Biot Poroelastic Model Research Articles

Characterization and estimation of the acoustical and elastic parameters in stone wool samples

Mineral wool is widely used in acoustic applications. The properties of glass wool have been broadly investigated, thus resulting in a good knowledge of its structure and parameters. This information is used to describe mineral wool in general. Nevertheless, other types, such as stone wool, involve a different production process and therefore have a different fibre structure. e.g. stone wool is not transversally isotropic as glass wool. These differences can have important effects depending on the use of the material; therefore, assuming the same parameter values for both materials may lead to inaccurate results in simulations and calculations. Currently there are few studies that characterize both the acoustic and elastic properties of stone wool. This contribution reports on the characterization of two stone wool samples with different densities (~70 kg/m3 and ~140 kg/m3). The characterization aimed at obtaining the parameters that are required for the description of the propagation of sound in the material through the Biot poroelastic model, i.e. the acoustic and elastic parameters. The obtained knowledge enables the adequate modelling of the investigated poroelastic materials in different configurations by numerical methods such as finite elements and the transfer matrix method.

A posteriori error estimates for Biot system using a mixed discretization for flow

We first derive convergence and a priori stability, and next reliability and efficiency of a posteriori error indicators for a Biot poroelastic model coupled with an elastic model in R3, solved by a continuous Galerkin scheme (CG) for the displacement and a mixed finite element scheme for the flow. The coupled system is decoupled by a fixed stress splitting algorithm. The numerical implementation of the residual based error indicators is simple, even for the mixed discretization, but at the expense of two suboptimal bounds. The scheme is tested on two benchmark problems via several numerical experiments.

Fractured formation evaluation by seismic attenuation derived from array acoustic log waves based on modified spectral ratio method and an extended Biot's poroelastic model

A comprehensive application of the seismic attenuation deduced from the array acoustic log waves by using modified spectral ratio method together with an extended Biot's poroelastic model has been developed for detection and evaluation of fractures in carbonate and sandstone formations. First, profile of amplitude ratio based on deconvolution interferometry (DCI) and modified spectral ratio method (MSRM) are introduced and applied on array acoustic waveforms to derive the seismic attenuation factors. Second, an extended Biot's poroelastic model with consideration of fracture parameters is described and derived for its expression of attenuation. Both the computational method and the theoretical model are tested on simulated sonic data and meaningful results are obtained which are connected with analysis of application on field data in final section. Finally, the MSRM and extended Biot's poroelastic model are applied to field waveforms acquired from carbonate and sandstone formation in oilfields of Petro China. In the present study, we mainly focused on the subject of fracture identification and evaluation in both carbonate and sandstone formation. Field application results show that within frequency range of 1–40 kHz, attenuation derived from sonic log waves by using MSRM has close relationship with fracture development. Meanwhile, the simulation responses based on the extended Biot's poroelastic model verify the fact that the attenuation magnitude is mainly sensitive to fracture porosity. Our results also suggest that profile of amplitude ratio can be a potentially useful indicator of fracture identification and evaluation, and, maybe also an interpreting toll of radial heterogeneity.

Numerical Algorithms for Simulation of a Fluid-Filed Fracture Evolution in a Poroelastic Medium

