Poroelastic Constants Research Articles

Impact of stress dependence of elastic moduli and poroelastic constants on earth surface uplift due to injection

Impact of stress dependence of elastic moduli and poroelastic constants on earth surface uplift due to injection

Multiscale chemo-mechanical modeling of concrete expansion with free lime-based expansive additives under restraint conditions

Expansion of concrete due to expansive additives (EAs) is greatly affected by the degree of external restraints in the form of reinforcements and supports. Furthermore, the presence of anisotropic restraints influences concrete expansion in different directions. In this study, a chemo-mechanical model was developed to estimate concrete expansion under the framework of poromechanics. The volume of expansive hydrates was obtained from the hydration model, which was used for estimating the stress on the cement paste using poromechanical formulations. An external non-isotropic state of stress affects the orientation of crystal precipitating inside the pores of the cement paste. Thus, anisotropy in concrete expansion due to external stress was considered by maintaining the chemo-mechanical equilibrium of crystals of expansive hydrates, which was reflected in the poro-elastic constants utilized to estimate the stress at the cement paste scale. Poroelastic coefficients were estimated using a micromechanical scheme reflecting the effect of crystal orientation. Dedicated experiments were performed to comprehend the anisotropic expansion of concrete under anisotropic external restraint conditions. Finally, the applicability of the model was verified by comparing the expansion in different directions estimated from the model and experiments performed in this study with available literature data. Thus, the developed chemo-mechanical model could benefit practitioners and engineers in evaluating the expansion behavior of restrained concrete for a given application.

Negative Unjacketed Pore Modulus in Limestones? Critical Examination of a Peculiar Laboratory Observation

AbstractThere is recent experimental evidence that the effective pressure coefficient for pore volume exceeds unity in limestones with dual porosity structure. Within the realm of linear poroelastcity, this means that the unjacketed pore modulus is negative. This would be a rather unusual rock behavior since it implies that the pore volume increases for a positive pressure increment during hydrostatic compression. Generally, any deviation of the unjacketed pore modulus from strictly positive mineral bulk modulus means that the deformation is inhomogeneous. We critically examine these observations by attempting to reconcile them with the poroelastic constitutive equations of micro‐inhomogeneous rocks that encompass inhomogeneous deformations. Consistency checks among measured poroelastic constants reveal that the measurement accuracy of pore volume changes concurring with the deformation experiments is not as high as desired. However, assigning moderate variations to the measured pore volume changes either due to random or systematic measurement errors, we show that the experimental observations are indeed reconcilable with the constitutive poroelastic equations. Hence, the negative unjacketed pore modulus observed is deemed plausible though overestimated. Within the poroelasticity framework for micro‐inhomogeneous rocks, a negative unjacketed pore modulus is the result of a partial localization of elastic strain energy in the fluid space. This in turn could be caused by dissolved calcite particles that block thin pore throats and thus generate temporary fluid pressure compartmentalization. A far‐reaching consequence of a negative unjacketed pore modulus is that description of the undrained bulk modulus in rocks with dual porosity structure is more complicated than previously thought.

Effects of fluid saturations on undrained poroelastic constants in layered media

SUMMARYUnderstanding the fluid dependence of the poroelastic stiffness constants of a layered porous package is of great importance in subsurface exploration and development. While the effects of the pore-fluid distribution caused by coarse-scale heterogeneities within an isotropic medium have been studied for several decades, the role of these heterogeneities on the poroelastic constants of a finely layered package is still largely unexplored. In this study, we apply the poroelastic upscaling methods to estimate the fluid-dependent poroelastic stiffness constants of a layered package at the coarse scale. The numerical results show that the refined Gassmann's fluid substitution formulae presented in this paper is applicable if a single fluid phase is uniformly saturated within a layered package. The stiffness constants (${c_{11}}$ and ${c_{33}}$) of the layered package with patchy saturations are always higher than or equal to those obtained for the medium with homogeneous saturations, the stiffness constants predicted by the refined fluid substitution formulae for the package simultaneously saturated with different fluids fall between them. Experimental results confirm the relationship between the undrained vertical stiffness constant and the effective pore-fluid bulk modulus for the patch saturated package, indicating that a reasonable result can be achieved by properly choosing an effective poroelastic model that accounts both for the fluid hydraulic communication and the anisotropy of the medium. The results improve the understanding of the coarse-scale fluid dependence of the poroelastic stiffness constants of a layered package, and therefore, it can be used to interpret the seismically inverted elastic parameter for the petrophysical properties in heterogeneous reservoirs.

