Kirill D. Nikitin

A monotone nonlinear finite volume method for diffusion equations and multiphase flows

We present a new nonlinear monotone finite volume method for diffusion equation and its application to two-phase flow model. We consider full anisotropic discontinuous diffusion or permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which provides the conventional seven-point stencil for the discrete diffusion operator on cubic meshes. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water breakthrough time. We compare two two-point flux approximations (TPFA), the proposed nonlinear TPFA and the conventional linear TPFA, and multipoint flux approximation (MPFA). The new nonlinear scheme has a number of important advantages over the traditional linear discretizations. Compared to the linear TPFA, the nonlinear TPFA demonstrates low sensitivity to grid distortions and provides appropriate approximation in case of full anisotropic permeability tensor. For nonorthogonal grids or full anisotropic permeability tensors, the conventional linear TPFA provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional TPFA, yet it is rather competitive. Compared to MPFA, the new scheme provides sparser algebraic systems and thus is less computational expensive. Moreover, it is monotone which means that the discrete solution preserves the nonnegativity of the differential solution.

A semi-Lagrangian method on dynamically adapted octree meshes

Abstract The paper develops a semi-Lagrangian method for the numerical integration of the transport equation discretized on adaptive Cartesian cubic meshes. We use dynamically adaptive graded Cartesian grids. They allow for a fast grid reconstruction in the course of numerical integration. The suggested semi- Lagrangian method uses a higher order interpolation with a limiting strategy and a back-and-forth correction of the numerical solution. The interpolation operators have compact nodal stencils. In a series of experiments with dynamically adapted meshes, we demonstrate that the method has at least the second-order convergence and acceptable conservation and monotonicity properties.

A Splitting Method for Numerical Simulation of Free Surface Flows of Incompressible Fluids with Surface Tension

Abstract The paper studies a splitting method for the numerical time-integration of the system of partial differential equations describing the motion of viscous incompressible fluid with free boundary subject to surface tension forces. The method splits one time step into a semi-Lagrangian treatment of the surface advection and fluid inertia, an implicit update of viscous terms and the projection of velocity into the subspace of divergence-free functions. We derive several conservation properties of the method and a suitable energy estimate for numerical solutions. Under certain assumptions on the smoothness of the free surface and its evolution, this leads to a stability result for the numerical method. Efficient computations of free surface flows of incompressible viscous fluids need several other ingredients, such as dynamically adapted meshes, surface reconstruction and level set function re-initialization. These enabling techniques are discussed in the paper as well. The properties of the method are illustrated with a few numerical examples. These examples include analytical tests and the oscillating droplet benchmark problem.

Nonlinear Monotone FV Schemes for Radionuclide Geomigration and Multiphase Flow Models

We present applications of the nonlinear monotone finite volume method to radionuclide transport and multiphase flow in geological media models. The scheme is applicable for full anisotropic discontinuous permeability or diffusion tensors and arbitrary conformal polyhedral cells. We consider two versions of the nonlinear scheme: two-point flux approximation preserving positivity of the solution and compact multi-point flux approximation that provides discrete maximum principle. We compare the new nonlinear schemes with the conventional linear two-point and multi-point (O-scheme) flux approximations. Both new nonlinear schemes have compact stencils and a number of important advantages over the traditional linear discretizations. Two industrial applications are discussed briefly: radionuclides transport modeling within the radioactive waste safety assessment and multiphase flow modeling of oil recovery process.

Nonlinear finite volume method with discrete maximum principle for the two-phase flow model

The discrete maximum principle is a meaningful requirement for numerical schemes used in multiphase flow models. It eliminates numerical pressure overshoots and undershoots, which may cause unnatural Darcy velocities and wrong numerical saturations. In this paper we study the application of the nonlinear finite volume method with discrete maximum principle [1] to the two-phase flow model. The method satisfies the discrete maximum principle for numerical pressures of incompressible fluids with neglected capillary pressure. For non-zero capillary pressure and constant phase viscosities the discrete maximum principle holds for numerical global pressure.

A splitting method for free surface flows over partially submerged obstacles

Abstract The paper proposes a stable time-splitting method for the numerical simulation of free-surface viscous flows. The key features of the method are a semi-Lagrangian scheme for the level-set function transport improved with MacCormack predictor–corrector step with limiting strategy and an adaptive volume-correction procedure. The spatial discretization is done by a hybrid finite volume/finite difference method on dynamically adaptive hexahedral meshes. Numerical verification is done by comparing full-scale 3D numerical simulations of the sloshing tank and the coastal wave run-up with other numerical and experimental results known from the literature.

Application of the Parallel INMOST Platform to Subsurface Flow and Transport Modelling

INMOST (Integrated Numerical Modelling and Object-oriented Supercomputing Technologies) is a tool for supercomputer simulations characterized by a maximum generality of supported computational meshes, distributed data structure flexibility and cost-effectiveness, as well as crossplatform portability. INMOST is a software platform for developing parallel numerical models on general meshes. User guides, online documentation, and the open-source code of the library is available at http://www.inmost.org.

Enhanced Nonlinear Finite Volume Scheme for Multiphase Flows

We present the latest enhancement of the nonlinear monotone finite volume method for the near-well regions. The original nonlinear method is applicable for diffusion, advection-diffusion and multiphase flow model equations with full anisotropic discontinuous permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which reduces to the conventional two-point flux approximation (TPFA) on cubic meshes but has much better accuracy for the general case of non-orthogonal grids and anisotropic media. The latest enhancement of the nonlinear method takes into account the nonlinear (e.g. logarithmic) singularity of the pressure in the near-well region and introduces the nonlinear correction to improve accuracy of the pressure and the flux calculation. The new method is generalized for anisotropic media, polyhedral grids and nontrivial wells cases. Numerical experiments show the noticeable reduction of the numerical errors compared to the original monotone nonlinear FV scheme with the conventional Peaceman well model and even with the given analytical well rate.

Two methods of surface tension treatment in free surface flow simulations

Abstract We describe our approach to treatment of surface tension in free surface flow simulations on adaptive octree-type grids. The approach is based on the semi-Lagrangian method for the transport and momentum equations and the pressure projection method to enforce the incompressibility constrain. The surface tension contributes to the Dirichlet boundary condition for the pressure equation at the projection step. The treatment of surface tension is based either on accurate finite difference calculation of the mean curvature or on a curvature estimation by the implicit solution of conservative mean curvature flow problem. The first method provides almost the second order accuracy in space for surface tension forces. The second method is characterized by greater stability and essentially larger time steps. Numerical experiments illustrate the main features of the methods.

Dynamic Optimization of Linear Solver Parameters in Mathematical Modelling of Unsteady Processes

The optimization of linear solver parameters in unsteady multiphase groundflow modelling is considered. Two strategies of dynamic parameters setting for the linear solver are proposed when the linear systems properties are modified during simulation in the INMOST framework. It is shown that the considered algorithms for dynamic selection of linear solver parameters provide a more efficient solution than any prescribed set of parameters. The results of numerical experiments on the INM RAS cluster are presented.