Modelling the capillary imbibition kinetics in sedimentary rocks: Role of petrographical features

The kinetics of capillary imbibition into sedimentary rocks has been measured experimentally and calculated with a model that has been described previously by Hammecker and colleagues (1993). The validity of this model has been discussed and compared to other models. Three limestones, two clean sandstones and three clayey sandstones have been studied. The capillary processes are discussed as a function of their petrography and the pore structures. The role of the grain surface, described by the specific surface area, has been especially studied. The influence of clay coating on detrital grains on capillary processes has been quantified.Nomenclature A weight increase rate by capillary imbibition (A=W/(S ₁t)) - B capillary rise rate (B=l/t) - l height of the capillary fringe or the meniscus over the free water level - g gravitational constant - L height of the geometrical elements - P _a pressure in the air - P _c capillary pressure - P _w pressure in the water - Q flow rate - N _l free porosity - Nt total porosity - r radius of the pore (meniscus) - r ₁ pore access radius (neck) - R pore radius (widening) - R _s particle radius - S specific surface area - S ₁ macroscopical area of the sample surface throughout which imbibition occurs - t time - W weight - z height - surface tension - dynamic viscosity - contact angle - density - _r true density     

Methodology to Assess the Effects of Rice Cultivation Under Flooded Conditions on van Genuchten’s Model Parameters and Pore Size Distribution

The effects of rice (Oryza sativa L.) cultivation under flooded conditions on soil’s physical-hydraulic properties were studied in this article, using a new methodology based on a combined analysis on soil water retention curve (WRC) and pore size distribution (PSD). WRC analysis was carried out through the changes of van Genuchten’s model parameters, the characteristics of WRC at the inflection point, and the specific water capacity curve. Analysis of PSD was performed on the volume changes of porosity fractions through a detailed pore size classification, while different pore size classifications based on their hydraulic and structural characteristics were also used. The methodology was applied using a small dataset obtained from fine-textured Entisol soils which were subjected to rice cultivation under flooded conditions in Axios River plain (Northern Greece). Measurements of WRC were obtained at four depths of the soil profile from two fields, before and after the growing season of rice. The analysis indicated that the van Genuchten’s model parameters (θ _s, a, and n) and the WRC characteristics at the inflection point (pressure head h _i , pore equivalent diameter D _i , and slope S _i ) significantly changed after the growing season following similar patterns, along the soil profile in both fields. The parameters θ _s, a, D _i , and S _i were decreased, while n and h _i were increased. The h _i and a were the most sensitive parameters, while the values of (h _i and 1/a) in each layer before and after the growing season for each field were linearly correlated and shifted to higher values because of compaction, indicating that it could be applied as a tool to evaluate the degree of soil compaction to similarly textured soils. The peaks of the specific water capacity curves were compressed (lower values of slope S _i ) and shifted to lower water potentials (h _i ) that corresponded to pores of equivalent diameter D _i between 2 and 6 μm. The soils had few structural pores (>9 μm) and low air-filled porosity (>30 μm) before the growing season, which presented accessory reduction after the growing season in both fields. Total porosity was reduced at the expense of structural porosity along the soil profile, while the pore size class of 5–3 μm was identified as the threshold where the smaller pores’ volume started to increase in all layers of both fields. The results indicated that the changes in the WRC and the PSD follow specific trends, which can be used in future studies to model temporal variability of soil’s physical-hydraulic properties.     

Some asymptotic results concerning the buckling of a spherical shell of arbitrary thickness

For a spherical shell of arbitrary thickness which is subjected to an external hydrostatic pressure, symmetrical buckling takes place at a value of μ₁ which depends on and the mode number, where A₁ and A₂ are the undeformed inner and outer radii, and μ₁ is the ratio of the deformed inner radius to the undeformed inner radius. In the large mode number limit, we find that the dependence of μ₁ on has a boundary layer structure: it is a constant over almost the entire region of and decreases sharply from this constant value to unity as tends to unity (the thin-shell limit). Simple asymptotic expressions for the bifurcation condition are obtained. The classical result for thin shells is recovered directly from the equations of finite elasticity, and an asymptotic critical neutral curve (which envelops the neutral curves corresponding to different mode numbers) is obtained.     

Flow Laws in Metal Foams: Compressibility and Pore Size Effects

The aim of our experimental work was to establish a simple relation between the flow parameters and the morphological parameters of metallic foam. We used foam samples made from different metals or alloys (Cu, Ni, Ni-Cr, etc) and of various thicknesses. Pore size ranged between 500 and 5000 μm. We measured the pressure profiles in foam samples using a specific experimental set-up of 12 pressure sensors distributed 1 cm apart along the main flow axis. The experimental loop made it possible to use indifferently water or air as working fluid. For the study of the gas (air) flow, velocities ranged roughly from 0 up to 20 m/s and for the liquid (water) flow, velocities ranged between 0 and 0.1 m/s. The measurements of the pressure gradients were performed systematically. We validated the Forchheimer flow model. The influence of the compressibility effects on permeability and inertia coefficient was emphasized. We demonstrated that the pore size Dp in itself is sufficient to describe flow laws in such high porosity material: K and β are respectively proportional to Dp² and Dp⁻¹.     

