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Collaborating Authors
posteriori error estimate
Summary This work focuses on the development of adaptive timesteps, stopping criteria, and error control strategies for reservoir simulations with fully implicit (FIM) solvers. Using a rigorous error control framework, an adaptive time selector combined with nonlinear stopping criteria is used to control nonlinear iterations as well as to balance accuracy and robustness for challenging nonlinear simulations. In reservoir simulation, efficiently solving a system of nonlinear equations arising from the FIM method can be computationally burdensome for complex recovery processes. Theoretically, an FIM reservoir simulator has no stability limit on the timestep size. In practice, standard Newton’s method often fails to converge for large timestep sizes and must therefore cut the timestep multiple times to achieve convergence, resulting in a large number of unnecessary iterations. Another cause of nonlinear convergence issues is the presence of wells, which are often presented as singular point/line sources strongly coupled to the reservoir model, posing additional restrictions on the timestep choice. Here, we use a posteriori error estimators to avoid unnecessary nonlinear iterations and timestep cuts when solving immiscible multiphase flow. First, we estimate error components (e.g., spatial, temporal, and nonlinear) and then apply these to balancing criteria, providing us with dynamic and adaptive strategies to control timestep and nonlinear iterations. The error estimators are fully and locally computable, inexpensive to use, and target the various error components, including well singularities. The method provides an adaptive criterion for stopping the nonlinear iteration process whenever the linearization error does not significantly affect the overall error. Simultaneously, timesteps are adapted to maintain a constant size of the temporal discretization error with respect to the total error. Altogether, this avoids using unnecessary linearization iterations, wasteful timestep cuts, and too small timesteps. To demonstrate the effectiveness of these adaptive features, we present results for a suite of cases, covering both standard benchmarks and conceptual problems incorporating highly heterogeneous media with multiple wells. The proposed timestep selector cooperates with the new stopping criteria to improve nonlinear solver performance and increases robustness for cases with high nonlinearity. Perhaps most important, the adaptive features ensure balanced temporal and spatial errors while maintaining sufficiently small nonlinear errors, which ensures solution accuracy by accurately reproducing saturation fronts, production plateau, and breakthrough times.
- Europe (1.00)
- North America > United States (0.93)
- Europe > United Kingdom > North Sea > Central North Sea > Ness Formation (0.99)
- Europe > Norway > North Sea > Central North Sea > South Viking Graben > PL 046 > Utsira Formation (0.99)
- Europe > Norway > North Sea > Central North Sea > Ness Formation (0.99)
Abstract Flow coupled with geomechanics problems has gathered increased research interest due to its resemblance to engineering applications, such as unconventional reservoir development, by incorporating multiple physics. Computations for the system of such a multiphysics model is often costly. In this paper, we introduce a posteriori error estimators to guide dynamic mesh adaptivity and to determine a novel stopping criterion for the fixed-stress split algorithm to improve computational efficiency. Previous studies for flow coupled with geomechanics have shown that local mass conservation for the flow equation is critical to the solution accuracy of multiphase flow and reactive transport models, making mixed finite element method an attractive option. Such a discretization maintains local mass conservation by enforcing the constitutive equation in strong form and can be readily incorporated into existing finite volume schemes, that are standard in the reservoir simulation community. Here, we introduced a posteriori error estimators derived for the coupled system with the flow and mechanics solved by mixed method and continuous Galerkin respectively. The estimators are utilized to guide the dynamic mesh adaptivity. We demonstrate the effectiveness of the estimators on computational improvement by a fractured reservoir example. The adaptive method only requires 20% of the degrees of freedom as compared to fine scale simulation to obtain an accurate solution. To avoid solving enormous linear systems from the monolithic approach, a fixed-stress split algorithm is often adopted where the flow equation is resolved first assuming a constant total mean stress, followed by the mechanics equation. The implementation of such a decoupled scheme often involves fine tuning the convergence criterion that is case sensitive. Previous work regarding error estimators with the flow equation solved by Enriched Galerkin proposed a novel stopping criterion that balances the algorithmic error with the discretization error. The new stopping criterion does not require fine tuning and avoids over iteration. In this paper, we extend such a criterion to the flow solved by mixed method and further confirm its validity.
