History Matching Geostatistical Model Realizations Using a Geometrical Domain Based Parameterization Technique

Reservoir characterization needs the integration of various data through history matching, especially dynamic information such as production or 4D seismic data. Although reservoir heterogeneities are commonly generated using geostatistical models, random realizations cannot generally match observed dynamic data. To constrain model realizations to reproduce measured dynamic data, an optimization procedure may be applied in an attempt to minimize an objective function, which quantifies the mismatch between real and simulated data. Such assisted history matching methods require a parameterization of the geostatistical model to allow the updating of an initial model realization. However, there are only a few parameterization methods available to update geostatistical models in a way consistent with the underlying geostatistical properties. This paper presents a local domain parameterization technique that updates geostatistical realizations using assisted history matching. This technique allows us to locally change model realizations through the variation of geometrical domains whose geometry and size can be easily controlled and parameterized. This approach provides a new way to parameterize geostatistical realizations in order to improve history matching efficiency.     

A unified formulation for generalized oilfield development optimization

Oilfield development involves several key decisions, including the number, type (injection/production), location, drilling schedule, and operating control trajectories of the wells. Without considering the coupling between these decision variables, any optimization problem formulation is bound to find suboptimal solutions. This paper presents a unified formulation for oilfield development optimization that seeks to simultaneously optimize these decision variables. We show that the source/sink term of the governing multiphase flow equations includes all the above decision variables. This insight leads to a novel and unified formulation of the field development optimization problem that considers the source/sink term in reservoir simulation equations as optimization decision variables. Therefore, a single optimization problem is formulated to simultaneously search for optimal decision variables by determining the complete dynamic form of the source/sink terms. The optimization objective function is the project net present value (NPV), which involves discounted revenue from oil production, operating costs (e.g. water injection and recycling), and capital costs (e.g., cost of drilling wells). A major difficulty after formulating the generalized field development optimization problem is finding an efficient solution approach. Since the total number of cells in a reservoir model far exceeds the number of cells that are intersected by wells, the source/sink terms tend to be sparse. In fact, the drilling cost in the NPV objective function serves as a sparsity-promoting penalty to minimize the number of wells while maximizing the NPV. Inspired by this insight, we solve the optimization problem using an efficient gradient-based method based on recent algorithmic developments in sparse reconstruction literature. The gradients of the NPV function with respect to the source/sink terms is readily computed using well-established adjoint methods. Numerical experiments are presented to evaluate the feasibility and performance of the generalized field development formulation for simultaneous optimization of the number, location, type, controls, and drilling schedule of the wells.     

Model-reduced gradient-based history matching

Gradient-based history matching algorithms can be used to adapt the uncertain parameters in a reservoir model using production data. They require, however, the implementation of an adjoint model to compute the gradients, which is usually an enormous programming effort. We propose a new approach to gradient-based history matching which is based on model reduction, where the original (nonlinear and high-order) forward model is replaced by a linear reduced-order forward model and, consequently, the adjoint of the tangent linear approximation of the original forward model is replaced by the adjoint of a linear reduced-order forward model. The reduced-order model is constructed with the aid of the proper orthogonal decomposition method. Due to the linear character of the reduced model, the corresponding adjoint model is easily obtained. The gradient of the objective function is approximated, and the minimization problem is solved in the reduced space; the procedure is iterated with the updated estimate of the parameters if necessary. The proposed approach is adjoint-free and can be used with any reservoir simulator. The method was evaluated for a waterflood reservoir with channelized permeability field. A comparison with an adjoint-based history matching procedure shows that the model-reduced approach gives a comparable quality of history matches and predictions. The computational efficiency of the model-reduced approach is lower than of an adjoint-based approach, but higher than of an approach where the gradients are obtained with simple finite differences.     

