Deformation-dependent Permeability Research Articles

Cohesive phase-field model for dynamic fractures in coal seams

A novel rate-dependent cohesive phase-field model that can simulate fluid-driven dynamic quasibrittle fracture in soft coal rocks is presented. The coal matrix and fractures are described by a unified fluid continuity equation, and the Poiseuille flow in the fractures is modeled by deformation-dependent permeability. Dynamic evolution equations for porosity, permeability and the Biot coefficient during phase-field evolution are developed. The phase-field evolution equation is derived from the total energy generalization based on the Francfort–Marigo variational principle. To simulate the rate-dependent dynamic fracture problem accurately, the effects of the phase-field evolution dissipation energy and inertia energy are considered. The above multifield coupled system is solved via a staggered approach within the finite element framework via the Newton–Raphson iterative algorithm. The independence and convergence of the proposed model were verified, and its accuracy was validated via an analytical solution for crack width and a benchmark test for dynamic crack propagation. The propagation law of single/multicluster hydraulic fractures in coal seams with interlayers was investigated via the proposed model. The simulations revealed that the number of clusters and the mechanical properties of the interlayers significantly influence the fracture morphology. Finally, hydraulic fracturing of fracture-developed coal seams was simulated, confirming the potential of the proposed model.

Modeling flow and deformation in porous media from pore-scale to the Darcy-scale

In this paper we address the connections between the computational models of coupled flow and mechanical deformation in soils at the Darcy-scale and pore-scale. At the Darcy scale the Biot model requires data including permeability which is traditionally provided by experiments and empirical measurements. At the pore-scale we consider the Discrete Element Method (DEM) to generate physically realistic assemblies of the particles, and we follow up with the Stokes flow model. Next we apply upscaling to obtain the permeabilities which we find dependent on the deformation. We outline the workflow with its challenges and methods, and present results which show, e.g., hysteretic dependence of the permeability and porosity on the load. We also show how to incorporate the deformation dependent permeability in a nonlinear Biot model, and illustrate with computational results.

Deformation-dependent permeability of fibrous materials

Fluid flow in fibrous materials depends on volumetric deformation, distortional deformation and fibre reorientation. The latter two effects introduce anisotropy in the hydraulic permeability. Motivated by the study of biological tissues, this work aims to develop an analytical model for evaluating the deformation-dependent permeability of fibrous materials. The material consists of a three-dimensional assembly of long fibres of a single type, with the fluid flowing through the interstices between the fibres. The macroscopic deformations are related to the deformation of a cylindrical representative element of volume (REV), comprised of a fibre surrounded by a cylindrical jacket of fluid. The effect of fibre orientation is modelled using an orientational probability distribution function (OPDF), and the REV permeability tensor is then evaluated using previously developed methods, accounting for the macroscopic deformation. Finally, the overall permeability tensor is calculated via directional averaging of the REV permeability tensor, in which the fibre OPDF serves as the weighing function. The model is validated numerically via a uniaxial contraction (confined compression) test performed for the case of an OPDF that is isotropic in the reference (undeformed) configuration, and is found to qualitatively reproduce experimental findings. The proposed model offers an analytical tool for evaluating the permeability of fibrous materials under large deformations, based on structural information such as fibre volumetric fraction, diameter and spatial orientation.

Deformation-dependent polydimethylsiloxane permeability measured using osmotic microactuators.

In soft solids, large deformations significantly alter molecular structure and device geometry, which can impact other properties. In the case of mass transport, an interplay between flux and mechanical deformation results. Here we demonstrate a platform for the simultaneous characterization of mechano-permselectivity using the (slow) transport of water through polydimethylsiloxane (PDMS) as a challenging test case. The platform uses micron-sized, cylindrical, NaCl solution-filled PDMS chambers encapsulated by selectively-permeable PDMS thin film membranes. When placed in a high chemical potential environment (high water potential) the osmotic pressure difference between the chamber and environment induces water to flow through the PDMS membrane into the chamber, resulting in membrane bulging. A model combining membrane flux and nonlinear elasticity captures the time-dependent response well, but only when a deformation-dependent permeability is used. Notably, the permeability of water through PDMS decreases by nearly an order of magnitude, from 2 × 10-12 to 5 × 10-13 m2 s-1, due primarily to its thickness decreasing by nearly an order of magnitude as the average biaxial stretch increases from 1 to 2.75.