The poroelastic constants of multiple-porosity solids

Multiple-porosity models are commonly used to describe the poromechanical behavior of complex porous solids occurring in earth and life sciences. From the characterization standpoint, the poroelastic constants of such materials may be determined from the poroelastic constants of their single-porosity constituents. The uniqueness principle of the poroelastic energy density function is herein used to construct and solve a system of equations between the described sets of poroelastic constants in a multiple-porosity material. The method systematically applies to an arbitrary number of co-existing porosities. The presented energy-based approach resolves the perceived closure problem that was reported for solids with more than two porosity systems.

Five year drying of high performance concretes: Effect of temperature and cement-type on shrinkage

This experimental study imposes limited relative humidity (RH) gradients to small mature concrete samples, at a constant temperature T = 20, 50 or 80°C. Mass loss and shrinkage are recorded until stabilization at each RH and T, for up to 1991days. Firstly, our mass loss data are consistent with those presented in former research (on different samples of the same batch). After presenting and analyzing shrinkage kinetics, experimental data are fitted with usual models for shrinkage prediction, at each temperature of 20, 50 and 80°C. An adequate match is obtained by combining capillarity (i.e. Vlahinic's model coupling poro-elastic constants and water saturation level) and desorption (Bangham's equation).Subsequently, relative mass variation (RMV) is plotted against shrinkage εshdry data. Three distinct phases are obtained at 20 or 50°C and down to 30%RH; up to four distinct phases are observed at T = 80°C and down to 12%RH. The latter are confirmed by experiments on (60°C; 7%RH) dried concrete. The four phases in the (RMV,εshdry) diagram are interpreted against shrinkage data on mature cement paste dried at 60°C; 7%RH and against the literature.

Generalized poroelasticity framework for micro‐inhomogeneous rocks

ABSTRACTPoroelastic modelling of micro‐inhomogeneous rocks is of interest for applications in rock physics and geomechanics. Laboratory measurements from both communities indicate that the Biot poroelasticity framework is not adequate. For the case of a macroscopically homogeneous and isotropic rock, we present the most general poroelasticity framework within the scope of equilibrium thermodynamics that is able to capture the effects of micro‐inhomogeneities in a natural way. Within this generalized poroelasticity framework, the concept of micro‐inhomogeneity is generically related to partial localization of the deformational potential energy either in the solid phase, including the interfacial region or in the fluid phase. The former case can occur in the presence of surface roughness or multi‐mineralic frame and the latter case can be related to suspended particles residing in the fluid phase. A measure for micro‐inhomogeneity is the coefficient that governs the effective pressure dependence of porosity changes as described by the porosity perturbation equation of this framework. It can be therefore equivalently interpreted as porosity effective pressure coefficient or as micro‐inhomogeneity parameter. We show how this parameter and the other poroelastic constants embedded in this framework can be expressed in terms of experimentally accessible poroelastic constants.

Anisotropic Inhomogeneous Poroelastic Inclusions: With Application to Underground Energy-Related Problems

Understanding the stress change in a reservoir generated by fluid production/injection is important for field development purposes. In this paper, we provide the Eshelby solution for stress and strain distribution inside and outside of an anisotropic poroelastic inhomogeneity due to pore pressure changes inside the inhomogeneity. The term anisotropic inhomogeneity refers to an inhomogeneity with anisotropic poroelastic constants. Some graphical results for strain and stress ratios for different material properties and geometries are presented as well. Anisotropy in elastic properties has been studied extensively in the last century; however, anisotropy in poroelastic properties, despite its potential significant impact in different engineering problems, has not been explored thoroughly. The results show how neglecting the effect of anisotropic poroelastic properties may result in large differences in calculated stresses. Due to the authors' primary interest in geomechanical problems, the discussions and examples are chosen for applications involving fluid withdrawal/injection into hydrocarbon reservoirs.

Dynamics of Interstitial Fluid Pressure in Extracellular Matrix Hydrogels in Microfluidic Devices.