循环孔隙水压下混凝土常规三轴压缩损伤破坏特性分析

本文进行了孔隙水压不同循环次数(0次,10次,50次,100次和200次)以及不同应变速率下(10~(-5)/s,10~(-4)/s,10~(-3)/s和10~(-2)/s)混凝土常规三轴压缩试验,分析了混凝土峰值应变的变化规律、应力-应变曲线及损伤特性。结果表明:相同循环次数孔隙水压下,峰值应变随应变速率增加,整体呈现出增加的趋势;而相同应变速率下,峰值应变随孔隙水压循环次数的变化规律并不明显;在中低应变速率(10~(-5)~10~(-3)/s)下,混凝土的损伤变化受孔隙水压循环次数影响较大;当循环次数达到200次时,孔隙水压作用对混凝土产生较大的损伤。通过对循环孔隙水作用下混凝土动态损伤破坏机理的分析可知:混凝土的破坏过程实际上是内部裂纹不断形成、扩展、贯通,材料损伤不断产生、累积的过程;当损伤达到一定程度,混凝土发生宏观破坏,失去承载力。     

The double packer permeameter with narrow packers. Analytical solution for non steady flow

The hydraulic diffusivity and permeability of a rock mass can be measured with a double packer permeameter, placed in a long bore hole. If the flow is steady, the relation between supply pressure and flux is a measure of the permeability whereas if the flow is transient, the relation is a measure of both the hydraulic diffusivity of the rock and the elasticity of the equipment. If the equipment is effectively rigid and the distance between the packers is much bigger than the size of the packers, an approximate closed solution is characterized bytwo parameters: the distance between the packers divided by the radius of the hole and the width of the packers divided by the radius of the hole. The flow is fully 3-D if the hydraulic diffusion distance far exceeds the distance between the packers. If the elasticity of the equipment is not negligible, the transient solution is characterized byone parameter including the elasticity coefficient of the equipment the conductivity of the rock and the radius of the hole. The relations between pressure and flux is given as a closed formula, suitable for engineering practice.     

On the Helmholtz-Boltzmann thermodynamics of mechanical systems

In an 1884 paper, Boltzmann showed that for a one-dimensional mechanical system with a convex potential energy that depends on a parameter V, it is possible to define a temperature T, pressure p, and entropy S that satisfy the Gibbs relation TdS = de + p dV, where . In the paper we review the extension of the Boltzmann construction to general natural mechanical systems endowed with a fibration over the (possibly multidimensional) space of macroscopic parameters. Moreover, for certain discrete mechanical systems with non-convex potential energies, which are used as models for phase transitions in solids, we compare the thermodynamic pressure p = p(e,V) introduced above with the quasi-static, macroscopic, stress-strain relation.Received: 20 February 2002, Accepted: 19 May 2003PACS: 5.20.y, 5.45.a, 5.70.ce Correspondence to: F. Cardin     

Pore structures and transport properties of sandstone

To quantitatively analyze the macroscopic properties of the flow in porous media by means of the continuum approach, detailed information (velocity and pressure fields) on the microscopic scale is necessary. In this paper, the numerical solution for incompressible, Newtonian flow in a diverging-converging representative unit cell (RUC) is presented. A new solution procedure for the problem is introduced. A review of the accuracy of the computational method is given.Nomenclature A _ff ^* area of entrance and exit of RUC - A _fs ^* interfacial area between the fluid and solid phases - d throat diameter of RUC (m) - D pore diameter of RUC (m) - i, j unit vector for RUC - L ^* wave length of a unit cell - L _p pore length of RUC (m) - L _t throat length of RUC (m) - n unit outwardly directed vector for the fluid phase - p ^* fluid pressure - ^* cross-sectional mean pressure - _en ^* entrance cross-sectional mean pressure - Re _d Reynolds number - x ^*, r^* cylindrical coordinates - u ^*, v^* velocity - u _cl ^* centerline velocity - _d mean velocity at the throat of RUC (m/s) - _D mean velocity at the large segment of RUC (m/s) Greek viscosity coefficient (Ns/m²) - _p excess momentum loss factor defined in (4.1) - fluid density (kg/m³) - ^* stream function - ^* vorticity - dimensionless circulation defined in (2.7) Symbols - the mean value - * dimensionless quantities     

