Theory of Plasticity of Porous Media With Fluid Flow

1974 ◽  
Vol 14 (03) ◽  
pp. 263-270 ◽  
Author(s):  
Milos Kojic ◽  
J.B. Cheatham
Keyword(s):  
Plastic Deformation ◽  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
General Theory ◽  
Pressure Gradient ◽  
Yield Criterion ◽  
Solid Particles ◽  
Fluid Mixture ◽  
Moving Fluid

Abstract Plastic deformation of a porous medium containing moving fluid is analyzed as a motion of a solid-fluid mixture. The fluid is considered to be Newtonian, and the porous material consists of interconnected pore spaces and of solid particles that can deform pore spaces and of solid particles that can deform elastically. The effective stress principle and a general form of the yield function-including work-hardening characteristics-and general stress-strain relations are applied to describe the plastic deformation of the solid. The system of plastic deformation of the solid. The system of governing equations with the number of unknowns being equal to the number of equations is formed. A possible method of solution of a general problem is described. Some simplification such as problem is described. Some simplification such as the assumptions of quasi-static plastic deformation and incipient plastic deformation with the application of Darcy's law for the fluid flow are discussed. To illustrate an application of the theory, the problem of incipient plane plastic deformation of a Coulomb material is presented. Introduction The motion of fluid through a porous medium and the deformation of a porous medium containing fluid have been the subjects of many investigations. For problems concerning fluid flow through porous media in petroleum and civil engineering literature, the porous material is usually considered undeformable and Darcy's law is taken as the governing relation between the velocity and the pressure of the fluid. pressure of the fluid. Most of the effort concerning fluidization of porous media has been experimental; here the task porous media has been experimental; here the task is to find the critical pressure gradient or the critical velocity of the fluid that will cause fluidization. Only the one-dimensional equilibrium equation, which relates Ne pressure gradient of the fluid and densities of solid and fluid, has been analyzed in most fluidization studies. Recently, a more general theoretical approach has been taken and equations of motion of fluid and solid have been established. Some of the results of this theory are used in the present study. Previous investigations of the deformation of porous media containing fluid have been both porous media containing fluid have been both empirical and theoretical. In the domain of elastic deformation much of the published material has dealt with experimental work aimed at finding the relation between a change in fluid pressure and stresses and deformation of the solid phase. A general theory of elasticity of porous media containing moving fluid was established by Biot. However, that theory is approximate since Darcy's law is considered as a governing relation for the fluid, and the change of permeability with the deformation of the solid is neglected. A simplification of this theory was presented by Lubinski. Experimental work has been carried out in the domain of plastic deformation of porous media containing fluid. The effective stress principle has been established as a result of experiments using saturated sand and porous rocks with various pore pressures (fluid is static in these experiments. pressures (fluid is static in these experiments. This principle, which is considered as a fundamental principle in soil mechanics, states that the pore principle in soil mechanics, states that the pore pressure does not affect the yield criterion of the pressure does not affect the yield criterion of the solid. In other words, the yield condition of the solid depends only on stresses transmitted among the solid particles. The influence of fluid flow on plasticity of porous media was indicated by Lambe and Whitman porous media was indicated by Lambe and Whitman in the analysis of stability of an infinite slope of a soil. In the equilibrium equation of a so-called "free body" a term equal to the negative pressure gradient is added. There is no general theory for plasticity of porous media containing moving fluid. plasticity of porous media containing moving fluid. GENERAL THEORY Consider the motion of a solid-fluid mixture and suppose that the motion of the solid is a plastic deformation. Then the problem reduces to the following: define the motion of a solid-fluid mixture so that the yield criterion of the solid is satisfied. The mechanical model can be described as follows. 1. The system comprises one fluid and one should constituent. SPEJ P. 263

Analysis of the Influence of Fluid Flow on the Plasticity of Porous Rock Under an Axially Symmetric Punch

1974 ◽  
Vol 14 (03) ◽  
pp. 271-278 ◽  
Author(s):  
Milos Kojic ◽  
J.B. Cheatham
Keyword(s):  
Plastic Deformation ◽  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Pressure Gradient ◽  
Stress Tensor ◽  
Unit Volume ◽  
Fluid Pressure ◽  
The Body ◽  
Axially Symmetric