The paper deals with the computational framework for the numerical simulation of the three dimensional fluid-filled fracture evolution in a poroelastic medium. The model consists of several groups of equations including the Biot poroelastic model to describe a bulk medium behavior, Reynold’s lubrication equations to describe a flow inside fracture and corresponding bulk/fracture interface conditions. The geometric model of the fracture assumes that it is described as an arbitrary sufficiently smooth surface with a boundary. Main attention is paid to describing numerical algorithms for particular problems (poroelasticity, fracture fluid flow, fracture evolution) as well as an algorithm for the coupled problem solution. An implicit fracture mid-surface representation approach based on the closest point projection operator is a particular feature of the proposed algorithms. Such a representation is used to describe the fracture mid-surface in the poroelastic solver, Reynold’s lubrication equation solver and for simulation of fracture evolutions. The poroelastic solver is based on a special variant of X-FEM algorithms, which uses the closest point representation of the fracture. To solve Reynold’s lubrication equations, which model the fluid flow in fracture, a finite element version of the closet point projection method for PDEs surface is used. As a result, the algorithm for the coupled problem is purely Eulerian and uses the same finite element mesh to solve equations defined in the bulk and on the fracture mid-surface. Finally, we present results of the numerical simulations which demonstrate possibilities of the proposed numerical techniques, in particular, a problem in a media with a heterogeneous distribution of transport, elastic and toughness properties.

Justification for power laws and fractional models

Wave equations with non-integer derivative operators describe attenuation which increases with frequency with other powers than two, unlike ordinary wave equations. It is desirable to try to understand what properties of, e.g., biological tissue that give rise to this behavior. The main attenuation mechanisms of standard acoustics are heat conduction and relaxation, as well as structural and chemical relaxation. They have fractional parallels and the first one is heat relaxation described by fractional Newton cooling due to anomalous diffusion. The most important mechanism is however the fractional parallel to structural relaxation. Instead of one there are multiple relaxation processes with a distribution of relaxation times that follows a power-law distribution, possibly indicating fractal properties. The distribution also has a relationship to the Arrhenius equation, indicating a link to chemical relaxation, albeit a quite speculative one. The multiple relaxation model may also be formulated as a hierarchical polymer model. Time-varying non-Newtonian viscosity and a medium with a fractal distribution of scatterers can also give rise to power law behavior. Existing models in sediment acoustics such as the grain shearing model and the Biot poroelastic model can also be reformulated with fractional operators. These approaches are presented in the hope of progressing towards an understanding of whether fractional wave equations give clues to some deeper reality, or if they are just a compact phenomenological description.

Effect of Subsurface Microseisms on the Motion of Dispersed Droplets in Pores

AbstractHuman‐induced seismicity has drawn substantial attention in recent years. The effect of seismicity on subsurface structures has been studied extensively. However, the effect of seismicity, especially microseismicity, on surrounding immiscible fluids is rarely investigated. In porous media with two or more immiscible fluids, different amplitudes of vibration induced by seismicity have distinct effects on the dynamic behavior of the fluids. Three types of pore‐scale models are prevalent for analyzing the motion of immiscible droplets. The underlying assumptions and accuracy of these models are compared in this study in both the frequency and time domains. The frequency domain analysis shows that resonance can be addressed in all three of the models; the frequency response curves, however, present significant differences, which are attributed to the missing physics considered in some models. Time domain analysis is performed in both small‐amplitude and large‐amplitude oscillation. The nonlinearity in large‐amplitude oscillation is attributed to the constricted geometry of the capillary tube. Although the term “large‐amplitude” is used in this study, the corresponding amplitude is still within the amplitude range of microseisms. The momentum balance model is identified as the most accurate oscillatory model so far in comparison with computational fluid dynamics simulations. In addition, the potential approach to incorporate this pore‐scale model in seismic wave attenuation analysis is found to be possible. The frequency correction function and structural factor are calculated to embed the momentum balance model into Biot's poroelastic model. The resonance of the dispersed phase can also be addressed theoretically in porous media of random packed spheres.

