Poroelastic Domain Research Articles

Sound transmission reduction of double walls using porous and viscoelastic materials

Double-walled structures separated by an acoustic cavity are widely used in many applications for sound insulation purposes. Their sound transmission loss is estimated by calculating the ratio between the incident sound power on the first wall and the radiated sound power by the second. The introduction of a melamine foam associated with heavy masses brings a better sound insulation against external noise. In this context, this work deals with the numerical prediction of the sound transmission of a roller shutter box equipped with porous and viscoelastic materials. Such system can be modeled, at first approximation, as a double-walled structure. The proposed approach is based on the use of the finite element method for the modeling of the different solid and fluid domains of the problem. A mixed displacement pressure formulation of the Biot poroelasticity equations is used to model the poroelastic domain. The structure is excited by a diffuse field represented by a superposition of plane waves with random phases and directions. The acoustic power radiated by the second wall is calculated by applying the Rayleigh integral method. The results show the importance of optimizing the position of the porous and the heavy layers to maximize the sound insulation.

Hydromechanical behavior analysis of fractured vuggy carbonate rocks based on homogenization theory

Fractured vuggy carbonate rocks are important for underground water and geo-energy reservoirs due to their significant contribution on water and hydrocarbon reserves and production. A vug is a small cavity in a carbonate rock, which is relatively larger than the intergranular pore space. The presence of multiscale fractures and vugs makes the hydromechanical behavior of rocks different from that of most geological materials. The objective of this work is to develop an upscaling method to analyze the hydromechanical behavior of fractured vuggy carbonate rocks based on homogenization theory. To this end, at first a novel conceptual model named discrete fracture-vug network (DFVN) model was proposed to describe the hydromechanical behavior on the fine scale. The matrix and fractures are poroelastic domains in which Biot equations are applied. The vugs are free fluid domains governed by Stokes equations. Two domains are coupled with extended Beavers–Joseph–Saffman interface conditions. Then, an upscaled hydromechanical model was developed via two-scale asymptotic homogenization. The model consistent with classical Biot equations, but the model coefficients possess explicit formulations which can be determined by three periodicity cell problems. Subsequently, efficient numerical solutions of cell problems are provided using finite element methods. Herein, the discrete fractures are modeled as lower-dimensional interfaces between matrix elements. The proposed model and method are verified through several numerical examples and experimental data. The results show that the storage coefficient and Biot coefficient increase with the presence of fractures and vugs. The equivalent elastic stiffness of a fractured vuggy rock is majorly affected by the vugs' volume ratio. The connectivity of DFVN has an important impact on the equivalent permeability.

A hybrid PML formulation for the 2D three-field dynamic poroelastic equations

Simulation of wave propagation in poroelastic half-spaces presents a common challenge in fields like geomechanics and biomechanics, requiring Absorbing Boundary Conditions (ABCs) at the semi-infinite space boundaries. Perfectly Matched Layers (PML) are a popular choice due to their excellent wave absorption properties. However, PML implementation can lead to problems with unknown stresses or strains, time convolutions, or PDE systems with Auxiliary Differential Equations (ADEs), which increases computational complexity and resource consumption.This article introduces a novel hybrid PML formulation for arbitrary poroelastic domains. Instead of using ADEs, this formulation utilizes time-history variables to reduce the number of unknowns and mathematical operations. The modification of the PDEs to account for the PML is limited to the PML domain only, resulting in smaller matrices while maintaining the governing equations in the interior domain and preserving the temporal structure of the problem. The hybrid approach introduces three scalar variables localized within the PML domain.The proposed formulation was tested in three numerical experiments in geophysics using realistic parameters for soft sites with free surfaces. The results were compared with numerical solutions from extended domains and simpler ABCs, such as paraxial approximation, demonstrating the accuracy, efficiency, and precision of the proposed method. The article also discusses the applicability of the method to complex media and its extension to the Multiaxial PML formulation.The codes used for the simulations are available for download from https://github.com/hmella/POROUS-HYBRID-PML.