Introduction A number of problems occur in the fields of drilling and rock mechanics for which consideration must be given to the interaction of fluid flow and rock deformation. Such problems include those of borehole stability, chip removal from under a drill bit, drilling in the presence of a fluid pressure gradient between the drilling fluid and formation fluid, and drilling by use of hydraulic jets. We have recently developed a general theory of the influence of fluid pressure gradients and gravity on the plasticity of porous media. The solution of the problem considered here serves as an example of the application of that theory. The illustrative problem is to determine the load required on a flat problem is to determine the load required on a flat axially symmetric punch for incipient plasticity of the porous medium under the punch when fluid flows through the bottom face of the punch. The rock is assumed to behave as a Coulomb plastic material under the influence of body forces plastic material under the influence of body forces due to fluid pressure gradients and gravity. Numerical methods that have been used by Cox et al. for analyzing axially symmetric plastic deformation in soils with gravity force are applied to the problem considered here. Involved is an iterative process for determining the slip lines. The fluid flow field ‘used for calculating the fluid pressure gradient is based upon the work by Ham pressure gradient is based upon the work by Ham in his study of the potential distribution ahead of the bit in rotary drilling. The effective stresses in the porous rock and the punch force for incipient plasticity are computed in terms of the fluid plasticity are computed in terms of the fluid pressure and the cohesive strength and internal pressure and the cohesive strength and internal friction of the rock. PLASTICITY OF POROUS MEDIA PLASTICITY OF POROUS MEDIA A recently developed general theory of plasticity of porous media under the influence of fluid flow is summarized in this section. The equation of motion for the porous solid for the case of incipient plastic deformation reduces to the following equilibrium equation:(1) where Ts is the partial stress tensor of the solid; Fs is the body force acting on the solid per unit volume of the solid material; P is the interaction force between the solid and the fluid; and is the porosity, which is defined as the ratio of the pore porosity, which is defined as the ratio of the pore volume to the total volume of the solid-fluid mixture. The partial stress tensor Ts can be considered as the effective stress tensor that is used in sod mechanics. With the acceptance of the effective stress principle defined in Ref. 5, the yield function, f, in the following form is satisfied for plastic deformation of the porous medium. plastic deformation of the porous medium.(2) where EP is the plastic strain tensor and K and the work-hardening parameter. From the equation of motion for the fluid, the interaction force P can be expressed in the form(3) where is the inertial force of the fluid per unit volume of the mixture and F is the body force acting on the fluid per unit volume of fluid. For the case of incipient plastic deformation the solid can be considered static (velocities of the solid particles are zero), and the problem of determining particles are zero), and the problem of determining the fluid flow field is the one usually analyzed in petroleum engineering. petroleum engineering. Consider a flow of be fluid such that the inertial forces of the fluid can be neglected and assume that Darcy's law is applicable. SPEJ P. 271

Plastic Flow in Confined Porous Media of Complex Contours

1999 ◽  
Author(s):  
Mario F. Letelier ◽  
César E. Rosas
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Theoretical Study ◽  
Perturbation Parameter ◽  
Resistance Coefficient ◽  
Boundary Perturbation ◽  
Bingham Plastic ◽  
Flow Through ◽  
Central Plug

Abstract A theoretical study of the fully developed fluid flow through a confined porous medium is presented. The fluid is described by the Bingham plastic model for small values of the yield number. The analysis allows for many admissible shapes of the wall contour. The velocity field is computed for several combination of relevant parameters, i.e., the yield number, Darcy resistance coefficient and the boundary perturbation parameter. The wall effect is especially highlighted and the characteristics of the central plug region as well. Plots of isovel curves and velocity profiles are included for a variety of flow and geometry parameters.

Conservation of Mass

1997 ◽  
Author(s):  
David Jon Furbish
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Unsaturated Flow ◽  
Flow In Porous Media ◽  
Degree Of Saturation ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Conservation Of Mass ◽  
Special Case