Linear and half order fractional viscoelastic equivalents of the extended Biot model

The extended Biot poro-viscoelastic model [Chotiros and Isakson, JASA (2014)] like the standard Biot poroelastic model is biphasic and predicts two compressional waves (fast and slow) and one shear wave. Taking squirt flow into account, the model introduces two additional relaxation modes apart from the Biot crossover relaxation mode (global flow). The frequency dependent relaxation processes and a high number of independent parameters make the finite element implementation and the inversion of parameters for reconstruction a challenge. To overcome this challenge, we identify single phase viscoelastic equivalents for all three wave solutions of the extended Biot model from its dispersion relations. Considering squirt flow and high-frequency turbulent flow, the fast compressional wave and the shear wave solutions at low frequencies are equivalent to a non-standard solid model with eight parameters (four springs and four dashpots) and six parameters (Three springs and three dashpots), respectively. At high frequencies, the Kelvin-Voigt model used for tangential contact stiffness (shear relaxation) leads to non-physical equivalents. Neglecting this shear relaxation mode, the fast compressional wave and the shear wave solutions at high frequencies are equivalent to a fractional half-order solid model of order three and two, respectively. Alternative models for tangential contact stiffness are discussed.

On modelling of consolidation processes in geological materials

Low-permeablity materials may be seen as natural geological barriers for radioactive waste repositories. However, to ensure their safe performance, a good understanding of their mechanical properties is required. Although the standard Biot's poroelastic model is widely used to estimate the key properties of these materials, experimental observations differ from this mathematical formulation and suggest that a more complex rock deformation behaviour to include a creep effect is needed. In this study, the Biot's differential equations are modified to include a rheological skeleton. In comparison with other existing models, here we propose a formulation with a minimal parametric uncertainty: we show that with just one additional physically-based parameter, the experimental creep behaviour is properly described. This enhanced model is implemented within a finite element framework and employed in a fitting algorithm to extract the hydro-mechanical properties from experimental data. To illustrate its generality, we analyse laboratory tests performed on three different types of materials: (a) an unlithified lower Oligocene clay from Belgium (Boom Clay), (b) an indurated Jurassic mudrock (Callovo-Oxfordian mudstone) and (c) a Triassic siltstone (Mercia Mudstone Formation). Numerical fits to the data support the validity of this approach and demonstrate its applicability to a range of low-permeability materials regardless of mineralogy or burial history.

The Biot coefficient for a low permeability heterogeneous limestone

This paper presents the experimental and theoretical developments used to estimate the Biot coefficient for the heterogeneous Cobourg Limestone, which is characterized by its very low permeability. The coefficient forms an important component of the Biot poroelastic model that is used to examine coupled hydro-mechanical and thermo-hydro-mechanical processes in the fluid-saturated Cobourg Limestone. The constraints imposed by both the heterogeneous fabric and its extremely low intact permeability [ $$K \in (10^{-23},10^{-20})~\hbox {m}^{2}$$ ] require the development of alternative approaches to estimate the Biot coefficient. Large specimen bench-scale triaxial tests (150 mm diameter and 300 mm long) that account for the scale of the heterogeneous fabric are complemented by results for the volume fraction-based mineralogical composition derived from XRD measurements. The compressibility of the solid phase is based on theoretical developments proposed in the mechanics of multi-phasic elastic materials. An appeal to the theory of multi-phasic elastic solids is the only feasible approach for examining the compressibility of the solid phase. The presence of a number of mineral species necessitates the use of the theories of Voigt, Reuss and Hill along with the theories proposed by Hashin and Shtrikman for developing bounds for the compressibility of the multi-phasic geologic material composing the skeletal fabric. The analytical estimates for the Biot coefficient for the Cobourg Limestone are compared with results for similar low permeability rocks reported in the literature.

Numerical investigation of dynamic soil response around a submerged rubble mound breakwater

A better understanding of physical process of the fluid-seabed-structure interaction (FSSI) is beneficial for engineers involved in the design of marine infrastructures. Most previous studies for the problem of FSSI have considered wave-only conditions, despite the co-existence of wave and current in the real ocean environment. Unlike the previous studies, currents are included in the present study for the numerical modelling of FSSI, using an integrated FVM-FEM scheme, in which the VARANS equation is used to simulate fluid field, while Biot's poro-elastic model is used for porous flow in a seabed. Numerical examples show the important influences of currents on the local hydrodynamic process and the resulting dynamics of seabed foundation around a submerged rubble mound breakwater. The structure is relatively stable in the presence of counter-current waves, whereas the co-current waves would significantly compromise the instability of structure due to potential of shear failure and liquefaction in its sandy seabed foundation.

