Mandel's Problem Research Articles

Using a poroelastodynamic model to investigate the dynamic behaviour of articular cartilage

Background and objectiveThere is still a few studies about the poroelastic model that performed dynamic behaviour, especially for the case of the poroelastic cartilage model. Therefore, this study is aimed to use the poroelastodynamic model to simulate the dynamic behaviour of cartilage. MethodsThe governing equations of the poroelastodynamic model is firstly established. The validation of the model is initialised by modifying the equations into the static poroelastic model. The modified equations are then discretised using the finite element method. Mandel's problem is used to validate the discretised equations. The numerical solution calculated using FreeFEM++ is validated with the analytical solution for the quasi-static state and compared with the results generated using COMSOL Multiphysics software. Finally, the quasi-static solution is compared with the dynamic solution to discuss the difference in pore pressure and displacement variations of the poroelastic cartilage model. ResultsThe dynamic solution showed transient behaviour at the beginning of the excitation. When the compressive force acts on the cartilage, there are obvious fluctuations during the initial stage and then the dynamic numerical solution gradually approaches the quasi-static value over a period of time. The deduced results of the analytical solution were approximately the same as the numerical simulation results. ConclusionThis study was able to use the poroelastodynamics equation to simulate the dynamic behaviour of the poroelastic cartilage model. The comparison between the result coming from poroelastodynamics equation with that of the validated numerical solution was satisfactorily compared. The approximate similarity between the results of quasi-static and dynamic solutions underscored the importance of performing the dynamic solution for a more realistic simulation. This dynamic solution can be further used for the analysis of vibration or stress waves in future research.

A novel coupled NS‐PFEM with stable nodal integration and polynomial pressure projection for geotechnical problems

AbstractThe node‐based smoothed particle finite element method (NS‐PFEM) offers high computational efficiency but is numerically unstable due to possible spurious low‐energy mode in direct nodal integration (NI). Moreover, the NS‐PFEM has not been applied to hydromechanical coupled analysis. This study proposes an implicit stabilised T3 element‐based NS‐PFEM (stabilised node‐based smoothed particle finite element method [SNS‐PFEM]) for solving fully hydromechanical coupled geotechnical problems that (1) adopts the stable NI based on multiple stress points over the smooth domain to resolve the NI instability of NS‐PFEM, (2) implements the polynomial pressure projection (PPP) technique in the NI framework to cure possible spurious pore pressure oscillation in the undrained or incompressible limit and (3) expresses the NI for assembling coefficient matrices and calculating internal force in SNS‐PFEM with PPP as closed analytical expressions, guaranteeing computational accuracy and efficiency. Four classical benchmark tests (1D Terzaghi's consolidation, Mandel's problem, 2D strip footing consolidation and foundation on a vertical cut) are simulated and compared with analytical solutions or results from other numerical methods to validate the correctness and efficiency of the proposed approach. Finally, penetration of strip footing into soft soil is investigated, showing the outstanding performance the proposed approach can offer for large deformation problems. All results demonstrate that the proposed SNS‐PFEM with PPP is capable of tracking hydromechanical coupled geotechnical problems under small and large deformation with different drainage capacities.

Mandel's problem as a benchmark for two-dimensional nonlinear poroelasticity

In this paper, we consider Mandel's problem in the context of nonlinear single-phase poroelasticity, where it is assumed that the fluid is sightly compressible and porosity and permeability are given functions of the volume strain. In the first part of the paper we prove well-posedness of the time-discrete incremental problem by recasting the equations in an abstract form involving a pseudo-monotone operator. Further, we show existence of a Lyapunov functional yielding a global time discrete solution. In the second part, we investigate numerically the behavior of the poroelastic structure. In particular, we verify the assumptions leading to Mandel's solution. We also demonstrate some consequences of the proposed nonlinearities.

A two-grid simulation framework for fast monitoring of fault stability and ground deformation in multiphase geomechanics

We present a simulation framework for fast monitoring of fault stability and ground deformation in multiphase geomechanics and demonstrate its efficacy for a joint CO2 sequestration and enhanced oil recovery case study. Fault stability is estimated by tracking the time evolution of the change in Coulomb Failure Function at critical points on the fault. The staggered solution algorithm for the coupled problem is augmented with a feature that allows for the flow and geomechanics sub-problems to be solved on two different unstructured tetrahedral grids. We demonstrate the accuracy of the two-grid method on the classical Mandel's problem. For the field scale problem, the geomechanics grid extends to the ground surface while the flow grid is truncated at a depth above which no well activity occurs and the flow field can be assumed to remain unperturbed. This framework reduces the computational burden associated with initialization of pressure and stress fields in the overburden, allows for a study of the critical interaction between overburden and faults, allows for fast renditions of flow-induced ground deformation, and allows choosing flow and geomechanics grid resolutions independently to capture disparate length scales of the underlying physics.

