Poroelasticity

This paper outlines the theoretical framework and experimental validation of a fully Thermo-Hydro-Mechanical model, ThermalEngine, as part of the open source DEM software Yade. ThermalEngine is constructed on top of Yade's existing thoroughly validated and well optimized Pore Finite Volume (PFV) scheme, which enables a highly efficient upwind forward finite difference advection scheme that reuses existing pore scale mass-fluxes to conserve pore energy. Conductive heat transfer within phases and between phases is solved explicitly in time by conserving energy and modeling heat fluxes with classic heat transfer models. In addition to advection and conduction, the thermo-mechanical component of ThermalEngine incorporates the thermal expansion or contraction of particles and pore fluids by estimating pore volume changes and adding them to the existing PFV mass balance. In addition to the theoretical framework, the experimental and numerical validation of the coupled THM ThermalEngine model is presented by heating a cylindrical rock specimen and observing interior cavity fluid pressure changes. Cavity fluid pressure changes follow experimental data closely; as rock temperature increases the cavity fluid expands resulting in an increased cavity pressure until pressure gradients direct fluid out of cavity allowing fluid pressure to stabilize back to initial conditions. It is clear that the foundational, easily modifiable, and open sourced THM-DEM framework presented within is applicable to a wide range of disciplines including geomechanics and powder technologies.

"Quantitative methods are increasingly important in the earth sciences but there are few good books specifically written to help earth scientists develop the mathematical tools that they need. This book helps to fill the gap by providing a concise introduction to mathematical modelling with an emphasis on examples taken from the earth sciences. One of the major strengths of the book is that it is possible to get a good overview of a whole topic during an initial read of the relevant chapter without getting too bogged down in detail. Where additional detail is provided it is invariably worth further study. Another strength of the book is the large number of worked examples embedded in the text almost every other double-page spread seems to have one. With very few exceptions, these examples are of great help in providing both focus and clarity to what may sometimes seem a little obscure. The style of the book is informal - I wish that it had been available when I was an undergraduate to help counterbalance the theorem-and-proof approach that I was subjected to... In short, the book provides excellent value for money and is a well-crafted introduction to mathematical modelling for earth scientists. Armed with insights obtained from this book, the reader will be well placed for further study of selected topics at the next level of detail. "
--Geoscientist, January 2009


Mathematical modelling and computer simulations are an essential part of the analytical toolset used by earth scientists. Computer simulations based on mathematical models are routinely used to study geophysical, environmental and geological processes in many areas of work and research from geophysics to petroleum engineering and from hydrology to environmental fluid dynamics. Dr Yang has carefully selected topics which will be of most value to students and has recognised the need to be careful in his examples whilst being comprehensive enough to include important topics and popular algorithms. The book is designed to be theorem-free and yet to balance formality and practicality. Using worked examples and tackling each problem in a step-by-step manner the text is especially suitable for non-mathematicians approaching this aspect of earth sciences for the first time, The coverage and level, for instance in the calculus of variation and pattern formation, that even mathematicians will find the examples interesting. Topics covered include: vector and matrix analysis ordinary differential equations partial differential equations calculus of variations integral equations probability geostatistics numerical integration optimisation finite difference methods finite volume methods finite element methods reaction-diffusion system elasticity fracture mechanics poroelasticity, and flows in porous media. Mathematical Modelling for Earth Sciences introduces a wide range of mathematical modelling and numerical techniques, and is written for undergraduates and graduate students.

Using the method of complex potentials we present the closed form of analytical solution for the stresses and displacements around an elliptical cavity filled with fluid in the permeable poroelastic medium. The far-field pore pressure is different from the pressure inside cavity. The fluid is allowed to diffuse through the cavity's wall towards surrounding medium. The two-dimensional plane-strain and plane-stress models are considered with steady-state distribution of diffusive pore-fluid pressure. The diffusion of fluid is coupled to the deformation of a medium using a linear theory of poroelasticity. Since in the static case the diffusion of pore fluid is controlled by an ordinary Laplace equation, the problem is reduced to elastic problem with distribution of volume forces (known as seepage forces) along gradients of pore pressure. Due to similarity between poroelasticity and thermoelasticity, the present solution is applicable both to the poroelastic and thermoelastic problems with a steady-state distribution of pore pressure and temperature, respectively. Since it is getting more common to model the deformation coupled to Darcy-flow in geosciences and no clear criteria has been suggested for verification of numerical codes, we discuss the possible application of the analytical solution as the benchmark for testing of poroelastic and thermoelastic numerical codes. To stimulate a wider use of the solution for stresses, displacements and pore-fluid pressure around an elliptical cavity are implemented in MATLAB and can be downloaded from the web.