Investigation of nonlinear wave-induced seabed response around mono-pile foundation

Stability and safety of offshore wind turbines with mono-pile foundations, affected by nonlinear wave effect and dynamic seabed response, are the primary concerns in offshore foundation design. In order to address these problems, the effects of wave nonlinearity on dynamic seabed response in the vicinity of mono-pile foundation is investigated using an integrated model, developed using OpenFOAM, which incorporates both wave model (waves2Foam) and Biot's poro-elastic model. The present model was validated against several laboratory experiments and promising agreements were obtained. Special attention was paid to the systematic analysis of pore water pressure as well as the momentary liquefaction in the proximity of mono-pile induced by nonlinear wave effects. Various embedded depths of mono-pile relevant for practical engineering design were studied in order to attain the insights into nonlinear wave effect around and underneath the mono-pile foundation. By comparing time-series of water surface elevation, inline force, and wave-induced pore water pressure at the front, lateral, and lee side of mono-pile, the distinct nonlinear wave effect on pore water pressure was shown. Simulated results confirmed that the presence of mono-pile foundation in a porous seabed had evident blocking effect on the vertical and horizontal development of pore water pressure. Increasing embedded depth enhances the blockage of vertical pore pressure development and hence results in somewhat reduced momentary liquefaction depth of the soil around the mono-pile foundation.

Geophysical models of heat and fluid flow in damageable poro-elastic continua

A rather general model for fluid and heat transport in poro-elastic continua undergoing possibly also plastic-like deformation and damage is developed with the goal to cover various specific models of rock rheology used in geophysics of Earth’s crust. Nonconvex free energy at small elastic strains, gradient theories (in particular the concept of second-grade nonsimple continua), and Biot poro-elastic model are employed, together with possible large displacement due to large plastic-like strains evolving during long time periods. Also the additive splitting is justified in stratified situations which are of interest in modelling of lithospheric crust faults. Thermodynamically based formulation includes entropy balance (in particular the Clausius–Duhem inequality) and an explicit global energy balance. It is further outlined that the energy balance can be used to ensure, under suitable data qualification, existence of a weak solution and stability and convergence of suitable approximation schemes at least in some particular situations.

Evaluation of a model-based poroelastography algorithm for edema quantification

A critical component of end stage renal disease treatment is optimal fluid management. Patients with reduced kidney function risk developing fluid overload, and many dialysis patients exceed recommended levels of fluid retention, leading to frequent clinical intervention and higher mortality. Despite this, current clinical standards of care rely on reactive semi-quantitative tests to grade fluid overload and peripheral edema. Poroelastography has been proposed as a quantitative means of estimating fluid overload in peripheral edema, but clinical studies have met limited success, in part because of measurement noise. We developed a new poroelastography method based on an inverse problem formulation to address the shortcomings of current methods. Our technique iteratively minimizes the error between ultrasound displacement measurements and the solution of a Biot poroelastic model. This formulation does not depend on noise-amplifying derivatives and ratios. We tested our algorithm in a simulation study using synthetic displacement data with Gaussian noise and speckle tracking measurements from simulated sonograms. Our algorithm accurately estimated the Poisson’s ratio at all noise levels tested. Reconstruction errors for all cases were less than 10% and outperformed traditional methods. Although our method required 4-12 hours to run, it is easily parallelized. Future work will focus on benchtop validation.

