Porous Continuum Research Articles

Convergent flow observed in a laboratory‐scale unsaturated fracture system1

An experiment was designed to observe water flow in a simple analogue fractured rock network. The fracture network was modeled by the gaps between an array of limestone blocks, whose permutations generated different realizations of the network. Three out of five of these realizations displayed flow convergence, in contrast to that predicted by porous continuum models. Such convergence occurs because fracture intersections act as capillary barriers that integrate or focus flow within the network.

Estimating the hydraulic functions of dual‐permeability models from bulk soil data

Predictions of nonequilibrium preferential flow and transport are limited as long as the fracture and the matrix pore system cannot be represented by separate hydraulic functions. Previous approaches to describe hydraulic properties of soils with bimodal or multimodal pore size distributions are still restricted to soils with a single porous continuum. The objective of this study was to develop and test a fitting procedure for estimating retention and conductivity functions for dual‐permeability models using bulk soil data. The estimation is based on a set of van Genuchten‐Mualem functions for the two pore systems. A stepwise procedure is carried out that (1) assumes 3 out of 11 parameters to be known, (2) determines initial values for the remaining eight parameters to be fitted, and (3) fits the set of hydraulic functions simultaneously to θ(h) and K(h) data to obtain the parameter estimates. This procedure was evaluated using a synthetic data set, and it was applied to laboratory‐measured θ(h) and K(h) data from a loamy Dystric Gleysol soil. The results show that the effect of a fracture pore system on the unsaturated hydraulic conductivity function might be underestimated if only observations of water retention are considered. The fitted dual‐continuum hydraulic conductivity functions match the unsaturated soil's conductivity data better than a bimodal single‐domain approach that predicts hydraulic conductivity based on fitted water retention functions. Moreover, the dual‐continuum functions provide hydraulic parameters for dual‐permeability preferential flow models.

A constitutive equation for viscoelastic porous gels during syneresis and drying

In this investigation, a constitutive equation was developed for viscoelastic porous gels during drying process based on the mechanism of shrinkage. One important application for the analysis is to predict the residual stress developed during the drying of gels in the sol–gel fabrication process. The gels can be modeled as porous continuum material, where the solid phases are viscoelastic material, and the pores are filled with some kinds of liquid. During drying, the large capillary forces generated in the tiny (∼3–10 nm) pores serve as the main driving force for the shrinkage of gels, whereas the applied forces and hydrostatic pressure from the liquid and also contribute to the deformation process. Finally, the tensile residual stress in the thin film that developed in the drying was predicted based on the constitutive equation. It is hopeful that the developed viscoelastic constitutive equation can be adapted into a FEM code to predict the residual stress and optimum design the fabrication parameters without cracking of the specimen.

Models for computing geomechanical constants of double‐porosity materials from the constituents' properties

In previous work, phenomenological equations for the poroelastic behavior of a double‐porosity dual‐permeability medium (i.e., low‐permeability storage porosity and high‐permeability transport porosity both present in the same three‐dimensional (3‐D) volume) were formulated, and the coefficients in these equations were identified. In the present work, models are given instead that allow all of these coefficients to be determined from the underlying constituents' properties. Two different models are provided for the six geomechanical constants in the isotropic theory. In one model the low‐permeability storage porosity is assumed to be a mixture of two porous materials in nonwelded or partially welded contact, and the high‐permeability joint porosity develops as a misfit porosity at the interface between the two partially welded constituents. In this model the joint phase is 100% fluid. In the other model the low‐permeability storage porosity is modeled as a single uniform porous material, while the high‐permeability joint porosity is modeled as a second distinct porous material in welded contact with the first. In this model the joint phase is a porous continuum possessing a skeletal framework representing the asperities and gouge material that are present in real joints. The complete set of formulas needed for forward modeling the geomechanical constants is obtained for both scenarios. Examples are evaluated with these formulas, and comparisons are made to previous results on double‐porosity systems analysis.