In order to understand how interstitial fluid pressure and flow affect cell behavior, many studies use microfluidic approaches to apply externally controlled pressures to the boundary of a cell-containing gel. It is generally assumed that the resulting interstitial pressure distribution quickly reaches a steady-state, but this assumption has not been rigorously tested. Here, we demonstrate experimentally and computationally that the interstitial fluid pressure within an extracellular matrix gel in a microfluidic device can, in some cases, react with a long time delay to external loading. Remarkably, the source of this delay is the slight (∼100 nm in the cases examined here) distension of the walls of the device under pressure. Finite-element models show that the dynamics of interstitial pressure can be described as an instantaneous jump, followed by axial and transverse diffusion, until the steady pressure distribution is reached. The dynamics follow scaling laws that enable estimation of a gel's poroelastic constants from time-resolved measurements of interstitial fluid pressure.

Gassmann equations and the constitutive relations for multiple‐porosity and multiple‐permeability poroelasticity with applications to oil and gas shale

SummaryMicromechanical characterization of multiple‐porosity and multiple‐permeability fluid‐saturated porous materials from the properties of their single‐porosity constituents is, to date, an open problem in our poromechanics society. This paper offers an in‐depth view to this problem by considering the thermodynamic potential energy density, consistent with Biot's original definition, together with the general thought experiment, which allows for independent control of the sample's confining stress and distinct fluid pore pressures within its individual porosity networks. The complete set of well‐known poroelastic constants, namely, Biot–Willis effective stress, Skempton's pore pressure, and specific storage coefficients, as well as drained, undrained, and Biot moduli for a fluid‐saturated porous material, is herein identified with the reformulated theory. In particular, Gassmann relation for the bulk compressibility of the fluid‐saturated material is accordingly upgraded to the case being addressed in this study.The practical implications of the theory are showcased through a class of analytical solutions to the time‐dependent poroelastic responses of shale to compression, when the hierarchical structure of its porous networks are accounted for at different levels of complexity and inter‐porosity exchange effects. For this purpose, the laboratory setup of a quasi‐2D compression test is considered, in which disk‐shaped fluid‐saturated samples of shale are allowed to drain laterally, while being sealed and confined from the top and bottom. A general closed‐form solution to this problem is derived in the Laplace space, and the inverse numerical results for the cases of single‐porosity, double‐porosity, triple‐porosity, and quadruple‐porosity shale are discussed in the time domain. Copyright © 2015 John Wiley & Sons, Ltd.

Mechanical Stresses Induced by Evaporation in Consolidated Colloidal Suspensions of Hard Particles. Poroelasticity Theory Versus Experiments

Drying of colloidal suspensions often yields intriguing, complex and generally undesirable crack patterns. Despite a consensus on the main underlying physical mechanisms that lead to their formation, quantitative prediction seems at present out of reach. Notably, a prerequisite for this is the determination of the mechanical stress field. Here, our point of view is to provide a sound and well-established continuum mechanics model that can be used safely and easily by a large community and particularly by the fracture mechanics one. We show by comparison with beam deflection experiments, that (i) the use of Biot’s linear poroelasticity theory allows for the prediction of the stresses that lead to the crack formation and that (ii) the setup allows for the identification of some poroelastic constants. This evidenced that the hypothesis of incompressible constituents is reasonable for the nanolatex suspensions considered here, greatly simplifying the constitutive equations and reducing the number of material constants from four to two.

Combining analysis of random elastic polycrystals with poroelasticity for granular composites having orthotropic porous grains and fluid-filled pores

Analysis of random polycrystals has typically been applied to solid grains of anisotropic elastic materials. Poroelastic analysis has similarly been applied to otherwise isotropic systems with pores having a variety of shapes and filled with fluids. Present effort is focused on combining these two types of geomechanical analyses by treating anisotropic (specifically orthotropic) poroelastic grains jumbled together to form an overall isotropic polycrystalline poroelastic material. The resulting problem is approximately twice as difficult to solve as the typical elastic polycrystal problem because the polycrystal analysis must be carried through twice: once for the drained (pore-fluid free to escape) poroelastic constants, and again for the undrained (pore-fluid trapped) poroelastic constants. As should be anticipated, poroelastic effects induced by trapped fluid are significantly stronger for the effective bulk modulus than for the effective shear modulus.

