Finite-difference methods in multi-dimensional two-phase flow

2008 ◽  
pp. 149-168
Author(s):  
John R. Travis
Keyword(s):  
Finite Difference ◽  
Two Phase Flow ◽  
Finite Difference Methods ◽  
Phase Flow ◽  
Two Phase ◽  
Difference Methods

Magma dynamics using FD-PDE: a new, PETSc-based, finite-difference staggered-grid framework for solving partial differential equations

2020 ◽  
Author(s):  
Adina E. Pusok ◽  
Dave A. May ◽  
Richard F. Katz
Keyword(s):  
Free Surface ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Two Phase Flow ◽  
Flow Dynamics ◽  
Staggered Grid ◽  
Phase Flow ◽  
Two Phase

<p>All divergent plate boundaries are associated with magmatism, yet its role in their dynamics and deformation is not known. The RIFT-O-MAT project seeks to understand how magmatism promotes and shapes rifts in continental and oceanic lithosphere by using models that build upon the two-phase flow theory of magma/rock interaction. Numerical models of magma segregation from partially molten rocks are usually based on a system of equations for conservation of mass, momentum and energy. One key challenge of these problems is to compute a mass-conservative flow field that is suitable for advecting thermochemically active material that feeds back on the flow. This feedback tends to destabilise the coupled mechanics+thermochemical solver. </p><p>Staggered grid finite-volume/difference methods are: mimetic (i.e., discrete differential operators mimic the properties of the continuous differential operators); conservative by construction; inf-sup stable and "light weight" (small stencil) thus they are well suited to address these problems. We present a new framework for finite difference staggered grids for solving partial differential equations (FD-PDE) that allows testable and extensible code for single-/two-phase flow magma dynamics. We build the framework using PETSc (Balay et al., 2019) and make use of the new features for staggered grids, such as DMStag. The aim is to separate the user input from the discretization of governing equations, allow for extensible development, and implement a robust framework for testing. Any customized applications can be created easily, without interfering with previous work or tests.</p><p>Here, we present benchmark and performance results with our new FD-PDE framework. In particular, we focus on preliminary results of a two-phase flow mid-ocean ridge (MOR) model with a free surface and extensional boundary conditions. We compare flow calculations with previous work on MORs that either employed two-phase flow dynamics with kinematic boundary conditions (i.e., corner flow, Spiegelman and McKenzie, 1987), or single-phase flow dynamics with free surface (i.e., Behn and Ito, 2008). In the latter case, the effect of magma is parameterised according to a priori expectations of its role. </p><p> </p><p>Balay et al. (2019), PETSc Users Manual, ANL-95/11 - Revision 3.12, 2019.</p><p>Spiegelman and McKenzie (1987), EPSL, 83 (1-4), 137-152.</p><p>Behn and Ito (2008), Geochem. Geophys. Geosyst., 9, Q08O10.</p>

Research on Fluid Mechanics with the Finite Analytic/Grid Method for Two-Phase Turbulence Based on PDF Theory

2013 ◽  
Vol 345 ◽  
pp. 331-335
Author(s):  
Wei Wei Ye ◽  
Jiang Rong Xu ◽  
Su Juan Hu ◽  
Ying Tian
Keyword(s):  
Phase Space ◽  
Finite Difference ◽  
Analytical Solution ◽  
Finite Difference Method ◽  
Fluid Mechanics ◽  
Two Phase Flow ◽  
Grid Method ◽  
High Dimensional ◽  
Phase Flow ◽  
Two Phase

in this paper, a finite analysis/grid method for high-dimensional PDF equation of fluid mechanics of two phase flow is proposed. The particle high-dimensional scalar PDF transport equation in position-velocity phase space is reduced to velocity phase space and position phase space. An analytical solution of the particle velocity PDF is obtained by Fokker-Planck techniques, the particle position PDF is solved by finite difference method, which determines the distribution of particles in two-phase flow. It is shown that the Finite Analytic/Grid method can simulate the two-phase flow very efficiently.

Analysis of Pressure Transients in Two-Phase Radial Flow

1963 ◽  
Vol 3 (02) ◽  
pp. 116-126 ◽  
Author(s):  
J.C. Martin ◽  
D.M. James
Keyword(s):  
Finite Difference ◽  
Compressible Flow ◽  
Single Phase ◽  
Numerical Solutions ◽  
Two Phase Flow ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Short Term ◽  
Two Phase ◽  
Finite Difference Techniques

Abstract The results are presented of a study of the application of analytical methods to the solution of two phase flow into single wells. Approximate analytical expressions for the pressure distribution in two-phase flow are found for a number of conditions. The results obtained from the analytical solutions are found to be in good agreement with results obtained by finite difference techniques using a high speed digital computer. Mathematical solutions for four sets of boundary conditions are presented. All of these solutions are composed of a short-term transient plus a steady or quasi-steady state. The rates of decay of the short-lived transients are analyzed. It is found that the durations of the short-term transients may be characterized by a parameter defined as the time constant which can be determined from simple relations. It is shown also that if the outer radius is much greater than the radius of the well, the short term transients decay at rates which are proportional to the square of the exterior radius, and the rates of decay are only slightly dependent upon the radius ratio. Numerical solutions based on finite difference techniques are presented for a number of conditions. The numerical solutions are in good agreement with the predictions based on the theoretical analysis for small and moderate drawdowns. Examples involving large drawdowns indicate that the nonlinearities in the equations of flow do not appreciably alter the longevity of the short-term transients. In all cases the time required for the short-term transients to disappear is predicted satisfactorily. Introduction The mechanism by which oil and gas flow into a single well is of vital interest to the petroleum industry. The fundamental equations of two-phase flow which describe this mechanism are nonlinear partial differential equations. Numerical solutions of these equations describing pressure transients have been obtained with the aid of electronic computers. Although solutions obtained in this manner take into account a large number of effects, the reduction of this information to useful generalities is difficult. One method of obtaining generalities is the use of linearized approximations of the nonlinear equations. Since it is possible to obtain explicit solutions of the linearized equation, general properties of the role of pressure in the flow mechanism may be ascertained. Results obtained from this approach are limited to some extent by the linearizing assumptions. The severity of these limitations may be evaluated by comparing solutions of the linear equation with numerical solutions of the more exact nonlinear equations of two-phase flow. In the past considerable amount of work has been devoted to studying pressure build-up using the single-phase flow theory. Unfortunately, most pressure build-up tests involve multiphase flow. A small amount of work has been done studying pressure build-up where the flow is two-phase. The encouraging results of these studies suggest that useful results may be found from additional studies of not only pressure build-up, but also the rapid transients associated with placing a well on production. This paper presents the basic theory of the pressure transients associated with placing a well on production and with closing it in. The paper is concerned chiefly with two-phase compressible flow; however, the results also apply to single-phase flow, The results are based on analytical solutions of the flow equations, and they are verified by numerical solutions using finite difference techniques. Much of the previous work on compressible flow into wells has been confined to single-phase flow. Some work has been done on compressible two-phase steady state flow, and solutions of the equations of flow have been found by finite difference techniques using high-speed computers. Muskat presents some rather general solutions to the equations of single-phase compressible flow into wells. Much work has been done on pressure build-up in wells (see, for example, Refs. 2–6). Almost all of the work on pressure build-up concerns single-phase flow with the exception of Ref. 5 and part of Ref. 2. Some work has been done on pressure fall-off in injection wells. Muskat presents the solution of the equations for radial steady state two-phase compressible flow. SPEJ P. 116^

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