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Ellipticity of gradient poroelasticity

INTERNATIONAL JOURNAL OF ENGINEERING SCIENCE 2023年第1期184卷

作者： Eremeyev, Victor A. Univ Cagliari Via Marengo 2 I-09123 Cagliari Italy Gdansk Univ Technol Ul Gabriela Narutowicza 11-12 PL-80233 Gdansk Poland Natl Res Lobachevsky State Univ Nizhni Novgorod Pr Gagarin 23 Nizhnii Novgorod 603022 Russia

We discuss the ellipticity properties of an enhanced model of poroelastic continua called dilatational strain gradient elasticity. Within the theory there exists a deformation energy density given as a function of strains and gradient of dilatation. We show that the equilibrium equations are elliptic in the sense of Douglis-Nirenberg. These conditions are more general than the ordinary and strong ellipticity but keep almost all necessary properties of equilibrium equations. In particular, the loss of the ellipticity could be considered as a criterion of a strain localization or material instability.

关键词： Douglis-Nirenberg ellipticity Strong ellipticity Strain gradient elasticity poroelasticity Dilatational strain gradient elasticity

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Domain decomposition and partitioning methods for mixed finite element discretizations of the Biot system of poroelasticity

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COMPUTATIONAL GEOSCIENCES 2021年第6期25卷 1919-1938页

作者： Jayadharan, Manu Khattatov, Eldar Yotov, Ivan Univ Pittsburgh Dept Math Pittsburgh PA 15260 USA

We develop non-overlapping domain decomposition methods for the Biot system of poroelasticity in a mixed form. The solid deformation is modeled with a mixed three-field formulation with weak stress symmetry. The fluid flow is modeled with a mixed Darcy formulation. We introduce displacement and pressure Lagrange multipliers on the subdomain interfaces to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced to an interface problem for the Lagrange multipliers, which is solved by a Krylov space iterative method. We study both monolithic and split methods. In the monolithic method, a coupled displacement-pressure interface problem is solved, with each iteration requiring the solution of local Biot problems. We show that the resulting interface operator is positive definite and analyze the convergence of the iteration. We further study drained split and fixed stress Biot splittings, in which case we solve separate interface problems requiring elasticity and Darcy solves. We analyze the stability of the split formulations. Numerical experiments are presented to illustrate the convergence of the domain decomposition methods and compare their accuracy and efficiency.

关键词： Domain decomposition poroelasticity Biot system Splitting methods Mixed finite elements

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Accurate discretization of poroelasticity without Darcy stability Stokes-Biot stability revisited

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BIT NUMERICAL MATHEMATICS 2021年第3期61卷 941-976页

作者： Mardal, Kent-Andre Rognes, Marie E. Thompson, Travis B. Univ Oslo Dept Math Oslo Norway Simula Res Lab Fornebu Norway Simula Res Lab Dept Sci Comp & Numer Anal Fornebu Norway Univ Oxford Math Inst Oxford England

In this manuscript we focus on the question: what is the correct notion of Stokes-Biot stability? Stokes-Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot's equations of poroelasticity;such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes-Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we remark on the observation that the latter condition can be generalized to a wider class of discrete spaces. In particular: a parameter-uniform inf-sup condition for a mixed Darcy sub-problem is not strictly necessary to retain the practical advantages currently enjoyed by the class of Stokes-Biot stable Euler-Galerkin discretization schemes.

关键词： poroelasticity Biot’ s equations Mixed method Darcy stability Stokes– Biot stability

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Stability in Kelvin-Voigt poroelasticity

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BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA 2021年第2期14卷 357-366页

作者： Straughan, Brian Univ Durham Dept Math Durham DH1 3LE England

Holder continuous dependence of solutions upon the initial data is established for the linear theory of Kelvin-Voigt poroelasticity requiring only symmetry conditions upon the elastic coefficients. A novel functional ... 详细信息

关键词： Continuous dependence Kelvin– Voigt Improperly posed poroelasticity

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Finite Element Methods for Large-Strain poroelasticity/Chemotaxis Models Simulating the Formation of Myocardial Oedema

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JOURNAL OF SCIENTIFIC COMPUTING 2022年第3期92卷 92-92页

