J.D. Jansen

Hierarchical Long-Term and Short-Term Production Optimization

Model-based dynamic optimization of oil production has a significant potential to improve economic life-cycle performance, as has been shown in various studies. However, within these studies, short-term operational objectives are generally neglected. As a result, the optimized injection and production rates often result in a considerable decrease in short-term production performance. In reality, however, it is often these short-term objectives that dictate the course of the operational strategy. Incorporating short-term goals into the life-cycle optimization problem, therefore, is an essential step in model-based life-cycle optimization. We propose a hierarchical optimization structure with multiple objectives. Within this framework, the life-cycle performance in terms of net present value (NPV) serves as the primary objective and short-term operational performance is the secondary objective, such that optimality of the primary objective constrains the secondary optimization problem. This requires that optimality of the primary objective does not fix all degrees of freedom (DOF) of the decision variable space. Fortunately, the life-cycle optimization problem is generally ill-posed and contains many more decision variables than necessary. We present a method that identifies the redundant DOF in the life-cycle optimization problem, which can subsequently be used in the secondary optimization problem. In our study, we used a 3D reservoir in a fluvial depositional environment with a production life of 7 years. The primary objective is undiscounted NPV, while the secondary objective is aimed at maximizing short-term production. The optimal life-cycle waterflooding strategy that includes short-term performance is compared to the optimal strategy that disregards short-term performance. The experiment shows a very large increase in short-term production, boosting first-year production by a factor of 2, without significantly compromising optimality of the primary objective, showing a slight drop in NPV of only ?0.3%. Our method to determine the redundant DOF in the primary objective function relies on the computation of the Hessian matrix of the objective function with respect to the control variables. Although theoretically rigorous, this method is computationally infeasible for realistically sized problems. Therefore, we also developed a second, more pragmatic, method relying on an alternating sequence of optimizing the primary- and secondary-objective functions. Subsequently, we demonstrated that both methods lead to nearly identical results, which offers scope for application of hierarchical long-term and short-term production optimization to realistically sized flooding-optimization problems.

Closed Loop Reservoir Management

Closed-loop reservoir management is a combination of model-based optimization and data assimilation (computer-assisted history matching), also referred to as ‘real-time reservoir management’, ‘smart reservoir management’ or ‘closed-loop optimization’. The aim is to maximize reservoir performance, in terms of recovery or financial measures, over the life of the reservoir by changing reservoir management from a periodic to a near-continuous process. The key sources of inspiration for our work are measurement and control theory as used in the process industry and data assimilation techniques as used in meteorology and oceanography. We present results of a numerical example to illustrate the scope for closed-loop water flooding using real-time production data under uncertain reservoir conditions. The example concerns a 12-well water flood in a channelized reservoir. Optimization was performed using a reservoir simulator with functionality for adjoint-based life cycle optimization under rate and pressure constraints. Data assimilation was performed using the ensemble Kalman filter. Applying an optimization frequency of respectively once per 4 years, once per 2 years, once per year and once per 30 days resulted in an increase of net present value (NPV) with 6.68, 8.29, 8.30 and 8.71% compared to a conventional reactive control strategy. Moreover, the results for the 30-day cycle were very close (0.15% lower NPV) to those obtained by open-loop optimization using the ‘true’ reservoir model. We illustrate that for closed-loop reservoir management with a fixed well configuration, the use of considerably different reservoir models may lead to near-identical results in terms of NPV. This implies that in such cases the essential information may be represented with a much less complex model than suggested by the large number of grid blocks in typical reservoir models. We also illustrate that the optimal rates and pressures as obtained by openor closedloop optimization are often too irregular to be practically applicable. Fortunately, just as is the case for the data assimilation problem, the flooding optimization problem usually contains many more control variables than necessary, allowing for optimization of long-term reservoir performance while maintaining freedom to perform short-term production optimization. Introduction Our work aims at increased reservoir performance, in terms of recovery or financial measures, using a measurement and control approach to reservoir management. This idea has been around for many years in different forms, often centered around attempts to improve reservoir characterization from a geosciences perspective; see e.g. Chierici (1992). Moreover, recently ‘closed-loop’ or ‘real-time’ approaches to hydrocarbon production have received growing attention as part of various industry initiatives with names as ‘smart fields’, ‘i-fields’, ‘e-fields’, ‘self-learning reservoir management’ or ‘integrated operations’; see Jansen et al. (2005) for some further references. However, whereas the focus of most of these initiatives is primarily on optimization of short-term production, in our work we concentrate on life-cycle optimization, i.e. on processes at a timescale from years to tens of years. We perform reservoir flooding optimization, based on numerical simulation models, in combination with frequent model updating through data assimilation (computer-assisted history matching). This approach has lately also been referred to as ‘closed-loop reservoir modeling’ or ‘closed-loop production optimization’ and some recent references will be discussed below. In contrast to the geosciences-focused approach, we emphasize the need to focus on those elements of the modeling process that can both be verified from measurements and that bear relevance to controllable parameters such as well locations or, in particular, production parameter settings. The underlying hypothesis is that “It will be possible to significantly increase life-cycle value by changing reservoir management from a batch-type to a near-continuous model-based controlled activity.”