Simulation of flow in deformable fractures using a quasi-Newton based partitioned coupling approach

We introduce a partitioned coupling approach for iterative coupling of flow processes in deformable fractures embedded in a poro-elastic medium that is enhanced by interface quasi-Newton (IQN) methods. In this scope, a unique computational decomposition into a fracture flow and a poro-elastic domain is developed, where communication and numerical coupling of the individual solvers are realized by consulting the open-source library preCICE. The underlying physical problem is introduced by a brief derivation of the governing equations and interface conditions of fracture flow and poro-elastic domain followed by a detailed discussion of the partitioned coupling scheme. We evaluate the proposed implementation and undertake a convergence study to compare a classical interface quasi-Newton inverse least-squares (IQN-ILS) with the more advanced interface quasi-Newton inverse multi-vector Jacobian (IQN-IMVJ) method. These coupling approaches are verified for an academic test case before the generality of the proposed strategy is demonstrated by simulations of two complex fracture networks. In contrast to the development of specific solvers, we promote the simplicity and computational efficiency of the proposed partitioned coupling approach using preCICE and FEniCS for parallel computations of hydro-mechanical processes in complex, three-dimensional fracture networks.

Coupling Stokes Flow with Inhomogeneous Poroelasticity

Summary We investigate the behaviour of flux-driven flow through a single-phase fluid domain coupled to a biphasic poroelastic domain. The fluid domain consists of an incompressible Newtonian viscous fluid while the poroelastic domain consists of a linearly elastic solid filled with the same viscous fluid. The material properties of the poroelastic domain, that is permeability and elastic parameters, depend on the inhomogeneous initial porosity field. We identify the dimensionless parameters governing the behaviour of the coupled problem: the ratio between the magnitudes of the driving velocity and the Darcy flows in the poroelastic domain, and the ratio between the viscous pressure scale and the size of the elastic stresses in the poroelastic domain. We consider a perfusion system, where flow is forced to pass from the single-phase fluid to the biphasic poroelastic domain. We focus on a simplified two-dimensional geometry with small aspect ratio and perform an asymptotic analysis to derive analytical solutions. The slender geometry is divided in four regions, two outer domains that describe the regions away from the interface and two inner domains that are the regions across the interface. Our analysis advances the quantitative understanding of the role of heterogeneous material properties of a poroelastic domain on its mechanical response when coupled with a fluid domain. The analysis reveals that, in the interfacial zone, the fluid and the elastic behaviours of this coupled Stokes—poroelastic problem can be treated separately via (i) a Stokes–Darcy coupling and (ii) the solid skeleton being stress free. This latter finding is crucial to derive the coupling condition across the outer domains for both the elastic part of the poroelastic domain and the fluid flow. Via specification of heterogeneous material properties distribution, we reveal the effects of heterogeneity and deformability on the mechanics of the poroelastic domain.

Dual length scale non-local model to represent damage and transport in porous media

Recent experiments and physical evidence show that fractured porous media feature cracks and fluid capillary networks at various scales. We present a multi-physics macro-scale model that can distinguish between the mechanics and transport interactions. The porous media is represented by a poroelastic domain incorporating non-local damage and non-local transport. The evolution of each of these processes is governed by a unique length scale and driving force, which allows for better flexibility in modeling hydraulic-deformation network systems. For consistency the governing equations of the non-local multi-physics problem are derived from thermodynamics principles. Hence, a four-field (u,P,σ̃eq,D,κ̃) mixed finite element formulation is developed. The non-linear system of equations is linearized and solved using Newton’s method and a backward Euler scheme is used to evolve the system in time, for which a consistent Jacobian matrix and residual vector are derived analytically. Two benchmark examples are investigated: hydraulic fracturing of rocks and soil consolidation. The numerical examples show the viability of this model, and how the variation of the two length scales and damage parameters can be used to describe different physical phenomena.