Dynamics of coaxial torsional resonators consisting or segments with different radii,material properties and surrounding media

High-Q torsional resonators constitute the most sensitive transducers for high frequency dynamic viscoelastic measurements of dilute polymer solutions. Most such resonators described in the literature are segmented. Because of the need for torque and torsional displacement transducers the Q-value of the individual segments most often differ, but normally all segments have the same radius.A detailed analysis of the dynamics of such resonators when both the radii, material properties and surrounding media may be different for each segment, is presented. For resonators where all segment lengths equal an integer multiple of a quarter of the torsional wavelength, we find that the Q-value of the resonator as a whole is mainly determined by the Q-value of the segment with the smallest radius. We further find that reduction of the radius of the segment surrounded by polymer solution results in a stronger mechanical coupling between the resonator as a whole and the polymer solution. These findings suggest that the segment radii are important optimization parameters of segmented torsional resonators used to measure the high frequency dynamic viscoelastic properties of e.g. polymer solutions.     

Hydraulic radius and transport in reconstructed model three-dimensional porous media

Methods for reconstructing three-dimensional porous media from two-dimensional cross sections are evaluated in terms of the transport properties of the reconstructed systems. Two-dimensional slices are selected at random from model three-dimensional microstructures, based on penetrable spheres, and processed to create a reconstructed representation of the original system. Permeability, conductivity, and a critial pore diameter are computed for the original and reconstructed microstructures to assess the validity of the reconstruction technique. A surface curvature algorithm is utilized to further modify the reconstructed systems by matching the hydraulic radius of the reconstructed three-dimensional system to that of the two-dimensional slice. While having only minor effects on conductivity, this modification significantly improves the agreement between permeabilities and critical diameters of the original and reconstructed systems for porosities in the range of 25–40%. For lower porosities, critical pore diameter is unaffected by the curvature modification so that little improvement between original and reconstructed permeabilities is obtained by matching hydraulic radii.     

Two-Phase Flow Properties Prediction from Small-Scale Data Using Pore-Network Modeling

The objective of this work is to evaluate the prediction accuracy of network modeling to calculate transport properties of porous media based on the interpretation of mercury invasion capillary pressure curves only. A pore-scale modeling approach is used to model the multi-phase flow and calculate gas/oil relative permeability curves. The characteristics of the 3-D pore-network are defined with the requirement that the network model satisfactorily reproduces the capillary pressure curve (P_c curve), the porosity and the permeability. A sensitivity study on the effect of the input parameters on the prediction of capillary pressure and gas/oil relative permeability curves is presented. The simulations show that different input parameters can lead to similarly good reproductions of the experimental P_c, although the predicted relative permeabilities K_r are somewhat widespread. This means that the information derived from a mercury invasion P_c curve is not sufficient to characterize transport properties of a porous medium. The simulations indicate that more quantitative information on the wall roughness and the node/bond aspect ratio would be necessary to better constrain the problem. There is also evidence that in narrow pore size distributions pore body volume and pore throat radius are correlated while in broad pore size distributions they would be uncorrelated.     

Turbulent forced convection in a helically coiled square duct with one uniform temperature and three adiabatic walls

In this study, effects of geometrical parameters on the average convection heat transfer characteristics in helical square ducts were investigated both experimentally and numerically. The inner wall of the helical square duct was uniformly temperatured, and the top, bottom, and outer walls were adiabatic. The Renormalization Group (RNG) k–ε turbulence model was used to simulate turbulent flow and heat transfer. The governing equations were solved by a finite volume method. Numerical results were found to be in good agreement with the presented experimental data. The new correlation was proposed for the average heat transfer coefficient on the inner wall of the helical square duct. The results showed that the ratio of pitch to coil radius b/R has no obvious effect on the inner wall convective heat transfer coefficient but the ratio of hydraulic radius to coil radius a/R has considerable effect.     

Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space. The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation employing finite elements or analogous numerical procedures.     

Calculating parameters for infiltration equations from soil hydraulic functions

Simple equations for predicting infiltration of water into soil are valuable both for hydrological application and for investigating soil hydraulic properties. Their value is greatly enhanced if they involve parameters that can be related to more basic soil hydraulic properties. In this paper we extend infiltration equations developed previously for positive surface heads to negative heads. The equations are then used to calculate infiltration into a sand and a clay for a range of initial and surface conditions. Results show errors of less than three percent compared with accurate numerical solutions. Analytical approximations to parameters in the equations are developed for a Brooks and Corey power law hydraulic conductivity-water content relation combined with either a Brooks and Corey or a van Genuchten water retention function. These are compared with accurate numerical values for a range of hydraulic parameters encompassing the majority of soil types and a range of initial and boundary conditions. The approximations are excellent for a wide range of soil parameters.An important attribute of the infiltration equations is their use of dimensionless parameters that can be calculated from normalised water retention and hydraulic conductivity functions. These normalised functions involve only parameters that it may be possible to estimate from surrogate data such as soil particle size distribution. Application of the equations for predicting infiltration, or their use in inferring hydraulic properties, then involves only simple scaling parameters.     

