Effect of Mass Equations for Two-Phase Flow Analyses on Numerical Solutions

Abstract. In source regions of magmatic systems the temperature is above solidus and melt ascent is assumed to occur predominantly by two-phase flow which includes a fluid phase (melt) and a porous deformable matrix. Since McKenzie (1984) introduced his equations for two-phase flow, numerous solutions have been studied one of which predicts the emergence of solitary porosity waves. By now most analytical and numerical solutions for these waves used strongly simplified models for the shear- and bulk viscosity of the matrix, significantly overestimating the viscosity or completely neglecting the porosity-dependence of the bulk viscosity. Schmeling et al. (2012) suggested viscosity laws in which the viscosity decreases very rapidly for small melt fractions. They are incorporated into a 2D finite difference mantle convection code with two-phase flow (FDCON) to study the ascent of solitary porosity waves. The models show that, starting with a Gaussian shaped wave, they rapidly evolve into a solitary wave with similar shape and a certain amplitude. Despite the strongly weaker rheologies compared to previous viscosity laws the effect on dispersion curves and wave shape are only moderate as long as the background porosity is fairly small. The models are still in good agreement with semi-analytic solutions which neglect the shear stress term in the melt segregation equation. However, for higher background porosities and wave amplitudes associated with a viscosity decrease of 50% or more, the phase velocity and the width of the waves are significantly decreased. Our models show that melt ascent by solitary waves is still a viable mechanism even for more realistic matrix viscosities.

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The effect of effective rock viscosity on 2-D magmatic porosity waves

Solid Earth ◽

10.5194/se-10-2103-2019 ◽

2019 ◽

Vol 10 (6) ◽

pp. 2103-2113

Author(s):

Janik Dohmen ◽

Harro Schmeling ◽

Jan Philipp Kruse

Keyword(s):

Bulk Viscosity ◽

Numerical Solutions ◽

Two Phase Flow ◽

Fluid Phase ◽

Analytic Solutions ◽

Phase Flow ◽

Two Phase ◽

Source Regions ◽

The Matrix ◽

Melt Segregation

Abstract. In source regions of magmatic systems the temperature is above solidus, and melt ascent is assumed to occur predominantly by two-phase flow, which includes a fluid phase (melt) and a porous deformable matrix. Since McKenzie (1984) introduced equations for two-phase flow, numerous solutions have been studied, one of which predicts the emergence of solitary porosity waves. By now most analytical and numerical solutions for these waves used strongly simplified models for the shear- and bulk viscosity of the matrix, significantly overestimating the viscosity or completely neglecting the porosity dependence of the bulk viscosity. Schmeling et al. (2012) suggested viscosity laws in which the viscosity decreases very rapidly for small melt fractions. They are incorporated into a 2-D finite difference mantle convection code with two-phase flow (FDCON) to study the ascent of solitary porosity waves. The models show that, starting with a Gaussian-shaped wave, they rapidly evolve into a solitary wave with similar shape and a certain amplitude. Despite the strongly weaker rheologies compared to previous viscosity laws, the effects on dispersion curves and wave shape are only moderate as long as the background porosity is fairly small. The models are still in good agreement with semi-analytic solutions which neglect the shear stress term in the melt segregation equation. However, for higher background porosities and wave amplitudes associated with a viscosity decrease of 50 % or more, the phase velocity and the width of the waves are significantly decreased. Our models show that melt ascent by solitary waves is still a viable mechanism even for more realistic matrix viscosities.

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Analysis of Pressure Transients in Two-Phase Radial Flow

Society of Petroleum Engineers Journal ◽

10.2118/425-pa ◽

1963 ◽

Vol 3 (02) ◽

pp. 116-126 ◽

Cited By ~ 4

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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ON A HIGH-RESOLUTION GODUNOV METHOD FOR A CFD-PBM COUPLED MODEL OF TWO-PHASE FLOW IN LIQUID-LIQUID EXTRACTION COLUMNS

International Journal of Computational Methods ◽

10.1142/s0219876210002283 ◽

2010 ◽

Vol 07 (03) ◽

pp. 421-442 ◽

Cited By ~ 2

Author(s):

D. ZEIDAN ◽

M. ATTARAKIH ◽

J. KUHNERT ◽

S. TIWARI ◽

V. SHARMA ◽

...

Keyword(s):

Numerical Solutions ◽

Two Phase Flow ◽

Coupled Model ◽

Liquid Extraction ◽

Godunov Method ◽

Balance Model ◽

Extraction Column ◽

Phase Flow ◽

Two Phase ◽

Liquid Liquid Extraction

This paper is about the numerical solutions for a computational fluid dynamics-population balance model (CFD-PBM) coupled model of two-phase flow in a liquid-liquid extraction column. The model accounts for a complete description between both the dispersed and continuous phases, and constitutes a hyperbolic system of equations having a linearly degenerate nature. A numerical algorithm based on operator splitting approach for the numerical solution of the model is used. The homogeneous part is solved using the TVD MUSCL-Hancock scheme. Numerical results are presented, demonstrating the accuracy of the proposed methods and in particular, the accurate numerical description of the flow in the vicinity of the contact discontinuities.