Electrokinetic Contributions to Self‐Potential Signals From Magmatic Stressing

AbstractPre‐eruptive electrical signals at active volcanoes are generally interpreted in terms of electrokinetic processes. Spatio‐temporal self‐potential (SP) signals can be caused by strain‐induced fluid flow in volcanic aquifers, however, previous studies lack the quantitative assessments of these phenomena and the underpinning poroelastic responses. Here we use Finite‐Element Analysis to study poroelastic responses induced by subsurface stressing from sill and dike sources by jointly solving for ground displacements, pore pressure, and SP signals. We evaluate the influence of pressure source orientation on the poroelastic response in two different volcanic aquifers (pyroclastic and lava flow) to provide insights on emergent geodetic and SP signals and their sensitivity to governing parameters. Strain‐induced SP amplitudes deduced from a reference parameter set vary in both aquifer models and are of negative polarity (−0.35 and −22.6 mV) for a pressurized dike and of positive polarity (+4 and +20 mV) for a pressurized sill. Importantly, we find uniquely different SP and ground displacement patterns from either sill or dike intrusions. Our study shows that SP signals are highly sensitive to the subsurface Young's modulus, streaming potential coupling coefficient and electrical conductivity of the poroelastic domains. For the set of parameters tested, the dike model predicts SP amplitudes of up to −947 mV which are broadly representative of recorded amplitudes from active volcanoes. Our study demonstrates that electrokinetic processes reflect magma‐induced stress and strain variations and highlights the potential of joint geodetic and SP studies to gain new insights on causes of volcanic unrest.

Frequency response analysis of concrete seawall including soil-structure-seawater interaction

In this study, a frequency-domain finite element model is proposed to evaluate the seismic response of water-structure-sediment-foundation-backfill coupling system. It includes the seawater as a compressible fluid, seawall as viscoelastic solid, backfilled soil, sediment, and foundation as two-phase poroelastic domains with dynamic behavior described by u-p formulation of Biot’s equations. The dynamic interactions among various domains are fully considered and the PML is adopted as an absorbing boundary condition to truncate the computational domain, absorbing all out-going seismic waves. The dynamic responses in the structure, water and soil are assumed to be generated by the vertical propagating shear or compressional waves from the bedrock substrate under the foundation soil. Both the actions of horizontal and vertical seismic excitations at the bottom of foundation are respectively considered. After verifying the accuracy of PML and coupled model of soil-structure-water system, a series of parametric studies are conducted to investigate the effects of sediment thickness, water depth, foundation stiffness, foundation thickness, inclination of seawall, and properties of backfilled soil including soil permeability, shear modulus, and saturation degree, on the frequency response at the top of seawall.

Flow over natural or engineered surfaces: an adjoint homogenization perspective

Natural and engineered surfaces are never smooth, but irregular, rough at different scales, compliant, possibly porous, liquid impregnated or superhydrophobic. The correct numerical modelling of fluid flowing through and around them is important but poses problems. For media characterized by a periodic or quasi-periodic microstructure of characteristic dimensions smaller than the relevant scales of the flow, multiscale homogenization can be used to study the effect of the surface, avoiding the numerical resolution of small details. Here, we revisit the homogenization strategy using adjoint variables to model the interaction between a fluid in motion and regularly micro-textured, permeable or impermeable walls. The approach described allows for the easy derivation of auxiliary/adjoint systems of equations which, after averaging, yield macroscopic tensorial properties, such as permeability, elasticity, slip, transpiration, etc. When the fluid in the neighbourhood of the microstructure is in the Stokes regime, classical results are recovered. Adjoint homogenization, however, permits simple extension of the analysis to the case in which the flow displays nonlinear effects. Then, the properties extracted from the auxiliary systems take the name of effective properties and do not depend only on the geometrical details of the medium, but also on the microscopic characteristics of the fluid motion. Examples are shown to demonstrate the usefulness of adjoint homogenization to extract effective tensor properties without the need forad hocparameters. In particular, notable results reported herein include:(i)an original formulation to describe filtration in porous media in the presence of inertial effects;(ii)the microscopic and macroscopic equations needed to characterize flows through poroelastic media;(iii)an extended Navier’s condition to be employed at the boundary between a fluid and an impermeable rough wall, with roughness elements which can be either rigid or linearly elastic;(iv)the microscopic problems needed to define the relevant parameters for a Saffman-like condition at the interface between a fluid and a porous substrate; and(v)the macroscopic equations which hold at the dividing surface between a free-fluid region and a fluid-saturated poroelastic domain.