The poroviscoelastodynamic solution to Mandel's problem

Mandel's problem of quasi-static poroelasticity describes axial compression of a rectangular sample from a porous, fluid-saturated, and elastic material while allowing the pore fluid to drain from the lateral boundaries. This paper reports the closed-form analytical solution to an extension of this problem that considers dynamic response of a porous, fluid-saturated, and viscoelastic specimen with similar geometry and boundary conditions to harmonic excitation. Biot's theory of poroelastodynamics, along with frequency-domain formulation of the viscoelastic constitutive behavior, are used for this purpose. Results indicate that a quasi-static approximation of the dynamic Mandel's problem could generate significantly deviant results compared to the presented dynamic solution, even at frequencies that are substantially smaller than the first natural frequency of the specimen. In this aspect, the solution delineates the existence of four timescale categories pertaining to viscous energy dissipation within the solid and fluid phases of the tested material, as well as the specimen natural frequencies and the loading excitation frequency. Relative order of the described timescales is shown to shape the various forms that the sample frequency response may take. These variations are demonstrated via three case studies involving the brain tissue, asphalt mix, and shale. The presented poroviscoelastodynamic solution verifies previous findings from experimental studies suggesting that a disentangled exhibition of the poroelastic and viscoelastic energy dissipation mechanisms may be captured by the loss angle curves of a frequency sweep test on poroviscoelastic materials.

Fundamental solutions to the transversely isotropic poroelastodynamics Mandel's problem

AbstractMany geomaterials, such as shale, and biomaterials, including human bone and cartilage, are anisotropic. Understanding the dynamic responses of such anisotropic porous media is essential in field operations and laboratory measurements. This paper presents analytical solutions to the transversely isotropic poroelastodynamics Mandel's problem, using the Fourier transform method and Cardano formula to solve coupled six‐order partial differential equations. Potential applications of the solutions include explaining coupled fluid‐solid dynamics, validating numerical algorithms, and interpreting dynamic responses of anisotropic porous media. As an example, we apply the solutions to simulate a fluid‐saturated transversely isotropic Trafalgar shale specimen subjected to harmonic excitation. The simulation shows that pore pressure builds up as frequency increases. The buildup occurs at a lower frequency for a lower horizontal permeability. The amount of pore pressure buildup due to the Mandel‐Cryer effect is sensitive to material anisotropy. The porous material's effective stiffness reaches its periodic minimum and maximum at the resonant and anti‐resonant frequencies controlled by mechanical anisotropy and pore fluid.

Field-scale fully coupled simulation of fluid flow and geomechanics: Gas storage/recovery process in a depleted sandstone reservoir

This work addresses a field-scale model, in which the equations of multiphase/multicomponent fluid flow and geomechanics were fully coupled, to simulate natural gas successive storage/recovery processes in an Iranian sandstone depleted gas reservoir. First, a 1D mechanical earth model was developed for the reservoir wells and then converted to a geomechanical 3D model of the reservoir, through sequential Gaussian simulation. The fully coupled simulation of the fluid flow and geomechanics - validated by the analytical solution of the Mandel's problem - was then performed for ten years of gas injection/production cycles in a section of the reservoir between two wells. Variations of different parameters including porosity, permeability, subsidence and effective stress were investigated versus time and compared to those of the non-fully coupled model, within the 10-year period in the studied section of the reservoir. Moreover, the results of the fully coupled model were compared to that of the non-fully coupled one in terms of the gas production rate and bottom-hole pressure in a single well. The results showed that the maximum rate of gas production as well as the maximum bottom-hole pressure predicted by the fully coupled model were 9.5 and 15% lower than those of the non-fully coupled model, respectively.

Mathematical Proof of the Mandel--Cryer Effect in Poroelasticity

We consider Mandel's problem from poroelasticity, which describes the behavior of a water saturated porous sample being sandwiched between two rigid plates. It was observed, both computationally an...

Mandel's problem reloaded

Mandel's problem considers the time-dependent, quasi-static deformation of a rectangular, laterally pervious specimen of fluid-saturated, poroelastic material after confinement by a suddenly applied, steady load from the top and bottom. Biot's theory of poroelastodynamics is used to generalize the analytical solution to this problem by accounting for the inertial body forces when the specimen is subjected to harmonic excitation. The poroelastodynamic solution entails and extends the well-known Mandel-Cryer effect of coupled poroelasticity in frequency domain.

