems and which couples with a variety of local optimization routines. The algorithm first constructs a grid of objective function values over some feasible region. The region dimensions and grid spacings are based on specific problem considerations. First differences are then calculated for successive points along each grid line and monitored in sign only, which rapidly locates extrema. User interaction determines how many of these extrema will undergo further investigation, which is carried out by passing locations to a local optimization subroutine. The algorithm has proved successful on a number of problems. A geological example—determination of benthic mixing parameters in deep-sea sediments via minimization of stratigraphic offset between ¹⁸ O signals from two different species of planktonic foraminifera—is given. FORTRAN code is provided for the global optimization routine, a golden section search subroutine for one-dimensional objective functions, and a simplex subroutine for multidimensional problems.     

An enhanced stochastic optimization in fracture network modelling conditional on seismic events

This paper presents an approach to modelling fracture networks in hot dry rock geothermal reservoirs. A detailed understanding of the fracture network within a geothermal reservoir is critically important for assessments of reservoir potential and optimal production design. One important step in fracture network modelling is to estimate the fracture density and the fracture geometries, particularly the size and orientation of fractures. As fracture networks in these reservoirs can never be directly observed there is significant uncertainty about their true nature and the only feasible approach to modelling is a stochastic one. We propose a global optimization approach using simulated annealing which is an extension of our previous work. The fracture model consists of a number of individual fractures represented by ellipses passing through the micro-seismic points detected during the fracture stimulation process, i.e. the fracture model is conditioned on the seismic points. The distances of the seismic points from fitted fracture planes (ellipses) are, therefore, important in assessing the goodness-of-fit of the model. Our aims in the proposed approach are to formulate an appropriate objective function for the optimal fitting of a set of fracture planes to the micro-seismic data and to derive an efficient modification scheme to update the model parameters. The proposed objective function consists of three components: orthogonal projection distances of the seismic points from the nearest fitted fractures, the amount of fracturing (fitted fracture areas) and the volumes of the convex hull of the associated points of fitted fractures. The functions used in the model update scheme allow the model to achieve an acceptable fit to the points and to converge to acceptable fitted fracture sizes. These functions include two groups of proposals: one for updating fracture parameters and the other for determining the size of the fracture network. To increase the efficiency of the optimization, a spatial clustering approach, the Distance-Directional Transform, was developed to generate parameters for newly proposed fractures. A simulated dataset was used as an example to evaluate our approach and we compared the results to those derived using our previously published algorithm on a real dataset from the Habanero geothermal field in the Cooper Basin, South Australia. In a real application, such as the Habanero dataset, it is difficult to determine definitively which algorithm performs better due to the many uncertainties but the number of association points, the number of final fractures and the error are three important factors that quantify the effectiveness of our algorithm.     

Spectral Simulation and Conditioning to Local Continuity Trends in Geologic Modeling

Spectral simulation has gained application in building geologic models due to the advantage of better honoring the spatial continuity of petrophysical properties, such as reservoir porosity and shale volume. Distinct from sequential simulation methods, spectral simulation is a global algorithm in the sense that a global density spectrum is calculated once and the inverse Fourier transform is performed on the Fourier coefficient also only once to generate a simulation realization. The generated realizations honor the spatial continuity structure globally over the whole field instead of only within a search neighborhood, as with sequential simulation algorithms. However, the disadvantage of global spectral simulation is that it traditionally cannot account for the local information such as the local continuity trends, which are often observed in reservoirs and hence are important to be accounted for in geologic models. This disadvantage has limited wider application of spectral simulation in building geologic models. In this paper, we present ways of conditioning geologic models to the relevant local information. To account for the local continuity trends, we first scale different frequency components of the original model with local-amplitude spectrum ratios that are specific to the local trend. The sum of these scaled frequency components renders a new model that displays the desired local continuity trend. The implementation details of this new method are discussed and examples are provided to illustrate the algorithm.     

一维层状介质大地电磁模拟退火反演法

大地电磁模拟退火反演法是一种最优化的非线性反演方法,与传统的线性反演方法相比该方法具有：（１）不依赖于初始模型的选择;（２）能寻找全局最小点而不陷入局部极小;（３）在反演过程中不用计算雅可比偏数矩阵等优点;通过对各种类型的大地电磁测深理论曲线试算,结果表明模拟退火法能准确地自动反演地电参数（地层电阻率,厚度）最后对实际资料进行了处理,取得了较好的效果。     

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.     

互相关与最小二乘加权目标函数全波形反演

全波形反演是一个高度非线性的优化问题,当地震数据中缺少低频成分而初始速度与真实速度相差较远时,反演容易陷入局部极小值。笔者提出一种新的目标函数,将模拟地震记录和观测记录的归一化互相关与最小二乘结合。互相关侧重相位匹配,具有更强的线性,能减弱"跳周"现象。通过设置权重因子,在反演前期利用互相关先恢复低波数的背景速度模型,再加入最小二乘约束恢复高波数的模型细节。数值模拟试验结果表明,基于该目标函数的全波形反演不依赖精确的初始模型和低频信息,向全局极小值迅速收敛,能有效改善反演的稳定性,并获得比基于常规目标函数的全波形反演更精确的结果。     

A nonlinear multiparameter prestack seismic inversion method based on hybrid optimization approach

Multiparameter prestack seismic inversion is one of the most powerful techniques in quantitatively estimating subsurface petrophysical properties. However, it remains a challenging problem due to the nonlinearity and ill-posedness of the inversion process. Traditional regularization approach can stabilize the solution but at the cost of smoothing valuable geological boundaries. In addition, compared with linearized optimization methods, global optimization techniques can obtain better results regardless of initial models, especially for multiparameter prestack inversion. However, when solving multiparameter prestack inversion problems, the application of standard global optimization algorithms maybe limited due to the issue of high computational cost (e.g., simulating annealing) or premature convergence (e.g., particle swarm optimization). In this paper, we propose a hybrid optimization-based multiparameter prestack inversion method. In this method, we introduce a prior constraint term featured by multiple regularization functions, intended to preserve layered boundaries of geological formations; in particular, to address the problem of premature convergence existing in standard particle swarm optimization algorithm, we propose a hybrid optimization strategy by hybridizing particle swarm optimization and very fast simulating annealing to solve the nonlinear optimization problem. We demonstrate the effectiveness of the proposed inversion method by conducting synthetic test and field data application, both of which show encouraging results.     

Optimal pumping from Palmela water supply wells (Portugal) using simulated annealing

Aquifer systems are an important part of an integrated water resources management plan as foreseen in the European Union’s Water Framework Directive (2000). The sustainable development of these systems demands the use of all available techniques capable of handling the multidisciplinary features of the problems involved. The formulation and resolution of an optimization model is described for a planning and management problem based on the Palmela aquifer (Portugal), developed to supply a given number of demand centres. This problem is solved using one of the latest optimization techniques, the simulated annealing heuristic method, designed to find the optimal solutions while avoiding falling into local optimums. The solution obtained, providing the wells location and the corresponding pumped flows to supply each centre, are analysed taking into account the objective function components and the constraints. It was found that the operation cost is the biggest share of the final cost, and the choice of wells is greatly affected by this fact. Another conclusion is that the solution takes advantage of the economies of scale, that is, it points toward drilling a large capacity well even if this increases the investment cost, rather than drilling several wells, which together will increase the operation costs.