The concept of conservation of mass holds a fundamental role in most problems in fluid physics. For a given problem this concept is cast in the form of an equation of continuity. Such an equation describes a condition—conservation of mass—that must be satisfied in any formal analysis of a problem. Thus an equation of continuity often is one of several complementary equations that are solved simultaneously to arrive at a solution to a flow problem, for example, the flow velocity as a function of coordinate position in a flow field. (Typically these complementary equations, as we will see in later chapters, involve conservation of momentum or energy, or both.) Although we did not explicitly use this idea in analyzing the one-dimensional flow problems at the end of Chapter 3, it turns out that continuity was implicitly satisfied in setting up each problem. We will return to these problems to illustrate this point. We will develop equations of continuity for three general cases: purely fluid flow, saturated single-phase flow in porous media, and unsaturated flow in porous media. The most general of the three equations is that for unsaturated flow, where pores are partially filled with the fluid phase of interest, such that the degree of saturation with respect to that phase is less than one. We will then show that this equation reduces, in the special case in which the degree of saturation equals one, to a simpler form appropriate for saturated single-phase flow. Then, this equation for saturated flow could be reduced further, in the special case in which the porosity equals one, to a form appropriate for purely fluid flow. For pedagogical reasons, however, we shall reverse this order and consider purely fluid flow first. In addition we will consider conservation of a solid or gas dissolved in a liquid, and take this opportunity to introduce Fick’s law for molecular diffusion. For simplicity we will consider only species that do not react chemically with the liquid, nor with the solid phases of a porous medium. Most of the derivations below are based on the idea of a small control volume of specified dimensions embedded within a fluid or porous medium.

Cellular‐automaton fluids: A model for flow in porous media

Geophysics ◽  
1988 ◽  
Vol 53 (4) ◽  
pp. 509-518 ◽  
Author(s):  
Daniel H. Rothman
Keyword(s):  
Porous Media ◽  
Fluid Flow ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
Pressure Gradient ◽  
Cellular Automaton ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Cellular Automaton Fluids

Numerical models of fluid flow through porous media can be developed from either microscopic or macroscopic properties. The large‐scale viewpoint is perhaps the most prevalent. Darcy’s law relates the chief macroscopic parameters of interest—flow rate, permeability, viscosity, and pressure gradient—and may be invoked to solve for any of these parameters when the others are known. In practical situations, however, this solution may not be possible. Attention is then typically focused on the estimation of permeability, and numerous numerical methods based on knowledge of the microscopic pore‐space geometry have been proposed. Because the intrinsic inhomogeneity of porous media makes the application of proper boundary conditions difficult, microscopic flow calculations have typically been achieved with idealized arrays of geometrically simple pores, throats, and cracks. I propose here an attractive alternative which can freely and accurately model fluid flow in grossly irregular geometries. This new method solves the Navier‐Stokes equations numerically using the cellular‐automaton fluid model introduced by Frisch, Hasslacher, and Pomeau. The cellular‐ automaton fluid is extraordinarily simple—particles of unit mass traveling with unit velocity reside on a triangular lattice and obey elementary collision rules—but is capable of modeling much of the rich complexity of real fluid flow. Cellular‐automaton fluids are applicable to the study of porous media. In particular, numerical methods can be used to apply the appropriate boundary conditions, create a pressure gradient, and measure the permeability. Scale of the cellular‐automaton lattice is an important issue; the linear dimension of a void region must be approximately twice the mean free path of a lattice gas particle. Finally, an example of flow in a 2-D porous medium demonstrates not only the numerical solution of the Navier‐Stokes equations in a highly irregular geometry, but also numerical estimation of permeability and a verification of Darcy’s law.

Oscillatory forcing of flow through porous media. Part 2. Unsteady flow

2002 ◽  
Vol 465 ◽  
pp. 237-260 ◽  
Author(s):  
D. R. GRAHAM ◽  
J. J. L. HIGDON
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Pressure Gradient ◽  
Flow Rate ◽  
Flow Through Porous Media ◽  
Mean Flow ◽  
Two Fluids ◽  
Mean Pressure ◽  
Flow Through ◽  
The Mean

Numerical computations are employed to study the phenomenon of oscillatory forcing of flow through porous media. The Galerkin finite element method is used to solve the time-dependent Navier–Stokes equations to determine the unsteady velocity field and the mean flow rate subject to the combined action of a mean pressure gradient and an oscillatory body force. With strong forcing in the form of sinusoidal oscillations, the mean flow rate may be reduced to 40% of its unforced steady-state value. The effectiveness of the oscillatory forcing is a strong function of the dimensionless forcing level, which is inversely proportional to the square of the fluid viscosity. For a porous medium occupied by two fluids with disparate viscosities, oscillatory forcing may be used to reduce the flow rate of the less viscous fluid, with negligible effect on the more viscous fluid. The temporal waveform of the oscillatory forcing function has a significant impact on the effectiveness of this technique. A spike/plateau waveform is found to be much more efficient than a simple sinusoidal profile. With strong forcing, the spike waveform can induce a mean axial flow in the absence of a mean pressure gradient. In the presence of a mean pressure gradient, the spike waveform may be employed to reverse the direction of flow and drive a fluid against the direction of the mean pressure gradient. Owing to the viscosity dependence of the dimensionless forcing level, this mechanism may be employed as an oscillatory filter to separate two fluids of different viscosities, driving them in opposite directions in the porous medium. Possible applications of these mechanisms in enhanced oil recovery processes are discussed.