作者： Barnafi, N. A. Gomez-Vargas, B. Lourenco, W. J. Reis, R. F. Rocha, B. M. Lobosco, M. Ruiz-Baier, R. Weber dos Santos, R. Univ Milan Dept Math Federigo Enriques Via Saldini 50 I-20133 Milan Italy Univ Costa Rica Secc Matemat Sede Occidente San Ramon De Alajuela Costa Rica Univ Fed Juiz de Fora Grad Program Computat Modeling Rua Jose LourencoKelmer S-N BR-36036900 Juiz De Fora MG Brazil Monash Univ Sch Math Sci 9 Rainforest Walk Melbourne Vic 3800 Australia Monash Univ Victorian Heart Inst 9 Rainforest Walk Melbourne Vic 3800 Australia Sechenov First Moscow State Med Univ World Class Res Ctr Digital Biodesign & Personali Moscow Russia Univ Adventista Chile Casilla 7-D Chillan Chile

In this paper we propose a novel coupled poroelasticity-diffusion model for the formation of extracellular oedema and infectious myocarditis valid in large deformations, manifested as an interaction between interstitial flow and the immune-driven dynamics between leukocytes and pathogens. The governing partial differential equations are formulated in terms of skeleton displacement, fluid pressure, Lagrangian porosity, and the concentrations of pathogens and leukocytes. A five-field finite element scheme is proposed for the numerical approximation of the problem, and we provide the stability analysis for a simplified system emanating from linearisation. We also discuss the construction of an adequate, Schur complement based, nested preconditioner. The produced computational tests exemplify the properties of the new model and of the finite element schemes.

关键词： poroelasticity Reaction-diffusion Finite-strain regime Cardiac applications Oedema formation Finite element discretisation

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poroelasticity Contributes to Hydraulic-Stimulation Induced Pressure Changes

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GEOPHYSICAL RESEARCH LETTERS 2021年第6期48卷 e2020GL091468-e2020GL091468页

作者： Dutler, N. O. Valley, B. Amann, F. Jalali, M. Villiger, L. Krietsch, H. Gischig, V. Doetsch, J. Giardini, D. Swiss Fed Inst Technol Swiss Competence Ctr Energy Res Supply Elect SCCE Zurich Switzerland Univ Neuchatel Ctr Hydrogeol & Geotherm Neuchatel Switzerland Rhein Westfal TH Aachen Dept Engn Geol & Hydrogeol Aachen Germany Swiss Fed Inst Technol Dept Earth Sci Zurich Switzerland ILF Beratende Ingenieure AG Zurich Switzerland CSD Ingenieure AG Bern Switzerland

High-pressure fluid injections cause transient pore pressure changes over large distances, which may induce seismicity. The zone of influence for such an injection was studied at high spatial resolutions in six decameter-scaled fluid injection experiments in crystalline rock. Pore pressure time series revealed two distinct responses based on the lag time and magnitude of pressure change, namely, a near- and far-field response. The near-field response is due to pressure diffusion. In the far-field, the fast response time and decay of pressure changes are produced by effective stress changes in the anisotropic stress field. Our experiments confirm that fracture fluid pressure perturbations around the injection point are not limited to the near field and can extend beyond the pressurized zone.

关键词： anisotropy fractures hydraulics poroelasticity stressfield

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Intrinsic material constants of poroelasticity

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INTERNATIONAL JOURNAL OF ROCK MECHANICS AND MINING SCIENCES 2021年 142卷 104754-104754页

作者： Cheng, Alexander H. D. Univ Mississippi Oxford MS 38655 USA

Biot and Willis (1957) performed micromechanical analysis on isotropic poroelastic materials and presented three material constants, the jacketed compressibility, the unjacketed compressibility, and the coefficient of fluid content, to characterize the volumetric deformation. Brown and Korringa (1975) and Rice and Cleary (1976) recast them into a drained bulk modulus K, an unjacketed bulk modulus K-s', and an unjacketed pore modulus K-s ''. The constant K-s '' is tied to the storage capacity of porous medium;hence is highly important in petroleum and geosequestration applications. It is, however, difficult to measure in the laboratory due to the lack of a direct way to observe pore volume deformation. In this work, a physics based approach is used to explore these micromechanical constants. Three intrinsic material constants that isolate the physical mechanisms, an unjacketed solid bulk modulus K-s, a drained pore modulus K-p, and a micro inhomogeneity and anisotropy modulus K-psi, are presented to characterize the porous material. Bounds on material constants are developed based on thermodynamics and elastic stability requirements. Past attempts in measuring K-s '' using direct or indirect approaches are compiled. Most of the efforts failed to satisfy the theoretical bounds. Only those attempted to measure the change in pore volume passed the test. The intrinsic constants are used to interpret the bulk material constants, such as the drained and undrained bulk modulus, and the storage and Skempton pore pressure coefficient. The total volume change and the storage capacity can be partitioned into those attributed to solid, pore, fluid, and the microinhomogeneity compressibilities. A published data set on stress dependent Skempton pore pressure coefficient and bulk modulus is reanalyzed. The data are matched by the calibration of a single of K-psi value.