Numerical simulation of cooling-induced convective currents on a littoral slope

The transient development of cooling-induced flow in a triangular domain filled with water is studied by means of numerical simulation as a model for flow developing in the littoral region of lakes or coasts. The domain is fitted in polar co-ordinates; solutions are obtained for different values of the independent parameters of the model, which are the Rayleigh number (Ra), the Prandtl number (Pr) and the slope of the domain (S). Within the ranges examined, as Ra is increased, different regimes of the developing flow are observed; these are found qualitatively to be insignificantly influenced by changes in S, whereas the flow is found to be quantitatively insensitive to Pr for high enough values of Pr. Several interesting features of the flow are depicted and integral values useful in the analysis of flow in lakes are extracted.     

Natural convection from a vertical cylinder in a thermally stratified porous medium

The problem of non-Darcy natural convection adjacent to a vertical cylinder embedded in a thermally stratified porous medium has been analyzed. Nonsimilarity solutions are obtained for the case that the ambient temperature increases linearly with height of the cylinder. A generalized flow model was used in the present study to include the effects of the macroscopic viscous term and the microscopic inertial force. Also, the thermal dispersion effect is considered in the energy equation. Thus, the main aim of this work is to examine the effects of thermal stratification and non-Darcy flow phenomena on the free convection flow and heat transfer characteristics. It was found that the present problem depends on six parameters, namely, the local thermal stratification parameter ξ, the boundary effect parameter Bp, the modified Grashof number Gr^*, wall temperature exponent m, the curvature parameter ω, and the modified Rayleigh number based on pore diameter Ra _d . The impacts of these governing parameters on the local heat transfer parameter are discussed in great detail. Also, representative velocity and temperature profiles are presented at selected values of the thermal stratification parameter. In general, the local heat transfer parameter is increased with increasing the values of m, ω, and Ra _d ; while it is decreased with increasing the values of ξ, Bp, and Gr^*. Received on 19 May 1998     

The radii inversion problem associated with the Hg capillary pressure experiment

Today's practice of interpreting Hg capillary pressure curves — a widespread method in porosimetry — is generally unsatisfactory. This has already been demonstrated by Fatt. First, the saturation branch of such a curve is interpreted using the concept of a pore space model in which essential features of a network structure are disregarded. Second, the data provided by the desaturation branch are not used. Distributions of radii of capillaries within porous materials derived by this technique are usually incorrect in that the frequencies of occurrence of the greater radii turn out too small, those of the smaller radii too large.We present a more reliable approach which constrains radii frequency ranges for the Hg saturated pore space and for both the part of the pore space that desaturates and the part that traps mercury when Hg pressure is released. The pore space may be of an arbitrary geometrical structure, the radii distribution may be continuous. Also, the Hg desaturation may enable one to distinguish experimentally between structural and contact angle hysteresis.List of Symbols and Abbreviations a actual cross-section of a capillary - a hydrostatic equivalent circular crosssection of a capillary - c circumference ofa - des desaturation process, index - i index denoting radii, radii numbers, and radii frequencies - j number of saturated capillaries - k upper boundary fori - k, k ₀ (mean) hydrostatic shape factor of the capillary cross-section - T, l mean capillary lengths - n _i number of radii of sizer _i - p _s ^c capillary pressure of capillaries with radiusr _s - (-) p _s ^c difference in capillary pressure between capillaries with radiusr _s ,r _s-1 - (Hg/vacuum) path probability - r _i capillary radius - r radius ofa - s index denoting the state of the capillary pressure experiment - sat satuation process, index - BCT bundle of capillary tubes model - CTM chain-type model - D dimensional - Hg mercury - M number of meshes within a network - M _s measured value - N number of capillaries (radii) in a network - N _b number of border capillaries of a network in direct contact with the external Hg reservoir - N _i number of capillaries inside a network without direct contact with the external Hg reservoir - NS network structure - P point of intersection (indicated by sat or des) - ^k k-D space of representation - RD radii distribution - RP radii placement - S number of nodes (sites) within a network - SHN single hexagonal network model - SMN square mesh network model - V, measured (expected) network volume during saturation or desaturation, respectively (indicated by Hg, sat, 2, or des) - V _por pore volume of a network or a porous material - network constant - network constant (mesh density) - network constant (node density) - surface tension of mercury - contact angle of mercury - , (expected) saturation/desaturation belonging toV ( ) - 2 index denoting vacuum