The observed underprediction in the position of the front is due to the neglect of dynamic saturation gradients in. However, it appears that a general form of the volumeaveraged or macroscopic momentum equation, where the transient as well as the nonlinear inertial terms are included, has not yet been developed for a moving porous medium. Parlange no static citation data no static citation data cite. Sep 17, 2002 a new equation for macroscopic description of capillary rise in porous media. Capillary pressure can serve as both an opposing or driving force for fluid transport and is a significant property for research and industrial purposes namely microfluidic design and. Porous media fluid transport and pore structure 1st edition. A new capillary and thin film flow model for predicting the. In the interior of soils, tiny pores which are connected with each other fulfill the precondition of capillary rise 7, 8. In section 1, the washburn equation in porous media is derived and the conditions for its validity are stated. Background and problem description rigorous treatment of transport in porous media must begin with diffusion theory applied at the pore scale. The presented model constitutes a reduction of the conventional navierstokescahnhilliard phasefield model, valid in situations where interest is restricted to dynamical and equilibrium behavior in an aggregated sense, rather than a precise description of microscale flow phenomena. The nonuniform capillary converges and diverges at an angle of orientation.
Jan 27, 2009 we examine the effects of capillarity and gravity in a model of onedimensional imbibition of an incompressible liquid into a deformable porous material. An artificial neural network ann was used to analyse the capillary rise in porous media. The definition follows from rigorously averaged microscopic pressures in two fluid phases. This equation describes the local saturation under the influence of saturation and pressure gradients and is parametrized by the water release curve and the saturationdependent hydraulic conductivity, both are measured. So without actually modelling the structure, it is not possible to find the liquid rise. A multiscale diffuseinterface model for twophase flow in. Surface curvature in a fluid gives rise to an additional socalled. Capillary displacement and percolation in porous media. Extraction of poremorphology and capillary pressure. The relationship between capillary pressure, surface tension. A new macroscopic description of capillary transport of liquid and gas in porous materials is presented within the framework of multiphase continuum mechanics. If capillary rise of a wetting liquid in a homogeneous irregular packing of rotund particles is considered, however, it seems that none of the existing porespace models is capable of explaining. The usual macroscopic equations of motion for twophase immiscible displacement in porous media are known to be physically incomplete because they do not contain the surface tension and surface areas governing capillary phenomena.
Vadose zone journal original research a macroscopic. The main problem in this area is to change the scale of the description of the. Analytical, computational and experimental studies of capillary. The good image contrast achieved is represented by the relatively well separated peaks in the histogram plot fig. Capillary pinning and blunting of immiscible gravity currents. Request pdf a new equation for macroscopic description of capillary rise in porous media capillary rise in porous media is frequently modeled using the washburn equation. The anomalous dynamics of capillary rise in a porous medium discovered experimentally more than a decade ago delker et al. A lambert w functional form is proposed to describe the invasion dynamics in a single capillary tube, that predicts both earlytime washburntype behavior. Recent accurate measurements of advancing fronts clearly. Theoretical analysis of capillary rise in porous media. A macroscopic description of multiphase flow in porous.
Spe 77540 relative permeability from capillary pressure. A new equation for macroscopic description of capillary rise in porous media. Starting with the quasistatic motions of two compressible fluids, with zero surface tension, it is possible to construct a complete system of equations in which all parameters are clearly defined by. The governing equation for draining foam or a soil variant termed the soil foam drainage equation sfde obviates the need for macroscopic. The soil foam drainage equation an alternative model for. Therefore, a more general system of macroscopic equations is derived here that incorporates the spatiotemporal. Macroscopic equations for flow in unsaturated porous media. Capillary flow wicking may also occur between closely spaced surfaces, such as within fine brushes and fine dry powders such as in thin layer chromatography. On the concept of macroscopic capillary pressure in two. We extend the classical model of gravity currents in porous media huppert and woods, 1995.
The analogy between the geometry and dynamics of wet foam drainage and gravity drainage of unsaturated porous media expands modeling capabilities for capillary flows and supplements the standard richards equation representation. Therefore a more general system of macroscopic equations is derived here which incorporates the spa. Oct 15, 2004 equation describes the movement of fluids in capillary tubes and corresponds to the green and ampt model of macroscopic water absorption in porous media, which assumes an abrupt interface between the wet and dry regions. Dec 29, 2004 it is possible experimentally to determine the capillary potential. Porous media system composed of subdomains 1 and 122 with contrasting material properties that give rise to discontinuous characteristic transport coefficients at the inter face f. On the anomalous dynamics of capillary rise in porous. Nonlocal interface dynamics and pattern formation in gravitydriven unsaturated flow through porous media luis cuetofelgueroso and ruben juanes.