scholarly journals The Effect Of Reynolds Number At Fluid Flow In Porous Media

REAKTOR ◽  
2017 ◽  
Vol 6 (2) ◽  
pp. 48
Author(s):  
L. Buchori ◽  
M. D. Supardan ◽  
Y. Bindar ◽  
D. Sasongko ◽  
IGBN Makertihartha
Keyword(s):  
Porous Media ◽  
Fluid Flow ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Reynold Number ◽  
Packed Bed ◽  
Solid Particles ◽  
Flow In Porous Media ◽  
Viscous Term ◽  
Volume Method

In packed bed catalytic reactor, the fluid flow phenomena are very complicated because of the fluid and solid particles interaction to dissipate the energy. The governing equations need to be developed to the forms of specific models. Flows modeling of fluid flow in porous media with thw absence of the convection and viscous terms have been considerably developed such as Darcy, Brinkman, Forchheimer, Ergun, Liu, et.al and Liu and Masliyah models. These equations usually are called shear factor model. Shear factor is determined by the flow regime, porous media characteristics and fluid properties. It is true that these models are limited to condition whether the models can be applied. Analytical solution for the model types above is available only for simple one-dimentionalcases. For two or three-dimentional problem, numerical solution is the only solution. The present work is aimed to developed a two-dimentional numerical modeling flow in porous media by including the convective and viscous term. The momentum lost due  to flow and porous material interaction is modeled using the available Brinkman-Forchheimer and Liu and Masliyah equations. Numerical method to be used is finite volume method. This method is suitable for the characteristic of fluid flow in porous media which is averaged by a volume base. The effect of the solid and fluid interaction  in porous media is the basic principle of the flow model in porous media. The momentum and continuity  equations are solved for two-dimentional cylindrical coordinate. The result were validated with the experimental data . the result show a good agreement in their trend between Brinkman-Forchheimer equqtion with the Stephenson and Stewart (1986) and Liu and Masliyah equation with Kufner and Hoffman (1990) experimental data.Keywords : finite volume method, porous media, Reynold number, shear factor

Two-phase gravity currents in porous media

2011 ◽  
Vol 678 ◽  
pp. 248-270 ◽  
Author(s):  
MADELEINE J. GOLDING ◽  
JEROME A. NEUFELD ◽  
MARC A. HESSE ◽  
HERBERT E. HUPPERT
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Current Model ◽  
Gravity Current ◽  
Capillary Forces ◽  
Gravity Currents ◽  
Flow In Porous Media ◽  
Two Phase ◽  
Residual Trapping

We develop a model describing the buoyancy-driven propagation of two-phase gravity currents, motivated by problems in groundwater hydrology and geological storage of carbon dioxide (CO2). In these settings, fluid invades a porous medium saturated with an immiscible second fluid of different density and viscosity. The action of capillary forces in the porous medium results in spatial variations of the saturation of the two fluids. Here, we consider the propagation of fluid in a semi-infinite porous medium across a horizontal, impermeable boundary. In such systems, once the aspect ratio is large, fluid flow is mainly horizontal and the local saturation is determined by the vertical balance between capillary and gravitational forces. Gradients in the hydrostatic pressure along the current drive fluid flow in proportion to the saturation-dependent relative permeabilities, thus determining the shape and dynamics of two-phase currents. The resulting two-phase gravity current model is attractive because the formalism captures the essential macroscopic physics of multiphase flow in porous media. Residual trapping of CO2 by capillary forces is one of the key mechanisms that can permanently immobilize CO2 in the societally important example of geological CO2 sequestration. The magnitude of residual trapping is set by the areal extent and saturation distribution within the current, both of which are predicted by the two-phase gravity current model. Hence the magnitude of residual trapping during the post-injection buoyant rise of CO2 can be estimated quantitatively. We show that residual trapping increases in the presence of a capillary fringe, despite the decrease in average saturation.

