Pavel Tomin

Hybrid Multiscale Finite Volume method for two-phase flow in porous media

We present a novel hybrid (or multiphysics) algorithm, which couples pore-scale and Darcy descriptions of two-phase flow in porous media. The flow at the pore-scale is described by the Navier-Stokes equations, and the Volume of Fluid (VOF) method is used to model the evolution of the fluid-fluid interface. An extension of the Multiscale Finite Volume (MsFV) method is employed to construct the Darcy-scale problem. First, a set of local interpolators for pressure and velocity is constructed by solving the Navier-Stokes equations; then, a coarse mass-conservation problem is constructed by averaging the pore-scale velocity over the cells of a coarse grid, which act as control volumes; finally, a conservative pore-scale velocity field is reconstructed and used to advect the fluid-fluid interface. The method relies on the localization assumptions used to compute the interpolators (which are quite straightforward extensions of the standard MsFV) and on the postulate that the coarse-scale fluxes are proportional to the coarse-pressure differences. By numerical simulations of two-phase problems, we demonstrate that these assumptions provide hybrid solutions that are in good agreement with reference pore-scale solutions and are able to model the transition from stable to unstable flow regimes. Our hybrid method can naturally take advantage of several adaptive strategies and allows considering pore-scale fluxes only in some regions, while Darcy fluxes are used in the rest of the domain. Moreover, since the method relies on the assumption that the relationship between coarse-scale fluxes and pressure differences is local, it can be used as a numerical tool to investigate the limits of validity of Darcys law and to understand the link between pore-scale quantities and their corresponding Darcy-scale variables.

Local-global splitting for spatiotemporal-adaptive multiscale methods

We present a novel spatiotemporal-adaptive Multiscale Finite Volume (MsFV) method, which is based on the natural idea that the global coarse-scale problem has longer characteristic time than the local fine-scale problems. As a consequence, the global problem can be solved with larger time steps than the local problems. In contrast to the pressure-transport splitting usually employed in the standard MsFV approach, we propose to start directly with a local-global splitting that allows to locally retain the original degree of coupling. This is crucial for highly non-linear systems or in the presence of physical instabilities. To obtain an accurate and efficient algorithm, we devise new adaptive criteria for global update that are based on changes of coarse-scale quantities rather than on fine-scale quantities, as it is routinely done before in the adaptive MsFV method. By means of a complexity analysis we show that the adaptive approach gives a noticeable speed-up with respect to the standard MsFV algorithm. In particular, it is efficient in case of large upscaling factors, which is important for multiphysics problems. Based on the observation that local time stepping acts as a smoother, we devise a self-correcting algorithm which incorporates the information from previous times to improve the quality of the multiscale approximation. We present results of multiphase flow simulations both for Darcy-scale and multiphysics (hybrid) problems, in which a local pore-scale description is combined with a global Darcy-like description. The novel spatiotemporal-adaptive multiscale method based on the local-global splitting is not limited to porous media flow problems, but it can be extended to any system described by a set of conservation equations. Local-global splitting as a framework for multiscale multiphysics simulations.Spatiotemporal adaptivity for the Multiscale Finite Volume method.Adaptive criteria based on the changes of coarse-scale quantities.Self-correcting scheme to improve the quality of multiscale approximation.Multiscale multiphysics simulations of two-phase flow in porous media.

Unified thermo-compositional-mechanical framework for reservoir simulation

We present a reservoir simulation framework for coupled thermal-compositional-mechanics processes. We use finite-volume methods to discretize the mass and energy conservation equations and finite-element methods for the mechanics problem. We use the first-order backward Euler for time. We solve the resulting set of nonlinear algebraic equations using fully implicit (FI) and sequential-implicit (SI) solution schemes. The FI approach is attractive for general-purpose simulation due to its unconditional stability. However, the FI method requires the development of a complex thermo-compositional-mechanics framework for the nonlinear problems of interest, and that includes the construction of the full Jacobian matrix for the coupled multi-physics discrete system of equations. On the other hand, SI-based solution schemes allow for relatively fast development because different simulation modules can be coupled more easily. The challenge with SI schemes is that the nonlinear convergence rate depends strongly on the coupling strength across the physical mechanisms and on the details of the sequential updating strategy across the different physics modules. The flexible automatic differentiation-based framework described here allows for detailed assessment of the robustness and computational efficiency of different coupling schemes for a wide range of multi-physics subsurface problems.

General Implicit Coupling Framework for Multi-Physics Problems

Wednesday, October 4th, 1:45pm, Green Earth Sciences Building, Room 365 Growth in the complexity of processes combined with increased computational power, has created the need for a general simulation framework applicable to different physical processes. The availability of a generalized framework can decrease the development cost of incorporating additional physical mechanics and exploring new formulations and solution methods. In addition, a general simulation framework would aid in the transition from the existing paradigm, in which a small number of process models are coupled together in a limited number of predefined combinations, to a new paradigm, where many different processes are coupled in arbitrary permutations. The flexibility of such a framework allows us to investigate new nonlinear coupling configurations across the different physical processes and improve our understanding of the coupled nature of involved processes. This would enable us to improve the performance and stability of coupled systems by finding new coupling configurations and more efficient solvers for a given physics. The goal of this work is the development and implementation of such a multi-physics reservoir simulation platform. Shale-gas/oil, Oil shale Enhanced Steam Injection CO2 storage in saline aquifers

