Geometric multigrid methods for Darcy-Forchheimer flow in fractured porous media

In this paper, we present a monolithic multigrid method for the efficient solution of flow problems in fractured porous media. Specifically, we consider a mixed-dimensional model which couples Darcy flow in the porous matrix with Forchheimer flow within the fractures. A suitable finite volume discretization permits to reduce the coupled problem to a system of nonlinear equations with a saddle point structure. In order to solve this system, we propose a full approximation scheme (FAS) multigrid solver that appropriately deals with the mixed-dimensional nature of the problem by using mixed-dimensional smoothing and inter-grid transfer operators. Remarkably, the nonlinearity is localized in the fractures, and no coupling between the porous matrix and the fracture unknowns is needed in the smoothing procedure. Numerical experiments show that the proposed multigrid method is robust with respect to the fracture permeability, the Forchheimer coefficient and the mesh size.Comment: arXiv admin note: text overlap with arXiv:1811.0126

Mixed-dimensional geometric multigrid methods for single-phase flow in fractured porous media

This paper deals with the efficient numerical solution of single-phase flow problems in fractured porous media. A monolithic multigrid method is proposed for solving two-dimensional arbitrary fracture networks with vertical and/or horizontal possibly intersecting fractures. The key point is to combine two-dimensional multigrid components (smoother and intergrid transfer operators) in the porous matrix with their one-dimensional counterparts within the fractures, giving rise to a mixed-dimensional geometric multigrid method. This combination seems to be optimal since it provides an algorithm whose convergence matches the multigrid convergence factor for solving the Darcy problem. Several numerical experiments are presented to demonstrate the robustness of the monolithic mixed-dimensional multigrid method with respect to the permeability of the fractures, the grid size, and the number of fractures in the network.The work of the first and fourth authors was supported by Spanish project PGC2018-099536-A-I00 (MCIU/AEI/FEDER, UE). The work of the second author was supported by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement 705402, POROSOS. The work of the third author was partially supported by the Spanish project FEDER/MCYT MTM2016-75139-R. The work of the fourth author was supported by the DGA (Grupo de referencia APEDIF, ref. E24 17R)

Simulation Studies For Relative Importance Of Unconventional Reservoir Subgrid Scale Physics Parameters

In this endeavor we attempt to better understand gas transport in shale gas reservoirs, specifically the impact and effects of different physical phenomena. We start by documenting the nature of the reservoirs and the need for accurate modeling of various physical phenomena in multiple interconnected continua. The physical phenomena of interest include non-linear Forchheimer flow, Knudsen diffusion in the form of slip Klinkenberg flow and adsorption/desorption. The numerical methods used in the reservoir simulator are also introduced, along with a derivation of the main equations used. Various verification and validation results are compared against manufactured and analytical solutions and finally advanced features including mesh adaptivity and multi-block support are showcased. Several detailed parameter survey studies are conducted with realistic and exaggerated field values to identify the need for advanced physics models based on deviation from Darcy models. Recommendations as to the applicability of each model are presented along with suggested best practices of when to apply these models in real simulations. A redefinition of the SRV is proposed, based on the need to apply a non-Darcy flow model. This new definition would highlight the need for advanced (and costly) non-linear flow ow models near the wells and hydraulic fractures. The judicious application of computationally intensive physical models in the SRV and lower fidelity models further away is presented as an efficient alternative to large scale high fidelity simulations

Dual virtual element method for discrete fractures networks

Discrete fracture networks is a key ingredient in the simulation of physical processes which involve fluid flow in the underground, when the surrounding rock matrix is considered impervious. In this paper we present two different models to compute the pressure field and Darcy velocity in the system. The first allows a normal flow out of a fracture at the intersections, while the second grants also a tangential flow along the intersections. For the numerical discretization, we use the mixed virtual finite element method as it is known to handle grid elements of, almost, any arbitrary shape. The flexibility of the discretization allows us to loosen the requirements on grid construction, and thus significantly simplify the flow discretization compared to traditional discrete fracture network models. A coarsening algorithm, from the algebraic multigrid literature, is also considered to further speed up the computation. The performance of the method is validated by numerical experiments