Two Types of Nonlinear Pressure-Drop Versus Flow-Rate Relation Observed for Saturated Porous Media

1997 ◽  
Vol 119 (3) ◽  
pp. 700-706 ◽  
Author(s):  
J. L. Lage ◽  
B. V. Antohe ◽  
D. A. Nield
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Hydrostatic Pressure ◽  
Pressure Gradient ◽  
Fluid Velocity ◽  
Rate Of Change ◽  
Transition To Turbulence ◽  
Abnormal Behavior ◽  
Hydrostatic Pressure Gradient ◽  
Transition Speed

Previous reports of experiments performed with water (Fund et at., 1987 and Kececioglu and Jiang, 1994) indicated that beyond the Forchheimer regime the rate of change of the hydrostatic pressure gradient along a porous medium suddenly decreases. This abnormal behavior has been termed “transition to turbulence in a porous medium.” We investigate the relationship between the hydrostatic pressure gradient of a fluid (air) through a porous medium and the average seepage fluid velocity. Our experimental results, reported here, indicate an increase in the hydrostatic pressure rate beyond a certain transition speed, not a decrease. Physical arguments based on a consideration of internal versus external incompressible viscous flow are used to justify this distinct behavior, a consequence of the competition between a form dominated transition and a viscous dominated transition. We establish a criterion for the viscous dominated transition from consideration of the results of three porous media with distinct hydraulic characteristics. A theoretical analysis based on the semivariance model validation principle indicates that the pressure gradient versus fluid speed relation indeed departs from the quadratic Forchheimer-extended Darcy flow model, and can be correlated by a cubic function of fluid speed for the velocity range of our experiments.

scholarly journals Numerical Study of the Effect of Magnetic Field on Nanofluid Heat Transfer in Metal Foam Environment

Geofluids ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Hamid Shafiee ◽  
Elaheh NikzadehAbbasi ◽  
Majid Soltani
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Volume Fraction ◽  
Numerical Study ◽  
Computational Simulation ◽  
Thermodynamic Efficiency ◽  
Two Phase

The magnetic field can act as a suitable control parameter for heat transfer and fluid flow. It can also be used to maximize thermodynamic efficiency in a variety of fields. Nanofluids and porous media are common methods to increase heat transfer. In addition to improving heat transfer, porous media can increase pressure drop. This research is a computational simulation of the impacts of a magnetic field induced into a cylinder in a porous medium for a volume fraction of 0.2 water/Al2O3 nanofluid with a diameter of 10 μm inside the cylinder. For a wide variety of controlling parameters, simulations have been made. The fluid flow in the porous medium is explained using the Darcy-Brinkman-Forchheimer equation, and the nanofluid flow is represented utilizing a two-phase mixed approach as a two-phase flow. In addition, simulations were run in a slow flow state using the finite volume method. The mean Nusselt number and performance evaluation criteria (PEC) were studied for different Darcy and Hartmann numbers. The results show that the amount of heat transfer coefficient increases with increasing the number of Hartmann and Darcy. In addition, the composition of the nanofluid in the base fluid enhanced the PEC in all instances. Furthermore, the PEC has gained its highest value at the conditions relating to the permeable porous medium.

Experimental Investigation of Fluid Flow and Internal Convection Heat Transfer in Mini/Micro Porous Media

2009 ◽  
Author(s):  
Yu-Li Huang ◽  
Pei-Xue Jiang ◽  
Rui-Na Xu
Keyword(s):  
Heat Transfer ◽  
Carbon Dioxide ◽  
Porous Media ◽  
Fluid Flow ◽  
Convection Heat Transfer ◽  
Reynolds Numbers ◽  
Solid Particles ◽  
Convection Heat ◽  
Friction Factors ◽  
Internal Convection

The flow characteristics of different gases such as air, helium and carbon dioxide and internal convection heat transfer between the solid particles and the fluid in mini/micro porous media were studied experimentally. The test sections for fluid flow and heat transfer were made of sintered bronze particles with average diameters of 225 μm, 125 μm, 90 μm and 40 μm. The experimentally measured friction factors with consideration of compressibility for air, helium and carbon dioxide in the porous media with average diameters of 225 μm and 125 μm agree well with the known correlation for normal size porous media (the correlation of Aerov and Tojec), especially at the relatively high Reynolds numbers. The experimentally measured friction factors for air, helium and carbon dioxide in the porous media with average diameters of 90 μm are slightly less than the correlation of Aerov and Tojec at the relatively low Reynolds numbers. The experimental values for the friction factors for air, helium and carbon dioxide in the microporousmedia with 40 μm average diameters are much less than the correlation of Aerov and Tojec. The results show that rarefaction effects occur in air, helium and hydrogen flows in the microporous media with particle diameters less than 90 μm. The internal convection heat transfer coefficients between particles and fluid for air, helium and carbon dioxide in the micro porous media were determined experimentally.

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