Waterflooding optimization with the INSIM-FT data-driven model

In a recent paper, we developed a physics-based data-driven model referred to as INSIM-FT and showed that it can be used for history matching and future reservoir performance predictions even when no prior geological model is available. The model requires no prior knowledge of petrophysical properties. In this work, we explore the possibility of using INSIM-FT in place of a reservoir simulation model when estimating the well controls that optimize water flooding performance where we use the net present value (NPV) of life-cycle production as our cost (objective) function. The well controls are either the flowing bottom-hole pressure (BHP) or total liquid rates at injectors and producers on the time intervals which represent the prescribed control steps. The optimal well controls that maximize NPV are estimated with an ensemble-based optimization algorithm using the history-matched INSIM-FT model as the forward model. We compare the optimal NPV obtained using INSIM-FT as the forward model with the estimate of the optimal NPV obtained using the correct full-scale reservoir simulation model when performing waterflooding optimization.     

Optimization with the Gradual Deformation Method

Building reservoir models consistent with production data and prior geological knowledge is usually carried out through the minimization of an objective function. Such optimization problems are nonlinear and may be difficult to solve because they tend to be ill-posed and to involve many parameters. The gradual deformation technique was introduced recently to simplify these problems. Its main feature is the preservation of the spatial structure: perturbed realizations exhibit the same spatial variability as the starting ones. It is shown that optimizations based on gradual deformation converge exponentially to the global minimum, at least for linear problems. In addition, it appears that combining the gradual deformation parameterization with optimizations may remove step by step the structure preservation capability of the gradual deformation method. This bias is negligible when deformation is restricted to a few realization chains, but grows increasingly when the chain number tends to infinity. As in practice, optimization of reservoir models is limited to a small number of iterations with respect to the number of gridblocks, the spatial variability is preserved. Last, the optimization processes are implemented on the basis of the Levenberg–Marquardt method. Although the objective functions, written in terms of Gaussian white noises, are reduced to the data mismatch term, the conditional realization space can be properly sampled.     

Reservoir Description with Integrated Multiwell Data Using Two-Dimensional Wavelets

Inferring reservoir data from dynamic production data has long been done through matching the production history. However, proper integration of available production history has always been a challenge. Different production history data such as well pressure and water cut often occur at different scales making their joint inversion difficult. Furthermore, production data obtained from the same well or even the same reservoir are often correlated making a significant portion of the dataset redundant. Thirdly, the massiveness of the data recorded from wells in a large reservoir over a long period of time makes the nonlinear inversion of such data computational demanding. In this paper, we propose the integration of multiwell production data using wavelet transform. The method involves the use of a two-dimensional wavelet transformation of the data space in order to integrate multiple production data and reduce the correlation between multiwell data. Multiple datasets from different wells, representing different production responses (pressure, water cut, etc.), were treated as a single matrix of data rather than separate vectors that assume no correlation amongst datasets. This enabled us to transform the multiwell production data into a two-dimensional wavelet domain and subsequently select the most important wavelets for history match. By minimizing the square of the Frobenius norm of the residual matrix we were able to match the calculated response to the observed response. We derived the relationship that allows us to replace a conventional minimization of the sum of squares of the l ₂ norms of multi-objective functions with the minimization of the square of the Frobenius norm of the integrated data. The usefulness of the approach is demonstrated using two examples. The approach proved very effective at reducing correlation between multiwell data. In addition, the method helped to reduce the cost of computing sensitivity coefficients. However, the method gave poor prediction of water cut when the datasets were not scaled before inverse modeling.     

Closed-loop reservoir management on the Brugge test case

This paper proposes an augmented Lagrangian method for production optimization in which the cost function to be maximized is defined as an augmented Lagrangian function consisting of the net present value (NPV) and all the equality and inequality constraints except the bound constraints. The bound constraints are dealt with using a trust-region gradient projection method. The paper also presents a way to eliminate the need to convert the inequality constraints to equality constraints with slack variables in the augmented Lagrangian function, which greatly reduces the size of the optimization problem when the number of inequality constraints is large. The proposed method is tested in the context of closed-loop reservoir management benchmark problem based on the Brugge reservoir setup by TNO. In the test, we used the ensemble Kalman filter (EnKF) with covariance localization for data assimilation. Production optimization is done on the updated ensemble mean model from EnKF. The production optimization resulted in a substantial increase in the NPV for the expected reservoir life compared to the base case with reactive control.     