Influences of brain tissue poroelastic constants on intracranial pressure (ICP) during constant-rate infusion

A 3D finite element (FE) model has been developed to study the mean intracranial pressure (ICP) response during constant-rate infusion using linear poroelasticity. Due to the uncertainties in the poroelastic constants for brain tissue, the influence of each of the main parameters on the transient ICP infusion curve was studied. As a prerequisite for transient analysis, steady-state simulations were performed first. The simulated steady-state pressure distribution in the brain tissue for a normal cerebrospinal fluid (CSF) circulation system showed good correlation with experiments from the literature. Furthermore, steady-state ICP closely followed the infusion experiments at different infusion rates. The verified steady-state models then served as a baseline for the subsequent transient models. For transient analysis, the simulated ICP shows a similar tendency to that found in the experiments, however, different values of the poroelastic constants have a significant effect on the infusion curve. The influence of the main poroelastic parameters including the Biot coefficient , Skempton coefficient B, drained Young's modulus E, Poisson's ratio , permeability , CSF absorption conductance and external venous pressure was studied to investigate the influence on the pressure response. It was found that the value of the specific storage term is the dominant factor that influences the infusion curve, and the drained Young's modulus E was identified as the dominant parameter second to . Based on the simulated infusion curves from the FE model, artificial neural network (ANN) was used to find an optimised parameter set that best fit the experimental curve. The infusion curves from both the FE simulation and using ANN confirmed the limitation of linear poroelasticity in modelling the transient constant-rate infusion.

Poroelastic Response of Orthotropic Fractured Porous Media

An algorithm is presented for inverting either laboratory or field poroelastic data for all the drained constants of an anisotropic (specifically orthotropic) fractured poroelastic system. While fractures normally weaken the system by increasing the mechanical compliance, any liquids present in these fractures are expected to increase the stiffness somewhat, thus negating to some extent the mechanical weakening influence of the fractures themselves. The analysis presented in this article quantifies these effects and shows that the key physical variable needed to account for the pore-fluid effects is a factor of (1 − B), where B is Skempton’s second coefficient and satisfies 0 ≤ B < 1. This scalar factor uniformly reduces the increase in compliance due to the presence of communicating fractures, thereby stiffening the fractured composite medium by a predictable amount. One further aim of the discussion is to determine the number of the poroelastic constants that needs to be known by other means to determine the rest from remote measurements, such as seismic wave propagation data in the field. Quantitative examples arising in the analysis show that, if the fracture aspect ratio \({a_f \simeq 0.1}\) and the pore fluid is liquid water, then for several cases considered, Skempton’s \({B \simeq 0.9}\), and so the stiffening effect of the pore-liquid reduces the change in compliance due to the fractures by a factor \({1 - B \simeq 0.1}\), in these examples. The results do, however, depend on the actual moduli of the unfractured elastic material, as well as on the pore-liquid bulk modulus, so these quantitative predictions are just examples, and should not be treated as universal results. Attention is also given to two previously unremarked poroelastic identities, both being useful variants of Gassmann’s equations for homogeneous—but anisotropic—poroelasticity. Relationships to Skempton’s analysis of saturated soils are also noted. The article concludes with a discussion of alternative methods of analyzing and quantifying fluid-substitution behavior in poroelastic systems, especially for those systems having heterogeneous constitution.

Poroelastic relaxation indentation of thin layers of gels

We develop a method of poroelastic relaxation indentation (PRI) to characterize thin layers of gels. The solution to the time-dependent boundary-value problem is obtained in a remarkably simple form, so that the force-relaxation curve obtained by indenting a gel readily determines all the poroelastic constants of the gel—the shear modulus, Poisson’s ratio, and the effective diffusivity. The method is demonstrated with a layer of polydimethylsiloxane immersed in heptane.

Variation of unjacketed pore compressibility using Gassmann’s equation and an overdetermined set of volumetric poroelastic measurements

Gassmann’s original equation provides a means to relate bulk elastic parameters of a porous material with the compressibility of the pore fluid. The original analysis assumed microhomogeneity and isotropy, which assumed that pore compressibility was equal to grain compressibility. Although subsequent theoretical arguments have shown that Gassmann’s original assumption is violated for most rocks and that pore compressibility need not equal grain compressibility, few experimental studies have compared the two compressibilities; the assumption that pore compressibility equals grain compressibility is still commonly made. We measured hydrostatic poroelastic constants of Berea sandstone and Indiana limestone under drained, undrained, and unjacketed conditions over a range of confining and pore pressures to test the assumption that pore compressibility equals grain compressibility. These two rocks were chosen because they havesimilar values of porosity but different elastic behaviors: Berea sandstone is nonlinearly elastic, especially at low effective stresses, but Indiana limestone is linearly elastic at nearly all stresses. At low effective stresses below [Formula: see text], the pore compressibility for Berea sandstone does not equal grain compressibility but approaches fluid compressibility. Even at higher effective stresses, pore compressibility for Berea sandstone does not equal bulk grain compressibility but approaches a value approximately two to three times the bulk grain compressibility. In contrast, pore compressibility for Indiana limestone does seem to be equal to grain compressibility except perhaps at low effective stresses below [Formula: see text]. The difference between pore compressibilities of these two rocks is likely from the presence of more compliant clay minerals mixed with quartz grains with more microcracks in the Berea sandstone as compared to the well-cemented Indiana limestone.

