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We critically review the derivation of closed-form analytical expressions for elastic displacements, strains, and stresses inside a subsurface reservoir undergoing pore pressure changes using inclusion theory. Although developed decades ago, inclusion theory has been used recently by various authors to obtain fast estimates of depletion-induced and injection-induced fault stresses in relation to induced seismicity. We therefore briefly address the current geomechanical relevance of this method, and provide a numerical example to demonstrate its use to compute induced fault stresses. However, the main goal of our paper is to correct some erroneous assumptions that were made in earlier publications. While the final expressions for the poroelastic stresses in these publications were correct, their derivation contained conceptual mistakes due to the mathematical subtleties that arise because of singularities in the Green's functions. The aim of our paper is therefore to present the correct derivation of expressions for the strains and stresses inside an inclusion and to clarify some of the results of the aforementioned studies. Furthermore, we present two conditions that the strain field must satisfy, which can be used to verify the analytical expressions., Reservoir Engineering, Mathematical Physics

In fractured and stress-sensitive reservoirs and aquifers, hydromechanical coupling is important, in connection with their heat and solute transport properties, and because the fluid production or extraction leads to land subsidence and potentially to induced seismicity. Classical dual-porosity poroelasticity (DPP) models cannot upscale pressure diffusion and deformation in fractured porous media, which are characterized by anomalous behaviors that manifest in strong tailing in the temporal evolution of flow rate and subsidence. We study these behaviors using detailed numerical simulations of fluid production in naturally fractured formations characterized by multi-Gaussian distributions of the matrix permeability. We find that the tailing behaviors depend on the permeability contrast between fracture and matrix, on the permeability distribution in the matrix, and on the correlation length. We use a non–equilibrium, multi-porosity model to quantify the coupled behaviors of anomalous pressure diffusion, fluid flow and deformation. The model is parameterized by medium and fluid properties, which set the characteristic pressure diffusion time scales. It allows to identify the emerging scaling regimes and scaling behaviors of flow rate and subsidence. We propose a model implementation that captures the full anomalous evolution of flow rates and displacements observed in the detailed numerical simulations in terms of the permeability distribution and matrix length scales. The presented results shed new light on the controls of medium heterogeneity and geometry on pressure diffusion, fluid production and subsidence in highly heterogeneous fractured media.

A Mw 5.5 earthquake struck Pohang (South Korea) on November 2017, following a sequence of five hydraulic stimulations of an Enhanced Geothermal System (EGS). The processes that led to this earthquake, which nucleated two months after the end of the last stimulation in borehole PX2, are not well understood yet. We propose a hydromechanical model that integrates available data to understand the potential relationship between the earthquake and the EGS. Data scarcity is translated into model uncertainties, which we address with sensitivity analyses. Results show that the Mw 5.5 earthquake is linked to the high-injection overpressures of up to 90 MPa induced during the stimulations in borehole PX2 and highlight the usefulness of hydromechanical modeling to forecast the seismicity of an EGS and, more specifically, the need to integrate the low permeability fault core that hindered fluid pressure dissipation at Pohang, which explains the long delay of the mainshock.

We critically review the derivation of closed-form analytical expressions for elastic displacements, strains, and stresses inside a subsurface reservoir undergoing pore pressure changes using inclusion theory. Although developed decades ago, inclusion theory has been used recently by various authors to obtain fast estimates of depletion-induced and injection-induced fault stresses in relation to induced seismicity. We therefore briefly address the current geomechanical relevance of this method, and provide a numerical example to demonstrate its use to compute induced fault stresses. However, the main goal of our paper is to correct some erroneous assumptions that were made in earlier publications. While the final expressions for the poroelastic stresses in these publications were correct, their derivation contained conceptual mistakes due to the mathematical subtleties that arise because of singularities in the Green's functions. The aim of our paper is therefore to present the correct derivation of expressions for the strains and stresses inside an inclusion and to clarify some of the results of the aforementioned studies. Furthermore, we present two conditions that the strain field must satisfy, which can be used to verify the analytical expressions.

This contribution introduces and discusses a formulation of poro-hyperelasticity at finite strains. The prediction of the time-dependent response of such media requires consideration of their characteristic multi-scale and multi-physics parameters. In the present work this is achieved by formulating a non-dimensionalised fluid–solid interaction problem (FSI) at the pore level using an arbitrary Lagrange–Euler description (ALE). The resulting coupled systems of PDEs on the reference configuration are expanded and analysed using the asymptotic homogenisation technique. This approach yields three partially novel systems of PDEs: the macroscopic/effective problem and two supplementary microscale problems (fluid and solid). The latter two provide the microscopic response fields whose average value is required in real-time/online form to determine the macroscale response (a concurrent multi-scale approach). In order to overcome the computational challenges related to the above multi-scale closure, this work introduces a surrogate approach for replacing the direct numerical simulation with an artificial neural network. This methodology allows for solving finite strain (multi-scale) porohyperelastic problems accurately using direct automated differentiation through the strain energy. Optimal and reliable training data sets are produced from direct numerical simulations of the fully-resolved problem by including a simple real-time output density check for adaptive sampling step refinement. The data-driven approach is complemented by a sensitivity analysis of the RVE response. The significance of the presented approach for finite strain poro-elasticity/poro-hyperelasticity is shown in the numerical benchmark of a multi-scale confined consolidation problem. Finally, to show the robustness of the method, the system response is dimensionalised using characteristic values of soil and brain mechanics scenarios.