A multiobjective steepest descent method with applications to optimal well control

Multiobjective optimization deals with mathematical optimization problems where two or more objective functions (cost functions) are to be optimized (maximized or minimized) simultaneously. In most cases of interest, the objective functions are in conflict, i.e., there does not exist a decision (design) vector (vector of optimization variables) at which every objective function takes on its optimal value. The solution of a multiobjective problem is commonly defined as a Pareto front, and any decision vector which maps to a point on the Pareto front is said to be Pareto optimal. We present an original derivation of an analytical expression for the steepest descent direction for multiobjective optimization for the case of two objectives. This leads to an algorithm which can be applied to obtain Pareto optimal points or, equivalently, points on the Pareto front when the problem is the minimization of two conflicting objectives. The method is in effect a generalization of the steepest descent algorithm for minimizing a single objective function. The steepest-descent multiobjective optimization algorithm is applied to obtain optimal well controls for two example problems where the two conflicting objectives are the maximization of the life-cycle (long-term) net-present-value (NPV) and the maximization of the short-term NPV. The results strongly suggest the multiobjective steepest-descent (MOSD) algorithm is more efficient than competing multiobjective optimization algorithms.     

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.     

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.     

Approximate Derivative Computations for the Gradient-Based Optimization Methods in the Local Gradual Deformation for History Matching

Reservoir characterization needs the integration of various data through history matching, especially dynamic information such as production or four-dimensional seismic data. To update geostatistical realizations, the local gradual deformation method can be used. However, history matching is a complex inverse problem, and the computational effort in terms of the number of reservoir simulations required in the optimization procedure increases with the number of matching parameters. History matching large fields with a large number of parameters has been an ongoing challenge in reservoir simulation. This paper presents a new technique to improve history matching with the local gradual deformation method using the gradient-based optimizations. The new approach is based on the approximate derivative calculations using the partial separability of the objective function. The objective function is first split into local components, and only the most influential parameters in each component are used for the derivative computation. A perturbation design is then proposed to simultaneously compute all the derivatives with only a few simulations. This new technique makes history matching using the local gradual deformation method with large numbers of parameters tractable.     

Near-well upscaling for three-phase flows

Large-scale flow models constructed using standard coarsening procedures may not accurately resolve detailed near-well effects. Such effects are often important to capture, however, as the interaction of the well with the formation can have a dominant impact on process performance. In this work, a near-well upscaling procedure, which provides three-phase well-block properties, is developed and tested. The overall approach represents an extension of a recently developed oil–gas upscaling procedure and entails the use of local well computations (over a region referred to as the local well model (LWM)) along with a gradient-based optimization procedure to minimize the mismatch between fine and coarse-scale well rates, for oil, gas, and water, over the LWM. The gradients required for the minimization are computed efficiently through solution of adjoint equations. The LWM boundary conditions are determined using an iterative local-global procedure. With this approach, pressures and saturations computed during a global coarse-scale simulation are interpolated onto LWM boundaries and then used as boundary conditions for the fine-scale LWM computations. In addition to extending the overall approach to the three-phase case, this work also introduces new treatments that provide improved accuracy in cases with significant flux from the gas cap into the well block. The near-well multiphase upscaling method is applied to heterogeneous reservoir models, with production from vertical and horizontal wells. Simulation results illustrate that the method is able to accurately capture key near-well effects and to provide predictions for component production rates that are in close agreement with reference fine-scale results. The level of accuracy of the procedure is shown to be significantly higher than that of a standard approach which uses only upscaled single-phase flow parameters.     

An Initial Guess for the Levenberg–Marquardt Algorithm for Conditioning a Stochastic Channel to Pressure Data

A standard procedure for conditioning a stochastic channel to well-test pressure data requires the minimization of an objective function. The Levenberg–Marquardt algorithm is a natural choice for minimization, but may suffer from slow convergence or converge to a local minimum which gives an unacceptable match of observed pressure data if a poor initial guess is used. In this work, we present a procedure to generate a good initial guess when the Levenberg–Marquardt algorithm is used to condition a stochastic channel to pressure data and well observations of channel facies, channel thickness, and channel top depth. This technique yields improved computational efficiency when the Levenberg–Marquardt method is used as the optimization procedure for generating realizations of the model by the randomized maximum likelihood method.     