The flow is directly influenced by phase interfaces, i. Dense chlorinated solvents in porous and fractured media, model experiments. A new equation for macroscopic description of capillary. In this work, we study the dynamics of capillary driven fluid invasion in three different settings including. Capillary action is one of the most common fluid mechanical effects explored in the field of microfluidics. It is of some interest to point out that the combination of eqs. Keywords unsaturated porous materials macroscopic description capillary transport 1 introduction the description and analysis of liquid and gas transport in unsaturated porous materials is very. A macroscopic theory for capillarity in porous media is presented, challenging the established view that capillary pressure and relative permeability are constitutive parameter functions. Prediction of time of capillary rise in porous media using. Macroscopic equations, volume averaging, moving porous media, volumeaveraged velocity, lattice boltzmann equation 1. Further, they investigated constitutive relations among these parameters. In classic theory of twophase flow of immiscible fluids in porous media, capillary pressure p c is defined as scheidegger, 1963, bear, 1972 1 p c p n. The dependence of the capillary rise on the system parameters is given by the leverett scaling h cap 1. The established macroscopic equations of motion for two phase immiscible displacement in porous media are known to be physically incomplete because they do not contain the surface tension and surface areas governing capillary phenomena.
We consider an approximate solution of the governing macroscopic equation, which retains these gradients, and derive new analytical formulae for the position of the advancing front, its. Starting from the description of liquid flow through individual pores, a macroscopic equation for flow of a liquid in a porous medium in the presence of a gas is derived. Journal of colloid and interface science 2004, 278 2, 404409. The macroscopic pressure difference between two immiscible, incompressible fluid phases flowing through homogeneous porous media is considered.
A new macroscopic description of capillary transport of liquid and gas in porous. The developed theory is based on considering the principal modes of motion of the menisci that collectively form the wetting front on the darcy scale. A new equation for macroscopic description of capillary rise in porous. Macroscopic capillarity without a constitutive capillary. We focus primarily on a capillary rise problem but also discuss a capillary gravitational drainage configuration in which capillary and gravity forces act in the same direction.
Some limitations of the washburn equation have been discussed in the. A new equation for macroscopic description of capillary rise. Calculation of capillary rise height of soils by swcc model. Capillary action is the physical phenomenon arising due to surface tension on the interface of immiscible media. The capillary pressure function in the present theory is not an input parameter but an outcome. The most common method to determine these two parameters is through measurement of the capillary pressure generated by a reference liquid i. Capillary rise of a liquid into a deformable porous material. Extent of capillary rise in sands and silts semantic scholar. Although twophase fluid flow in porous media has been an established research field for decades, its theoretical background is still incomplete. A new equation for macroscopic description of capillary rise in porous media by d. However, in the experiments, the advancing front exceeded the predicted equilibrium height. Next, we regard the essential features of capillary ows on the basis of the the generic problem of capillary action of a uid in a narrow tube. For the glass bead filled columns, early time data are well fitted by the washburn equation. Micromechanical approach to swelling behavior of capillary.
A new equation for macroscopic description of capillary rise in porous media d. Jurins law, or capillary rise, is the simplest analysis of capillary actionthe induced motion of liquids in small channels and states that the maximum height of a liquid in a capillary tube is inversely proportional to the tubes diameter. A multiscale di useinterface model for twophase flow in. We examine the effects of capillarity and gravity in a model of onedimensional imbibition of an incompressible liquid into a deformable porous material. Measurement of capillary radius and contact angle within. A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media marc lebeau1 and jean. Lago and araujo 44 used this method a vertical capillary tube to obtain the capillary rise height in porous media. Pore geometry control of apparent wetting in porous media. For soils, we regard it as homogenous porous media, which means the pores inside are similar in diameter and the dispersion is small. Controlling macroscopic phase separation of aqueous twophase polymer systems in porous media david y.
Dynamic effects in capillary pressuresaturations relationships for twophase flow in 3d porous media. The derivation for capillary rise in a porous sample, such as the one represented in fig. Purchase porous media fluid transport and pore structure 1st edition. Capillary processes in porous media energy state of water in porous media the water potential watersolid properties surface tension, wettability, capillarity wetting of rough and heterogeneous surfaces capillarity in angular pores dynamics of capillary rise the role of inertia in capillary processes oscillations. Prediction of time of capillary rise in porous media using artificial. The capillary number for transition from the quasi. On the pore scale the standard equations for macroscopic.
Wetting experiments were performed with fifteen liquids and fifteer, different powders. Spe 77540 relative permeability from capillary pressure 3 so that, strictly speaking, pc is not a function of state. Influence of the dynamic contact angle on the characterization of porous media. The smaller the capillary, the higher the water can rise in it, as you see below. Macroscopic capillarity and hysteresis for flow in porous. Capillary rise in porous media is frequently modeled using the washburn equation. Lockington d, parlange jy 2004 a new equation for macroscopic description of capillary rise in porous media. Simulation results of drainage on a microct image of a carbonate rock are shown.
Enhancing wicking microflows in metallic foams springerlink. The washburn equation, also called the lucaswashburn equation, is based on the hagenpoiseuille equation and its assumptions, and has been applied to model capillarydriven flows in porous media, including paper. The motion is represented as a stepwise monte carlo process on a finite twodimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. In this paper we consider a multiscale phasefield model for capillaritydriven flows in porous media. The macroscopic flow of multiple fluid phases in porous media, for example soil, is often described by richards equation.