Three‐dimensional numerical inversion of pneumatic cross‐hole tests in unsaturated fractured tuff: 1. Methodology and borehole effects

We describe a three‐dimensional numerical inverse model for the interpretation of cross‐hole pneumatic tests in unsaturated fractured tuffs at the Apache Leap Research Site (ALRS) near Superior, Arizona. The model combines a finite volume flow simulator, FEHM, an automatic mesh generator, X3D, a parallelized version of an automatic parameter estimator, PEST, and a geostatistical package, GSTAT. The tests are simulated by considering single‐phase airflow through a porous continuum, which represents primarily interconnected fractures at the site. The simulator solves the airflow equations in their original nonlinear form and accounts directly for the ability of all packed‐off borehole intervals to store and conduct air through the system. Computations are performed in parallel on a supercomputer using 32 processors. We analyze pneumatic cross‐hole test data, previously conducted by our group at ALRS, in two ways: (1) by considering pressure records from individual borehole monitoring intervals one at a time, while treating the rock as being spatially uniform, and (2) by considering pressure records from multiple tests and borehole monitoring intervals simultaneously, while treating the rock as being randomly heterogeneous. The first approach yields a series of equivalent air permeabilities and air‐filled porosities for the rock volume being tested, having length scales of the order of meters to tens of meters. The second approach yields a high‐resolution geostatistical estimate of how air permeability and air‐filled porosity, defined on grid blocks having a length scale of 1 m, vary spatially throughout the tested rock volume. It amounts to three‐dimensional pneumatic “tomography” or stochastic imaging of the rock, a concept originally proposed by one of us in 1987. The first paper of this two‐part series describes the field data, the model, and the effect of boreholes on pressure propagation through the rock. The second paper implements our approach on selected cross‐hole test data from ALRS.

Type curve interpretation of a cross‐hole pneumatic injection test in unsaturated fractured tuff

Cross‐hole pneumatic injection tests have been conducted in 16 vertical and inclined boreholes in unsaturated fractured tuffs at the Apache Leap Research Site (ALRS) near Superior, Arizona. Their purpose was to characterize the bulk pneumatic properties and connectivity of fractures at the site on scales ranging from meters to several tens of meters. We describe the design, conduct, and type curve interpretation of one of these tests. Our cross‐hole type curves are modified after Hsieh and Neuman [1985] to consider single‐phase airflow and extended to consider the effects of storage and skin in monitoring intervals. Cross‐hole type curves of pressure derivatives and recovery are included for improved pneumatic characterization of the site. We analyze recorded pressures in each monitoring interval separately from those in other intervals while treating the fractured rock as a uniform, isotropic porous continuum. Each record yields an equivalent directional air permeability and air‐filled porosity for fractures that connect the corresponding monitoring and injection intervals. Both parameters are found to vary considerably from one monitoring interval to another, reflecting the nonuniform nature of pneumatic rock properties at the ALRS. The geometric mean of these equivalent permeabilities is found to be larger by a factor of 50 than that obtained from single‐hole pneumatic injection tests on a nominal scale of 1–3 m (single‐hole tests yield only limited information about porosities, which therefore cannot be meaningfully compared with cross‐hole results). Our results find support in numerical inverse modeling of cross‐hole tests at the site by Vesselinov [2000; see also Illman et al., 1998; Vesselinov et al, 2000]. Vesselinov [2000] has further demonstrated [see also Chen et al., 2000] that a similar scale effect is exhibited by fracture porosity at the ALRS and that both scale effects disappear when cross‐hole tests at the site are interpreted by means of a numerical inverse model, which resolves heterogeneities down to a scale of 1 m. When considered jointly with these and other studies of the site, our analysis implies that the pneumatic pressure behavior of unsaturated fractured tuffs at the ALRS can be described quite accurately by means of linearized single‐phase airflow equations; this behavior can be interpreted by treating the rock as a continuum on scales ranging from meters to tens of meters: the continuum is representative primarily of interconnected fractures; as these fractures are filled primarily with air, their pneumatic permeabilities and porosities are close to the bulk intrinsic properties of fractures at the site: these intrinsic properties vary randomly with location and direction across the ALRS, and they depend strongly on the scale at which they are determined.