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Free surface models of partially molten rock with visco-elasto–plastic rheology: numerical solutions using a staggered-grid finite-difference discretisation

10.5194/egusphere-egu21-7631 ◽

2021 ◽

Author(s):

Yuan Li ◽

Adina Pusok ◽

Dave May ◽

Richard Katz

Keyword(s):

Free Surface ◽

Finite Difference ◽

Phase Field ◽

Numerical Solutions ◽

Two Phase Flow ◽

Staggered Grid ◽

Phase Flow ◽

Two Phase ◽

Rock Interaction ◽

Pde Framework

<p>It is broadly accepted that magmatism plays a key dynamic role in continental and oceanic rifting. However, these dynamics remain poorly studied, largely due to the difficulty of consistently modelling liquid/solid interaction across the lithosphere. The RIFT-O-MAT project seeks to quantify the role of magma in rifting by using models that build upon the two-phase flow theory of magma/rock interaction. A key challenge is to extend the theory to account for the non-linear rheological behaviour of the host rocks, and investigate processes such as diking, faulting and their interaction. Here we present our progress in consistent numerical modelling of poro-viscoelastic&#8211;plastic modelling of deformation with a free surface.</p><p>Failure of rocks (plasticity) is an essential ingredient in geodynamics models because Earth materials cannot sustain unbounded stresses. However, plasticity represents a non-trivial problem even for single-phase flow formulations (Spiegelman et al. 2016). The elastic deformation of rocks can also affect the propagation of internal failure. Furthermore, deformation and plastic failure drives topographic change, which imposes a significant static stress field. Robustly solving a discretised model that includes this physics presents severe challenges, and many questions remain as to effective solvers for these strongly nonlinear systems.&#160;</p><p>We present a new finite difference staggered grid framework for solving partial differential equations (FD-PDE) for single-/two-phase flow magma dynamics (Pusok et al., 2020). Staggered grid finite-difference methods are mimetic, conservative, inf-sup stable and with small stencil &#8212; thus they are well suited to address these problems. The FD-PDE framework uses PETSc (Balay et al., 2020) and aims to separate the user input from the discretization of governing equations. The core goals for the FD-PDE framework is to allow for extensible development and implement a framework for rigorous code validation. Here, we present simplified model problems using the FD-PDE framework for two-phase flow visco-elasto-plastic models designed to characterise the solution quality and assess both the discretisation and solver robustness. We also present results obtained using the phase-field method (Sun and Beckermann, 2007) for representing the free surface. Verification of the phase-field approach will be shown via simplified problems previously examined in the geodynamics community (Crameri et al, 2012).</p><p>Balay et al. (2020), PETSc Users Manual, ANL-95/11 - Revision 3.13.</p><p>Pusok et al. (2020) https://doi.org/10.5194/egusphere-egu2020-18690&#160;</p><p>Spiegelman et al. (2016) https://doi.org/10.1002/2015GC006228</p><p>Sun and Beckermann (2007) https://doi.org/10.1016/j.jcp.2006.05.025</p><p>Crameri et al. (2012) https://doi.org/10.1111/j.1365-246X.2012.05388.x</p><div> <div><span></span><div></div> </div> <div></div> </div><div> <div><span></span><div></div> </div> <div></div> </div>

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Numerical Simulation of Acoustic Streaming in Gas-Liquid Two-Phase Flow

Volume 1: Symposia, Parts A and B ◽

10.1115/fedsm2005-77305 ◽

2005 ◽

Author(s):

Bo Lu ◽

Arthur E. Ruggles

Keyword(s):

Numerical Solutions ◽

Two Phase Flow ◽

Bubble Radius ◽

Acoustic Streaming ◽

Gas Bubbles ◽

Homogeneous Distribution ◽

Phase Flow ◽

Sinusoidal Vibration ◽

Two Phase ◽

Left Wall

Acoustic streaming phenomena pertaining to liquid-gas two-phase flow in a one-dimensional rigid duct is investigated numerically. The oscillatory bubbly flow is generated due to the sinusoidal vibration of the vertical left wall of the enclosure. Time-averaged streaming flow patterns exist in the duct as a consequence of interaction between gas bubbles and liquid which are similar to the Rayleigh-type acoustic streaming phenomena extensively investigated in single-phase flow. The liquid is treated as incompressible with a homogeneous distribution of non-condensable gas bubbles. The system is modeled with coupled nonlinear and flux-conservative partial differential equations combined with the Rayleigh-Plesset equation governing the bubble radius. The viscous interaction between bubbles and the surrounding incompressible liquid phase is the main mechanism for attenuation of the wave energy considered in this analysis. The numerical solutions are obtained by a control-volume based finite-volume Lagrangian method.

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