Efforts to diversify the energy portfolio and to alleviate environmental impacts are rendering the activity of injecting fluid into a geologic formation as important as extracting fluid from it. Fluid injection that is associated with a thermal effect can cause more complex geomechanical responses than fluid extraction does. To test these responses, we conducted in-depth investigations of pore-pressure buildup, temperature, and stress changes under isothermal or nonisothermal injection scenarios, using numerical simulations. The numerical simulation method combines fluid flow, poroelasticity, thermal diffusion, and thermal stress, and is based on the single-phase fluid flow condition. We also examined temporal evolutions of stress states and mobilized friction angles across base, injection zone, and caprock layers for two different stress regimes, normal-faulting and reverse-faulting. Results show that injecting cold fluid shifts an area in which the mobilized friction angle becomes maximum, which is where induced-seismic events are likely to be triggered first, for both stress regimes. We also tested a hypothetical stepwise injection of cold fluid; results show that this injection helps to improve the stability of the injection zone to some extent. Finally, we suggest using dimensionless parameters to determine a prevalence of the thermal-stress effect in the injection zone, and discuss the impact of the initial stress regime and implications of using the single-phase fluid flow condition.

Osmotic phenomena influence the intervertebral disc biomechanics. Their simulation is challenging and can be undertaken at different levels of complexity. Four distinct approaches to simulate the osmotic behaviour of the intervertebral disc (a fixed boundary pore pressure model, a fixed osmotic pressure gradient model in the whole disc or only in the nucleus pulposus, and a swelling model with strain-dependent osmotic pressure) were analysed. Predictions were compared using a 3D poroelastic finite element model of a L4–L5 spinal unit under three different loading conditions: free swelling for 8 h and two daily loading cycles: (i) 200 N compression for 8 h followed by 500 N compression for 16 h; (ii) 500 N for 8 h followed by 1000 N for 16 h. Overall, all swelling models calculated comparable results, with differences decreasing under greater loads. Results predicted with the fixed boundary pore pressure and the fixed osmotic pressure in the whole disc models were nearly identical. The boundary pore pressure model, however, cannot simulate differential osmotic pressures in disc regions. The swelling model offered the best potential to provide more accurate results, conditional upon availability of reliable values for the required coefficients and material properties. Possible fields of application include mechanobiology investigations and crack opening and propagation. However, the other approaches are a good compromise between the ease of implementation and the reliability of results, especially when considering higher loads or when the focus is on global results such as spinal kinematics.

A validated model of water transport in the cerebral environment is both an ambitious and timely task; many brain diseases relate to imbalances in water regulation. From tumours to strokes, chronic or acute, transport of fluid in the brain plays a crucial role. The importance and complexity of the brain, together with the range of unmet clinical needs that are connected with this organ, make the current research a high-priority.

One of the most paradoxical cerebral conditions, hydrocephalus, serves as an excellent metric for judging the success of anymodel developed. In particular, normal pressure hydrocephalus (NPH) is a paradoxical condition with no known cure and existing treatments display unacceptably high failure rates. NPH is considered to be a disease of old age, and like many such diseases, it is related to a change in the transport of fluid in the cerebral environment. This complex system ranges from organ-level transport to cellular membrane channels such as aquaporins; through integrating it in a novel mathematical framework, we suggest that the underlying logic of treatment methods may be misleading.

By modelling the transport of cerebrospinal fluid (CSF) between the ventricular system, cerebral tissue and blood networks, we find that changes to the biophysical properties of the brain (rather than structural changes such as aqueduct obstruction) are capable of producing clinically relevant ventriculomegaly in the absence of any obstruction to CSF flowthrough the ventricular system. Specifically, the combination of increased leakiness and compliance of the capillary bed leads to the development of enlarged ventricles with a normal ventricular pressure, replicating clinical features of the presentation of NPH. These results, while needing experimental validation, imply that treatment methods like shunting, that are focussed on structural manipulation, may continue to fail at unacceptably high rates.

Governing equations of poroelastodynamics in time and frequency domain are derived. The continuity equation complements the momentum balance equations. After reduction for spherical symmetry (geometry and loading), the governing equations in frequency domain are solved by introducing wave potentials. The wave propagation velocities are obtained as the real parts of the characteristic equation of the coupled ODE system. Time domain solution for Dirac type boundary pressure is obtained through numerical inversion of transformed solutions. The results are compared to the solution in classical elasticity theory found in the literature.

Growth, invasion and metastasis of cancers are directly linked to the cancer's stiffness and the solid stress (SSg) that develops inside the cancer. Currently, there are no non-invasive methods to assess the SSg distribution in cancers. In this paper, we develop an analytical model for the compression-induced solid stress (SSc) distribution that generates inside a tumor in a poroelastography experiment. We theoretically prove that the SSc has the same spatial distribution as the SSg inside the tumor. We also demonstrate that it is possible to estimate the spatial distribution parameter of SSg α and the ratio between vascular permeability (VP) and interstitial permeability (IP) from the computed SSc. The developed analytical model is validated using finite element and ultrasound simulations. The technical feasibility of the proposed technique is demonstrated in a small animal model study in vivo. Based on the influential role of solid stress in modulating the cancer microenvironment, the proposed methodology may be useful in cancer diagnosis, prognosis and treatment.