Type‐Curve Interpretation of Multirate Single‐Hole Pneumatic Injection Tests in Unsaturated Fractured Rock

AbstractMore than 270 multirate single‐hole pneumatic injection tests were conducted by Guzman et al. (1996) within six shallow vertical and inclined boreholes in unsaturated fractured tuff at the Apache Leap Research Site (ALRS) near Superor, Arizona. The authors used steady‐state formulae to obtain air permeability values for the rock based solely on late pressure data from each test. We developed pressure and pressure‐derivative type‐curves for the interpretation of transient data from these tests, which account for storage in the test interval due to air compressibility and skin effect due to damage to the surrounding rock. We applied our type‐curves to pressure data from more than 40 of the tests to obtain information about air permeability, skin factor, phenomenology, and dimensionality of the flow regime on a nominal scale of 1 m in the immediate vicinity of each test interval. Our air permeabilities agreed well with those determined previously by means of steady‐ state formulae but correlated poorly with fracture density data. Nonlinear effects due to air compressibility were pronounced, but skin effect was generally small or nonexistent. Nonlinear two‐phase flow and inertia were evident but did not have a significant impact on type‐curve interpretation of the test data. Most of these data fit a continuum model of spherical airflow around the test interval, but a few fit a continuum model of radial flow or suggest the dominance of a single fracture. We take this to imply that fractures at the site are pneumatically well connected and can, for the most part, be treated as a porous continuum when analyzing airflow on scales of 1 m or more. This is supported by the finding that 1 m scale air permeability data from single‐hole tests at the ALRS are amenable to geostatistical analysis (Chen et al. 2000), which views them as a sample from a random field defined over a continuum. That the same holds true on larger scale is indicated by our ability to interpret cross‐hole pneumatic injection tests at the site by means of type‐curves and inverse methods (Illman et al. 1998), which treat the entire rock mass as a porous continuum.

Experimental Study of Bimolecular Reaction Kinetics in Porous Media

The representation of multispecies chemical transformations into models for porous media flow and transport is investigated. Since the rate of transformation is dependent on the actual concentrations of the reactants inside the pore-spaces (rather than the averaged concentration that is amenable to a porous continuum model) such models may show a lesser or greater rate of transformation than is actually occurring, depending on the pore-scale correlation of the reactant concentrations. To examine this experimentally, a bimolecular reaction (A + B → Product) in a saturated porous media flow in a column was studied using a spectrophotometer. The second-order rate constant of the reaction was independently determined in well-mixed batch tests. In the reactive transport experiments, the porous medium column was initially saturated with one reactant, the other reactant was injected at one longitudinal end of the column, and the effluent concentrations were measured. It is documented that a reactive transport model that does not account for the pore-scale segregation of reactants can substantially overpredict the product concentration.

A Mixture Theory for Microstructured Porous Media

AbstractA porous continuum with very large lacunae, partially saturated by a fluid inclusion, as described by means of a new immiscible mixture which consists of a medium with ellipsoidal microstructure and of a classical fluid.

Inclusion of microscopic rotation of powder particles during compaction in finite element method using Cosserat continuum theory

The effect of microscopic rotation of powder particles in compaction is included in the rigid-plastic finite element method on the basis of the Cosserat continuum theory. In the Cosserat continuum theory, couple stress induced from the microscopic rotation is introduced, and the equilibrium equations of moment for the couple stress are solved simultaneously with those of force. A yield criterion for the Cosserat porous continuum is proposed by taking the effect of the couple stress into consideration, and constitutive equations for the rigid-plastic porous material are derived from the yield criterion on the basis of the associated flow rule. The equilibrium equations of force and moment for the Cosserat continuum are formulated by the use of the Galerkin method. The effect of microscopic rotation of powder particles in plane-strain closed-die compaction is examined. In addition, the calculated result is compared with that for the conventional continuum without the microscopic rotation. © 1998 John Wiley & Sons, Ltd.

Creep and plasticity due to chemo-mechanical couplings

The paper explores the theory of reactive porous media for the modelling of creep and plasticity due to chemo-mechanical couplings at the macro-level of material description. The formulation is based upon thermodynamics of open porous media composed of a skeleton and several fluid phases saturating the porous space. This theoretical framework allows to introduce the kinetics of a chemical reaction directly at the macro-level of material description. In turn, it is used to model creep due to chemo-mechanical couplings within a closed reactive porous continuum, as well as ageing creep due to two chemical reactions, one associated with the apparent creep phenomenon, the other with the apparent ageing phenomenon. Furthermore, it is shown how the modelling can be extended to account for plastic (i.e. permanent) phenomena, including hardening/softening and damage phenomena, coupled with a chemical reaction (chemical hardening).