Hydro-mechanical continuum modelling of fluid percolation through rock salt

Salt caverns for storing renewably produced hydrogen underground pose a promising option to reduce CO2 emissions by improving the utilization of volatile renewable energy sources. Rock salt limits the infiltration of hydrogen into the storage cavern walls due to its favourable low permeability. However, pressure-driven percolation may create discrete pathways for the fluid and damage the cavern. Depicting this effect accurately in simulations may be crucial for assessing the long-term stability and integrity adequately. By introducing an extension to a deformation-dependent permeability model by Alonso, Olivella and Arnedo (2006) we can capture the non-linear feedback between salt deformation, permeability development and fluid migration using continuum models. Based on a quasi-isotropic arrangement of potential flow paths the model is able to reproduce pressure-driven percolation of hydrofrac experiments on rock salt specimens in terms of breakthrough pressure as well as preferential path orientation and locality. The model is implemented in the scientific open-source framework OpenGeoSys.

A moving finite element framework for fast infiltration in nonlinear poroelastic media

Poroelasticity theory can be used to analyse the coupled interaction between fluid flow and porous media (matrix) deformation. The classical theory of linear poroelasticity captures this coupling by combining Terzaghi’s effective stress with a linear continuity equation. Linear poroelasticity is a good model for very small deformations; however, it becomes less accurate for moderate to large deformations. On the other hand, the theory of large-deformation poroelasticity combines Terzaghi’s effective stress with a nonlinear continuity equation. In this paper, we present a finite element solver for linear and nonlinear poroelasticity problems on triangular meshes based on the displacement-pressure two-field model. We then compare the predictions of linear poroelasticity with those of large-deformation poroelasticity in the context of a two-dimensional model problem where flow through elastic, saturated porous media, under applied mechanical oscillations, is considered. In addition, the impact of introducing a deformation-dependent permeability according to the Kozeny-Carman equation is explored. We computationally show that the errors in the displacement and pressure fields that are obtained using the linear poroelasticity are primarily due to the lack of the kinematic nonlinearity. Furthermore, the error in the pressure field is amplified by incorporating a constant permeability rather than a deformation-dependent permeability.

Modelling fluid flow in complex natural fault zones: Implications for natural and human-induced earthquake nucleation

Pore fluid overpressures in active fault systems can drive fluid flow and cause fault weakening and seismicity. In return, deformation accommodated by different modes of failure (e.g. brittle vs. ductile) also affects fault zone permeability and, hence, fluid flow and pore fluid pressure distribution. Current numerical simulation techniques model how fluid flow controls fault reactivation and associated seismicity. However, the control exerted by pore fluid pressure on the transition from slow aseismic fault sliding to fast seismic sliding, during the earthquake nucleation phase, is still poorly understood. Here, we model overpressured, supercritical CO2 fluid flow in natural faults, where non-linear, complex feedback between fluid flow, fluid pressure and fault deformation controls the length of the nucleation phase of an earthquake and the duration of the interseismic period. The model setup is an analogue for recent seismic source events in the Northern Apennines of Italy (e.g. Mw 6.0 1997-98 Colfiorito and Mw 6.5 2016 Norcia earthquakes). Our modelling results of Darcy fluid flow show that the duration of the nucleation phase can be reduced by orders of magnitude, when realistic models of fault zone architecture and pore pressure- and deformation-dependent permeability are considered. In particular, earthquake nucleation phase duration can drop from more than 10 years to a few days/minutes, while the seismic moment can decrease by a factor of 6. Notably, the moment of aseismic slip (M0=109Nm) obtained during the nucleation phase modelled in our study is of the same order as the detection limit of local strain measurements using strain meters. These findings have significant implications for earthquake early warning systems, as the duration and moment of the nucleation phase will affect the likelihood of timely precursory signal detection. Interestingly, aseismic slip has been measured up to a few months before some recent large earthquakes, although in a different tectonic context than the model developed here, rekindling interest in the nucleation phase of earthquakes. In addition, our results have important implications for short and long term earthquake forecasting, as crustal fluid migration during the interseismic period may control fault strength and earthquake recurrence intervals.

A multi-physics process model for simulating the manufacture of resin-infused composite aerostructures

The increasing demand for large, complex and low-cost composite aerostructures has motivated advances in the simulation of liquid composite moulding techniques with textile reinforcement materials. This work outlines the development and validation of a multi-physics process model that better simulates infusion behaviour through a complex preform compared with traditional models used in industry that do not account for fabric deformation. By combining the results of a preform draping model with deformation-dependent permeability properties, the shape and local flow characteristics of a deformed textile reinforcement have been more realistically defined for infusion. Simulated shear deformation results were used to define the distributed permeability properties across the fabric domain of the infusion model. Full-scale vacuum infusion experiments were conducted for a complex double dome geometry using a plain weave carbon fibre material. The multi-physics process model showed significant improvement over basic models, since it is able to account for the change in flow behaviour that results from local fabric deformation.