Accurate prediction of reservoir production in structurally weak geologic areas requires both mechanical deformation and fluid flow modeling. Loose staggered-in-time coupling of two independent flow and mechanics simulators captures much of the complex physics at a substantially reduced cost. Two 3-D finite element simulators—Integrated Parallel Accurate Reservoir Simulator (IPARS) for flow and JAS3D for mechanics—together model multiphase fluid flow in reservoir rocks undergoing deformation ranging from linear elasticity to large, nonlinear inelastic compaction. The loose coupling algorithm uses a high-level driver to call the flow simulator for a set of time steps with fixed reservoir properties. Pore pressures from flow are used as loads for the geomechanics code in the determination of stresses, strains, and displacements. The mechanics-derived strain is used to calculate changes to the reservoir parameters (porosity and permeability) for the next set of flow time steps. Mass is conserved in the coupled code despite dynamically changing reservoir parameters via a modification to the Newton system for the flow equations, and an approximate rock compressibility becomes a useful preconditioner to help with convergence of the modified flow equations. Two numerical experiments illustrate the accuracy of the coupled code. The first example is a quarter-five-spot waterflood undergoing poroelastic deformation, which is validated against a fully coupled simulator. Vertical displacements at the well locations match to within 10%. Moreover, experimentation shows that 13 mechanics time steps (taken over the course of 5 years of simulation time) were sufficient to achieve this result (a substantial cost savings over full coupling in which both the mechanics and flow equations must be solved at each time step). The second numerical example is based on real data from the Belridge Field in California, which illustrates one of the complex plastic constitutive relationships available in the coupled code. The results mimic behavior which was observed in the field. The coupled code serves as a prototype for loosely coupling together any two preexisting simulators modeling diverse physics. This technique produces a coupled code relatively quickly and inexpensively and has the advantage of accurately modeling complex nonlinear phenomena often observed in a real field but difficult to capture with a fully coupled simulator. Further, the code has produced promising results when used for time-lapse studies of compactible reservoirs.

Numerical studies of the intervertebral disc (IVD) are important to better understand the load transfer and the mechanobiological processes within the disc. Among the relevant calculations, fluid-related outputs are critical to describe and explore accurately the tissue properties. Porohyperelastic finite element models of IVD can describe accurately the disc behaviour at the organ level and allow the inclusion of fluid effects. However, results may be affected by numerical instabilities when fast load rates are applied. We hypothesized that such instabilities would appear preferentially at material discontinuities such as the annulus-nucleus boundary and should be considered when testing mesh convergence. A L4-L5 IVD model including the nucleus, annulus and cartilage endplates were tested under pure rotational loads, with different levels of mesh refinement. The effect of load relaxation and swelling were also studied. Simulations indicated that fluid velocity oscillations appeared due to numerical instability of the pore pressure spatial derivative at material discontinuities. Applying local refinement only was not enough to eliminate these oscillations. In fact, mesh refinements had to be local, material-dependent, and supplemented by the creation of a material transition zone, including interpolated material properties. Results also indicated that oscillations vanished along load relaxation, and faster attenuation occurred with the incorporation of the osmotic pressure. We concluded that material discontinuities are a major cause of instability for poromechanical calculations in multi-tissue models when load velocities are simulated. A strategy was presented to address these instabilities and recommendations on the use of IVD porohyperelastic models were given.

This study details the acceleration techniques and associated performance gains in the time integration of coupled poromechanical problems using the Discrete Element Method (DEM) and a Pore scale Finite Volume (PFV) scheme in Yade open DEM software. Specifically, the model is tailored for accuracy by reducing the frequency of costly matrix factorizations (matrix factor reuse), moving the matrix factorizations to background POSIX threads (multithreaded factorization), factorizing the matrix on a GPU (accelerated factorization), and running PFV pressure and force calculations in parallel to the DEM interaction loop using OpenMP threads (parallel task management). Findings show that these four acceleration techniques combine to accelerate the numerical poroelastic oedometer solution by 170x, which enables more frequent triangulation of large scale time-dependent DEM+PFV simulations (356 thousand+ particles, 2.1 million DOFs).

