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

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

scholarly journals An Extended Finite Element Model for Fluid Flow in Fractured Porous Media

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Fei Liu ◽  
Li-qiang Zhao ◽  
Ping-li Liu ◽  
Zhi-feng Luo ◽  
Nian-yin Li ◽  
...  
Keyword(s):  
Finite Element ◽  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Fluid Pressure ◽  
Element Model ◽  
Fractured Porous Media ◽  
Extended Finite Element ◽  
Fluid Exchange ◽  
Volume Fracturing

This paper proposes a numerical model for the fluid flow in fractured porous media with the extended finite element method. The governing equations account for the fluid flow in the porous medium and the discrete natural fractures, as well as the fluid exchange between the fracture and the porous medium surrounding the fracture. The pore fluid pressure is continuous, while its derivatives are discontinuous on both sides of these high conductivity fractures. The pressure field is enriched by the absolute signed distance and appropriate asymptotic functions to capture the discontinuities in derivatives. The most important advantage of this method is that the domain can be partitioned as nonmatching grid without considering the presence of fractures. Arbitrarily multiple, kinking, branching, and intersecting fractures can be treated with the new approach. In particular, for propagating fractures, such as hydraulic fracturing or network volume fracturing in fissured reservoirs, this method can process the complex fluid leak-off behavior without remeshing. Numerical examples are presented to demonstrate the capability of the proposed method in saturated fractured porous media.

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.

On sound generated aerodynamically I. General theory

1952 ◽  
Vol 211 (1107) ◽  
pp. 564-587 ◽  
Keyword(s):  
Fluid Flow ◽  
Mach Number ◽  
General Theory ◽  
Stress Tensor ◽  
High Power ◽  
Velocity Vector ◽  
Unit Volume ◽  
Equations Of Motion ◽  
Sound Field ◽  
Velocity Of Sound

A theory is initiated, based on the equations of motion of a gas, for the purpose of estimating the sound radiated from a fluid flow, with rigid boundaries, which as a result of instability contains regular fluctuations or turbulence. The sound field is that which would be produced by a static distribution of acoustic quadrupoles whose instantaneous strength per unit volume is ρv i v j + p ij - a 2 0 ρ δ ij , where ρ is the density, v i the velocity vector, p ij the compressive stress tensor, and a 0 the velocity of sound outside the flow. This quadrupole strength density may be approximated in many cases as ρ 0 v i v j . The radiation field is deduced by means of retarded potential solutions. In it, the intensity depends crucially on the frequency as well as on the strength of the quadrupoles, and as a result increases in proportion to a high power, near the eighth, of a typical velocity U in the flow. Physically, the mechanism of conversion of energy from kinetic to acoustic is based on fluctuations in the flow of momentum across fixed surfaces, and it is explained in § 2 how this accounts both for the relative inefficiency of the process and for the increase of efficiency with U . It is shown in § 7 how the efficiency is also increased, particularly for the sound emitted forwards, in the case of fluctuations convected at a not negligible Mach number.

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 Effect of Inclined Magnetic Field on Peristaltic Flow of Carreau Fluid through Porous Medium in an Inclined Tapered Asymmetric Channel

2019 ◽  
Vol 29 (3) ◽  
pp. 94
Author(s):  
Tamara Sh. Ahmed
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Fluid Flow ◽  
Perturbation Technique ◽  
Weissenberg Number ◽  
The Body ◽  
Two Dimensional ◽  
Asymmetric Channel ◽  
Carreau Fluid ◽  
Slip Boundary

During this article, we have a tendency to show the peristaltic activity of magnetohydrodynamics flow of carreau fluid with heat transfer influence in an inclined tapered asymmetric channel through porous medium by exploitation the influence of non-slip boundary conditions. The tapered asymmetric channel is often created because of the intrauterine fluid flow induced by myometrial contraction and it had been simulated by asymmetric peristaltic fluid flow in an exceedingly two dimensional infinite non uniform channel, this fluid is known as hereby carreau fluid, conjointly we are able to say that one amongst carreau's applications is that the blood flow within the body of human. Industrial field, silicon oil is an example of carreau fluid. By exploitation, the perturbation technique for little values of weissenberg number, the nonlinear governing equations in the two-dimensional Cartesian coordinate system is resolved under the assumptions of long wavelength and low Reynolds number. The expressions of stream function, temperature distribution, the coefficient of heat transfer, frictional forces at the walls of the channel, pressure gradient are calculated. The effectiveness of interesting parameters on the inflow has been colluded and studied.

Interstitial fluid pressure gradient measured by micropuncture in excised dog lung

1984 ◽  
Vol 56 (2) ◽  
pp. 271-277 ◽  
Author(s):  
J. Bhattacharya ◽  
M. A. Gropper ◽  
N. C. Staub
Keyword(s):  
Fluid Flow ◽  
Pressure Gradient ◽  
Pressure Difference ◽  
Airway Pressure ◽  
Interstitial Fluid ◽  
Fluid Pressure ◽  
Interstitial Fluid Pressure ◽  
Pleural Pressure ◽  
Alveolar Wall ◽  
Fluid Filtration

We have directly measured lung interstitial fluid pressure at sites of fluid filtration by micropuncturing excised left lower lobes of dog lung. We blood-perfused each lobe after cannulating its artery, vein, and bronchus to produce a desired amount of edema. Then, to stop further edema, we air-embolized the lobe. Holding the lobe at a constant airway pressure of 5 cmH2O, we measured interstitial fluid pressure using beveled glass micropipettes and the servo-null method. In 31 lobes, divided into 6 groups according to severity of edema, we micropunctured the subpleural interstitium in alveolar wall junctions, in adventitia around 50-micron venules, and in the hilum. In all groups an interstitial fluid pressure gradient existed from the junctions to the hilum. Junctional, adventitial, and hilar pressures, which were (relative to pleural pressure) 1.3 +/- 0.2, 0.3 +/- 0.5, and -1.8 +/- 0.2 cmH2O, respectively, in nonedematous lobes, rose with edema to plateau at 4.1 +/- 0.4, 2.0 +/- 0.2, and 0.4 +/- 0.3 cmH2O, respectively. We also measured junctional and adventitial pressures near the base and apex in each of 10 lobes. The pressures were identical, indicating no vertical interstitial fluid pressure gradient in uniformly expanded nonedematous lobes which lack a vertical pleural pressure gradient. In edematous lobes basal pressure exceeded apical but the pressure difference was entirely attributable to greater basal edema. We conclude that the presence of an alveolohilar gradient of lung interstitial fluid pressure, without a base-apex gradient, represents the mechanism for driving fluid flow from alveoli toward the hilum.

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