Well placement optimization subject to realistic field development constraints

This work considers the well placement problem in reservoir management and field development optimization. In particular, it emphasizes embedding realistic and practical constraints into a mathematical optimization formulation. Such constraints are a prerequisite for the wider use of mathematical optimization techniques in well placement problems, since constraints are a way to incorporate reservoir engineering knowledge into the problem formulation. There are important design limitations that are used by the field development team when treating the well placement problem, and these limitations need to be articulated and eventually formalized within the problem before conducting the search for optimal well placements. In addition, these design limitations may be explicit or implicit. In this work, various design limitations pertaining to well locations have been developed in close collaboration with a field operator on the Norwegian Continental Shelf. Moreover, this work focuses on developing constraint-handling capability to enforce these various considerations during optimization. In particular, the Particle Swarm Optimization (PSO) algorithm is applied to optimize for the well locations, and various practical well placement constraints are incorporated into the PSO algorithm using two different constraint-handling techniques: a decoder procedure and the penalty method. The decoder procedure maps the feasible search space onto a cube and has the advantage of not requiring parameter tuning. The penalty method converts the constrained optimization problem into an unconstrained one by introducing an additional term, which is called a penalty function, to the objective function. In contrast to the penalty method, only feasible solutions are evaluated in the decoder method. Through numerical simulations, a comparison between the penalty method and the decoder technique is performed for two cases. We show that the decoder technique can easily be implemented for the well placement problem, and furthermore, that it performs better than the penalty method in most of the cases.     

A method for automatic history matching of a compositional reservoir simulator with multipoint flux approximation

A method for history matching of an in-house petroleum reservoir compositional simulator with multipoint flux approximation is presented. This method is used for the estimation of unknown reservoir parameters, such as permeability and porosity, based on production data and inverted seismic data. The limited-memory Broyden–Fletcher–Goldfarb–Shanno method is employed for minimization of the objective function, which represents the difference between simulated and observed data. In this work, we present the key features of the algorithm for calculations of the gradients of the objective function based on adjoint variables. The test example shows that the method is applicable to cases with anisotropic permeability fields, multipoint flux approximation, and arbitrary fluid compositions.     

Optimization of well placement by combination of a modified particle swarm optimization algorithm and quality map method

Determining the optimum placement of new wells in an oil field is a crucial work for reservoir engineers. The optimization problem is complex due to the highly nonlinearly correlated and uncertain reservoir performances which are affected by engineering and geologic variables. In this paper, the combination of a modified particle swarm optimization algorithm and quality map method (QM + MPSO), modified particle swarm optimization algorithm (MPSO), standard particle swarm optimization algorithm (SPSO), and centered-progressive particle swarm optimization (CP-PSO) are applied for optimization of well placement. The SPSO, CP-PSO, and MPSO algorithms are first discussed, and then the modified quality map method is discussed, and finally the implementation of these four methods for well placement optimization is described. Four example cases which involve depletion drive model, water injection model, and a real field reservoir model, with the maximization of net present value (NPV) as the objective function are considered. The physical model used in the optimization analyses is a 3-dimensional implicit black-oil model. Multiple runs of all methods are performed, and the results are averaged in order to achieve meaningful comparisons. In the case of optimizing placement of a single producer well, it is shown that it is not necessary to use the quality map to initialize the position of well placement. In other cases considered, it is shown that the QM + MPSO method outperforms MPSO method, and MPSO method outperforms SPSO and CP-PSO method. Taken in total, the modification of SPSO method is effective and the applicability of QM + MPSO for this challenging problem is promising     

Calculating derivatives for automatic history matching

Automatic history matching is based on minimizing an objective function that quantifies the mismatch between observed and simulated data. When using gradient-based methods for solving this optimization problem, a key point for the overall procedure is how the simulator delivers the necessary derivative information. In this paper, forward and adjoint methods for derivative calculation are discussed. Procedures for sensitivity matrix building, sensitivity matrix and transpose sensitivity matrix vector products are fully described. To show the usefulness of the derivative calculation algorithms, a new variant of the gradzone analysis, which tries to address the problem of selecting the most relevant parameters for a history matching, is proposed using the singular value decomposition of the sensitivity matrix. Application to a simple synthetic case shows that this procedure can reveal important information about the nature of the history-matching problem.