Understanding and characterizing the mechanical and structural properties of brain tissue is essential for developing and calibrating reliable material models. Based on the Theory of Porous Media, a novel nonlinear poro-viscoelastic computational model was recently proposed to describe the mechanical response of the tissue under different loading conditions. The model contains parameters related to the time-dependent behavior arising from both the viscoelastic relaxation of the solid matrix and its interaction with the fluid phase. This study focuses on the characterization of these parameters through indentation experiments on a tailor-made polyvinyl alcohol-based hydrogel mimicking brain tissue. The material behavior is adjusted to ex vivo porcine brain tissue. An inverse parameter identification scheme using a trust region reflective algorithm is introduced and applied to match experimental data obtained from the indentation with the proposed computational model. By minimizing the error between experimental values and finite element simulation results, the optimal constitutive model parameters of the brain tissue-mimicking hydrogel are extracted. Finally, the model is validated using the derived material parameters in a finite element simulation., We gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through grants BU 3728/3-1 and STE 544/70-1 and financial support by the Austrian Science Fund (FWF, Project-Nr. I 4828-N)., Peer Reviewed, Postprint (published version)

Global-scale periodic deformation has been studied using the (visco)elastic gravitational theory, which assumes a planetary body consists of solid or liquid layers. Recent planetary exploration missions, however, suggest that a global layer of a mixture of solid and liquid exists in several planetary bodies. This study provides a theory of periodic deformation of such a layer unifying the viscoelastic gravitational theory with the theory of poroelasticity without introducing additional constraints. The governing equation system and a formulation suitable for numerical calculation are given. Equations used to calculate the energy dissipation rate are also given. The analytical solutions for a homogeneous sphere are obtained using an eigenvalue approach. Simple numerical calculations assuming a homogeneous sphere reveal that a numerical instability occurs if a thick porous layer, a low permeability, or a high frequency is assumed. This instability can be avoided by choosing an appropriate interior structure model that is numerically equivalent. Different simple numerical calculations adopting a multilayered, radially varying interior profile reveal that the radial profile of the tidal heating rate for a fluid-saturated porous layer and that for a low-viscosity solid layer are completely different. In addition, the radial variation in porosity can lead to a factor of similar to 100 increase in the local heating rate. These results indicate that future studies should consider a wider variety of detailed interior structure models.

The ultimate goal in volcanology is to precisely forecast volcanic eruptions. One step towards this objective is to discriminate between processes of magmatic and non-magmatic origin. However, geophysical and geochemical anomalies are ambiguous and can be attributed to a variety of subsurface processes. This thesis aims to advance the understanding of unrest signals and underpinning processes in different volcanic settings, specifically two crater lake volcanoes in New Zealand. For this, I use Multiphysics modelling and forward model ground displacement, gravity, and self-potential anomalies to identify measurable signals from magmatic and hydrothermal unrest. First, I study magmatic unrest signals and test the sensitivity of self-potential to strain-induced fluid flow in volcanic aquifers by using a novel modelling approach. I was able to determine distinct patterns of self-potential anomalies and surface displacements from magmatic stressing of dike and sill intrusions. Then, I investigate magmatic and non-magmatic unrest signals at the crater lake volcano, Mt. Ruapehu in New Zealand. Model results show unique sets of measurable geophysical signals from either magmatic or hydrothermal perturbation, providing valuable information for monitoring efforts of the high-risk volcano. Lastly, I compare surface displacements from satellite observations with hydrothermal unrest models at Whakaari/White Island; with some models matching observed displacement magnitudes. Furthermore, I find detectable self-potential and gravity changes from hydrothermal fluid fluxes which advances the understanding of potential geophysical signals at Whakaari/White Island. The results demonstrate that magmatic and non-magmatic processes induce markedly different spatio-temporal unrest signals. I demonstrate that self-potential and gravity changes are useful indicators for interrogating source mechanisms behind volcanic unrest, especially when combined with common monitorin

This note provides the derivation of closed-form expressions for elastic displacements, strains, and stresses inside an inclusion. Jansen et al. (2019) and Wu et al. (2021) obtained correct expressions for the stresses inside an inclusion, but their derivation of these expressions contained mistakes. In this note, the correct derivation of expressions for the stresses inside an inclusion is presented and some of the results of the aforementioned studies are clarified., This note provides corrections to results published in: J.D. Jansen, P. Singhal, and F.C. Vossepoel. Insights from closed-form expressions for injection- and production-induced stresses in displaced faults. Journal of Geophysical Research - Solid Earth, 124:7193{7212, 2019. URL https://doi.org/10.1029/2019JB017932. H. Wu, V. Vilarrasa, S. De Simone, M. Saaltink, and F. Parisio. Analytical solution to assess the induced seismicity potential of faults in pressurized and depleted reservoirs. Journal of Geophysical Research - Solid Earth, 126: e2020JB020436, 2021. URL https://doi.org/10.1029/2020JB020436., Reservoir Engineering, Mathematical Physics, Civil Engineering & Geosciences

Both field injectivity and induced seismicity were reported to be inversely correlated with the temperature of re-injected fluids at the Hellisheiði geothermal field in Iceland. This observation has led to a hypothesis that transient cooling-induced permeability enhancement is a novel mechanism for induced seismicity, in addition to elevated fluid pressure, poroelastic stressing, and thermoelastic stressing in geothermal environments. In this work, a 3D calibrated coupled THM model was developed to model the colder fluids re-injection process over a 1-year period and evaluate the potential for induced seismicity in terms of Coulomb stress changes at the Hellisheiði geothermal field. Three modelling scenarios taking into account respectively the poroelastic effect, thermoporoelastic effect, and thermoporoelastic effect with permeability enhancement, were examined and compared to identify the dominant mechanism for the recorded seismicity and examine the contribution from each individual mechanism. Results have shown that, under normal fluid re-injection pressure and temperature conditions, the permeability enhancement effect is the dominant mechanism for induced seismicity at the Hellisheiði geothermal field. Specifically, the contribution to Coulomb stress changes from the permeability enhancement effect is almost twice of that from the thermoelastic stressing, which is in turn two orders of magnitude larger than that from the poroelastic stressing. It has also been noted that, when reducing temperature of re-injected fluids from 120°C to 20°C, the temperature change is increased by 2.1 times at 1,000 m depth, while the amount of mass flow by around 4 times. Thus, the amount of heat transferred can be increased 8.4 times by lowering temperature of the injected fluids, which explains the high sensitivity of induced seismicity to temperature. Outcomes of this work suggest temperature control of injected fluids as a feasible regulation method to mitigate against inject