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

An adjoint formulation for the gradient-based optimization of oil–gas compositional reservoir simulation problems is presented. The method is implemented within an automatic differentiation-based compositional flow simulator (Stanford’s Automatic Differentiation-based General Purpose Research Simulator, AD-GPRS). The development of adjoint procedures for general compositional problems is much more challenging than for oil–water problems due to the increased complexity of the code and the underlying physics. The treatment of nonlinear constraints, an example of which is a maximum gas rate specification in injection or production wells, when the control variables are well bottom-hole pressures, poses a particular challenge. Two approaches for handling these constraints are presented—a formal treatment within the optimizer and a simpler heuristic treatment in the forward model. The relationship between discrete and continuous adjoint formulations is also elucidated. Results for four example cases of increasing complexity are presented. Improvements in the objective function (cumulative oil produced) relative to reference solutions range from 4.2 to 11.6 %. The heuristic treatment of nonlinear constraints is shown to offer a cost-effective means for obtaining feasible solutions, which are, in some cases, better than those obtained using the formal constraint handling procedure.

A systems description of flow through porous media

This text forms part of material taught during a course in advanced reservoir simulation at Delft University of Technology over the past 10 years. The contents have also been presented at various short courses for industrial and academic researchers interested in background knowledge needed to perform research in the area of closed-loop reservoir management, also known as smart fields, related to e.g. model-based production optimization, data assimilation (or history matching), model reduction, or upscaling techniques. Each of these topics has connections to system-theoretical concepts. The introductory part of the course, i.e. the systems description of flow through porous media, forms the topic of this brief monograph. The main objective is to present the classic reservoir simulation equations in a notation that facilitates the use of concepts from the systems-and-control literature. Although the theory is limited to the relatively simple situation of horizontal two-phase (oil-water) flow, it covers several typical aspects of porous-media flow. The first chapter gives a brief review of the basic equations to represent single-phase and two-phase flow. It discusses the governing partial-differential equations, their physical interpretation, spatial discretization with finite differences, and the treatment of wells. It contains well-known theory and is primarily meant to form a basis for the next chapter where the equations will be reformulated in terms of systems-and-control notation. The second chapter develops representations in state-space notation of the porous-media flow equations. The systematic use of matrix partitioning to describe the different types of inputs leads to a description in terms of nonlinear ordinary-differential and algebraic equations with (state-dependent) system, input, output and direct-throughput matrices. Other topics include generalized state-space representations, linearization, elimination of prestart from escribed pressures, the tracing of stream lines, lift tables, computational aspects, and the derivation of an energy balance for porous-media flow. The third chapter first treats the analytical solution of linear systems of ordinary differential equations for single-phase flow. Next it moves on to the numerical solution of the two-phase flow equations, covering various aspects like implicit, explicit or mixed (IMPES) time discretizations and associated stability issues, Newton-Raphson iteration, streamline simulation, automatic time-stepping, and other computational aspects. The chapter concludes with simple numerical examples to illustrate these and other aspects such as mobility effects, well-constraint switching, time-stepping statistics, and system-energy accounting. The contents of this text should be of value to students and researchers interested in the application of systems-and-control concepts to oil and gas reservoir simulation and other applications of subsurface flow simulation such as CO2 storage, geothermal energy, or groundwater remediation.

