In production optimization, we seek to determine the well settings (bottomhole pressures, flow rates) that maximize an objective function such as net present value. In this paper we introduce and apply a new approximate dynamic programming (ADP) algorithm for this optimization problem. ADP aims to approximate the global optimum using limited computational resources via a systematic set of procedures that approximate exact dynamic programming algorithms. The method is able to satisfy general constraints such as maximum watercut and maximum liquid production rate in addition to bound constraints. ADP has been used in many application areas, but it does not appear to have been implemented previously for production optimization. The ADP algorithm is applied to twodimensional
problems involving primary production and water injection. We demonstrate that the algorithm is able to provide clear improvement in the objective function compared to baseline strategies. It is also observed that, in cases where the global optimum is known (or surmised), ADP provides a result within 1-2% of the global optimum. Thus the ADP procedure may be appropriate for practical production optimization problems.
The detailed interactions between the reservoir and the wellbore are especially important in thermal processes such as steamflooding and in-situ upgrading. These linkages, therefore, must be captured in thermal simulations. Although fully coupled thermal wellbore-/reservoir-flow simulators have been developed, the implementation of the thermal well model is somewhat complicated, and the simulations are computationally demanding. In this paper, we present a semianalytical treatment that enables the extension of existing isothermal wellbore-flow models to the nonisothermal case. The procedure entails the use of analytical solutions for wellbore temperature applied in conjunction with numerical solutions of the reservoir mass- and energy-balance equations coupled with wellbore mass- and momentum-balance equations. The approach thus enables a degree of decoupling between the wellbore flow and energy problems. We proceed by first presenting analytical solutions for wellbore temperature, developed under various assumptions (these basic solutions have been obtained previously). We then describe the use of one of these solutions, which allows for general variation of in-situ phase fraction and other properties along the wellbore, within the semianalytical context. The implementation of the overall method into a general purpose research simulator is also described. Results are presented for several cases involving multiphase flow in monobore and multilateral wells. Close agreement with reference solutions, obtained from a fully coupled thermal wellbore/reservoir model, is demonstrated for all of the examples. The semianalytical treatment is additionally shown to provide comparable or improved computational efficiency relative to the fully coupled model.
Advanced wells are being used more and more frequently to improve the economics of oil field operations. Such wells can contact larger regions of the reservoir than is possible with standard wells. Because these wells can be very long, pressure drop along the well and temperature effects (in thermal problems) can became quite important. Modeling of advanced wells where thermal effects need to be included is a difficult task, and available models are often limited in their capabilities or are inefficient.
A suitable model should be able to handle complicated well trajectories, heat transfer to and from the well, the hold-up of fluids due to slip between phases, and pressure change due to friction. This paper presents a comprehensive multisegmented well model that includes these features. The model has been implemented in Stanford's General Purpose Research Simulator (GPRS). It calculates both pressure and temperature profiles along the well for standard wells and wells with complicated trajectories. One of the advantages of our model is that both homogeneous (no slip) and drift-flux flow models can be used to determine phase distribution within the well. The hydrocarbon fluids can be described either through a black oil representation or using a general compositional model with an arbitrary number of components. The proposed multisegmented well model was tested with a number of challenging and realistic examples. Our results show that the model is stable and in most cases converges in only a few iterations even with reasonably large time steps. The capabilities of the model are demonstrated using three examples. The first example involves a vertical production well with the fluid represented in terms of six hydrocarbon components and water. For this case we demonstrate the importance of slip between phases. The second example is for a thermal multilateral well with two branches and a fluid with three hydrocarbon components. The third example shows our ability to model phase appearance and disappearance in the wellbore and demonstrates the impact of temperature on oil and gas rates during production.
The use of horizontal and multilateral wells has become a standard practice in the oil and gas industry. However, modeling of such wells is still a challenge in many cases. This is because many complex phenomena must be taken into account for the correct representation of such wells; among them are the complicated trajectory of the well, pressure drop due to friction, slip between phases, and the temperature distribution in the well. Modeling of thermal effects is necessary for simulating oil recovery from unconventional resources such as oil sands and shales. Further, such models can be used for the interpretation of data from distributed temperature sensors (DTS).
Significant effort has been directed towards the modelling of thermal wells, starting from analytical models proposed by, e.g., Ramey (1962) and Hasan and Kabir (2002), to complicated numerical models, such as those proposed by Shirdel and Sepehrnoori (2009) and Stone et al. (2002). All available models have advantages and disadvantages: some cannot model pressure loss due to friction, slip between phases or complicated well trajectories; some are not accurate or fast enough, and some are too complicated for routine use.
Growing interest in understanding, predicting, and controlling advanced oil-recovery methods emphasizes the importance of numerical methods that exploit the nature of the underlying physics. The fully implicit method offers unconditional stability of the discrete approximations. This stability comes at the expense of transferring the inherent physical stiffness onto the coupled nonlinear residual equations that are solved at each timestep. Current reservoir simulators apply safeguarded variants of Newton's method that can neither guarantee convergence nor provide estimates of the relation between convergence rate and timestep size. In practice, timestep chops become necessary and are guided heuristically. With growing complexity, such as in thermally reactive compositional flows, convergence difficulties can lead to substantial losses in computational effort and prohibitively small timesteps. We establish an alternative class of nonlinear iteration that converges and associates a timestep to each iteration. Moreover, the linear solution process within each iteration is performed locally.
