A Fractional Generalised Finite Difference Method to Linear Porous Media Dynamics

2016 ◽  
Vol 846 ◽  
pp. 403-408 ◽  
Author(s):  
Y.P. Zhang ◽  
D.M. Pedroso ◽  
L. Li
Keyword(s):  
Porous Media ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Theoretical Background ◽  
Saturated Porous Media ◽  
Mesh Free ◽  
Mesh Free Method ◽  
Dynamics Simulations ◽  
Interpolation Functions

The generalised finite difference method (GFDM) is a mesh-free method for solving partial differential equations (PDEs) in non-structured grids. Due to its strong theoretical background and simplicity, hence efficiency, it has been introduced to handle interesting and sophisticate engineering problems. However, the GFDM has not been applied to problems associated to dynamics of porous media yet. In these problems, the strong coupling between solid displacements and liquid pressures may cause large numerical oscillations if equal order interpolation functions are used for both variables. Nevertheless, some fractional steps techniques can be introduced in order to minimise these problems. In this contribution, a fractional steps scheme is developed and applied to the GFDM in order to model fully saturated porous media dynamics. Simulations of 1D and 2D wave propagation are performed in order to reveal the advantages, drawbacks and capabilities of the proposed method.

scholarly journals Extraction of density-layered fluid from a porous medium

2020 ◽  
Vol 61 ◽  
pp. C137-C151
Author(s):  
Jyothi Jose ◽  
Graeme Hocking ◽  
Duncan Farrow
Keyword(s):  
Porous Media ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Axisymmetric Flow ◽  
Density Stratification ◽  
Comsol Multiphysics ◽  
Approximate Theory ◽  
Hodograph Method ◽  
Water Coning

We consider axisymmetric flow towards a point sink from a stratified fluid in a vertically confined aquifer. We present two approaches to solve the equations of flow for the linear density gradient case. Firstly, a series method results in an eigenfunction expansion in Whittaker functions. The second method is a simple finite difference method. Comparison of the two methods verifies the finite difference method is accurate, so that more complicated nonlinear, density stratification can be considered. Such nonlinear profiles cannot be considered with the eigenfunction approach. Interesting results for the case where the density stratification changes from linear to almost two-layer are presented, showing that in the nonlinear case there are certain values of flow rate for which a steady solution does not occur. References Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, 9th ed. National Bureau of Standards, Washington, 1972. Bear, J. and Dagan, G. Some exact solutions of interface problems by means of the hodograph method. J. Geophys. Res. 69(8):1563–1572, 1964. doi:10.1029/JZ069i008p01563 Bear, J. Dynamics of fluids in porous media. Elsevier, New York, 1972. https://store.doverpublications.com/0486656756.html COMSOL Multiphysics. COMSOL Multiphysics Programming Reference Manual, version 5.3. https://doc.comsol.com/5.3/doc/com.comsol.help.comsol/COMSOL_ProgrammingReferenceManual.pdf Farrow, D. E. and Hocking, G. C. A numerical model for withdrawal from a two layer fluid. J. Fluid Mech. 549:141–157, 2006. doi:10.1017/S0022112005007561 Henderson, N., Flores, E., Sampaio, M., Freitas, L. and Platt, G. M. Supercritical fluid flow in porous media: modelling and simulation. Chem. Eng. Sci. 60:1797–1808, 2005. doi:10.1016/j.ces.2004.11.012 Lucas, S. K., Blake, J. R. and Kucera, A. A boundary-integral method applied to water coning in oil reservoirs. ANZIAM J. 32(3):261–283, 1991. doi:10.1017/S0334270000006858 Meyer, H. I. and Garder, A. O. Mechanics of two immiscible fluids in porous media. J. Appl. Phys., 25:1400–1406, 1954. doi:10.1063/1.1721576 Muskat, M. and Wycokoff, R. D. An approximate theory of water coning in oil production. Trans. AIME 114:144–163, 1935. doi:10.2118/935144-G GNU Octave. https://www.gnu.org/software/octave/doc/v4.2.1/ Yih, C. S. On steady stratified flows in porous media. Quart. J. Appl. Maths. 40(2):219–230, 1982. doi:10.1090/qam/666676 Yu, D., Jackson, K. and Harmon, T. C. Disperson and diffusion in porous media under supercritical conditions. Chem. Eng. Sci. 54:357–367, 1999. doi:10.1016/S0009-2509(98)00271-1 Zhang, H. and Hocking, G. C. Axisymmetric flow in an oil reservoir of finite depth caused by a point sink above an oil-water interface. J. Eng. Math. 32:365–376, 1997. doi:10.1023/A:1004227232732 Zhang, H., Hocking, G. C. and Seymour, B. Critical and supercritical withdrawal from a two-layer fluid through a line sink in a bounded aquifer. Adv. Water Res. 32:1703–1710, 2009. doi:10.1016/j.advwatres.2009.09.002 Zill, D. G. and Wright, W. S. Differential Equations with Boundary-value problems, 8th Edition. Brooks Cole, Boston USA, 2013.

scholarly journals A Moving Mesh Finite Difference Method for Non-Monotone Solutions of Non-Equilibrium Equations in Porous Media

2017 ◽  
Vol 22 (4) ◽  
pp. 935-964 ◽  
Author(s):  
Hong Zhang ◽  
Paul Andries Zegeling
Keyword(s):  
Porous Media ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Moving Mesh ◽  
Richards Equation ◽  
Time Step ◽  
Central Difference ◽  
Monitor Function ◽  
Non Equilibrium

AbstractAn adaptive moving mesh finite difference method is presented to solve two types of equations with dynamic capillary pressure effect in porous media. One is the non-equilibrium Richards Equation and the other is the modified Buckley-Leverett equation. The governing equations are discretized with an adaptive moving mesh finite difference method in the space direction and an implicit-explicit method in the time direction. In order to obtain high quality meshes, an adaptive monitor function with directional control is applied to redistribute the mesh grid in every time step, then a diffusive mechanism is used to smooth the monitor function. The behaviors of the central difference flux, the standard local Lax-Friedrich flux and the local Lax-Friedrich flux with reconstruction are investigated by solving a 1D modified Buckley-Leverett equation. With the moving mesh technique, good mesh quality and high numerical accuracy are obtained. A collection of one-dimensional and two-dimensional numerical experiments is presented to demonstrate the accuracy and effectiveness of the proposed method.

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