Sound transmission in two-dimensional periodic poroelastic structures

To obtain excellent sound reduction performance, in this paper we introduce a novel periodic poroelastic composite structure, which combines poroelastic material and periodic structure and aims at using the remarkable acoustic performance of these two. This periodic composite structure comprises three parts, i.e. the poroelastic domain, the elastic domain (thin plate), and the periodic resonators, which can be simple single-degree-of-freedom resonators (SRs) or composite two-degree-of-freedom resonators (CRs). A theoretical model is established by using Biot theory for the poroelastic domain, and by using the effective medium method for the resonator-plate coupling system, which is considered as an isotropic plate with an equivalent dynamic density. This method is validated with degenerated model in the literature; the results obtained by this method are in excellent consistence with the results in the literature. Parameter analyses are performed to test the influences of poroelastic addition and periodic resonator on the sound transmission loss (STL) of this periodic composite structure under two kinds of boundary conditions. The poroelastic addition is found to increase the STL while the influences of resonators are complicated. The STL increases notably in the frequency range bounded by the characteristic frequencies of these resonators, however, a decrease just follows when it exceeds these frequencies, which can be observed in both SR case and CR case under the two boundary conditions. In the meantime, when multiple SR is placed in a periodic lattice, it is found that different resonators with ascending mass and characteristic frequencies have superior STL to those with ascending characteristic frequencies but have equal mass. The case with CR, which is more complicated as expected, shows less STL decrease than the case with SR, but wider frequency range where the STL increases than a poroelastic composite structure without resonators. This results from the fact that the frequency band of vibration suppression in the CR case is wider than in the SR case. As a result, to achieve the desired STL performance in a frequency range, the proposed composite structure using SR with tuned characteristic frequencies is enough; however, if a wider frequency band is expected even if there is a slight STL tradeoff, the CR case is a better option. Though the method proposed is only valid in the low-to-medium frequency range, the results obtained can benefit theoretical development of low-to-medium sound modulation applications, they are also valuable and illuminating for investigating the broadband sound modulation.

A fast multipole accelerated indirect boundary element method for broadband scattering of elastic waves in a fluid‐saturated poroelastic domain

SummaryA fast multipole accelerated indirect boundary element method is developed to efficiently solve the scattering of broadband waves by inhomogeneity in a fluid‐saturated 3D poroelastic space. Based on the single layer potential theory, poroelastic free‐space Green's functions of point force and fluid source are distributed on the scatterer surface at fictitious densities to construct the scattered waves. By using the plane wave expansion of 3D potential functions of compressional and shear waves, the multipole expansion and the local expansion coefficients are derived. Numerical results demonstrated that this proposed method can greatly improve the efficiency of traditional indirect boundary element method and reduce the memory requirement for 3D broadband wave scattering problems in an unbounded poroelastic medium. Problems of wave scattering by a spherical cavity, a group of cavities, and a canyon in a semiinfinite poroelastic space are investigated. Several notable scattering characteristics are revealed.

