Abstract: Quantifying uncertainty effects of coefficients that exhibit heterogeneity at multiple scales is among many outstanding challenges in subsurface flow models. Typically, the coefficients are modeled as functions of random variables governed by certain statistics. To quantify their uncertainty in the form of statistics (e.g., average fluid pressure or concentration) Monte-Carlo methods have been used. In a separate direction, multiscale numerical methods have been developed to efficiently capture spatial heterogeneity that otherwise would be intractable with standard numerical techniques. Since heterogeneity of individual realizations can differ drastically, a direct use of multiscale methods in Monte-Carlo simulations is problematic. Furthermore, Monte-Carlo methods are known to be very expensive as a lot of samples are required to adequately characterize the random component of the solution. In this study, we utilize a stochastic representation method that exploits the solution structure of the random process in order to construct a problem dependent stochastic basis. Using this stochastic basis representation a set of coupled yet deterministic equations is constructed. To reduce the computational cost of solving the coupled system, we develop a multiscale domain decomposition method utilizing Robin transmission conditions. In the proposed method, enrichment of the solution space can be performed at multiple levels that offer a balance between computational cost, and accuracy of the approximate solution.

Abstract: The standard nodal Lagrangian based continuous Galerkin finite element method (FEM) and control volume finite element method (CVFEM) are well known techniques for solving partial differential equations. Both of these methods have a common shortcoming in that the first derivative of the approximate solution of both methods is discontinuous. Further shortcomings of nodal Lagrangian bases arise when considering time dependent problems. For instance, increasing the degree of the basis in an effort to improve the accuracy of the approximate solution prohibits the use of common techniques such as mass matrix lumping. We introduce a $\mu^{\mathrm{th}}$ degree clamped basis-spline (B-spline) based analog of both the control volume finite element method and the continuous Galerkin finite element method in conjunction with a post processing technique which shall impose local conservation. The advantage of these techniques is that the B-spline basis is not only non-negative for any order $\mu$, and thus lends itself to mass matrix lumping for higher order basis functions, but also, for $\mu>2$, each basis function is smooth on the domain. We implement both the B-spline based CVFEM and FEM techniques as well as the post processing technique as they pertain to solving various two-point boundary value problems. A comparison of the convergence rates and properties of the error associated with satisfying local conservation is presented.

Abstract: Computational Fluid Dynamics (CFD) utilizes numerical solutions of Partial Differential Equations (PDE) on discretized volumes. These sets of discretized volumes, grids, can often contain tens of mil-lions, or billions of volumes. The analysis time of these large unstructured grids can take weeks to months to complete even on large computer clusters. For CFD solvers utilizing the Finite Volume Method (FVM) with implicit time stepping or a segregated pressure solver, a large portion of the computation time is spent solving a large linear system with a sparse coefficient matrix. In an effort to improve the performance of these CFD codes, in effect decreasing the time to solution of engineering problems, a conjugate gradient solver for a Finite Volume Method Solver Graphics Processing Units (GPU) was implemented to solve a model Poisson’s equation. Utilizing the improved memory throughput of NVIDIA’s Tesla K20 GPU a 2.5 times improvement was observed compared to a parallel CPU implementation on all 10 cores of an Intel Xeon E5-2670 v2. The parallel CPU implementation was constructed using the open source CFD toolbox, Open-FOAM.

Abstract: DarcyLite is a Matlab toolbox developed for numerical simulations of flow and transport in porous media in two dimensions. This paper focuses on the finite element methods and the corresponding code modules for solving the Darcy equation. Specifically, four major types of finite element solvers are presented: the continuous Galerkin (CG), the discontinuous Galerkin (DG), the weak Galerkin (WG), and the mixed finite element methods (MFEM). We further discuss the main design ideas and implementation strategies in DarcyLite. Numerical examples are included to demonstrate the usage and performance of this toolbox.

Abstract: In the present paper, a finite element model is developed based on a semi-discrete Streamline Upwind Petrov-Galerkin method to solve the fully-coupled two-dimensional shallow water and contaminant transport equations on a non-flat bed. The algorithm is applied on fixed computational meshes. Linear triangular elements are used to decompose the computational domain and a second-order backward differentiation implicit method is used for the time integration. The resulting nonlinear system is solved using a Newton-type method where the linear system is solved at each step using the Generalized Minimal Residual method. In order to examine the accuracy and robustness of the present scheme, numerical results are verified by different test cases.

Abstract: In this paper, we develop a two-scale reduced model for simulating the Darcy flow in two-dimensional porous media with conductive fractures. We apply the approach motivated by the embedded fracture model (EFM) to simulate the flow on the coarse scale, and the effect of fractures on each coarse scale grid cell intersecting with fractures is represented by the discrete fracture model (DFM) on the fine scale. In the DFM used on the fine scale, the matrix-fracture system are resolved on unstructured grid which represents the fractures accurately, while in the EFM used on the coarse scale, the flux interaction between fractures and matrix are dealt with as a source term, and the matrix-fracture system can be resolved on structured grid. The Raviart-Thomas mixed finite element methods are used for the solution of the coupled flows in the matrix and the fractures on both fine and coarse scales. Numerical results are presented to demonstrate the efficiency of the proposed model for simulation of flow in fractured porous media.

Abstract: Reduced-order model by proper orthogonal decomposition of Navier-Stokes equation can be established in different manners. After careful screening under different sampling intervals and numbers of basis vectors, it has been found that the model can achieve high precision only when it is constructed on collocated grid with the samples still on the staggered grid. The model straight-forward established on the staggered grid may lose accuracy apparently. To precisely capture the dynamic behavior of flow field, the sampling interval should be small enough while the number of basis vectors should be moderate. These conclusions can be a valuable principle for future modeling of the dynamics of fluid flow.