Poroelastic measurement schemes resulting in complete data sets for granular and other anisotropic porous media

Poroelastic analysis usually progresses from assumed knowledge of dry or drained porous media to the predicted behavior of fluid-saturated and undrained porous media. Unfortunately, the experimental situation is often incompatible with these assumptions, especially when field data (from hydrological or oil/gas reservoirs) are involved. The present work considers several different experimental scenarios typified by one in which a set of undrained poroelastic (stiffness) constants has been measured using either ultrasound or seismic wave analysis, while some or all of the dry or drained constants are normally unknown. Drained constants for such a poroelastic system can be deduced for isotropic systems from available data if a complete set of undrained compliance data for the principal stresses are available – together with a few other commonly measured quantities such as porosity, fluid bulk modulus, and grain bulk modulus. Similar results are also developed here for anisotropic systems having up to orthotropic symmetry if the system is granular ( i.e., composed of solid grains assembled into a solid matrix, either by a cementation process or by applied stress) and the grains are known to be elastically homogeneous. Finally, the analysis is also fully developed for anisotropic systems with nonhomogeneous (more than one mineral type), but still isotropic, grains – as well as for uniform collections of anisotropic grains as long as their axes of symmetry are either perfectly aligned or perfectly random.

The estimated elastic constants for a single bone osteonal lamella

Micromechanical estimates of the elastic constants for a single bone osteonal lamella and its substructures are reported. These estimates of elastic constants are accomplished at three distinct and organized hierarchical levels, that of a mineralized collagen fibril, a collagen fiber, and a single lamella. The smallest collagen structure is the collagen fibril whose diameter is the order of 20 nm. The next structural level is the collagen fiber with a diameter of the order of 80 nm. A lamella is a laminate structure, composed of multiple collagen fibers with embedded minerals and consists of several laminates. The thickness of one laminate in the lamella is approximately 130 nm. All collagen fibers in a laminate in the lamella are oriented in one direction. However, the laminates rotate relative to the adjacent laminates. In this work, all collagen fibers in a lamella are assumed to be aligned in the longitudinal direction. This kind of bone with all collagen fibers aligned in one direction is called a parallel fibered bone. The effective elastic constants for a parallel fibered bone are estimated by assuming periodic substructures. These results provide a database for estimating the anisotropic poroelastic constants of an osteon and also provide a database for building mathematical or computational models in bone micromechanics, such as bone damage mechanics and bone poroelasticity.

An estimate of anisotropic poroelastic constants of an osteon

The anisotropic poroelastic constants of an osteon are estimated by micromechanical analysis. Two extreme cases are examined, the drained and the undrained elastic constants. The drained elastic constants are the porous medium's effective elastic constants when the fluid in the pores easily escapes and the pore fluid can sustain no pore pressure. The undrained elastic constants are the porous medium's effective elastic constants when the medium is fully saturated with pore fluid and the fluid cannot escape. The drained and undrained elastic constants at the lacunar and canalicular porosity tissue levels are estimated by using an effective moduli model consisting of the periodic distribution of ellipsoidal cavities. These estimated anisotropic poroelastic constants provide a database for the development of an accurate anisotropic poroelastic model of an osteon.

Identification of Poroelastic Constants of “Tight” Rocks from Laboratory Tests

This paper discusses the identification of poroelastic constants and hydraulic conductivity from two transient laboratory tests performed on cylindrical core: drained isotropic confinement and pulse test. The combination of the two tests allows us to estimate all poroelastic parameters with the exception of the shear modulus. Despite the lack of analytical solutions for the fully coupled case, closed form solutions of the forward problems are obtained for a slender specimen. The validity of these solutions for a realistic aspect ratio of the core is assessed by a comparison with a finite element model. The identification problem is solved by minimizing a least square functional using an explicit gradient computed using the direct differentiation of the closed form solution and a Levenberg–Marquardt algorithm. The uniqueness of this inverse problem as well as the effect of noise on input data are fully discussed. The identification procedure is then applied to tests performed on a deep argillaceous rock (argillite of Meuse Haute–Marne).