关键词： poroelasticity Micromechanics Constitutive equation Intrinsic material constant Unjacketed pore modulus Effective stress Stress dependent constitutive law

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On the Role of poroelasticity in the Propagation Mode of Natural Fractures in Reservoir Rocks

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ROCK MECHANICS AND ROCK ENGINEERING 2020年第5期53卷 2419-2438页

作者： Kamali, Amirhossein Ghassemi, Ahmad Univ Oklahoma Reservoir Geomech & Seism Res Grp Norman OK 73071 USA

Almost all reservoir rocks contain natural fractures. The presence of such discontinuities affects reservoir stimulation to a certain degree. The interaction between the natural and hydraulic fractures determines the overall fracture trajectory and the stimulated reservoir volume. In contrast to conventional hydraulic fracturing, natural fractures do not propagate in a tensile-dominated regime. Propagation of mechanically closed fractures lies in the mixed-mode propagation regime which differs from that of open fractures. Therefore, the response of natural fractures to pore pressure and stress perturbations should be carefully addressed within the context of reservoir stimulation. This study aims at investigating the response of closed natural fractures to water injection in poroelastic rocks. A poroelastic displacement discontinuity (DD) model is developed and used to shed light on the propagation of closed natural fractures. Mohr-Coulomb contact elements are used in this study to identify the contact status of the elements in the transverse direction. Moreover, fracture propagation is effectively accounted for based on linear elastic fracture mechanics. Hydro-mechanical coupling is achieved by combining an implicit finite difference scheme with the poroelastic DD model. Our simulation results indicate that natural fracture shear slip and propagation are likely to occur when the closed natural fractures are subject to direct water injection. The injection pressure required to maintain the propagation in poroelastic rocks was found to be consistently higher than the pressure in its elastic counterpart. The propagation trajectory of the wing cracks was, on the other hand, found to be similar to that in elastic rocks. It was found in our results that the formation of wing cracks and shear (secondary) cracks is an integral part of the reservoir stimulation in geothermal systems where natural fractures are in direct contact with the injection wells. This point is often

关键词： poroelasticity Displacement discontinuity Contact elements Natural fracture Wing crack

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Seismic AVO statistical inversion incorporating poroelasticity

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Petroleum Science 2020年第5期17卷 1237-1258页

作者： Kun Li Xing-Yao Yin Zhao-Yun Zong Hai-Kun Lin School of Geosciences China University of Petroleum(East China)Qingdao 266580ShandongChina Laboratory for Marine Mineral Resources Qingdao National Laboratory for Marine Science and TechnologyQingdao 266071ShandongChina

Seismic amplitude variation with offset(AVO) inversion is an important approach for quantitative prediction of rock elasticity,lithology and fluid *** Biot-Gassmann's poroelasticity,an improved statistical AVO inversion approach is *** distinguish the influence of rock porosity and pore fluid modulus on AVO reflection coefficients,the AVO equation of reflection coefficients parameterized by porosity,rock-matrix moduli,density and fluid modulus is initially derived from Gassmann equation and critical porosity *** the analysis of the influences of model parameters on the proposed AVO equation,rock porosity has the greatest influences,followed by rock-matrix moduli and density,and fluid modulus has the least influences among these model ***,a statistical AVO stepwise inversion method is implemented to the simultaneous estimation of rock porosity,rock-matrix modulus,density and fluid ***,the Laplace probability model and differential evolution,Markov chain Monte Carlo algorithm is utilized for the stochastic simulation within Bayesian *** and field data examples demonstrate that the simultaneous optimizations of multiple Markov chains can achieve the efficient simulation of the posterior probability density distribution of model parameters,which is helpful for the uncertainty analysis of the inversion and sets a theoretical fundament for reservoir characterization and fluid discrimination.

关键词： poroelasticity AVO inversion Statistical inversion Bayesian inference Seismic fluid discrimination

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Finite Element Solvers for Biot's poroelasticity Equations in Porous Media

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MATHEMATICAL GEOSCIENCES 2020年第8期52卷 977-1015页

作者： Kadeethum, T. Lee, S. Nick, H. M. Tech Univ Denmark Lyngby Denmark Cornell Univ Ithaca NY 14853 USA Florida State Univ Tallahassee FL 32306 USA

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.

关键词： poroelasticity Biot's equations Local mass conservative Finite element Two-field formulation Three-field formulation

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