From arteries to boreholes: steady-state response of a poroelastic cylinder to fluid injection.

The radially outward flow of fluid into a porous medium occurs in many practical problems, from transport across vascular walls to the pressurization of boreholes. As the driving pressure becomes non-negligible relative to the stiffness of the solid structure, the poromechanical coupling between the fluid and the solid has an increasingly strong impact on the flow. For very large pressures or very soft materials, as is the case for hydraulic fracturing and arterial flows, this coupling can lead to large deformations and, hence, to strong deviations from a classical, linear-poroelastic response. Here, we study this problem by analysing the steady-state response of a poroelastic cylinder to fluid injection. We consider the qualitative and quantitative impacts of kinematic and constitutive nonlinearity, highlighting the strong impact of deformation-dependent permeability. We show that the wall thickness (thick versus thin) and the outer boundary condition (free versus constrained) play a central role in controlling the mechanics.

Hydraulic fracture in poro-hydro-elastic media

The prediction of fluid-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing, the so-called “fracking”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developing cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media that provides a rigorous approach to a diffusive crack modeling based on the introduction of a regularized crack surface. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. It shows that the quasi-static problem of elastically deforming, fluid-saturated porous media at fracture is related to a minimization principle for the evolution problem. The existence of this minimization principle for the coupled problem is advantageous with regard to a new unconstrained stable finite element design, while previous space discretizations of the saddle point principles are constrained by the LBB condition. This proposed formulation includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length.

Large Deformations of a Soft Porous Material

Compressing a porous material will decrease the volume of the pore space, driving fluid out. Similarly, injecting fluid into a porous material can expand the pore space, distorting the solid skeleton. This poromechanical coupling has applications ranging from tissue mechanics to hydrogeology. The classical theory of linear poroelasticity captures this coupling by combining Darcy's law with Terzaghi's effective stress and linear elasticity in a linearized kinematic framework. Linear poroelasticity is a good model for very small deformations, but it becomes increasingly inappropriate for moderate to large deformations, which are common in the context of phenomena such as swelling and damage, and for soft materials such as gels and tissues. The well-known theory of large-deformation poroelasticity combines Darcy's law with Terzaghi's effective stress and nonlinear elasticity in a rigorous kinematic framework. This theory has been used extensively in biomechanics to model large elastic deformations in soft tissues, and in geomechanics to model large elastoplastic deformations in soils. Here, we first provide an overview and discussion of this theory with an emphasis on the physics of poromechanical coupling. We present the large-deformation theory in an Eulerian framework to minimize the mathematical complexity, and we show how this nonlinear theory simplifies to linear poroelasticity under the assumption of small strain. We then compare the predictions of linear poroelasticity with those of large-deformation poroelasticity in the context of two uniaxial model problems: Fluid outflow driven by an applied mechanical load (the consolidation problem) and compression driven by a steady fluid throughflow. We use these problems to explore the steady and dynamical errors associated with the linear model in both situations, as well as the impact of introducing a deformation-dependent permeability.

Finite element modeling of finite deformable, biphasic biological tissues with transversely isotropic statistically distributed fibers: toward a practical solution.

The distribution of collagen fibers across articular cartilage layers is statistical in nature. Based on the concepts proposed in previous models, we developed a methodology to include the statistically distributed fibers across the cartilage thickness in the commercial FE software COMSOL which avoids extensive routine programming. The model includes many properties that are observed in real cartilage: finite hyperelastic deformation, depth-dependent collagen fiber concentration, depth- and deformation-dependent permeability, and statistically distributed collagen fiber orientation distribution across the cartilage thickness. Numerical tests were performed using confined and unconfined compressions. The model predictions on the depth-dependent strain distributions across the cartilage layer are consistent with the experimental data in the literature.

Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media

The prediction of fluid- and moisture-driven crack propagation in deforming porous media has achieved increasing interest in recent years, in particular with regard to the modeling of hydraulic fracturing, the so-called “fracking”. Here, the challenge is to link at least three modeling ingredients for (i) the behavior of the solid skeleton and fluid bulk phases and their interaction, (ii) the crack propagation on not a priori known paths and (iii) the extra fluid flow within developed cracks. To this end, a macroscopic framework is proposed for a continuum phase field modeling of fracture in porous media. It provides a rigorous geometric approach to a diffusive crack modeling based on the introduction of a constitutive balance equation for a regularized crack surface and its modular linkage to a Darcy–Biot-type bulk response of hydro-poro-elasticity. The approach overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies including branching. A modular concept is outlined for linking of the diffusive crack modeling with the hydro-poro-elastic response of the porous bulk material. This includes a generalization of crack driving forces from energetic definitions towards threshold-based criteria in terms of the effective stress related to the solid skeleton of a fluid-saturated porous medium. Furthermore, a Poiseuille-type constitutive continuum modeling of the extra fluid flow in developed cracks is suggested based on a deformation-dependent permeability, that is scaled by a characteristic length. This proposed modular model structure is exploited in the numerical implementation by constructing a robust finite element method, based on an algorithmic decoupling of updates for the crack phase field and the state variables of the hydro-poro-elastic bulk response. We demonstrate the performance of the phase field formulation of fracture for a spectrum of model problems of hydraulic fracture. A slight modification of the framework allows the simulation of drying-caused crack patterns in partially saturated capillar-porous media.

Confined compression of collagen hydrogels

Reconstituted collagen hydrogels are often used for in vitro studies of cell–matrix interaction and as scaffolds for tissue engineering. Understanding the mechanical and transport behaviours of collagen hydrogels is therefore extremely important, albeit difficult due to their very high water content (typically >99.5%). In the present study the mechanical behaviour of collagen hydrogels in confined compression was investigated using biphasic theory (J Biomechemical Engineering 102 (1980) 73), to ascertain whether the technique is sufficiently sensitive to determine differences in the characteristics of hydrogels of between 0.2% and 0.4% collagen. Peak stress, equilibrium stress, aggregate modulus and hydraulic permeability of the hydrogels exhibited sensitivity to collagen content, demonstrating that the technique is clearly able to discriminate between hydrogels with small differences in collagen content and may also be sensitive to factors that affect matrix remodelling. The results also offer additional insight into the deformation-dependent permeability of collagen hydrogels. This study suggests that confined compression, together with biphasic theory, is a suitable technique for assessing the mechanical properties of collagen hydrogels.

Numerical Simulation of 3D Hydraulic Fracturing Based on an Improved Flow-Stress-Damage Model and a Parallel FEM Technique

The failure mechanism of hydraulic fractures in heterogeneous geological materials is an important topic in mining and petroleum engineering. A three-dimensional (3D) finite element model that considers the coupled effects of seepage, damage, and the stress field is introduced. This model is based on a previously developed two-dimensional (2D) version of the model (RFPA2D-Rock Failure Process Analysis). The RFPA3D-Parallel model is developed using a parallel finite element method with a message-passing interface library. The constitutive law of this model considers strength and stiffness degradation, stress-dependent permeability for the pre-peak stage, and deformation-dependent permeability for the post-peak stage. Using this model, 3D modelling of progressive failure and associated fluid flow in rock are conducted and used to investigate the hydro-mechanical response of rock samples at laboratory scale. The responses investigated are the axial stress–axial strain together with permeability evolution and fracture patterns at various stages of loading. Then, the hydraulic fracturing process inside a rock specimen is numerically simulated. Three coupled processes are considered: (1) mechanical deformation of the solid medium induced by the fluid pressure acting on the fracture surfaces and the rock skeleton, (2) fluid flow within the fracture, and (3) propagation of the fracture. The numerically simulated results show that the fractures from a vertical wellbore propagate in the maximum principal stress direction without branching, turning, and twisting in the case of a large difference in the magnitude of the far-field stresses. Otherwise, the fracture initiates in a non-preferred direction and plane then turns and twists during propagation to become aligned with the preferred direction and plane. This pattern of fracturing is common when the rock formation contains multiple layers with different material properties. In addition, local heterogeneity of the rock matrix and macro-scale stress fluctuations due to the variability of material properties can cause the branching, turning, and twisting of fractures.

Modeling coupled processes of non-steady seepage flow and non-linear deformation for a concrete-faced rockfill dam

A coupled non-linear elastic deformation and non-steady seepage flow model was established for performance assessment of the concrete-faced rockfill (CFR) dams. The deformation behaviors of the rockfills and the soil–structure interfaces were modeled with the Duncan–Chang model and Desai’s thin-layer elements, respectively. The seepage flow process was characterized with a newly-developed parabolic variational inequality formulation of the Signorini’s condition. The deformation-dependent permeability of the rockfills was characterized by a calibrated Kozeny–Carman model that considered the influence of confining pressure on porosity. The proposed model was applied to capture the seepage flow and deformation behaviors of the Shuibuya CFR dam in China.

