Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data

This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methods show convergence properties to a continuous solution of the problem in a weak sense, through the change of variable u = ψ(v), where ψ is a well chosen diffeomorphism between (−1, 1) and R, and v is valued in (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove the existence of a discrete solution, and, under standard regularity conditions, we prove its convergence to a weak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2D and 3D cases where the solution does not belong to H 1(Ω), show that this method can provide accurate results. We then construct a numerical scheme which presents a convergence property to the entropy weak solution of the problem in the case where the right-hand side belongs to L1 . This is achieved owing to a nonlinear control volume finite element (CVFE) method, keeping the same nonlinear reformulation, and adding an upstream weighting evaluation and a nonlinear p−Laplace vanishing stabilisation term.

Semi-Elimination Methodology for Simulating High Flow Features in a Reservoir

Despite the rapid rise of computing power and advances in computational techniques in past decades, it is still challenging in reservoir simulation to model complex and detailed features that are represented by small cells with large permeability values, for example, fractures, multi-segment wells, etc. While those features may carry a large amount of flow and thus have a significant impact on the performance prediction, the combination of small volume and large permeability unfortunately leads to well-known time stepping and convergence difficulties during Newton iteration. We address this issue of high flow through small cells by developing a new semi-elimination computational technique. At the beginning of simulation, we construct a set of pressure basis which is a mapping from pressures at surrounding cells in the bulk of reservoir to pressures at those small cells. Next, we start the time-stepping scheme. For each time step or iteration within a time step, small cells are first employed to provide an accurate computation of flow rates and derivatives using upstream weighting and a flow partitioning scheme. Afterwards, small cells are eliminated and a linear system of equations is assembled and solved involving only bulk cells. This semi-elimination technique allows us to fundamentally avoid the drawbacks caused by including small cells in the global system of equations, while capturing their effect on the flow of hydrocarbon in the reservoir. One of the advantages of the proposed techniques over other existing methods is that it is fully implicit and preserves upstream weighting and compositions of the flow field even after small cells are eliminated, which enhances numerical stability and accuracy of simulation results. Application of this technique to several synthetic and field models demonstrates significant performance and accuracy improvement over standard approaches. This method thus offers a practical way to model complex and dynamic flow behaviors in important features without incurring penalties in speed and robustness of the simulation.

Groundwater Flow-Modeling and Sensitivity Analysis in a Hyper Arid Region

Groundwater modelling is particularly challenging in arid regions where limited water recharge is available. A fault zone will add a significant challenge to the modelling process. The Western Desert in Iraq has been chosen to implement the modelling concept and calculate the model sensitivity to the changes in aquifer hydraulic properties and calibration by researching 102 observations and irrigation wells. MODFLOW-NWT, which is a Newtonian formulation for MODFLOW-2005 approaches, have been used in this study. Further, the simulation run has been implemented using the Upstream-Weighting package (UPW) to treat the dry cells. The results show sensitivity to the change of the Kx value for the major groundwater discharge flow. Only about 7% of the models from the region can be irrigated utilizing greenhouses supported by external recharge.

Analysis of Finite Element Discretisation Schemes for Multi-Phase Darcy Flow

We demonstrate that care must be taken in a finite-element discretisation of multi-phase compressible Darcy flow, otherwise constraints of non-negativity of fluid mass can become violated. Generalising a technique pioneered by Dalen, a lumped, finite-difference-inspired approximation with upstream weighting is described. This is numerically cheap, physically elegant, and the fluid mass at all nodes remains non-negative.

Computing Internodal Conductivities in Numerical Modeling of two Dimensional Unsaturated Flow on Rectangular Grid

This paper compares numerical solutions of transient two-dimensional unsaturated flow equation by using different averaging schemes for internodal conductivities. Averaging methods such as arithmetic mean, geometric mean, upstream weighting, and integrated mean are taken into account, as well as a recent approach based on steady-state approximation. The latter method proved the most flexible, producing relatively accurate solutions for both downward and upward flow cases.