Advanced Strategies of Forward Simulation for Adjoint-Based Optimization

Abstract Adjoint-based simulation is one of the most efficient methods for reservoir simulation optimization. The gradient information of the objective function and constraints is used to generate a sequence of quadratic programming subproblems converging to the extremum of non-linear problem. The adjoint method provides accurate gradients that help to converge to the optimal solution using the least number of iterations, where each iteration is a forward simulation. The quality and stability of the gradients play important roles in the optimization process. In this paper we present analysis of adjoint-gradients based on different aspects of the forward simulation. We demonstrate that in the presence of compressibility, gradients evaluated using bottom hole pressure (BHP) controls are less consistent with respect to time step refinement, and less stable compared with gradients evaluated using rate controls. Using simple examples, we demonstrate that adjoint-based gradients for rate-controls converge with refinement of the time step while gradients for BHP-controls suffer from convergence problem. Another important aspect of our study is the effect of different nonlinear constraints in the optimization process. In forward simulation, nonlinear constraints often introduce additional complexities due to the discontinuous nature of the switching procedure. Switching can occur at control points in time, or between two controls, and depends strongly on the time-stepping strategy and the truncation error. We compare strategies where individual well constraints are applied directly during the forward simulations and as nonlinear constraints in the optimization process. We demonstrate using two practical examples the advantages and disadvantages of both strategies. We also study the effect of time-truncation error and time-stepping strategy on the quality of the adjoint-gradients. For the time scale, we propose coarsening in both simulation time and redundant control time steps. With larger time steps and smaller numbers of control switches, we can improve efficiency of forward simulation by several fold. Next, the optimal controls of coarse time-step simulation are used as the initial guess for forward simulation of finer time-step resolution. We show how all of these issues affect the optimization of a full-field model.</jats:p

Compositional Space Parameterization: Multicontact Miscible Displacements and Extension to Multiple Phases

Summary We generalize the compositional space parameterization (CSP) approach, which was originally developed for immiscible two-phase multicomponent problems, to multicontact miscible displacements. The tie-line based parameterization method improves both the accuracy of the phase-behavior representation as well as the efficiency of equation of state (EOS) computations in compositional flow simulation. For immiscible compositional simulation, compositional space adaptive tabulation (CSAT) can be used to avoid most of the redundant EOS calculations. Because the supercritical region cannot be parameterized using tie-lines, the original CSAT approach is not effective for modeling multicontact miscible gas injection processes. To deal with supercritical compositions, a supercritical state criteria (SSC) algorithm based on adaptive tabulation of the minimal critical pressure (MCP) tie-lines is proposed. For general-purpose simulation of miscible and immiscible compositional displacement processes, we combined the adaptive CSAT strategy in the region of tie-line extensions and the adaptive SSC scheme; we refer to the overall framework as CSAT. Results of several challenging tests of practical interest indicate that the general CSAT strategy is quite robust and that it leads to an order of magnitude gain in computational efficiency. We also describe the extension of the CSP framework for mixtures that form more than two phases.</jats:p

Multi-level discrete fracture model for carbonate reservoirs

The main challenge for predictive simulation of carbonate reservoirs is associated with large uncertainties in the geological characterization with multiple features including fractures and cavities. This type of reservoirs requires robust and efficient forward-simulation capabilities to apply data assimilation or optimization technique under uncertainties. The interaction between reservoir matrix and various features introduces a complex multi-scale flow response driven by global boundary conditions. The Discrete Fracture Models (DFM), which represent fractures explicitly, is capable to accurately depict all important features of flow behavior. However, these models are constrained by many degrees of freedom when the fracture network becomes complicated. The Embedded DFM, which represents the interaction between matrix and fractures analytically, is an efficient approximation. However, it cannot accurately reproduce the effect of local flow conditions, especially when the secondary fractures are present. In this study, we applied a numerical upscaling of DFM a triple continuum model where large features are represented explicitly using the numerical EDFM and small features are upscaled as a third continuum. In this approach, we discretize the original geo-model with unstructured grid based on DFM and associate the mesh geometry with large features in the model. Using the global solution, we generate local boundary conditions for the model capturing the response of primary features to the flow. Applying local boundary conditions, we resolve all secondary features using a fine scale solution and update the local boundary conditions. This procedure is applied iteratively using the local-global-upscaling formalism. To demonstrate the accuracy of the Multi-Level Discrete Fracture Model, several realistic cases have been tested. By comparing with fine scale DFM solution and the traditional EDFM technique, we demonstrate that the proposed model is accurate enough to capture the flow behavior in complex fractured systems with advanced computational efficiency.</p

Compositional Space Parameterization: Theory and Application for Immiscible Displacements

Summary Thermodynamic equilibrium calculations in compositional flow simulators are used to find the partitioning of components among fluid phases, and they can be a time consuming kernel in a compositional flow simulation. We describe a tie-line-based compositional space parameterization (CSP) approach for dealing with immiscible gas-injection processes with large numbers of components. The multicomponent multiphase equilibrium problem is recast in terms of this parameterized compositional space, in which the solution path can be represented in a concise manner. This tie-line-based parameterization approach is used to speed up the phase behavior calculations of standard compositional simulation. Two schemes are employed. In the first method, the parameterization of the phase behavior is computed in a preprocessing step, and the results are stored in a table. During the course of a simulation, the flash calculation procedure is replaced by the solution of a multidimensional optimization problem in terms of the parameterized space. For processes where significant changes in pressure and temperature take place, this optimization procedure is combined with linear interpolation in tie-line space. In the second method, compositional space adaptive tabulation (CSAT) is used to accelerate the equation of state (EOS) computations associated with standard compositional reservoir simulation. The CSAT strategy takes advantage of the fact that, in gas injection processes, the solution path involves a limited number of tie-lines. The adaptively collected tie-lines are used to avoid redundant phase-stability checks in the course of a flow simulation. Specifically, we check if a given composition belongs to one of the tie-lines (or its extension) already in the table. If not, a new tie-line is computed and added to the table. The CSAT technique was implemented in a general-purpose research simulator (GPRS), which is designed for compositional flow simulation on unstructured grids. Using a variety of challenging models, we show that, for immiscible compositional processes, CSAT leads to significant speed up (at least a several-fold improvement) of the EOS calculations compared with standard techniques.</jats:p