Effect of non-uniform pore pressure fields on hydraulic fracture propagation

In elastic materials, it is evident that the path of a propagating fracture is deflected from its normal course in presence of imperfections of the material or loading conditions. This paper aims at investigating hydraulic fracture deviation induced by non-uniformity of pore pressure fields with the use of a fully coupled poroelastic model based on the finite element method. The model includes a poroelastic domain in which pressurized hydraulic fractures are explicitly embedded, thus allowing to realistically model the fluid flow inside the fracture and to intrinsically consider the fracturing fluid load on the fracture walls as well as fluid leak-off into the formation. The latter process (fluid leak-off into the formation) controls both the length and the orientation of the fracture by changing the local pore pressure which in turn leads to a change in magnitude and direction of local principal stresses around the fracture tip. An innovative method, Mean Rotation Angle (MRA) is utilised for post-processing of evolving stress data at the vicinity of the fracture tip. The MRA predicts the potential growth path of pressurized fractures. In this paper pore pressure induced fracture reorientation is studied for a single fracture as well as closely spaced fractures. Results of this study indicate that presence of a pore pressure anomaly changes the growth path of a hydraulic fracture, towards or away from the anomaly. A higher than average pore pressure zone attracts the fracture while a lower pressure anomaly zone repulses the growing fracture. The fracture growth direction depends on the differential pressure and the distance between the anomaly and the fracture tip. Also in case of two simultaneously growing transverse fractures pressurized by injected fluid, it has been observed that the fluid leak-off controls the potential deviation angle of the fractures through changing the local pore pressure distribution pattern. It is shown that there are three distinct trends for the change of potential deviation angle due to fluid leak-off and that these three trends are linked to three corresponding stages of hydraulic communication between the two fractures. Furthermore, this study shows that change in matrix permeability, stress anisotropy, fracture half-length, spacing and the rate of leak-off influence the timing of each of the stages to the extent to which the corresponding stage of hydraulic communication of the two fractures are affected. This new understanding gives a better insight into the mechanism by which closely spaced hydraulic fractures interact and help optimize the design of multi-stage hydraulic fracture treatments.

Derivation of an averaged model of isothermal acoustics in a heterogeneous medium in the case of two different poroelastic domains

We consider some mathematical model of isothermal acoustics in a composite medium consisting of two different porous soils (poroelastic domains) separated by a common boundary. Each of the domains has its own characteristics of the solid skeleton; the liquid filling the pores is the same for both domains. The differential equations of the exactmodel contain some rapidly oscillating coefficients. The averaged equations (i.e., without rapidly oscillating coefficients) are derived.

A direct hybrid finite element–wave based modelling technique for efficient analysis of poroelastic materials in steady-state acoustic problems

This work proposes a hybrid modelling technique for efficient analysis of poroelastic materials, which are widely used for noise reduction in acoustic problems. By combining the finite element method and the wave based method in a direct manner, the proposed hybrid technique maximises the advantages and compensates the drawbacks of both numerical methods. The considered poroelastic domain described by Biot’s theory is divided into two groups of domains according to their geometrical characteristics and boundary conditions. The group with complex geometries and/or boundary conditions leading to singularities is discretised into a large number of small finite elements. The other group consisting of large, geometrically moderate poroelastic domains is partitioned into wave based subdomains where the field variables are expanded with analytical poroelastic wave functions. Both groups modelled by the finite element method and the wave based method, respectively, are combined in a hybrid framework in this work to ensure their interacting dynamic behaviours. The properties of the hybrid model are investigated and are compared to existing modelling methods for some numerical examples. The proposed direct hybrid modelling technique provides stable predictions and exhibits fast convergence performances for the analysis of poroelastic materials, especially when singularities arise in the poroelastic domain.

Ground subsidence due to uniform fluid extraction over a circular region within an aquifer

This paper examines the ground subsidence caused by the uniform extraction of fluids from a disc-shaped region located within a poroelastic halfspace. Biot’s theory of poroelasticity is used to examine the problem. The fluid extraction over the circular region at the interior of the poroelastic domain induces time-dependent axisymmetric surface settlements. The theoretical results are also used to examine the accuracy of a multi-physics computational scheme that can be used to examine more complex geological settings.

Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method

We describe an algorithm for modeling saturated fractures in a poroelastic domain in which the reservoir simulator is coupled with a boundary element method. A fixed stress splitting is used on the underlying fractured Biot system to iteratively couple fluid and solid mechanics systems. The fluid system consists of Darcy’s law in the reservoir and is computed with a multipoint flux mixed finite element method, and a Reynolds’ lubrication equation in the fracture solved with a mimetic finite difference method. The mechanics system consists of linear elasticity in the reservoir and is computed with a continuous Galerkin method, and linear elasticity in the fracture is solved with a weakly singular symmetric Galerkin boundary element method. This algorithm is able to compute both unknown fracture width and unknown fluid leakage rate. An interesting numerical example is presented with an injection well inside of a circular fracture.

A finite element approach combining a reduced‐order system, Padé approximants, and an adaptive frequency windowing for fast multi‐frequency solution of poro‐acoustic problems

SUMMARYIn this work, a solution strategy is investigated for the resolution of multi‐frequency structural‐acoustic problems including 3D modeling of poroelastic materials. The finite element method is used, together with a combination of a modal‐based reduction of the poroelastic domain and a Padé‐based reconstruction approach. It thus takes advantage of the reduced‐size of the problem while further improving the computational efficiency by limiting the number of frequency resolutions of the full‐sized problem. An adaptive procedure is proposed for the discretization of the frequency range into frequency intervals of reconstructed solution. The validation is presented on a 3D poro‐acoustic example. Copyright © 2013 John Wiley & Sons, Ltd.

Efficient treatment of stress singularities in poroelastic wave based models using special purpose enrichment functions

The Finite Element Method is the most commonly used prediction technique to model the harmonic behavior of poroelastic materials. A major disadvantage of this method is its practical frequency limitation in that the computational efforts become prohibitively large at higher frequencies. A recently developed Wave Based Method is an efficient alternative deterministic prediction technique which aims at relaxing this frequency limitation by using exact solutions of the governing equations to approximate the field variables. This paper discusses the application of the Wave Based Method for the particular case that stress singularities are present in corners of the poroelastic domain. Based on an asymptotic analysis, the paper derives a criterion to predict the presence of stress singularities and proposes a suitable set of enrichment functions to extend the conventional set of expansion functions. The beneficial effect of incorporating these functions on the convergence of the Wave Based Method is illustrated by means of a numerical validation study.

Three‐dimensional models of reservoir sediment and effects on the seismic response of arch dams

AbstractThe important effects of bottom sediments on the seismic response of arch dams are studied in this paper. To do so, a three‐dimensional boundary element model is used. It includes the water reservoir as a compressible fluid, the dam and unbounded foundation rock as viscoelastic solids, and the bottom sediment as a two‐phase poroelastic domain with dynamic behaviour described by Biot's equations. Dynamic interaction among all those regions, local topography and travelling wave effects are taken into account. The results obtained show the important influence of sediment compressibility and permeability on the seismic response. The former is associated with a general change of the system response whereas the permeability has a significant influence on damping at resonance peaks. The analysis is carried out in the frequency domain considering time harmonic excitation due to P and S plane waves. The time‐domain results obtained by using the Fourier transform for a given earthquake accelerogram are also shown. The possibility of using simplified models to represent the bottom sediment effects is discussed in the paper. Two alternative models for porous sediment are tested. Simplified models are shown to be able to reproduce the effects of porous sediments except for very high permeability values. Copyright © 2004 John Wiley & Sons, Ltd.

Enhanced weak integral formulation for the mixed (u_,p_) poroelastic equations

Recently Atalla et al. [J. Acoust. Soc. Am. 104, 1444–1452 (1998)] and Debergue et al. [J. Acoust. Soc. Am. 106, 2383–2390 (1999)] presented a weak integral formulation and the general boundary conditions for a mixed pressure-displacement version of the Biot’s poroelasticity equations. Finite element discretization was applied to the formulation to solve 3D vibro-acoustic problems involving elastic, acoustic, and poroelastic domains. In this letter, an enhancement of the weak integral formulation is proposed to facilitate its finite element implementation. It is shown that this formulation simplifies the assembly process of the poroelastic medium, the imposition of its boundary conditions, and its coupling with elastic and acoustic media.