Cylindrical Borehole Failure in a Poroelastic Medium

Drilling a cylindrical borehole is the first and important step in oil mining. Borehole design and strength check are big problems of utmost importance. Biot introduced a poroelastic constitutive theory for porous rock with freely moving fluid inside. In this paper, by using Biot poroelastic model, we analyze a borehole with drilling fluid in an infinite porous rock with three-dimensional in situ stresses and obtain whole domain solutions for instantaneous, short-time, and long-time stress distributions. Maximum and minimum allowable drilling pressures are given for tensile failure and shear failure criterions, and allowable drilling pressure regions are drawn in the space of in situ hydrostatic stress P0, deviatoric stress S0, and pore pressure p0. By comparing with classical elastic constitutive relations, or Hooke's model, the necessity of Biot poroelastic constitutive relations is shown.

G0310302 IPMCを用いた力学センサの計算モデリング(材料力学部門一般セッション:解析・シミュレーション)

The present study newly attempts the numerical simulation of mechanical sensors using IPMCs (ionic polymer-metal composites). The generated electric potential of mechanical sensing mode is very much smaller than the supplied electric potential of actuating mode with respect to the same displacement and structure. Furthermore, the mechanical sensors show the relaxation and time lag of reaction force and electric potential, The non-invertible and time-dependent behaviors of the mechanical sensors are numerically modeled with a set of basic equations with multi-fields such as solid stress, fluid pressure, ion concentration and electric potential. The basic equations consist of Biot poroelastic model, layered Timoshenko beam model and Poisson-Nernst-Plank model. Instantaneous pore pressure is estimated on undrained condition, Poisson effect of pore pressure on uniaxial condition is added, and the significant changes of volume and mechanical stiffness due to hydration are considered with water uptake. The related models are integrated and simplified into numerical simulation, and the numerical results are compared with experimental results.

線形逆散乱解析による多孔質弾性体中の不均質部の形状再構成

The Linearized Inverse Scattering Analysis using Born approximation is applied to the Biot's Poroelastic model for the visualization of scatterers in porous media. Spatial and frequency distribution of the scattering amplitude is used for the shape reconstruction of a scatterer in which solid frame and/or pore fluid properties are different from surrounding media. The inversion method is based on the field integral expression of the scattered field, its linearization by Born approximation, and a direct inversion method by using the Inverse Fourier Transform. In this paper, theoretical background and numerical formulations of the method are described, and some examples of the reconstructed form of scatterers using simulated scattered field are shown. The examples show the applicability of the inversion method to the fluid filled porous material. The limited aperture problem is also discussed using the numerical results. It is shown that the reconstruction of scatterer geometry is still available with some degree of the lack of information.

Comparison of sound speed and attenuation measured in a sandy sediment to predictions based on the Biot theory of porous media

During the sediment acoustics experiment in 1999 (SAX99), several researchers measured sound speed and attenuation. Together, the measurements span the frequency range of about 125 Hz-400 kHz. The data are unique both for the frequency range spanned at a common location, and for the extensive environmental characterization that was carried out as part of SAX99. Environmental measurements were sufficient to determine or bound the values of almost all the sediment and pore water physical property input parameters of the Biot poroelastic model for sediment. However, the measurement uncertainties for some of the parameters result in significant uncertainties for Biot-model predictions. Here, measured sound-speed and attenuation results are compared to the frequency dependence predicted by Biot theory and a simpler "effective density" fluid model derived from Biot theory. Model/data comparisons are shown where the uncertainty in Biot predictions due to the measurement uncertainties for values of each input parameter are quantified. A final set of parameter values, for use in other modeling applications e.g., in modeling backscattering (Williams et al., 2002) are given, that optimize the fit of the Biot and effective density fluid models to the sound-speed dispersion and attenuation measured during SAX99. The results indicate that the variation of sound speed with frequency is fairly well modeled by Biot theory but the variation of attenuation with frequency deviates from Biot theory predictions for homogeneous sediment as frequency increases. This deviation may be due to scattering from volume heterogeneity. Another possibility for this deviation is shearing at grain contacts hypothesized by Buckingham; comparisons are also made with this model.