Abstract: In this paper, we introduce a mathematical model to describe the nanoparticles transport carried by a two-phase flow in a porous medium including gravity, capillary forces and Brownian diffusion. Nonlinear iterative IMPES scheme is used to solve the flow equation, and saturation and pressure are calculated at the current iteration step and then the transport equation is solved implicitly. Therefore, once the nanoparticles concentration is computed, the two equations of volume of the nanoparticles available on the pore surfaces and the volume of the nanoparticles entrapped in pore throats are solved implicitly. The porosity and the permeability variations are updated at each time step after each iteration loop. Numerical example for regular heterogenous permeability is considered. We monitor the changing of the fluid and solid properties due to adding the nanoparticles. Variation of water saturation, water pressure, nanoparticles concentration and porosity are presented graphically.

Abstract: In this work, an efficient coupling between Monte Carlo (MC) molecular simulation and Darcy-scale flow in porous media is presented. The cell centered finite difference method with non-uniform rectangular mesh were used to discretize the simulation domain and solve the governing equations. To speed up the MC simulations, we implemented a recently developed scheme that quickly generates MC Markov chains out of pre-computed ones, based on the reweighting and reconstruction algorithm. This method astonishingly reduces the required computational times by MC simulations from hours to seconds. To demonstrate the strength of the proposed coupling in terms of computational time efficiency and numerical accuracy in fluid properties, various numerical experiments covering different compressible single-phase flow scenarios were conducted. The novelty in the introduced scheme is in allowing an efficient coupling of the molecular scale and the Darcy's one in reservoir simulators. This leads to an accurate description of thermodynamic behavior of the simulated reservoir fluids; consequently enhancing the confidence in the flow predictions in porous media.

Abstract: This research addresses a sequential convex splitting method for numerical simulation of multicomponent two-phase fluids mixture in a single-pore at constant temperature, which is modeled by the gradient theory with Peng-Robinson equation of state. The gradient theory of thermodynamics and variational calculus are utilized to obtain a system of chemical equilibrium equations which are transformed into a transient system as a numerical strategy on which the numerical scheme is based. The proposed numerical algorithm avoids computing Hessian matrix arising from the second-order derivative of homogeneous contribution of free energy; it is also quite robust. This scheme is proved to be unconditionally component-wise energy stable. The Raviart-Thomas mixed finite element method is applied to spatial discretization.

Abstract: Relaxation process of magnetohydrodynamics (MHD) inside a sphere is investigated by a newly developed spherical grid system, Yin-Yang-Zhong grid. An MHD fluid with low dissipation rates is confined by a perfectly conducting, stress-free, and thermally insulating spherical boundary. The Reynolds number Re and the magnetic Reynolds number Rm is the same: Re=Rm=8600. Starting from a simple and symmetric state in which a ring-shaped magnetic flux without flow, a dynamical relaxation process of the magnetic energy is numerically integrated. The relaxed state has a characteristic structure of the magnetic field and the flow field with four vortices.

Abstract: Nanoparticles are particles that are between 1 and 100 nanometers in size. They present possible dangers to the environment due to the high surface to volume ratio, which can make the particles very reactive or catalytic. Furthermore, rapid increase in the implementation of nanotechnologies has released large amount of the nanowaste into the environment. In the last two decades, transport of nanoparticles in the subsurface and the potential hazard they impose to the environment have attracted the attention of researchers. In this work, we use numerical simulation to investigate the problem regarding the transport phenomena of nanoparticles in anisotropic porous media. We consider the case in which the permeability in the principal direction components will vary with respect to time. The interesting thing in this case is the fact that the anisotropy could disappear with time. We investigate the effect of the degenerating anisotropy on various fields such as pressure, porosity, concentration and velocities.

Abstract: Fully-Implicit (FI) Methods are often employed in the numerical simulation of large-scale subsurface flows in porous media. At each implicit time step, a Newton-like method is used to solve the FI discrete nonlinear algebraic system. The linear solution process for the Newton updates is the computational workhorse of FI simulations. Empirical observations suggest that the computed Newton updates during FI simulations of multiphase flow are often sparse. Moreover, the level of sparsity observed can vary dramatically from iteration to the next, and across time steps. In several large scale applications, it was reported that the level of sparsity in the Newton update can be as large as 99\%. This work develops a localization algorithm that conservatively predetermines the sparsity pattern of the Newton update. Subsequently, only the flagged nonzero components of the system need be solved. The localization algorithm is developed for general FI models of two phase flow. Large scale simulation results of benchmark reservoir models show a 10 to 100 fold reduction in computational cost for homogeneous problems, and a 4 to 10 fold reduction for strongly heterogeneous problems.

Abstract: Unconventional reservoirs are typically comprised of a multicontinuum stimulated formation, with complex fracture networks that have a wide range of length scales and geometries. A timely topic in the simulation of unconventional petroleum resources is in coupling the geomechanics of the fractured media to multiphase fluid flow and transport. We propose a XFEM-EDFM method which couples geomechanics with multiphase flow in fractured tight gas reservoirs. A proppant model is developed to simulate propped hydraulic fractures. The method is verified by analytical solutions. A simulation example with the configuration of two multiple-fractured horizontal wells is investigated. The influence of stress-dependent fracture permeability on cumulative production is analyzed.