Improving the convergence behaviour of a fixed-point-iteration solver for multiphase flow in porous media

A new method to admit large Courant numbers in the numerical simulation of multiphase flow is presented. The governing equations are discretized in time using an adaptive θ-method. However, the use of implicit discretizations does not guarantee convergence of the nonlinear solver for large Courant numbers. In this work, a double-fixed point iteration method with backtracking is presented, which improves both convergence and convergence rate. Moreover, acceleration techniques are presented to yield a more robust nonlinear solver with increased effective convergence rate. The new method reduces the computational effort by strengthening the coupling between saturation and velocity, obtaining an efficient backtracking parameter, using a modified version of Anderson's acceleration and adding vanishing artificial diffusion

A Cell-Centred CVD-MPFA Finite Volume Method for Two-Phase Fluid Flow Problems with Capillary Heterogeneity and Discontinuity

A novel finite-volume method is presented for porous media flow simulation that is applicable to discontinuous capillary pressure fields. The method crucially retains the optimal single of freedom per control-volume being developed within the flux-continuous control-volume distributed multi-point flux approximation (CVD-MPFA) framework (Edwards and Rogers in Comput Geosci 02(04):259–290, 1998; Friis et al. in SIAM J Sci Comput 31(02):1192–1220, 2008) . The new methods enable critical subsurface flow processes involving oil and gas trapping to be correctly resolved on structured and unstructured grids. The results demonstrate the ability of the method to resolve flow with oil/gas trapping in the presence of a discontinuous capillary pressure field for diagonal and full-tensor permeability fields. In addition to an upwind approximation for the saturation equation flux, the importance of upwinding on capillary pressure flux via a novel hybrid formulation is demonstrated

Nonlinear Acceleration of Sequential Fully Implicit (SFI) Method for Coupled Flow and Transport in Porous Media

The sequential fully implicit (SFI) method was introduced along with the development of the multiscale finite volume (MSFV) framework, and has received considerable attention in recent years. Each time step for SFI consists of an outer loop to solve the coupled system, in which there is one inner Newton loop to implicitly solve the pressure equation and another loop to implicitly solve the transport equations. Limited research has been conducted that deals with the outer coupling level to investigate the convergence performance. In this paper we extend the basic SFI method with several nonlinear acceleration techniques for improving the outer-loop convergence. Specifically, we consider numerical relaxation, quasi-Newton (QN) and Anderson acceleration (AA) methods. The acceleration techniques are adapted and studied for the first time within the context of SFI for coupled flow and transport in porous media. We reveal that the iterative form of SFI is equivalent to a nonlinear block Gauss-Seidel (BGS) process. The effectiveness of the acceleration techniques is demonstrated using several challenging examples. The results show that the basic SFI method is quite inefficient, suffering from slow convergence or even convergence failure. In order to better understand the behaviors of SFI, we carry out detailed analysis on the coupling mechanisms between the sub-problems. Compared with the basic SFI method, superior convergence performance is achieved by the acceleration techniques, which can resolve the convergence difficulties associated with various types of coupling effects. We show across a wide range of flow conditions that the acceleration techniques can stabilize the iterative process, and largely reduce the outer iteration count

A finite volume approach for the numerical analysis and solution of the Buckley-Leverett equation including capillary pressure

The study of petroleum recovery is significant for reservoir engineers. Mathematical models of the immiscible displacement process contain various assumptions and parameters, resulting in nonlinear governing equations which are tough to solve. The Buckley-Leverett equation is one such model, where controlling forces like gravity and capillary forces directly act on saturation profiles. These saturation profiles have important features during oil recovery. In this thesis, the Buckley-Leverett equation is solved through a finite volume scheme, and capillary forces are considered during this calculation. The detailed derivation and calculation are also illustrated here. First, the method of characteristics is used to calculate the shock speed and characteristics curve behaviour of the Buckley-Leverett equation without capillary forces. After that, the local Lax-Friedrichs finite-volume scheme is applied to the governing equation (assuming there are no capillary and gravity forces). This mathematical formulation is used for the next calculation, where the cell-centred finite volume scheme is applied to the Buckley- Leverett equation including capillary forces. All calculations are performed in MATLAB. The fidelity is also checked when the finite-volume scheme is computed in the case where an analytical solution is known. Without capillary pressure, all numerical solutions are calculated using explicit methods and smaller time steps are used for stability. Later, the fixed-point iteration method is followed to enable the stability of the local Lax-Friedrichs and Cell-centred finite volume schemes using an implicit formulation. Here, we capture the number of iterations per time-steps (including maximum and average iterations per time-step) to get the solution of water saturation for a new time-step and obtain the saturation profile. The cumulative oil production is calculated for this study and illustrates capillary effects. The influence of viscosity ratio and permeability in capillary effects is also tested in this study. Finally, we run a case study with valid field data and check every calculation to highlight that our proposed numerical schemes can capture capillary pressure effects by generating shock waves and providing single-valued saturation at each position. These saturation profiles help find the amount of water needed in an injection well to displace oil through a production well and obtains good recovery using the water flooding technique