Study of Explicit Finite Difference Method in One Dimensional System

Authors: Azizollah Khormali, Seyyed Shahab Tabatabaee Moradi, Dmitry Petrakov

Abstract:

One of the most important parameters in petroleum reservoirs is the pressure distribution along the reservoir, as the pressure varies with the time and location. A popular method to determine the pressure distribution in a reservoir in the unsteady state regime of flow is applying Darcy’s equation and solving this equation numerically. The numerical simulation of reservoirs is based on these numerical solutions of different partial differential equations (PDEs) representing the multiphase flow of fluids. Pressure profile has obtained in a one dimensional system solving Darcy’s equation explicitly. Changes of pressure profile in three situations are investigated in this work. These situations include section length changes, step time changes and time approach to infinity. The effects of these changes in pressure profile are shown and discussed in the paper.

Keywords: Explicit solution, Numerical simulation, Petroleum reservoir, Pressure distribution.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1091384

References:

[1] B. Guo and X. Wang, "Testing of New High-Order Finite Difference Methods for Solving the Convection-Diffusion Equation,” E-Journal of Reservoir Engineering, pp. 1-14, 2005.
[2] D. K. E. De Carvalho, P. R. M. Lyra and R. B. Willmersdorf, "A First Step Towards a Petroleum Reservoir Simulator Using an Edge-Based Unstructured Finite Volume Formulation,” The 2nd Brazilian Congress on R&D in Petroleum and Gas, 2003.
[3] T. Pal Singh, "An Improved Finite Difference Method for Well Models in Numerical Reservoir Simulation,” 2nd International Conference on Current Trends in Technology, 08–10 Dec. 2011.
[4] K. Hvistendahl Karlsen, K. A. Lieb and N. H. Risebro, "A Fast Marching Method for Reservoir Simulation,” Computational Geosciences, vol. 4, pp. 185–206, 2000.
[5] G. J. Moridis, D. A. Mc Vay, D. L. Reddell and T. A. Blasingame, "The Laplace Transform Finite Difference (LTFD) Numerical Method for the Simulation of Compressible Liquid Flow in Reservoirs,” SPE Advanced Technology Series, vol. 2, pp. 122–131, Apr. 1994.
[6] Y. Liu, K. Mrinal Sen, A. John and G. Katherine, "Acoustic VTI Modeling by a New Time-Space Domain High-Order Finite-Difference Method,” SEG Annual Meeting, Houston, Texas, Oct. 2009.
[7] F. Liu and W. Shi, "Numerical Solutions of Two-Dimensional Burgers’ Equations by Lattice Boltzmann Method,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 150–157, 2011.