- Reservoir Description and Dynamics > Reservoir Simulation (1.00)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Flow in porous media (1.00)
- Reservoir Description and Dynamics > Reservoir Characterization (1.00)
- Production and Well Operations > Well & Reservoir Surveillance and Monitoring > Downhole and wellsite flow metering (0.89)
Summary This work focuses on the development of adaptive timesteps, stopping criteria, and error control strategies for reservoir simulations with fully implicit (FIM) solvers. Using a rigorous error control framework, an adaptive time selector combined with nonlinear stopping criteria is used to control nonlinear iterations as well as to balance accuracy and robustness for challenging nonlinear simulations. In reservoir simulation, efficiently solving a system of nonlinear equations arising from the FIM method can be computationally burdensome for complex recovery processes. Theoretically, an FIM reservoir simulator has no stability limit on the timestep size. In practice, standard Newton’s method often fails to converge for large timestep sizes and must therefore cut the timestep multiple times to achieve convergence, resulting in a large number of unnecessary iterations. Another cause of nonlinear convergence issues is the presence of wells, which are often presented as singular point/line sources strongly coupled to the reservoir model, posing additional restrictions on the timestep choice. Here, we use a posteriori error estimators to avoid unnecessary nonlinear iterations and timestep cuts when solving immiscible multiphase flow. First, we estimate error components (e.g., spatial, temporal, and nonlinear) and then apply these to balancing criteria, providing us with dynamic and adaptive strategies to control timestep and nonlinear iterations. The error estimators are fully and locally computable, inexpensive to use, and target the various error components, including well singularities. The method provides an adaptive criterion for stopping the nonlinear iteration process whenever the linearization error does not significantly affect the overall error. Simultaneously, timesteps are adapted to maintain a constant size of the temporal discretization error with respect to the total error. Altogether, this avoids using unnecessary linearization iterations, wasteful timestep cuts, and too small timesteps. To demonstrate the effectiveness of these adaptive features, we present results for a suite of cases, covering both standard benchmarks and conceptual problems incorporating highly heterogeneous media with multiple wells. The proposed timestep selector cooperates with the new stopping criteria to improve nonlinear solver performance and increases robustness for cases with high nonlinearity. Perhaps most important, the adaptive features ensure balanced temporal and spatial errors while maintaining sufficiently small nonlinear errors, which ensures solution accuracy by accurately reproducing saturation fronts, production plateau, and breakthrough times.
- Europe (1.00)
- North America > United States (0.93)
- Europe > United Kingdom > North Sea > Central North Sea > Ness Formation (0.99)
- Europe > Norway > North Sea > Central North Sea > South Viking Graben > PL 046 > Utsira Formation (0.99)
- Europe > Norway > North Sea > Central North Sea > Ness Formation (0.99)
Abstract We present in this paper a-posteriori error estimators for multiphase flow with singular well sources. The estimators are fully and locally computable, distinguish the various error components, and target the singular effects of wells. On the basis of these estimators we design an adaptive fully-implicit solver that yields optimal nonlinear iterations and efficient time-stepping, while maintaining the accuracy of the solution. A key point is that the singular nature of the solution in the near-well region is explicitly captured and efficiently estimated using the adequate norms. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive algorithm.