Three-Dimensional Micro Flow-Stress-Damage (FSD) Model and Application in Hydraulic Fracturing in Brittle and Heterogeneous Rocks

In order to investigate the hydraylic fracture development of the specimens and simulate the cracks drived by fluid flow in rocks, a flow-stress-damage (FSD) model, implemented with parallel Rock Failure Process Analysis code (parallel -RFPA3D), is presented. The numerical code is based on linear elastic damage mechanics on mesoscopic scale and FEM. For simulating the complete progressive 3D failure and macroscopic mechanical behaviors of rock materials, rock properties such as elastic constants, peak strength, and poisson ratio are randomly distributed to reflect the initial random distributed weakness in mesoscopic scale. The FSD model is used to represent the permeability variation at the two stages, that is stress-dependent permeability for pre-failure and deformation-dependent permeability for post-peak stage of rock at the elemental scale. The results of the simulation with 680,000-element cylindrical rock specimen coincide well with reported experimental results and the process of crack from initiation to the instability extensions is captured vividly. The results and the process indicate that the FSD model works well and parallel-RFPA3D incorporated with FSD model is a valid tool of understanding the physical essence of the evolution of fracture with large-scale elements and fluid flows in rocks.

The consolidation behavior of silk hydrogels

Hydrogels have mechanical properties and structural features that are similar to load-bearing soft tissues including intervertebral disc and articular cartilage, and can be implanted for tissue restoration or for local release of therapeutic factors. To help predict their performance, mechanical characterization and mathematical modeling are the available methods for use in tissue engineering and drug delivery settings. In this study, confined compression creep tests were performed on silk hydrogels, over a range of concentrations, to examine the phenomenological behavior of the gels under a physiological loading scenario. Based on the observed behavior, we show that the time-dependent response can be explained by a consolidation mechanism, and modeled using Biot’s poroelasticity theory. Two observations are in strong support of this modeling framework, namely, the excellent numerical agreement between increasing load step creep data and the linear Terzaghi theory, and the similar values obtained from numerical simulations and direct measurements of the permeability coefficient. The higher concentration gels (8% and 12% w/v) clearly show a strain-stiffening response to creep loading with increasing loads, while the lower concentration gel (4% w/v) does not. A nonlinear elastic constitutive formulation is employed to account for the stiffening. Furthermore, an empirical formulation is used to represent the deformation-dependent permeability.

An extended biphasic model for charged hydrated tissues with application to the intervertebral disc

Finite element models for hydrated soft biological tissue are numerous but often exhibit certain essential deficiencies concerning the reproduction of relevant mechanical and electro-chemical responses. As a matter of fact, singlephasic models can never predict the interstitial fluid flow or related effects like osmosis. Quite a few models have more than one constituent, but are often restricted to the small-strain domain, are not capable of capturing the intrinsic viscoelasticity of the solid skeleton, or do not account for a collagen fibre reinforcement. It is the goal of this contribution to overcome these drawbacks and to present a thermodynamically consistent model, which is formulated in a very general way in order to reproduce the behaviour of almost any charged hydrated tissue. Herein, the Theory of Porous Media (TPM) is applied in combination with polyconvex Ogden-type material laws describing the anisotropic and intrinsically viscoelastic behaviour of the solid matrix on the basis of a generalised Maxwell model. Moreover, other features like the deformation-dependent permeability, the possibility to include inhomogeneities like varying fibre alignment and behaviour, or osmotic effects based on the simplifying assumption of Lanir are also included. Finally, the human intervertebral disc is chosen as a representative for complex soft biological tissue behaviour. In this regard, two numerical examples will be presented with focus on the viscoelastic and osmotic capacity of the model.

Biomechanics of the brain; some remarks on Biot\'s equations of consolidation theory with deformation-dependent permeability

We revisit models of hydrocephalus in the literature. In particular, we examine the class of models based on Biot\'s theory of consolidation with fixed boundary forcing. Instead of fixed boundaries we take free boundaries. We prove existence and uniqueness of solutions. As in the fixed boundary forcing, we show that in a free boundary, the pressure is higher when the permeability depends on deformation. On the other hand, the total filtration is lower. Unlike the fixed forcing, the effect of the deformation on permeability reduces over time: Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 511-516