We develop a model which couples the flow in a discrete fracture to a deformable porous medium. To account for the discrete representation of the fracture, a dimensionally-reduced fluid flow model is proposed. The fluid flow model incorporates both a reduced permeability of the fracture walls due to the skin effect, and a slip of fluid flowing along the permeable fracture walls. Biot's model for poroelastic media is coupled to a fracture flow model based on a thin-film approximation of the compressible Navier-Stokes equations. The fracture flow model incorporates a fluid entry resistance parameter to relate the leak-off through the fracture walls to a pressure jump across the fracture walls, and the Beavers-Joseph-Saffman slip rate coefficient to represent the fluid slip along the fracture walls. The numerical model is based on a thermodynamic framework in which all energy storage and dissipative mechanisms in the problem are identified, including the mechanisms related to the interface effects. The thermodynamic framework is employed to solve the nonlinear coupled problem up to a specified energy range through a Picard iteration technique and to study the model and its results. Studies are presented for a range of fluid entry resistance parameters and Beavers-Joseph-Saffman slip rate coefficients, showing the capability of the model to simulate skin and slip effects in a dimensionally-reduced fracture setting.

In the field of tissue engineering, in order to restore tissue functionality hydrogels that closely mimic biological and mechanical properties of the extracellular matrix are intensely developed. Mechanical properties including relaxation of the surrounding microenvironment regulate essential cellular processes. However, the mechanical properties of engineered hydrogels are particularly complex since they involve not only a nonlinear elastic behavior but also time-dependent responses. An accurate determination of these properties at microscale, i.e. as probed by cells, becomes an essential step to further design hydrogel-based biomaterials able to induce specific cellular responses. Atomic Force Microscopy (AFM) with contact sizes of the order of few micrometers constitutes an appropriate technique to determine the origin of relaxation mechanisms occurring in hydrogels. In the present study, AFM force relaxation experiments are conducted on chemically and physically crosslinked hydrogels respectively based on a synthetic polymer, polyacrylamide and a natural polymer, a bacterial exopolysaccharide infernan, produced by the deep-sea hydrothermal vent bacterium, Alteromonas infernus. Two distinct relaxation mechanisms are clearly evidenced depending on the nature of hydrogel network crosslinks. Chemically crosslinked hydrogel exhibits poroelastic relaxations, whereas physically crosslinked hydrogel shows time-dependent responses arising from viscoelastic effects. In addition, two relaxation processes are revealed in ionic physical hydrogel originating from chain rearrangement and breaking/reforming of the ionic crosslinks. The effect of the ionic strength on both the long-term elastic modulus and relaxation times of physical hydrogels was also shown. These findings highlight that physical hydrogels with well-defined time-dependent mechanical properties could be tuned for an optimized response of cells.

The Chapter provides a wide overview of the experimental techniques currently used in the field of orthopedic biomechanics. The Chapter is organized into three main sections. The first section is dedicated to “experimental tests at the organ and tissue levels, " including the techniques meant to characterize the mechanical properties of hard (i.e. bone), soft (i.e. ligaments), and biphasic and poroelastic tissues (i.e. cartilage) at different scales. The second section is focused on the “experimental tests on implants and prostheses, " often including the integration of implants with dedicated transducers (i.e. pressure sensors) and/or strain measurements (i.e. surface analysis with digital image correlation), which are used during in vitro and in vivo tests. The third section presents the state-of-the-art on the major joint simulators (hip, knee, spine). For each methodology, the basic principles are briefly recalled, and their utility is explained providing extensive references and examples from the orthopedic field., SCOPUS: ch.b, info:eu-repo/semantics/published

This paper presents a mixed finite element framework for coupled hydro-mechanical–chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic., Petroleum Engineering

We considerL-scheme and Newton-based solvers for Biot model under large deformation. The mechanical deformation follows the Saint Venant-Kirchoff constitutive law. Furthermore, the fluid compressibility is assumed to be non-linear. A Lagrangian frame of reference is used to keep track of the deformation. We perform an implicit discretization in time (backward Euler) and propose two linearization schemes for solving the non-linear problems appearing within each time step: Newton's method andL-scheme. Each linearization scheme is also presented in a monolithic and a splitting version, extending the undrained split methods to non-linear problems. The convergence of the solvers, here presented, is shown analytically for cases under small deformation and numerically for examples under large deformation. Illustrative numerical examples are presented to confirm the applicability of the schemes, in particular, for large deformation.

Non-destructive testing (NDT) plays an important role in the engineering, construction, and geophysical fields. The application of NDT in civil engineering is broad from quality control, structural health monitoring of infrastructure, geophysical and geotechnical field investigation and material characterization to detection of underground anomaly, among others. One of the frequently used NDT techniques for the characterization of geomaterials is based on the propagation of stress waves generated by an excitation source. However, the existing signal interpretation methods still predominantly rely on empirical relations or subjective judgements that are insufficient for the characterization of multiphase complex geomaterials. This research aims to develop novel physics-based signal interpretation methods to characterize physical and mechanical properties of multiphase geomaterials in both field and laboratory investigation scales. Several hybrid inverse and poromechanical models are developed to characterize dry, saturated, and frozen geomaterials subject to stress waves. First, a highly-efficient semi-analytical elastodynamic forward solver was proposed for the Multichannel Analysis of Surface Waves using the spectral element technique to determine effectively and efficiently the soil stratigraphy as well as soil properties. Next, a coupled piezoelectric and solid mechanics model is proposed to study the real response of the bender element (BE) and its interaction with soil samples in the BE test. A comprehensive laboratory investigation is also performed to better understand the response of the BEs inside different soil types. Then, a two-phase poromechanics-based signal interpretation model is developed for laboratory-scale ultrasonic testing to determine the physical and mechanical properties of saturated soil samples based on the distribution of stress waves. Subsequently, a three-phase poromechanical transfer function model is developed using the spectral eleme