Current diagnosis of bone loss and osteoporosis is based on the measurement of the bone mineral density (BMD) or the apparent mass density. Unfortunately, in most clinical ultrasound densitometers: 1) measurements are often performed in a single anatomical direction, 2) only the first wave arriving to the ultrasound probe is characterized, and 3) the analysis of bone status is based on empirical relationships between measurable quantities such as speed of sound (SOS) and broadband ultrasound attenuation (BUA) and the density of the porous medium. However, the existence of a second wave in cancellous bone has been reported, which is an unequivocal signature of poroelastic media, as predicted by Biot’s poroelastic wave propagation theory. In this paper, the governing equations for wave motion in the linear theory of anisotropic poroelastic materials are developed and extended to include the dependence of the constitutive relations upon fabric—a quantitative stereological measure of the degree of structural anisotropy in the pore architecture of a porous medium. This fabric-dependent anisotropic poroelastic approach is a theoretical framework to describe the microarchitectural-dependent relationship between measurable wave properties and the elastic constants of trabecular bone, and thus represents an alternative for bone quality assessment beyond BMD alone.

Aporoelastic model for porous materials with a nested pore space structure is developed to represent the interstitial fluid flow in bone tissue. The nested porosity model is applied to the problem of determining the exchange of pore fluid between the vascular porosity (PV) and the lacunar–canalicular porosity (PLC) in bone tissue in a ramp loading in the case where the fluid and solid constituents are assumed to be compressible. The compressibility assumption is appropriate for hard tissues while the incompressibility assumption is appropriate for soft tissues. The influence of blood pressure in the PV is included in the analysis. A formula for the fluid that moves between the two porosities is developed. The analysis showed the coupling of the two porosities and their influence on each other and concluded that the PV pore pressure has an influence less than 3% on the PLC pore pressure while the absence of the PV pore pressure will affect the fluid exchange between the PV and PLC by less than 6% (the blood pressure range is 40–60 mmHg). Also the analysis has shown that the draining time of the PLC is inversely proportional to its permeability. The significance of the result is basic to the understanding of interstitial flow in bone tissue that, in turn, is basic to understanding of nutrient transport from the vasculature to the bone cells buried in the bone tissue and to the process of mechanotransduction by these cells.

For Citations:
Salem, H.S., 2001. Determination of the acoustic coupling factor of Biot's theory of elasticity, using in situ seismic measurements. Energy Sources, 23(10) December: 917-936. (A Scholarly Paper Published in an International Journal).

Abstract:
Besides other elastic, petrophysical, and hydrophysical parameters, the acoustic coupling factor (coupling factor or structure factor, μ) is a critical micro-geometrical parameter in the Biot’s theory of elasticity. The coupling factor exerts a
speci®c in¯uence on the mechanisms of propagation and attenuation of vibrations (seismic waves and acoustic signals) in porous media. It is a measure of the internal structure (geometry and shape of pores and pore tubes) of a porous medium, and
thus it is a good indicator of the degree of coupling between pore ¯uid and solid grains. It is unity for no coupling and in®nity for perfect coupling. In this study, the coupling factor is obtained for surface soils and shallow sediments, using compressional
wave velocity (¸p) determined from in situ seismic refraction measurements. The investigated soils and underlying sediments (aeration zone and aquifer) exhibit a general μ range of 1.22±3.19 (average = 1.82), corresponding to a general ¸f range of 134±2060 m/s (average = 1134m/s). The μ values are correlated to the
bulk modulus-shear modulus ratio (K/·) and the porosity (¿). A direct linear correlation between μ and K/· and an inverse polynomial correlation between μ and ¿ are obtained, with
coefficients of correlation of 0.91 and 0.98, respectively.
Also, various items related to the coupling factor, Biot’s theory, attenuation mechanisms, fast and slow compressional waves (®rst and second kinds), and shear waves are discussed.

Keywords:
Acoustic coupling factor (structure factor), attenuation mechanisms, Biot’s theory, compressibility, compressional and shear wave velocities, fast and slow compressional waves, frequency, in situ seismic refraction measurements,
permeability, porosity, rigidity, shallow sediments (aeration zone and aquifer), surface soils, tortuosity, viscosity.

Salt rock or rock with salty ground water are often encountered as host media for the underground disposal of radioactive waste. The nuclear waste, contained in a metallic canister, is usually placed inside a tunnel or a shaft excavated in the rock deposit together with a buffer of compacted bentonite inserted between the host rock and the canister to provide hydro-mechanical sealing. Due to the very low permeability and rich clay content, the bentonite acts as an osmotic semi-permeable membrane under a gradient of concentration of salt dissolved in the ground water. In addition, chemically induced expansion or shrinkage of the bentonite is generated by changes in the concentration of dissolved salt. By including such important chemical aspects, the hydro-mechanical governing equations are derived for this particular boundary value problem within the framework of a linear Biot-like isotropic poroelastic consolidation. The equations are solved analytically and a parametric study is undertaken to highlight the influence of chemical osmosis and chemical deformation on the flow and mechanical response of the bentonite buffer.