What is macroscopic interfacial tension in porous media. Washburn equation was derived originally for a liquid rising in a cylindrical capillary tube by the effects of capillary forces. Recent accurate measurements of advancing fronts clearly illustrate its failure to describe the phenomenon in the long term. For the three fluid constituents of the medium, balance equations of mass. Already 65 years ago leverett introduced the capillary pressuresaturation relation p c. However, the complexity in the morphology of the sample makes it difficult to. The movement of water through capillaries depends on how hydrophilic, or wettable, a surface is and on how small the capillary is. Commonly, capillary phenomena occur in liquid media and are brought about by the curvature of their surface that is adjacent to another liquid, gas, or its own vapor. Konrad 2010, a new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media, water resour. It is assumed that unsaturated porous material form a macroscopic continuum composed of three constituents. A new capillary and thin film flow model for predicting. Recent accurate measurements of the advancing front clearly illustrate the failure of the ga model to describe the phenomenon in the long time.
A new approach is proposed, similar to that used in rational thermodynamics, based on entropy inequality analysis and the lagrange multipliers method. Capillary rise in porous media is frequently described using the greenampt model the washburn equation. Introduction the motion of permeable particles in a. Nonlocal interface dynamics and pattern formation in. Recent accurate measurements of advancing fronts clearly illustrate its failure to describe the phenomenon in the. This equation describes the local saturation under the influence of saturation and pressure gradients and is parametrized by the water release curve and the saturationdependent hydraulic conductivity, both are measured experimentally. Macroscopic description of capillary transport of liquid. If a porous material has one end sitting in water, the pores, or capillaries, will begin filling with water. Pce penetration into unsaturated porous medium if small amount of napl is spilled to the ground top figure, it will soak into the soil by capillary action and due to gravity. Using the capillary bundle model, we gain a deterministic representation of a porous medium, in which different water contents are established by assessing. Schematic diagram of a capillary rise experiment in a porous sample. The wetted region is presumed to be completely saturated. Parlange journal of colloid and interface science 2004 278 2, 404409. It is assumed that unsaturated porous material form a macroscopic.
Controlling macroscopic phase separation of aqueous twophase. A macroscopic description of a twophase flow in a porous medium is given by writing, firstly, mass and momentumbalance equations and, secondly, phenomenological equations derived from the theory of irreversible thermodynamic processes. Capillary rise experiments were performed in columns filled with glass beads and berea sandstones, using visual methods to register the advance of the water front. The capillary pressure curve consists of three branches. The impact of porescale flow regimes on upscaling of. In fluid statics, capillary pressure is the pressure between two immiscible fluids in a thin tube see capillary action, resulting from the interactions of forces between the fluids and solid walls of the tube.
In this work, a clear and rigorous definition of the macroscopic capillary pressure is proposed. Introduction to the physics of water in porous materials. Macroscopic description of capillary transport of liquid and. Equation describes the movement of fluids in capillary tubes and corresponds to the green and ampt model of macroscopic water absorption in porous media, which assumes an abrupt interface between the wet and dry regions.
Macroscopic description of capillary transport of liquid and gas in. The impact of pore microstructure jinyu tang1, michiel smit1, sebastien vincent. Imbibition, porous media, lucaswashburns equation, artificial neural network. But for my case modelling of copper foam is a very difficult only option i have is finding the properties of porous media and applying to the porous region. A molecular dynamics simulation of capillary imbibition. The equation is also extended for application to a porous sample supported by a plug, and a velocitydependent capillary pressure. A macroscopic description of multiphase flow in porous media involving spacetime evolution of fluidfluid interface journal, december 1987. Capillary transport processes in porous materials flow3d. Apr 20, 2006 we consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate.
Although the capillary rise phenomenon does hold for very small diameter capillaries, it is not clear that it can scale down to nanometer or molecularly sized capillaries. A new definition of the macroscopic capillary pressure is presented. Macroscopic equations of motion for twophase flow in. The total capillary gasliquid area in each cross section can be expressed as follows. By including these pore scale phenomena in a continuum description of fluid transport in. Despite its wellknown limitations, this capillary pressure function has, in various disguises, remained the cornerstone of the theory of two phase. The stress description presented in murad and moyne 7 was extended later on to include micropores and partially saturated conditions in mainka, murad, moyne, and lima. Rossen1 1department of geoscience and engineering, delft university of technology, delft, netherlands, 2shell global solutions international, rijswijk, netherlands abstract a new capillary number n. A multiscale perspective on capillary ow in porous media we consider the ow of a binaryuid con ned by a rigid porous solid. A new equation for macroscopic description of capillary rise in porous media lockington, d. In particular, while a universal definition of capillary pressure exists at the microscale, its upscaling to the macroscale is still rather vague. The capillary bundle approach is commonly used for understanding the relationship between unsaturated hydraulic properties and the geometry of porous media phases jury and horton, 2004. A macroscopic theory for capillarity in porous media is presented challenging the established. A new equation for macroscopic description of capillary rise in.