Brain tissue is one of the softest tissues in the human body and the quantification of its mechanical properties has challenged scientists over the past decades. Associated experimental results in the literature have been contradictory as characterizing the mechanical response of brain tissue not only requires well-designed experimental setups that can record the ultrasoft response, but also appropriate approaches to analyze the corresponding data. Due to the extreme complexity of brain tissue behavior, nonlinear continuum mechanics has proven an expedient tool to analyze testing data and predict the mechanical response using a combination of hyper-, visco-, or poro-elastic models. Such models can not only allow for personalized predictions through finite element simulations, but also help to comprehensively understand the physical mechanisms underlying the tissue response. Here, we use a nonlinear poro-viscoelastic computational model to evaluate the effect of different intrinsic material properties (permeability, shear moduli, nonlinearity, viscosity) on the tissue response during different quasi-static biomechanical measurements, i.e., large-strain compression and tension as well as indentation experiments. We show that not only the permeability but also the properties of the viscoelastic solid largely control the fluid flow within and out of the sample. This reveals the close coupling between viscous and porous effects in brain tissue behavior. Strikingly, our simulations can explain why indentation experiments yield that white matter tissue in the human brain is stiffer than gray matter, while large-strain compression experiments show the opposite trend. These observations can be attributed to different experimental loading and boundary conditions as well as assumptions made during data analysis. The present study provides an important step to better understand experimental data previously published in the literature and can help to improve experimental setups, Peer Reviewed, Postprint (published version)

This contribution addresses an experimental study on the poroelastic behavior of a water-saturated limestone. Samples come from Sarvak limestone, a major hydrocarbon-producing reservoir in Iran. Three back-saturated intact specimens were subjected to hydrostatic compression with different boundary pressure conditions. A coherent workflow from testing to data interpretation was established to efficiently deal with possible sources of uncertainty and provide accurate results. Obtained results revealed that drained and undrained bulk moduli, as well as the Biot and Skempton coefficients, strongly depend on the applied Terzaghi effective stress. Conversely, calculated values for the unjacketed solid modulus K′s were found to be constant in the range of applied stresses. Variations of the poroelastic moduli with the effective stress were modeled and used to extract depth-dependent poroelastic properties of the studied lithotype along a well. Besides, evaluated poroelastic moduli were utilized to verify the linear theory of poroelasticity. The indirectly calculated unjacketed pore modulus K′′s appeared to be highly variable with changes in the effective stress and possesses even negative values when the Skempton coefficient is lower than a critical value. A comparison between unjacketed bulk moduli and the bulk deformation modulus of the main mineral constituents Ks also disclosed that they are basically uncorrelated, and the assumption of ideal porous rock, K′′s=K′s=Ks, does not hold in general for the examined rock samples. Nevertheless, consideration of the ideality assumption in the theory of poroelasticity may result in reliable estimates for the Skempton coefficient in certain circumstances, particularly at high effective stresses.

A simulation tool capable of speeding up the calculation for linear poroelasticity problems in heterogeneous porous media is of large practical interest for engineers, in particular, to effectively perform sensitivity analyses, uncertainty quantification, optimization, or control operations on the fluid pressure and bulk deformation fields. Towards this goal, we present here a non-intrusive model reduction framework using proper orthogonal decomposition (POD) and neural networks based on the usual offline-online paradigm. As the conductivity of porous media can be highly heterogeneous and span several orders of magnitude, we utilize the interior penalty discontinuous Galerkin (DG) method as a full order solver to handle discontinuity and ensure local mass conservation during the offline stage. We then use POD as a data compression tool and compare the nested POD technique, in which time and uncertain parameter domains are compressed consecutively, to the classical POD method in which all domains are compressed simultaneously. The neural networks are finally trained to map the set of uncertain parameters, which could correspond to material properties, boundary conditions, or geometric characteristics, to the collection of coefficients calculated from an L2 projection over the reduced basis. We then perform a non-intrusive evaluation of the neural networks to obtain coefficients corresponding to new values of the uncertain parameters during the online stage. We show that our framework provides reasonable approximations of the DG solution, but it is significantly faster. Moreover, the reduced order framework can capture sharp discontinuities of both displacement and pressure fields resulting from the heterogeneity in the media conductivity, which is generally challenging for intrusive reduced order methods. The sources of error are presented, showing that the nested POD technique is computationally advantageous and still provides comparable accuracy to the classical PO

This paper presents a mixed finite element framework for coupled hydro-mechanical–chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic.

Modeling of hydraulic fracturing processes is of great importance in computational geosciences. In this paper, a phase-field model is developed and applied for investigating the hydraulic fracturing propagation in saturated poroelastic rocks with pre-existing fractures. The phase-field model replaces discrete, discontinuous fractures by continuous diffused damage field, and thus is capable of simulating complex cracking phenomena such as crack branching and coalescence. Specifically, hydraulic fracturing propagation in a rock sample of a single pre-existing natural fracture or natural fracture networks is simulated using the proposed model. It is shown that distance between fractures plays a significant role in the determination of propagation direction of hydraulic fracture. While the rock permeability has a limited influence on the final crack topology induced by hydraulic fracturing, it considerably impacts the distribution of the fluid pressure in rocks. The propagation of hydraulic fractures driven by the injected fluid increases the connectivity of the natural fracture networks, which consequently enhances the effective permeability of the rocks., QC 20200629