Determining Identifiable Parameterizations for Large-scale Physical Models in Reservoir Engineering ⋆

In this paper identifiable parameterizations are determined for models of flow in porous media as applied in the field of petroleum reservoir engineering. Starting from a large-scale, physics-based model parameterization with an extensive parameter space, the best identifiable reduced dimensional parameterization is constructed. This is achieved through the development of an analytical expression for the (finite-time) information matrix of the problem. It is shown that the information matrix can be expressed in terms of controllability and observability properties of the model and the sensitivity of the state space matrices w.r.t. the parameter vector. A reduced dimensional subspace is then obtained after a singular value decomposition of the information matrix, leading to the use of basis functions (spatial patterns) in the original parameter space. The approach is applied to two reservoir models: a siso model with 49 parameters and a mimo model with 441 parameters.

Whirl and Chaotic Motion of Stabilized Drill Collars

Stabilized bottomhole assemblies (BHAs) for oil and gas wells often exhibit complicated lateral vibrations. This paper aims to clarify this behavior with the theory of rotor dynamics. The motion of the collars varies from simple whirling motion, like that of an unbalanced centrifuge, to highly irregular motion caused by nonlinear effects of fluid forces, stabilizer clearance, and borehole-wall contact. In the extreme case, the vibrations can be classified mathematically as chaotic. The results presented in this paper are important in the interpretation of field measurements and indicate the limits of large-scale computer simulations for the prediction of directional tendencies.

Robust ensemble-based multi-objective optimization

We consider robust ensemble-based multi-objective optimization using a hierarchical switching algorithm for combined long-term and short term water flooding optimization. We apply a modified formulation of the ensemble gradient which results in improved performance compared to earlier formulations. We also apply multi-dimensional scaling to visualize projections of the high-dimensional search space, to aid in understanding the complex nature of the objective function surface and the performance of the optimization algorithm. This provides insights into the quality of the gradient, and confirms the presence of ridges in the objective function surface which can be exploited for multi-objective optimization. We used a 18553-gridblock reservoir model of a channelized reservoir with 4 producers and 8 injectors. The controls were the flow rates in the injectors, and the long-term and short-term objective functions were undiscounted net present value (NPV) and highly discounted (25%) NPV respectively. We achieved an increase of 15.2% in the secondary objective for a decrease of 0.5% in the primary objective, averaged over 100 geological realizations. The total number of reservoir simulations was around 20000, which indicates the potential to use the ensemble optimization method for robust multi-objective optimization of medium-sized reservoir models.

Identifiability: From Qualitative Analysis to Model Structure Approximation

Abstract The question whether a physical model structure is identifiable is usually considered in a qualitative way, i.e. it is answered with a yes/no answer. However when considering parameters in large scale (nonlinear) physical models it is relevant to raise the question how the notion of identifiability can be quantified. This implies addressing the question how the model structure can be approximated so as to achieve identifiability, while retaining the interpretation of the physical parameters. In this paper this problem is addressed in a prediction error setting, and it is shown how the construction of best locally identifiable model structure approximations relates to notions of controllability and observability. Additionally the analysis in terms of an prediction error approach relates to iterative optimization algorithms (like Gauss-Newton and Steepest-Descent) and to Bayesian parameter estimation.

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.

Ensemble-Based Multi-Objective Optimization of On-Off Control Devices Under Geological Uncertainty

We consider robust ensemble-based (EnOpt) multi-objective production optimization of on-off inflow control devices (ICDs) for a sector model inspired on a real-field case. The use of on-off valves as optimization variables leads to a discrete control problem. We propose a re-parameterization of such discrete controls in terms of switching times, i.e. we optimize the time at which a particular valve is either open or closed. This transforms the discrete control problem into a continuous control problem which can be efficiently handled with the EnOpt method. Additionally this leads to a significant reduction in the number of controls which is expected to be beneficial for gradient quality when using approximate gradients. We consider an ensemble of sector models where the uncertainty is described by different permeability, porosity, net-to-gross and initial water saturation fields. The controls are the ICD settings over time in the three horizontal injection wells, with approximately 15 ICDs per well. Different optimized strategies resulting from different initial strategies were compared. We achieved a mean 4.2% increase in expected NPV at a 10% discount rate compared to a traditional pressure maintenance strategy. Next, we performed a sequential bi-objective optimization, and achieved an increase of 9.2% in the secondary objective (25% discounted NPV to emphasize short-term production gains) for a minimal decrease of 1% in the primary objective (0% discounted NPV to emphasize long-term recovery gains), as averaged over the 100 geological realizations. The workflow was repeated for alternative numbers of ICDs showing that having fewer control options lowers the expected value for this particular case. The results demonstrate that ensemble-based optimization workflows are able to produce improved robust recovery strategies for realistic field sector models against acceptable computational cost. Copyright