- Europe > United Kingdom > North Sea > Central North Sea > Ness Formation (0.99)
- Europe > Norway > North Sea > Central North Sea > Ness Formation (0.99)
Dynamic Adaptivity for Coupled Flow and Geomechanics in Unconventional Reservoirs Using A Posteriori Error Estimation with Enriched Galerkin Method
Lu, Xueying (The University of Texas at Austin) | Girault, Vivette (Université Pierre et Marie Curie - Paris VI) | Wheeler, Mary F. (The University of Texas at Austin)
Abstract Coupled fluid ow and geomechanics is receiving growing research interests for applications in unconventional oil and gas recovery and geological CO2 sequestration. In this work, we first demonstrated the advantage of using the Enriched Galerkin (EG) method for ow discretization in unconventional scenarios where the permeability of fractures/faults is orders of magnitudes higher than the permeability of matrix. Then we demonstrate algorithmic improvements built upon a posteriori error estimates for coupled ow and geomechanics simulations. A posteriori error estimates are derived for the Biot's system solved with a fixed-stress split, the EG approximation for the ow equation, and the conforming Galerkin (CG) approximation for the mechanics equation. An upper bound is derived for the error equation, distinguishing different error components, namely the fixed-stress algorithmic error, the time error, the ow error, the penalty jump, and the errors arising from the mechanics equation. Using these error indicators, we propose a novel stopping criterion for the fixed-stress iterations that does not require tuning of the convergence threshold. Namely, the new stopping criterion balances the error from the fixed-stress split and the errors from the spatial and temporal discretizations. Numerical results demonstrated the efficiency and accuracy of the new stopping criterion, in that it achieves same accuracy with on average fewer fixed-stress iterations per time step. We also demonstrated the dynamic mesh adaptivity guided by the error estimators in simulating unconventional reservoirs. The adaptive solutions achieve comparable accuracy to the solutions on a very fine mesh. 1. INTRDOCTION Numerical and algorithmic studies of coupled fluid ow and poromechanics are gaining more research interests in the community of petroleum and environmental engineering, for the study of the potential for faults to reactivate during large-scale geologic carbon sequestration operations, and hydraulic fracturing during unconventional hydrocarbon recoveries (Jha and Juanes (2014); Rinaldi et al. (2014); Rutqvist et al. (2015); White and Foxall (2016); Lu et al. (2018)). In large-scale engineering applications, this coupled ow-poromechanics model is usually solved by iterative splitting techniques, among which the fixed-stress split algorithm is one of the most widely used. Kim et al. (2011) proved that the fixed-stress split is unconditionally stable by von Neuman stability analysis. Mikelific and Wheeler established its geometric convergence by showing that it is a contraction mapping for the mean stress (Mikelific and Wheeler (2013); Mikelific et al. (2014)). A major bottleneck for field-scale simulations of coupled owmechanics models is that it is computationally prohibitive and most of the computational time is spent on mechanics updates (Lu et al. (2019)). Several algorithms are developed to speed up the computations. Almani et al. (2016) and Kumar et al. (2016) studied a multi-rate fixed-stress iterative scheme where the ow equation takes multiple fine time steps within one coarse mechanics time step. Bause et al. (2017) and Borregales et al. (2018) explored space-time methods of the corresponding iterative coupling schemes. A multiscale extension of the fixed-stress split scheme to a poroelastic-elastic system where the mechanics problem is solved on a larger domain with coarse mesh is studied by Dana et al. (2018) and Dana and Wheeler (2018). Lu and Wheeler (2020) developed an asynchronous adaptive coupling scheme for poroelasticity that utilizes a tolerance in fixed stress splitting as an error indicator to determine when geomechanics step can be eliminated. Ahmed et al. (2020) demonstrated adaptive poromechanics computations based on a posteriori error estimates derived for two mixed variational formulations for both pressure and displacement. However, their approach required solving auxiliary local problems that could be computationally expensive. The numerical results presented in this work are built upon the work of Girault et al. (2020), where a posteriori error estimates are derived for the Biot's system solved with a fixed-stress split, the Enriched Galerkin (EG) approximation for the ow equation, and the conforming Galerkin (CG) approximation for the mechanics equation. EG is adopted for ow equation because it preserves local mass conservation (Lee et al. (2016)). Its advantage in the applications to coupled fluid ow and mechanical deformations has recently been demonstrated by Choo and Lee (2018) and Kadeethum et al. (2019) in that local mass conservation can also be crucial to the accurate simulation of deformation processes in fluid-infiltrated porous materials.
- Reservoir Description and Dynamics > Unconventional and Complex Reservoirs (1.00)
- Reservoir Description and Dynamics > Reservoir Simulation (1.00)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Flow in porous media (1.00)
- Reservoir Description and Dynamics > Storage Reservoir Engineering > CO2 capture and sequestration (0.87)