Damage and localisation in elastic materials with voids

An extension of the theory of elastic material with voids to the case where the material undergoes an irreversible void growth is presented. The particularity of this theory is that the continuum is described by two kinematic variables: the displacements and the variation of the volume fraction of material in the porous continuum. Motion is controlled by two governing equations, the classical one involving the displacement or stresses and another one that involves the other kinematic variable, similar to the governing equation in heat conduction problems. The degradation of the elastic moduli is described in the model by a damage scalar variable. A simplified model where the damage variable is proportional to the irreversible variation of volume fraction of material is discussed. From the governing equations, it is deduced that the equation which governs the growth of damage involves the second gradient of damage and a material parameter which plays the role of an internal length according to the analysis of strain localisation. The finite element implementation of the theory is briefly presented. The two variables are discretised separately and the form of the equations to be solved is similar to those obtained in coupled thermoelasticity. One dimensional finite element results of strain localisation show that a proper convergence upon mesh refinement is obtained. The equation which governs the irreversible variation of volume fraction (or the damage growth) acts as a localisation limiter.

Damage and localisation in elastic materials with voids

An extension of the theory of elastic material with voids to the case where the material undergoes an irreversible void growth is presented. The particularity of this theory is that the continuum is described by two kinematic variables: the displacements and the variation of the volume fraction of material in the porous continuum. Motion is controlled by two governing equations, the classical one involving the displacement or stresses and another one that involves the other kinematic variable, similar to the governing equation in heat conduction problems. The degradation of the elastic moduli is described in the model by a damage scalar variable. A simplified model where the damage variable is proportional to the irreversible variation of volume fraction of material is discussed. From the governing equations, it is deduced that the equation which governs the growth of damage involves the second gradient of damage and a material parameter which plays the role of an internal length according to the analysis of strain localisation. The finite element implementation of the theory is briefly presented. The two variables are discretised separately and the form of the equations to be solved is similar to those obtained in coupled thermoelasticity. One dimensional finite element results of strain localisation show that a proper convergence upon mesh refinement is obtained. The equation which governs the irreversible variation of volume fraction (or the damage growth) acts as a localisation limiter.

A Study on Flow Phenomena in Mixed Lubrication Regime by Porous Medium Model

A porous medium model is basically a macroscopic description of the fluid behavior and is considered more appropriate to the mixed lubrication regime. The present work has successfully combined the flow factor method, porous medium model and continuum percolation theory to investigate the flow phenomena in the mixed lubrication regime. Not only the lubricant flow can be solved but also the anisotropic percolation threshold can be obtained. Instead of the flow factor which is singular near h = 0, the lubricant flux itself is considered. A new definition of the pressure flow factor which is more appropriate to the mixed lubrication realm is suggested.

Numerical simulation of a fluid-saturated soil under a dynamically loaded pile

The behaviour of the soil under a dynamically loaded pile toe is studied. The soil is modelled as a fluid saturated porous continuum. The constitutive behaviour of the solid skeleton is described by the elasto-plastic model of Drucker-Prager. The wave propagation is simulated with a dynamical finite-element program.

The influence of porosity on the deformation and fracture of alloys

The influence of porosity on the deformation and fracture behavior of two alloys, powder-fabricated Ti and Ti-6Al-4V, with differing levels of matrix strain hardening has been examined both experimentally and analytically. A large strain elastoplastic finite element model based on a regular array of equal-sized spherical voids is used to predict bulk porosity effects; the analysis is in good agreement with the experimentally observed rates of void growth but underestimates the degradation of strength with increasing porosity. The effects of porosity on a local scale, especially as regards fracture, are examined by a model of a porous continuum which contains imperfections whose magnitude depends upon the maximum porosity path within the continuum. At critical values of strain these imperfections cause localization of plastic flow. The predicted values for the strains at localization are in good agreement with measured fracture strains. The analysis thus explicitly recognizes that a primary effect of pores on fracture is to localize deformation into narrow regions of high porosity (“imperfections”) which are present even in random distributions of pre-existing pores and which are the sites of macrofracture initiation.

Structural and mechanical properties and filtration in elastic fissured and porous material

Within the framework of the model of two coexisting homogeneous porous continua the article examines the influence of the state of stress of a material and of the liquid pressure on the effective parameters of the continua, and a system of equations of filtration is obtained that differs from the known system.

Sublimation in a Porous Continuum Developed From Nonequilibrium Thermodynamics

The case of sublimation dehydration of a porous continuum is analyzed from the viewpoint of nonequilibrium thermodynamics. The general energy and mass transport equations are shown and the application of sublimation is illustrated.