Generalized solution to the anisotropic Mandel's problem

SummaryExisting solutions to Mandel's problem focus on isotropic, transversely isotropic, and orthotropic materials, the last two of which have one of the material symmetry axes coincide with the vertical loading direction. The classical plane strain condition holds for all these cases. In this work, analytical solution to Mandel's problem with the most general matrix anisotropy is presented. This newly derived analytical solution for fully anisotropic materials has all the three nonzero shear strains. Warping occurs in the cross sections, and a generalized plane strain condition is fulfilled. This solution can be applied to transversely isotropic and orthotropic materials whose material symmetry axes are not aligned with the vertical loading direction. It is the first analytical poroelastic solution considering mechanical general anisotropy of elasticity. The solution captures the effects of material anisotropy and the deviation of the material symmetry axes from the vertical loading direction on the responses of pore pressure, stress, strain, and displacement. It can be used to match, calibrate, and simulate experimental results to estimate anisotropic poromechanical parameters. This generalized solution is capable of reproducing the existing solutions as special cases. As an application, the solution is used to study the responses of transversely isotropic and orthotropic materials whose symmetry axes are not aligned with the vertical loading direction. Examples on anisotropic shale rocks show that the effects of material anisotropy are significant. Mandel‐Cryer's effects are highly impacted by the degree of material anisotropy and the deviation of the material symmetry axes from the vertical loading direction.

Thermo-hydro-mechanical modeling of fault discontinuities using zero-thickness interface element

Abstract In this paper, a coupled thermo-hydro-mechanical (THM) simulation in a faulted deformable porous medium is presented. This model involves solving the mass conservation, linear momentum balance, and energy balance equations which are derived from the Biot's consolidation theory. Fluid pore pressure, solid displacement, and temperature are chosen as initial variables in these equations, and the finite element method in combination with the interface element is used for spatial discretization of continuous and discontinuities (fault) parts of the medium to solve the equations. The main purpose of this study is providing precise formulations, applicability, and ability of the triple-node zero-thickness interface element in THM modeling of faults. It should be noted that the system of equations is solved using a computer code written in Matlab program. In order to verify the developed method, simulations of index problems such as Mandel's problem, and coupled modeling of a faulted porous medium and a faulted aquifer are presented. The modeling results obtained from the developed method show a very good agreement with those by other modeling methods, which indicates its accuracy.

Three-way coupling of multiphase flow and poromechanics in porous media

We formulate a three-way coupling for both single phase flow and equation-of-state (EOS) compositional flow in a poroelastic medium. The algorithm is inspired by the previous work of Dean et al. [1]. An error indicator is calculated at each time step to determine if the mechanics equation must be solved and whether the fixed-stress iterative coupling is necessary; otherwise, only the flow equation is solved with an extrapolated mean stress. The convergence of three-way coupling is established by extending the a priori analyses of fixed-stress iterative coupling by Girault et al. [2]. Numerical results for the Mandel's problem confirm these theoretical results for single phase flow. Three-way coupling achieves speedup with a factor 2.7 and 6.6 for Mandel's problem and field-scale coupled compositional flow and geomechanics simulations based on Cranfield field data respectively. Specifically, field-scale simulations of CO2 sequestration and surfactant-alternating-gas (SAG) process show that the three-way coupling substantially reduces mechanics solving time by 99.4% and 97.5% respectively compared to the fixed-stress split.

A numerical method for analyzing fault slip tendency under fluid injection with XFEM

We propose a numerical method for analyzing fault slip tendency under fluid injection using the extended finite element method (XFEM) both for fluid flow and poroelasticity. The fault is modeled as a zero-thickness interface, and we use a reduced model for the fluid flow in the fault to account for its hydraulic behavior. We use the rate- and state-dependent friction model as the fault friction model, and Biot’s theory of poroelasticity to study the coupling between fluid flow and mechanical deformation in the surrounding porous media. Since a fully coupled method between fluid flow and poromechanics is computationally expensive, we have investigated the use of the so-called fixed-stress split in this context. In such a scheme, the fluid flow problem is solved firstly by freezing the total means stress field, and then the results are used to solve the mechanical problem. The fixed-stress split is unconditionally stable, consistent and more accurate for a given number of iterations compared with other type of splitting strategies. In order to verify our method, some test cases are presented. For the coupling between fluid flow and poromechanics, we consider the Terzaghi Problem and the Mandel Problem, comparing our results with those of previously published works. While, for the mechanic problem, we compare the results with those obtained using the software Pylith.