A coupled approach, combining the theory of rate- and state-dependent friction and methods from poroelasticity, forms the basis for a quantitative relationship between displacements and fluid leak-off from a growing fracture and changes in the rate of seismic events in the region surrounding the fracture. Poroelastic Green's functions link fracture aperture changes and fluid flow from the fracture to changes in the stress field and pore pressure in the adjacent formation. The theory of rate- and state-dependent friction provides a connection between Coulomb stress changes and variations in the rate of seismic events. Numerical modeling indicates that the Coulomb stress changes can vary significantly between formations with differing properties. The relationship between the seismicity rate changes and the changes in the formation stresses and fluid pressure is nonlinear, but a transformation produces a quantity that is linearly related to the aperture changes and fluid leak-off from the fracture. The methodology provides a means for mapping changes in seismicity into fracture aperture changes and to image an evolving fracture. An application to observed microseismicity associated with a hydrofracture reveals asymmetric fracture propagation within two main zones, with extended propagation in the upper zone. The time-varying volume of the fracture agrees with the injected volume, given by the integration of rate changes at the injection well, providing validation of the estimated aperture changes.

Modelling dynamic rupture is essential to correctly describe the process of induced seismicity. Defmod, an open-source finite-element code featuring quasi-static loading, co-seismic volumetric strain, and dynamic rupture, is used to simulate the entire chain of induced seismicity, from pressure evolution due to fluid injection and extraction, building up of stress, and nucleation of dynamic faulting, to wavefield propagation towards the surface. To study induced earthquakes caused by fluid extraction, we modelled the behaviour of a 2-D poroelastic medium including a predefined fault by assigning a fluid source, either constant or varying, in a homogeneous reservoir layer to induce a pressure-field change. For each quasi-static step, the pressure field difference generates a displacement field that in turn affects the pressure through a coupling matrix, depending on Biot's coefficient. The rate of pressure variation is subject to the fluid source as well as the material properties, e.g., porosity and fluid mobility, which affect the speed and distribution of the stress build-up on the fault and thus the pattern of rupture nucleation. In addition, we implemented a predefined pressure profile to simulate the induced rupture in case of a uniform depletion of the reservoir to allow for a comparison with other studies. The results provide useful insights on the causality between reservoir-pressure behaviour and the induced seismicity., Accepted Author Manuscript, Applied Geophysics and Petrophysics

The theory of morpho-poroelasticity is applied to tumor growth. This allows for the modeling of permanent deformation in the tissue as a result of the presence of tumor cells. The work done in this project can be divided into two parts. In the first part we review existing models for elasticity and build corresponding finite element models. The aim of the first part is to gain understanding in the behaviour of morphoelasticity and poroelasticity. In morphoelasticity the deformation tensor is decomposed into an elastic and plastic component, allowing for permanent deformations to occur in the tissue. The theory of morphoelasticity describes the interplay between a fluid in a porous tissue and the tissue's elastic properties. A common problem in poroelastic models is the occurence of non-physical oscillations in the pressure. The second part of this project contains novel work. First, a rigorous mathematical derivation is presented of the tuning parameter found in the diffusive stabilization method for poroelastic systems. Secondly, we present a finite element model for the novel combination of morpho- and poroelasticity. The derivation concerning the diffusive stabilization is also applicable to the morpho-poroelastic finite element model. We obtain a promising tumor growth model by combining a biochemcial tumor growth model with morpho-poroelasticity. This biochemical model is based on a nutrient transport equation and a tumor cell density evolution equation. Some example output of the tumor growth finite element model is shown. It can be used as a starting point for more elaborate tumor growth models.

We present the macroscale three-dimensional numerical solution of anisotropic Biot's poroelasticity, with coefficients derived from a micromechanical analysis as prescribed by the asymptotic homogenisation technique. The system of partial differential equations (PDEs) is discretised by finite elements, exploiting a formal analogy with the fully coupled thermal displacement systems of PDEs implemented in the commercial software Abaqus. The robustness of our computational framework is confirmed by comparison with the well-known analytical solution of the one-dimensional Therzaghi's consolidation problem. We then perform three-dimensional numerical simulations of the model in a sphere (representing a biological tissue) by applying a given constant pressure in the cavity. We investigate how the macroscale radial displacements (as well as pressures) profiles are affected by the microscale solid matrix compressibility (MSMC). Our results suggest that the role of the MSMC on the macroscale displacements becomes more and more prominent by increasing the length of the time interval during which the constant pressure is applied. As such, we suggest that parameter estimation based on techniques such as poroelastography (which are commonly used in the context of biological tissues, such as the brain, as well as solid tumours) should allow for a sufficiently long time in order to give a more accurate estimation of the mechanical properties of tissues.

We study and compare five different combinations of finite element spaces for approximating the coupled flow and solid deformation system, so-called Biot’s equations. The permeability and porosity fields are heterogeneous and depend on solid displacement and fluid pressure. We provide detailed comparisons among the continuous Galerkin, discontinuous Galerkin, enriched Galerkin, and two types of mixed finite element methods. Several advantages and disadvantages for each of the above techniques are investigated by comparing local mass conservation properties, the accuracy of the flux approximation, number of degrees of freedom (DOF), and wall and CPU times. Three-field formulation methods with fluid velocity as an additional primary variable generally require a larger number of DOF, longer wall and CPU times, and a greater number of iterations in the linear solver in order to converge. The two-field formulation, a combination of continuous and enriched Galerkin function space, requires the fewest DOF among the methods that conserve local mass. Moreover, our results illustrate that three out of the five methods conserve local mass and produce similar flux approximations when conductivity alteration is included. These comparisons of the key performance indicators of different combinations of finite element methods can be utilized to choose the preferred method based on the required accuracy and the available computational resources.