A Study of the Smackover Limestone Formation and the Reservoir Behavior of Its Oil and Condensate Pools

Abstract Studies of reservoir behavior of 12 Smackover limestone oil and condensatepools are presented. Buckner, Midway, McKamie, Magnolia, Mt. Holly, Schuler(Reynolds) and Village are treated in considerable detail and subjected toanalysis with an electrical device for analyzing reservoir behavior. Resultsand comparisons of these analyses are presented. In addition to these reservoir-behavior studies, an attempt is made to assembletransmissibility and storage data on the Smackover limestone formation fromcores and logs taken in dry holes and producers. This together with thebehavior of individual reservoirs leads to a better understanding of theformation as a porous aquifer. The ultimate purpose of this type of study is to improve predictions concerningthe future behavior of these pools in such a system. Predictions are presentedfor several pools. Introduction In 1937 the Phillips Petroleum Co. drilled its J. D. Reynolds No. 1 to the Smackover limestone formation at Snow Hill, Arkansas. This well found oil inthe porous, permeable, upper part of the Smackover thereafter referred to asthe Reynolds lime. Since then, six oil pools and six condensate pools have beendiscovered in the Reynolds lime, and more than 75 dry holes have explored partsof this formation. By the first of 1943, 44 million barrels of oil and 83 billion cubic feet ofgas had been withdrawn from the various reservoirs. The original reservesestimated by the Arkansas Oil and Gas Commission were 342 million barrels ofoil and 1200 billion cubic feet of gas. So far all of the production has been from anticlinal structures, most of whichare asymmetric and have their axes roughly perpendicular to the dip of theformation top. This is indicated in Fig. 1, which shows the pools, dry holesdrilled, and contours on top of the Smackover. These pools are of particular interest for reservoir-behavior studies becausethe production and drilling has been controlled, and in most cases the recordsof reservoir data are complete. In addition to the excellent production andpressure information available, more than one half of the wells drilled werecored and all of them were electrically logged. All of the pools show evidenceof water drive. While this paper is in a sense a collection of reservoir-behavior studies forseveral different pools, there has been an attempt to go one step further andto show the importance of the formation as a porous continuum or a singleaquifer in which the withdrawals of fluids from any pool has an. importanteffect on the behavior of all of the pools. An effort will be made to show thatthe water drives of these pools are dependent upon a much larger area of theSmackover limestone than that occupied by the petroleum accumulations. T.P. 1728

Pressure Prediction for Oil Reservoirs

Abstract This paper presents the essentials of a mathematical method of studying thepressure beha vior of an oil reservoir as the fluids are withdrawn. Methods areshown whereby the behavior of a reservoir can be used to predict the futurerelationship between withdrawals and pressure. The oil reservoir is consideredas a small part of a large porous continuum, which contains for the most partwater. The expansion of this water resulting from a decrease of pressure in theoil zone causes a movement of water that is an important factor in therelationship between fluid withdrawals and reservoir pressure. Examples aregiven to illustrate the methods of pressure prediction for a circular reservoircontaining undersaturated crude, the pressure distribution over a field that has a uniform production rate, theeffect of a fault line near a field, the mutual interference of near-by fields, and the perturbing influence of a gas cap. Applications of the analysis towell-spacing problems are shown. Introduction The pressure-prediction methods to be described here are based upon theconception of an oil pool as a small part of a large porous continuum, containing for the most part water. The formation containing this water may bethought of as extending over an area of many square miles and having one ormore small bodies of oil or gas trapped in high spots or reservoirs. It will betreated as having the general aspect of a thin, flat sheet with an over-alltilt to the horizontal and with numerous small high spots, tight zones, andirregularities. This porous continuum will be treated as being much moreextensive than the oil or gas zones, but will be recognized as terminating atsuch large boundaries as a general fault, outcrop, unconformity, orpinch-out. Throughout the greater part of this limited porous continuum it is likelythat there is a hydraulic connection. If this is true, the removal of fluid inone place will cause a disturbance that eventually will travel to all parts ofthe continuum. This picture has led to the recognition of water as the major fluid in theporous continuum and in many cases the expansive nature of this water as themain driving force producing the oil. The next step is the solution of thegeneral problem of compressible liquid behavior in porous media. T.P. 1454