Spatiotemporal adaptive multiphysics simulations of drainage-imbibition cycles

We present a spatiotemporal adaptive multiscale algorithm, which is based on the Multiscale Finite Volume method. The algorithm offers a very efficient framework to deal with multiphysics problems and to couple regions with different spatial and temporal resolution. We employ the method to simulate two-phase flow through porous media. At the fine scale, we consider a pore-scale description of the flow based on the Volume Of Fluid method. In order to construct a global problem that describes the coarse-scale behavior, the equations are averaged numerically with respect to auxiliary control volumes, and a Darcy-like coarse-scale model is obtained. The spatial adaptivity is based on the idea that the fine-scale description is only required in the front region, whereas the resolution can be coarsened elsewhere. Temporal adaptivity relies on the fact that the fine-scale and the coarse-scale problems can be solved with different temporal resolution (longer time steps can be used at the coarse scale). By simulating drainage-imbibition cycles for different flow regimes, we show that the method is able to capture the coarse-scale behavior outside the front region and to reproduce complex fluid patterns in the front region.

Robust And Accurate Formulation For Modeling Of Acid Stimulation

Summary Accurate representation of processes associated with energy extraction from subsurface formations often requires models which account for chemical interactions between different species in the presence of multiphase flow. In this study, we focus on modeling of acid stimulation in the near-well region. For the chemical processes which include a dissolution of rock material, an issue arises with the predictive representation of flow. Taking into account the spatial scale of discretization, some of simulation control volumes can have values of porosity close to 1, which makes an application of Darcy’s law inconsistent and requires employing a true momentum equation such as the Darcy-Brinkman-Stokes (DBS) equation. The DBS equation automatically switches the description between Darcy equation in control volumes with low porosity and Stokes equation in grid blocks with high porosity. For chemical reactions, we propose a local nonlinear solution technique that allows solving the balance of solid species separately yet retaining the full coupling with rest of the equations. Finally, we study the impact of multiphase flow. The DBS approach is not well established for multiphase flow description. Therefore we employ a hybrid approach, where we assume that the single-phase DBS flow and the multiphase Darcy flow occur in separate regions. We test the accuracy and performance of both approaches on realistic models of practical interest.

Spatiotemporal Adaptive Multiscale Multiphysics Simulations of Two-phase Flow

SUMMARY We present a spatiotemporal adaptive multiscale algorithm, which is based on the Multiscale Finite Volume method. The algorithm offers a very efficient framework to deal with multiphysics problems and to couple regions with different spatial resolution. We employ the method to simulate two-phase flow through porous media. At the fine scale, we consider a pore-scale description of the flow based on the Volume Of Fluid method. In order to construct a global problem that describes the coarse-scale behavior, the equations are averaged numerically with respect to auxiliary control volumes, and a Darcy-like coarsescale model is obtained. The space adaptivity is based on the idea that a fine-scale description is only required in the front region, whereas the resolution can be coarsened elsewhere. Temporal adaptivity relies on the fact that the fine-scale and the coarse-scale problems can be solved with different temporal resolution (longer time steps can be used at the coarse scale). By simulating drainage under unstable flow conditions, we show that the method is able to capture the coarse-scale behavior outside the front region and to reproduce complex fluid patterns in the front region.

A Framework for Hybrid Simulations of Two-phase Flow in Porous Media

In the last decade multiscale methods have proven efficient in solving large reservoir-scale problems with satisfactory accuracy. Computational efficiency is achieved by splitting the original problem into a set of local problems coupled through a global coarse problem. Although these techniques are usually employed for problems in which the fine-scale processes are described by Darcy’s law, they can also be applied to pore-scale simulations and used as a mathematical framework for hybrid methods that couples a Darcy and pore scales. In this work, we consider a pore-scale description of fine-scale processes. The Navier-Stokes equations are numerically solved in the pore geometry to compute the velocity field and obtain generalized permeabilities. In the case of two-phase flow, the dynamics of the phase interface is described by the volume of fluid method with the continuum surface force model. The MsFV method is employed to construct an algorithm that couples a Darcy macro-scale description with a pore-scale description at the fine scale. The hybrid simulations results presented are in good agreement with the fine-scale reference solutions. As the reconstruction of the fine-scale details can be done adaptively, the presented method offers a flexible framework for hybrid modeling.

Multiscale Method for Numerical Simulation of Multiphase Flows in Giant Production Fields

The problem of multiphase flow modeling for giant oil and gas fields partitioned into several areas is considered. The aggregation of essential number of input fine grid cells forms the cell of coarse grid. According to ideas of I.Babuska, one can show that the pore pressure at each cell of the coarse grid may be approximated using linear combination of special basis functions. These functions are solutions of single-phase flow problem in the cell of the coarse grid with special piecewise multilinear functions used as boundary condition. The support operator method (SOM) by A.A.Samarsky is used to calculate the basis functions. Using R.P.Fedorenko idea of the superelement method (SEM), the calculated basis functions are used for approximation in SOM instead of ordinary linear function. Compared to SEM, the number of basis functions for method developed is substantially smaller: not 8 but 3 for the hexahedron grid. Finally the distribution of pressure and saturation evolution is calculated with Neumann boundary conditions for governing system. Method developed is high-resolution one and allows effective simulation of the processes for giant production fields. We gratefully acknowledge the financial support of the RFBR (grant 09-01-00823).