A multiscale fixed stress split iterative scheme for coupled flow and poromechanics in deep subsurface reservoirs

In coupled flow and poromechanics phenomena representing hydrocarbon production or CO2 sequestration in deep subsurface reservoirs, the spatial domain in which fluid flow occurs is usually much smaller than the spatial domain over which significant deformation occurs. The typical approach is to either impose an overburden pressure directly on the reservoir thus treating it as a coupled problem domain or to model flow on a huge domain with zero permeability cells to mimic the no flow boundary condition on the interface of the reservoir and the surrounding rock. The former approach precludes a study of land subsidence or uplift and further does not mimic the true effect of the overburden on stress sensitive reservoirs whereas the latter approach has huge computational costs. In order to address these challenges, we augment the fixed-stress split iterative scheme with upscaling and downscaling operators to enable modeling flow and mechanics on overlapping nonmatching hexahedral grids. Flow is solved on a finer mesh using a multipoint flux mixed finite element method and mechanics is solved on a coarse mesh using a conforming Galerkin method. The multiscale operators are constructed using a procedure that involves singular value decompositions, a surface intersections algorithm and Delaunay triangulations. We numerically demonstrate the convergence of the augmented scheme using the classical Mandel's problem solution.

Generalized Biot's theory and Mandel's problem of multiple‐porosity and multiple‐permeability poroelasticity

AbstractThis paper finds in Biot's theory of poroelasticity a complete and consistent extension to the general case of multiple‐porosity and multiple‐permeability, fluid‐saturated, and linearly elastic media. The constitutive stress‐strain relations for a medium identified with this extension are presented, and the coefficient matrix of mechanical properties associated with these relations is derived from the corresponding intrinsic properties of its single‐porosity constituents. The closed form analytical solution to Mandel's problem is upgraded to the case being considered in this study. This problem addresses the transient consolidation of a porous elastic slab of rectangular geometry, when confined from the top and bottom. A numerical example solution for shale with laboratory setup of Mandel's problem is provided. Results are compared for the cases of single‐, double‐, and triple‐porosity solutions.

Anisotropic porochemoelectroelastic Mandel's problem solutions for applications in reservoir modeling and laboratory characterization

The porochemoelectroelastic analytical models have been used to describe the response of chemically active and electrically charged saturated porous media such as clay soils, shales, and biological tissues. In this work, the porochemoelectroelastic theory is applied to derive the solution for stress, pore pressure, strain, and displacement of the Mandel's problem for orthotropic charged saturated porous media. Numerical examples are given to demonstrate applications of the present solutions to laboratory characterization and hydraulic fracturing analysis of shale formation. The analysis shows that ignoring either the porochemoelectroelastic effects or the material anisotropy can leads to inaccurate prediction of the pore pressure and effective stress distributions in the shale sample or formations during these processes.

Mandel's problem revisited

The classic Mandel solution associated with the well-known Mandel–Cryer effect is re-examined. Using Biot theory of poroelasticity, the solution of the Mandel problem is extended to include material transverse isotropy, as well as the compressibility of the pore fluid and the solid constituents of the soil–rock skeleton. Le résultat classique de Mandel, associé à l'effet bien comnnu de Mandel—Cryer, est réexamineé. En utilisant la théorie poro-é1astique de Blot, la solution du probléme de Mandel est étendue afin d'introduire l'anisotropie transverse du matériau, ainsi que la compressibilité des fluides présents dans les pores et celle des constituants solides du squelette du sol ou de la roche.

FINITE ELEMENT ANALYSES OF ANISOTROPIC POROELASTICITY: A GENERALIZED MANDEL'S PROBLEM AND AN INCLINED BOREHOLE PROBLEM

The finite element equations for non-linear, anisotropic poroelasticity are cast in the form of measurable engineering constants. Two problems of importance to the rock and petroleum industry are analysed by the FEM. First, the classical Mandel's problem with an extension to transversely isotropic case is investigated. Second, the problem of an inclined borehole is explored. In particular, the effect of material anisotropy on stress concentration near the wall with implication to borehole instability is examined in detail.

A direct boundary element method for plane strain poroelasticity

AbstractThis paper presents a direct boundary element method (BEM), formulated in the Laplace transform space, for plane strain poroelasticity. The paper expands on work by Cheng and Liggett by recasting the theoretical foundation of BEM within the framework of Rice and Cleary's formulation of the Biot theory of poroelasticity. Furthermore, the numerical algorithm is generalized to deal with both interior and exterior domain problems, and a method for indirectly calculating the Cauchy principal value of the singular integrals is presented. Formulae for the stress and flux inside the domain are also derived. Finally, the algorithm is validated by comparing the numerical results with the analytical solution of a borehole subject to a far‐field deviatoric stress (exterior domain) and with the solution of Mandel's problem (interior domain). These two examples provide a critical test of the algorithm.