A CFD and experimental approach for simulating the coupled flow dynamics of near wellbore and reservoir

The modeling of simultaneous flow behavior through a reservoir and wellbore is important and an integrated model is needed which accounts for the transient multiphase flow in the wellbore and its surrounding region. In addition, reservoir and wellbore interface modeling and cost-effective Computational Fluid Dynamics (CFD) methodology are required to simulate the flow behavior in that region. The study outlines the development of an experimental prototype to study multiphase flow in the near wellbore region. To the best of my knowledge, this facility has the capability to accommodate a larger length scale compared to similar facilities available in the research organizations. This experimental setup can be used for investigating a wide variety of multiphase flow problems which have been considered in the present research. A CFD methodology has been developed using the 3D Navier-Stokes equations to simulate an integrated wellbore-reservoir flow. The CFD methodology has been verified for the fluid flow mechanism at near wellbore. The simulation results have been compared to the analytical solutions. Then, this model is extended to establish a coupled wellbore-reservoir framework which is based on 3D Navier-Stokes equations. The simulations have been performed to validate the newly developed CFD algorithm and various scenarios of a reservoir have been taken into consideration. The same process has been applied to investigate flow through a perforated tunnel and a new method of perforation has been discussed. The study indicates standard CFD techniques use a “numerical approach” such as the volume of fluid accounts for capillary pressure and surface tension force needs to be improved for more understanding of the flow through porous media. In this regards, Allen-Chan phase-field method has been combined with the Navier-Stokes equations to simulate multiphase flow in porous media. The simulations performed with the phase-field method have been verified with the experimental data. The experimental and CFD approach of this thesis make a unique contribution in the field of the petroleum industry and multiphase flow in porous media

Numerical Simulation of Multiphase Flow in Fractured Porous Media

Fractures provide preferred paths for flow and transport in many porous media. They have a significant influence on process behavior in groundwater remediation, reservoir engineering and safety analysis for waste repositories. We present a finite volume method for the numerical solution of the multiphase flow equations in fractured porous media. The capillary pressure is treated by an extended capillary pressure interface condition. The method is fully coupled and fully implicit and employs a mixed-dimensional formulation with lower dimensional elements in the fractures. The method features unstructured grids, adaptive refinement and multigrid methods. It is implemented for twodimensional and threedimensional complex problems with several million unknowns. Additionally, a discontinuous Galerkin method for the groundwater flow equation and its multigrid treatment is presented

Numerical simulation of miscible fluid flows in porous media

The study of miscible ow in porous media is an important topic in many disciplines of science and engineering, especially in the field of petroleum engineering. For example, Carbon dioxide (CO₂) may be injected into an oil reservoir in order to improve the oil recovery rates, which is called enhanced oil recovery (EOR). This thesis focuses on the study of a miscible displacement of two fluids, such as CO₂ and oil, in a porous medium. An upscaling methodology for modeling multiscale features of the ow and the porous medium has been studied, where the overall pressure drag and skin friction exerted on the porous medium has been modelled by combining the Darcy's law with a statistical mechanical theory of viscosity, which is an important contribution of this thesis. A numerical methodology for capturing the multiphysics and multiscale nature of the governing motion has been studied. The temporal discretization employs the second order Crank-Nicolson scheme for viscous and diffusive phenomena, and an explicit method for all other terms. The nonlinear advection terms in the momentum equation has been treated with an Euler explicit flux form central finite difference method; however, the advection of the CO2 mass flux has been treated with a streamline based Lagrangian method. In order to implement the Marker-and-Cell (MAC) scheme for resolving the incompressibility, a staggered arrangement of the velocity and pressure has been presented on a collocated grid. This approach enhances the implementation of a multigrid solver, and is a novel computational model for simulating miscible displacement processes. The performance of the Lagrangian method has been assessed with respect to an equivalent flux form upwind method. The results indicate that the viscous forces play a signicant role compared to the effect of permeability on miscible displacement of CO₂ and oil, where the injected CO₂ displaces the residual oil without being distorted, thereby enhancing the recovery of hydrocarbon. Although the present results with an idealized model lacks from verifications with field measurements, findings of this thesis provide useful feedback to further investigations on CO₂ based EOR techniques