Multiscale homogenization represents a powerful tool to treat certain fluid-structure interaction problems involving porous, elastic, fibrous media. This is shown here for the case of the interaction between a Newtonian fluid and a poroelastic, microstructured material. Microscopic problems are set up to determine effective tensorial properties (elasticity, permeability, porosity, bulk compliance of the solid skeleton) of the homogenized medium, both in the interior and at its boundary with the fluid domain, and an extensive description is provided of such properties for varying porosity. The macroscopic equations which are derived by homogenization theory employ such effective properties thus permitting the computation of velocities and displacements within the poroelastic mixture for two representative configurations of standing and travelling waves., QC 20190109

Induced seismicity linked to geothermal resource exploitation, hydraulic fracturing, and wastewater disposal is evolving into a global issue because of the increasing energy demand. Moderate to large induced earthquakes, causing widespread hazards, are often related to fluid injection into deep permeable formations that are hydraulically connected to the underlying crystalline basement. Using injection data combined with a physics-based linear poroelastic model and rate-and-state friction law, we compute the changes in crustal stress and seismicity rate in Oklahoma. This model can be used to assess earthquake potential on specific fault segments. The regional magnitude-time distribution of the observed magnitude (M) 3+ earthquakes during 2008-2017 is reproducible and is the same for the 2 optimal, conjugate fault orientations suggested for Oklahoma. At the regional scale, the timing of predicted seismicity rate, as opposed to its pattern and amplitude, is insensitive to hydrogeological and nucleation parameters in Oklahoma. Poroelastic stress changes alone have a small effect on the seismic hazard. However, their addition to pore-pressure changes can increase the seismicity rate by 6-fold and 2-fold for central and western Oklahoma, respectively. The injection-rate reduction in 2016 mitigates the exceedance probability of M5.0 by 22% in western Oklahoma, while that of central Oklahoma remains unchanged. A hypothetical injection shut-in in April 2017 causes the earthquake probability to approach its background level by ∼2025. We conclude that stress perturbation on prestressed faults due to pore-pressure diffusion, enhanced by poroelastic effects, is the primary driver of the induced earthquakes in Oklahoma.

Author Posting. © American Geophysical Union, 2018. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Journal of Geophysical Research: Solid Earth 123(11), (2018): 9376-9406. doi: 10.1029/2018JB015985., Improved constraints on the mechanical behavior of magma chambers is essential for understanding volcanic processes; however, the role of crystal mush on the mechanical evolution of magma chambers has not yet been systematically studied. Existing magma chamber models typically consider magma chambers to be isolated melt bodies surrounded by elastic crust. In this study, we develop a physical model to account for the presence and properties of crystal mush in magma chambers and investigate its impact on the mechanical processes during and after injection of new magma. Our model assumes the magma chamber to be a spherical body consisting of a liquid core of fluid magma within a shell of crystal mush that behaves primarily as a poroelastic material. We investigate the characteristics of time‐dependent evolution in the magma chamber, both during and after the injection, and find that quantities such as overpressure and tensile stress continue to evolve after the injection has stopped, a feature that is absent in elastic (mushless) models. The time scales relevant to the postinjection evolution vary from hours to thousands of years, depending on the micromechanical properties of the mush, the viscosity of magma, and chamber size. We compare our poroelastic results to the behavior of a magma chamber with an effectively viscoelastic shell and find that only the poroelastic model displays a time scale dependence on the size of the chamber for any fixed mush volume fraction. This study demonstrates that crystal mush can significantly influence the mechanical behaviors of crustal magmatic reservoirs., We thank James Rice, Tushar Mittal, Chris Huber and Helge Gonnerman for useful discussions in the early stages of this work. S. Adam Soule was supported by National Science Foundation Grant OCE‐1333492. Meghan Jones was supported by the U.S. Department of Defense through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program. The numerical codes used for computing the results in the work can be found at https://github.com/YangVol/MushyChamber., 2019-03-30

Poroelastic problems require multiscale and multiphysics techniques that are expensive and time-consuming, which result in either several simplifications or costly experimental tests. The latter motivates us to develop a more efficient approach to address more complex problems with an acceptable computational cost. In this manuscript, first, the necessary equations derived from Asymptotic homogenisation for poroelastic media are mentioned. Then, the variational formulation of the cell problems is carried out and solved by the open-source FE package FEniCS. This is followed by presenting the advantages and downsides of macroscale properties identification via asymptotic homogenisation and the application of Artificial Neural Network (ANN) to solve the issues stated as its downsides by means of bypassing the process of solving the cell problems. Finally, we study a practical example, namely, spatial dependent porosity (in macroscale) to demonstrate the feasibility of using the provided framework to include more details. Further applications, including growth and remodelling, are subjects of future articles.

Recently, an accurate coupling between subsurface flow and reservoir geomechanics has received more attention in both academia and industry. This stems from the fact that incorporating a geomechanics model into upstream flow simulation is critical for accurately predicting wellbore instabilities and hydraulic fracturing processes. One of the recently introduced iterative coupling algorithms to couple flow with geomechanics is the undrained split iterative coupling algorithm as reported by Kumar et al. (2016) and Mikelic and Wheeler (Comput. Geosci. 17: 455–461 2013). The convergence of this scheme is established in Mikelic and Wheeler (Comput. Geosci. 17:455–461 2013) for the single rate iterative coupling algorithm and in Kumar et al. (2016) for the multirate iterative coupling algorithm, in which the flow takes multiple finer time steps within one coarse mechanics time step. All previously established results study the convergence of the scheme in homogeneous poroelastic media. In this work, following the approach in Almani et al. (2017), we extend these results to the case of heterogeneous poroelastic media, in which each grid cell is associated with its own set of flow and mechanics parameters for both the single rate and multirate schemes. Second, following the approach in Almani et al. (Comput. Geosci. 21:1157–1172 2017), we establish a priori error estimates for the single rate case of the scheme in homogeneous poroelastic media. To the best of our knowledge, this is the first rigorous and complete mathematical analysis of the undrained split iterative coupling scheme in heterogeneous poroelastic media.

