Research On Some Problems Of Adaptive Mesh Refinement For Reservoir Numerical Simulation

In the oil production industry, the crude oil was displacement by water or other fluids which injected into the underground, in order to improve the recovery percent. Numerical analysis tools are essential for forecasting the oil production and improving the recovery percent. Oil-water displacement is an important problem often encountered in the oil field development.The difficulty for the numerical simulations of multi-phase flow in porous media is the existence of very sharp temperature and saturation fronts around the phase change regions. Adaptive mesh refinement technique is a kind of numerical algorithm dealing with jumpy frontal surface, which is capable of using fine grids in the area with steep gradients but coarse grids where the variations of variables are slower, to track the moving fronts inside the calculation domain. The fine grids use short time step, and the coarse grids use long time step.In this paper, we applied adaptive mesh refinement(AMR) technique into two-dimensional homogeneous steam assisted gravity flooding(SAGD) process, and compared the results with STARS software, which was widely used in the oil department. The numerical results show that, the accuracy of AMR technique is the same as STARS software, but AMR technique takes less calculation time than STARS software. Therefore, it enables us to implement large amount of simulations rapidly with different parameters. It is of help to find out problems and make better decisions for petroleum recovery. In this paper, we researched on some problems of AMR for reservoir numerical simulation.First of all, in the AMR technique, we need to calculate the equivalent permeability of the multi-grid. In the isotropy porous media, the equivalent permeability can be calculated easily by arithmetic mean or harmonic mean. But in the anisotropy porous media, arithmetic mean and harmonic mean may no longer be valid, and we must find a new numerical algorithm to calculate the equivalent permeability. Renormalization is a simple and fast method for calculating the equivalent permeability.The statistical properties for the renormalized permeability obtained from the renormalization of the correlated permeability field are investigated. In contrast to the uncorrelated porous media, scaling behavior of the variance of the renormalized permeability exhibits a crossover behavior. When the correlation lengths are larger compared with the domain scale covered by the renormalization procedure, the variance of the renormalized permeability will decrease slowly and the scaling exponent will be close to zero. As the renormalization number increases, the covered domain scale will eventually become larger than the correlation lengths, and the scaling property will transit to be the same as the uncorrelated case. The convergent values of the renormalized permeability for isotropic and anisotropic correlated media are also investigated. Both the theoretical analysis and the simulation results show that larger correlation length in one direction will lead to a larger convergent value in the corresponding direction. For the log-normal permeability field, numerical simulations show that the crossover scaling and the convergent value for the renormalized permeability can be fitted very well by simple mathematical functions.Based on renormalization methods under two dimensions, two upscaling methods, one obtaining scalar equivalent permeability and the other leading to tensorial equivalent permeability, are generalized to three dimensions. The numerical results indicate that both methods are close to the solutions from finite difference method, while the tensorial one is suggested to be more accurate compared with the scalar one.Secondly, Adaptive mesh refinement technique was also applied to the non-isothermal multiphase flow in the complex reservoir containing different types of rocks. The reservoir has rocks with different relative permeability curves, and the saturation is disconnected at the interface across different types of rocks. In order to apply AMR technique, we make the following assumptions:the capillary force can be ignored; the change of flows for each phase in coarse grid is very small; each phase saturation with the same rock type in coarse grid is the same. Under these assumptions, we have a new coarsening criterion and a new refining method in the AMR process which has solved the complex reservoir containing different types of rocks and achieved very good numerical simulation results.At last, we investigate the numerical algorithm of two-phase isothermal flow. The continuity equation is corresponding to the hyperbolic equation, and we usually use upwind scheme in the numerical algorithm for the purpose of avoiding unrealistic physical oscillation. But the first-order accuracy of upwind scheme may result in significant numerical dissipation. And five-point scheme may bring serious grid orientation effect in high-dimensional space. In order to reduce the numerical dissipation and decrease the grid orientation effect, we investigate the numerical algorithm of two-phase isothermal flow. Based upon the analysis of Buckley-Leverett equation by the characteristic line method, a mathematical judging condition is proposed to distinguish between the piston-like displacement and the non-piston-like displacement. Due to that the spatial distributions of the phase saturations keep the continuity, for the non-piston-like displacement, it is suggested that the internodal transmissibility be evaluated by the arithmetic average of the transmissibilities on its neighbor grids rather than by the upstream value. Numerical examples show that the proposed algorithm for the non-piston-like displacement can not only improve the numerical accuracy but also decrease the grid orientation effect.In summary, several basic problems of AMR technique have been discussed in this paper. The numerical results indicate a good proving for the theory. We expect the research to have a practical help to the reservoir simulation in the future.