By casting the nonlinear residual equations for a given timestep as an initial-value problem, we formulate a continuation-based solution process that associates a timestep size with each iteration. Subsequently, no iterations are wasted and a solution is always attainable. Moreover, we show that the rate of progression is as rapid as that for a convergent standard Newton method. Moreover, by exploiting the local nature of nonlinear wave propagation typical to multiphase-flow problems, we establish a linear solution process that performs computation only where necessary. That is, given a linear convergence tolerance, we identify a minimal subset of solution components that will change by more than the specified tolerance. Using this a priori criterion, each linear step solves a reduced system of equations. Several challenging examples are presented, and the results demonstrate the robustness and computational efficiency of the proposed method.
As many fields around the world are reaching maturity, the need to develop new tools that allow reservoir engineers to optimize reservoir performance is becoming more urgent. One of the more challenging and important problems along these lines is the well placement optimization problem. In this problem, there are many variables to consider: geological variables like reservoir architecture, permeability and porosity distributions, and fluid contacts; production variables, such as well placement, well number, well type, and production rate; and economic variables like fluid prices and drilling costs. Furthermore, availability of complex well types, such as multilateral wells (MLWs) and maximum reservoir contact (MRC) wells, aggravate this challenge. All these variables, together with reservoir geological uncertainty, make the determination of an optimum development plan for a given field difficult.
The objective of this work was to employ an optimization technique that can efficiently address the aforementioned challenges. Based on the success and versatility of Genetic Algorithms (GAs) in problems of high complexity with high dimensionality and nonlinearity, it is used here as the main optimization engine. Both binary GA (bGA) and continuous GA (cGA) were tested in the optimization of well location and design in terms of well type, number of laterals, and well and lateral trajectories in a channelized synthetic model. Both GA variants showed significant improvement over initial solutions but comparisons between the two types showed that the cGA was more robust for the problem under consideration. The cGA was, thereafter, applied to a real field located in the Middle East to investigate its robustness in optimizing well location and design in more complex reservoir models. The model is an upscaled version for an offshore carbonate reservoir, which is mildly heterogeneous with low and high permeability areas scattered over the field.
After choosing the optimization technique to achieve our objective, considerable work was performed to study the sensitivity of the different algorithm parameters on converged solutions. Then, multiple optimization runs were performed to obtain a sound development plan for this field. An attempt was made to quantify how solutions were affected by some of the assumptions and preconditioning steps taken during optimization. Finally, an optimization run was performed on the fine model using optimized solutions from the coarse model.
Thermal recovery processes are widely used for heavy oil production and are under investigation for other unconventional resources. The management and optimization of these processes require accurate representations of relevant physical phenomena in simulation models. An important aspect of the simulation capability is the modeling of flow in the wellbore and the coupling of wellbore and reservoir flows. In previous work, we developed and applied a black-oil thermal multiphase wellbore model and linked it to Stanford's General Purpose Research Simulator (GPRS). In this paper, we extend this formulation to treat thermal compositional systems. GPRS already contains a thermal compositional capability for reservoir flow; here we develop and test a thermal compositional wellbore model. This wellbore model includes an energy conservation equation, mass conservation equations for each component, and a general pressure drop relationship. The multiphase wellbore flow is represented using a drift-flux model, which includes slip between the three phases. The model determines the temperature, pressure, mixture flow velocity and component fractions as functions of time and axial position along the well. We apply the coupled thermal compositional wellbore-reservoir model to the simulation of several cases involving thermal effects. These include a verification example, in which we compare results from the compositional formulation to those of a black-oil model, and an example with seven components. Different well geometries (e.g., dual-lateral and deviated with varying inclination angle) are considered in these applications.
Growing interest in understanding, predicting, and controlling advanced oil recovery methods emphasizes the importance of numerical methods that exploit the nature of the underlying physics. The Fully Implicit Method offers unconditional stability in the sense of discrete approximations. This stability comes at the expense of transferring the inherent physical stiffness onto the coupled nonlinear residual equations which need to be solved at each time-step. Current reservoir simulators apply safe-guarded variants of Newton's method, and often can neither guarantee convergence, nor provide estimates of the relation between convergence rate and time-step size. In practice, time-step chops become necessary, and are guide heuristically. With growing complexity, such as in thermally reactive compositional models, this can lead to substantial losses in computational effort, and prohibitively small time-steps. We establish an alternate class of nonlinear iteration that both converges, and associates a time-step to each iteration. Moreover, the linear solution process within each iteration is performed locally.
By casting the nonlinear residual for a given time-step as an initial-value-problem, we formulate a solution process that associates a time-step size with each iteration. Subsequently, no iterations are wasted, and a solution is always attainable. Moreover, we show that the rate of progression is as rapid as a standard Newton counterpart whenever it does converge. Finally, by exploiting the local nature of nonlinear waves that is typical to all multi-phase problems, we establish a linear solution process that performs computation only where necessary. That is, given a linear convergence tolerance, we identify the minimal subset of solution components that will change by more than the specified tolerance. Using this a priori criterion, each linear step solves a reduced system of equations. Several challenging examples are presented, and the results demonstrate the robustness of the proposed method as well as its performance.