We consider single rate and multirate explicit schemes for the Biot system modeling coupled flow and geomechanics in a poro-elastic medium. These schemes are widely used in practice that follows a sequential procedure in which the flow and mechanics problems are fully decoupled. In such a scheme, the flow problem is solved first with time-lagging the displacement term followed by the mechanics solve. The multirate explicit coupling scheme exploits the different time scales for the mechanics and flow problems by taking multiple finer time steps for flow within one coarse mechanics time step. We provide fully discrete schemes for both the single and multirate approaches that use Backward Euler time discretization and mixed spaces for flow and conformal Galerkin for mechanics. We perform a rigorous stability analysis and derive the conditions on reservoir parameters and the number of finer flow solves to ensure stability for both schemes. Furthermore, we investigate the computational time savings for explicit coupling schemes against iterative coupling schemes.

Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic-poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct.

Monitoring the properties of a CO2 storage reservoir is important for two main reasons: firstly, to verify that the injected CO2 is safely contained in the reservoir rock as planned, and secondly, to provide data which can be used to update the existing reservoir models and support eventual mitigation measures in case of deviation from the CO2 storage plan. Reliable quantitative monitoring of reservoir rocks and pore-filling fluids remains a challenging task in geophysical prospecting. Typically, geophysical electrical and seismic surveys are used to predict reservoir properties. Electrical properties of rocks are strongly controlled by the chemistry of the fluids that fill the pore space. Thus, electrical surveys can provide accurate estimates of pore-filling fluid composition and porosity of reservoir rock. Seismic methods are particularly sensitive to elastic heterogeneities in the subsurface. Elastic properties of reservoir rocks can be extracted from recorded seismic data. Potentially, the simultaneous use of electrical and seismic geophysical surveys can reduce the uncertainty in the quantitative characterization of reservoir rocks. In this thesis, theoretical and experimental research is conducted to show the feasibility of such an integrated approach for a CO2 storage reservoir., Applied Geophysics and Petrophysics

Bacterial cellulose (BC) has several current and potential future uses in the food industry because of its ability to form hydrogels with distinctive properties. The texture of BC hydrogels is determined by both the cellulose fibre network and the internal dispersed water. In this study, mechanical properties of hydrated BC synthesised by six different strains of Komagataeibacter genus were investigated with regards to their extensibility, compressive strength, relaxation ability, viscoelasticity and poroelasticity. The stress/strain at failure and Young's modulus were assessed by uniaxial tensile testing. The compressive strength, relaxation ability and viscoelasticity were measured via a series of compression and small amplitude oscillatory shear steps. A poroelastic constitutive modelling simulation was used to investigate the mechanical effects of water movement. The morphology of the BC fibril network under compression was observed via scanning electron microscopy. Results showed that the mechanics of BC were highly dependent on the cellulose concentration, as well as the morphology of the fibril network. BC synthesised by ATCC 53524 was the most concentrated (0.71 wt%), and exhibited high tensile properties, stiffness and storage moduli; whereas comparatively low mechanical properties were noted for BC produced by ATCC 700178 and ATCC 10245, which contained the lowest cellulose concentration (0.18 wt%). Small deformation responses (normal stress, G′) scaled with cellulose concentration for all samples, whereas larger deformation responses (Young's modulus, poroelasticity) depended on both cellulose concentration and additional factors, presumably related to network morphology. Increasing concentration and compressive coalescence of fibres in the integrated BC network reduced both the relaxation of the normal stress and the movement of water. This research aids the selection of bacterial strains to modulate the texture and mechanical properties of hydrated BC-b

The permeability of the oceanic crust exerts a primary influence on the vigor of hydrothermal circulation at mid-ocean ridges, but it is a difficult to measure parameter that varies with time, space, and geological setting. Here we develop an analytical model for the poroelastic response of hydrothermal exit-fluid velocities and temperatures to ocean tidal loading in a two-layered medium to constrain the discharge zone permeability of each layer. The top layer, corresponding to extrusive lithologies (e.g., seismic layer 2A) overlies a lower permeability layer, corresponding to intrusive lithologies (e.g., layer 2B). We apply the model to three basalt-hosted hydrothermal fields (i.e., Lucky Strike, Main Endeavour and 9 degrees 46N L-vent) for which the seismic stratigraphy is well-established, and for which robust exit-fluid temperature data are available. We find that the poroelastic response to tidal loading is primarily controlled by layer 2A permeability, which is about 3 orders of magnitude higher for the Lucky Strike site (approximate to 10(-10) m(2)) than the 9 degrees 46N L-vent site (approximate to 10(-13) m(2)). By contrast, layer 2B permeability does not exert a strong control on the poroelastic response to tidal loading, yet strongly modulates the heat output of hydrothermal discharge zones. Taking these constraints into account, we estimate a plausible range of layer 2B permeability between approximate to 10(-15) m(2) and an upper-bound value of approximate to 10(-14) (9 degrees 46N L-vent) to approximate to 10(-12) m(2) (Lucky Strike). These permeability structures reconcile the short-term response and long-term thermal output of hydrothermal sites, and provide new insights into the links between permeability and tectono-magmatic processes along the global mid-ocean ridge.

In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based on a point-wise Vanka smoother that couples the pressure variable at a grid-point with the displacements around it. A semi-algebraic mode analysis is proposed to theoretically analyze the convergence of the multigrid waveform relaxation algorithm. This analysis is novel since it combines the semi-algebraic analysis, suitable for parabolic problems, with the non-standard analysis for overlapping smoothers. The practical utility of the method is illustrated through several numerical experiments in one and two dimensions.

In this work, we consider the popular P1–RT0–P0 discretization of the three-field formulation of Biot's consolidation problem. Since this finite-element formulation is not uniformly stable with respect to the physical parameters, several issues arise in numerical simulations. For example, when the permeability is small with respect to the mesh size, volumetric locking may occur. To alleviate such problems, we consider a well-known stabilization technique with face bubble functions. We then design a perturbation of the bilinear form, which allows for local elimination of the bubble functions. We further prove that such perturbation is consistent and the resulting scheme has optimal approximation properties for both Biot's model as well as the Stokes’ equations. For the former, the number of degrees of freedom is the same as for the classical P1–RT0–P0 discretization and for the latter (Stokes’ equations) the number of degrees of freedom is the same as for a P1–P0 discretization. We present numerical tests confirming the theoretical results for the poroelastic and the Stokes’ test problems.

Improved constraints on the mechanical behavior of magma chambers is essential for understanding volcanic processes; however, the role of crystal mush on the mechanical evolution of magma chambers has not yet been systematically studied. Existing magma chamber models typically consider magma chambers to be isolated melt bodies surrounded by elastic crust. In this study, we develop a physical model to account for the presence and properties of crystal mush in magma chambers and investigate its impact on the mechanical processes during and after injection of new magma. Our model assumes the magma chamber to be a spherical body consisting of a liquid core of fluid magma within a shell of crystal mush that behaves primarily as a poroelastic material. We investigate the characteristics of time‐dependent evolution in the magma chamber, both during and after the injection, and find that quantities such as overpressure and tensile stress continue to evolve after the injection has stopped, a feature that is absent in elastic (mushless) models. The time scales relevant to the postinjection evolution vary from hours to thousands of years, depending on the micromechanical properties of the mush, the viscosity of magma, and chamber size. We compare our poroelastic results to the behavior of a magma chamber with an effectively viscoelastic shell and find that only the poroelastic model displays a time scale dependence on the size of the chamber for any fixed mush volume fraction. This study demonstrates that crystal mush can significantly influence the mechanical behaviors of crustal magmatic reservoirs.

Biofilm formation in membrane systems negatively impacts the filtration system performances. This study evaluated how biofilm compression driven by permeate flow increases the hydraulic resistance and leads to reduction in permeate flux. We analysed the effect of biofilm compression on hydraulic resistance and permeate flux through computational models supported by experimental data. Biofilms with homogeneous surface structure were subjected to step-wise changes in flux and transmembrane pressure during compression and relaxation tests. Biofilm thickness under applied forces was measured non-invasively in-situ using optical coherence tomography (OCT). A numerical model of poroelasticity, which couples water flow through the biofilm with biofilm mechanics, was developed to correlate the structural deformation with biofilm hydraulics (permeability and resistance). The computational model enabled extracting mechanical and hydrological parameters corresponding to the experimental data. Homogeneous biofilms under elevated compression forces experienced a significant reduction in thickness while only a slight increase in resistance was observed. This shows that hydraulic resistance of homogeneous biofilms was affected more by permeability decrease due to pore closure than by a decrease in thickness. Both viscoelastic and elastoplastic models could describe well the permanent biofilm deformation. However, for biofilms under study, a simpler elastic model could also be used due to the small irreversible deformations. The elastic moduli fitting the measured data were in agreement with other reported values for biofilm under compression. Biofilm stiffening under larger flow-driven compression forces was observed and described numerically by correlating inversely the elastic modulus with biofilm porosity. The importance of this newly developed method lies in estimation of accurate biofilm mechanical parameters to be used in numerical models for both membrane filtration syst, BT/Environmental Biotechnology

Over the past decade, a steep increase in the number of seismic events has been observed in the Groningen gas field, the Netherlands. It is generally accepted that these are induced by compaction of the reservoir rock due to extensive depletion, causing a buildup of strain energy in faults that may be released seismically. We address the possible triggering of fault slip by the transient pressure field surrounding a well that has undergone a sudden rate change. Assuming a unilateral decoupling between displacement and pressure, numerical experiments are conducted using a sequential finite volume-finite element solution strategy that fully incorporates second-order terms in the radial flow equation. We investigate an idealized Groningen-like geometry to discover whether the hypothesized possibility of triggering-induced seismicity exists, explore how some controllable and uncontrollable variables influence its severity, and possibly provide clues on how production strategy might be able to avoid its occurrence. The results demonstrate that sudden production changes can indeed trigger near-well seismic events, but that the effect is very small compared to other potential causes. Changing from sudden to gradual changes in well rates is therefore not expected to lead to a significant reduction in the number or magnitude of production-induced earthquakes in the Groningen field., Civil Engineering and Geosciences, Geoscience and Engineering

Study designDisc recovery behavior under hypo- and hyperosmotic pressure.ObjectiveTo evaluate the effect of osmotic pressure on the unloaded recovery response of healthy discs.Summary of background dataThe intervertebral disc is a poroviscoelastic material that experiences large fluctuations in water composition throughout a diurnal loading cycle. Fluid flow out of the disc occurs through mechanical loading, whereas fluid flow into the disc occurs through passive diffusion because of an imbalance of ions between the disc and its surrounding environment. Osmotic pressure has been used to alter water uptake and tissue hydration.MethodsMotion segments were prepared from the caudal spine sections of the skeletally mature bovines. A 300-N compressive load was applied for 2 hours before unloaded recovery for 12 hours. Hypo- and hyperosmotic pressure was used to alter the rate of water uptake and disc height recovery during unloaded recovery. A 5-parameter rheological model was used to describe the disc's time-dependent recovery behavior.ResultsThe elastic response was not altered by changes in osmotic pressure; however, viscoelastic recovery was highly dependent on saline osmolarity and recovery time. The fast response of viscoelastic recovery was not dependent on osmotic pressure. The time constant for the slow response decreased whereas the slow response stiffness increased as osmotic pressure increased.ConclusionThe fast response of viscoelastic recovery is governed by flow-independent recovery, whereas the slow response is related to flow-dependent recovery. The rate and magnitude of flow-dependent recovery are highly sensitive to changes in osmotic pressure of the saline bath. There is an osmotic pressure that reduces disc recovery behavior to an elastic response or flow-independent recovery.Level of evidenceN/A.