Field development optimization is a computationally intensive task due to the large number of reservoir simulation runs required. These simulations can be expensive, especially for large and complex reservoir models. Proxies can be used to efficiently estimate the objective function value for new scenarios and can act to reduce the number of simulations required. Thus they can be very useful for speeding up field development optimization.
In this paper a procedure that combines an optimization algorithm (in this case a genetic algorithm or GA) and a new statistical proxy is described. The statistical proxy has the following key elements. First, a new selection procedure called individual-based selection is applied to decide which individuals (scenarios) are to be simulated. Second, the new approach uses multiple proxies for optimization problems involving multiple reservoir models, which are needed to account for geological uncertainty. Third, the statistical proxy is modified to work efficiently in distributed computing environments. Finally, the proxy procedure is successfully incorporated into an existing general field development optimization package (Williams et al., 2004; Litvak et al., 2007a).
In the individual-based selection method, for each scenario the proxy estimate of the objective function is compared to a threshold. If the estimate exceeds the threshold, then the case is simulated (otherwise it is not simulated). The threshold corresponds to a specified percentile of the cumulative distribution function constructed from previously simulated cases and therefore changes during the course of the optimization. In cases with multiple reservoir models, each model has its own corresponding proxy. This eliminates the problem of duplicate objective function estimates for different reservoir models, which may occur with previous proxy-based methods. The individual-based selection method is shown to perform better for a particular example than the population-based method published previously.
The overall procedure is applied to the optimization of infill drilling where we maximize the incremental net present value (NPV) by optimizing new well locations, well type and rig schedule, subject to field development constraints. We demonstrate the capabilities of the proxy using synthetic reservoir models and a real field in the Gulf of Mexico. In the first example, two optimization cases are considered, corresponding to the use of single and multiple reservoir models. In the case with one reservoir model, the hybrid procedure found the same field development scenario compared to GA only, and required 85% fewer simulations. In the case with multiple reservoir models, the hybrid procedure found a slightly different field development scenario than the pure GA approach, though the NPV from the hybrid procedure was within 1% of that using only GA. The hybrid approach, however, required 91% fewer simulations for this case. In the field application, a better field development scenario with 45% fewer simulations was found using the hybrid algorithm (GA and proxy) compared to using only GA. These examples clearly demonstrate the effectiveness of the statistical proxy procedure for accelerating field development optimization.
The detailed interactions between the reservoir and the wellbore are especially important in thermal processes such as steam flooding and in situ upgrading. These linkages must therefore be captured in thermal simulations. Although fully-coupled thermal wellbore-reservoir flow simulators have been developed, the implementation of the thermal well model is somewhat complicated and the simulations are computationally demanding. In this paper, we present a semianalytical treatment that enables a fairly straightforward extension of existing isothermal wellbore flow models to the nonisothermal case. The procedure entails the use of analytical solutions for wellbore temperature applied in conjunction with numerical solutions of the reservoir mass and energy balance equations coupled with wellbore mass and momentum balance equations. The approach thus enables a degree of decoupling between the wellbore flow and energy problems. We proceed by first presenting analytical solutions for wellbore temperature, developed under various assumptions (some of these solutions have been obtained previously). We then describe the use of one of these solutions, which allows for general variation of in situ phase fraction and other properties along the wellbore, within the semianalytical context. The implementation of the overall method into a general purpose research simulator is also described. Results are presented for several cases involving multiphase flow in monobore and multilateral wells. Close agreement with reference solutions, obtained from a fully-coupled thermal wellbore-reservoir model, is demonstrated for all of the examples. The semianalytical treatment is additionally shown to provide comparable or improved computational efficiency relative to the fully-coupled model. The overall procedure is therefore very well suited for use in general thermal reservoir simulation.
Thermal recovery processes are widely applied for the production of heavy oil and oil sands. Thermal reservoir simulation models, however, often lack a comprehensive well modeling capability. Such a capability is required to capture the detailed thermal effects that occur in the wellbore. These effects can be important as they impact wellbore pressure and temperature and thus production and injection. We recently developed a fully-coupled black-oil thermal multiphase wellbore flow model and implemented it into Stanford's General Purpose Research Simulator (GPRS). The model computes pressure, temperature, and oil, water and gas phase fractions along the wellbore as a function of time and includes treatments for slip between fluid phases, heat losses to the reservoir, and general variations of fluid properties with temperature and pressure. The purpose of this paper is to validate and test the coupled wellbore-reservoir model for challenging and realistic cases. The thermal wellbore model is first validated through comparison to field data for three-phase flow in a long well with both vertical and inclined sections. Close agreement between the model and field data is obtained. Complex wells containing multiple branches are then simulated, including a steam-water case with vaporization and condensation. The general conclusion from this work is that the new model is capable of simulating a wide variety of complex coupled reservoir-wellbore phenomena.