Biot's Theory Research Articles

Time-lapse amplitude variation with offset difference inversion based on the reformulated Biot-Squirt model

Time-lapse AVO (Amplitude variation with offset) inversion is a significant technique for the estimation of dynamic reservoir changes. The main theoretical foundation is Zoeppritz equations in single-phase media. At present, there are studies on the inversion method of using Biot's theory. Although it has some improvements in accuracy compared with the method of using Zoeppritz equations, the squirt-flow mechanism is not considered. Furthermore, the precision of difference data equations still needs to be improved in difference inversion. In view of the above problems, the reformulated BISQ (Biot-Squirt) model is introduced into the inversion, and the iterative optimization is used to increase the precision of difference data equations continually in this paper. First, we derive the difference data equations about the reservoir-parameter changes. The reformulated BISQ model is applied, and can describe the characteristics of P-wave reflection accurately and expediently. Secondly, the objective function about the reservoir-parameter changes is constructed. The Bayesian theory and the model smoothing constraint are used, and are helpful to obtain accurate and stable solutions. Thirdly, we derive the equation for estimating the reservoir-parameter changes, and the inversion is iterative for the continuous improvement of accuracy. The difference inversion method is applied, and is helpful to enhance the data repeatability and reduce the calculation amount. Finally, synthetic and real data are applied for the inversion and testing to prove the effectiveness and feasibility of the method.

Response of offshore large-diameter pipe piles in layered poroelastic soil to lateral dynamic loading

The analytical solution for the lateral dynamic response of offshore large-diameter pipe piles embedded in seawater and layered saturated seabed soil is proposed. The hydrodynamic pressures of the outer and inner seawater are derived based on the inviscid compressible assumption of fluid, and the dynamic soil resistances of the outer and inner layered saturated seabed soil are derived based on Biot's poroelastic theory. The analytical expressions of the lateral dynamic response of the pile are obtained by using the transfer matrix method, and the dynamic characteristics of the lateral response of offshore pipe piles are parametrically investigated. It is concluded from the analysis results that the outer and inner hydrodynamic pressures significantly affect the natural frequency of the pile–water–soil system. Additionally, employing a homogeneous soil model with the same average modulus to represent the layered seabed soil can lead to significantly inaccurate estimations of the natural frequency and pile-head displacement of offshore pipe piles.

On propagation characteristics of ultrasonic guided waves in layered fluid-saturated porous media using spectral method.

Fluid-saturated porous media plays an increasingly important role in emerging fields such as lithium batteries and artificial bones. Accurately solving the governing equations of guided wave is the key to the successful application of ultrasonic guided wave nondestructive testing technology in fluid-saturated porous media. This paper derives the Lamb wave equation in layered fluid-saturated porous materials based on Biot theory and proposes the spectral method suitable for solving complex wave equations. The spectral method reconstructs the fundamental wave equations in the form of a matrix eigenvalue problem using spectral differentiation matrices. It introduces boundary conditions by replacing corresponding rows in the wave equation matrix with stress or displacement in matrix form. For complex differential equations, such as the governing equations of guided waves in porous media, the spectral method has the significant advantages of faster computation speed, less root loss, and easier encoding process. The spectral method is used to calculate the acoustic field characteristics under different boundary conditions and environments of the layer fluid-saturated porous media. Results show that the surface treatment details and environment of fluid-saturated porous materials play an important role in the propagation of guided waves.

A novel hybrid PD-FEM-FVM approach for simulating hydraulic fracture propagation in saturated porous media

To enhance the computational efficiency of fluid–solid coupling in peridynamics (PD), a hybrid modeling approach based on the classical Biot theory is proposed for simulating hydraulic crack propagation in saturated porous media. The deformation and damage of solids are described by the coupling of the finite element method (FEM) and PD. Based on Darcy’s law, the finite volume method (FVM) is used to describe fluid seepage and calculate pore water pressure. The mutual transfer of fluid pressure and solid deformation is realized through the transition layer between the solid layer and the fluid layer. Firstly, the effectiveness of the proposed method is verified by a porous media seepage simulation example. Secondly, the ability and efficiency of this method to simulate crack propagation in saturated porous media are verified by several examples of hydraulic fracturing of rock with a single pre-existing crack. Finally, the synchronous hydraulic fracturing process of rock with double cracks is simulated. The ability of this method to simulate the simultaneous propagation of multiple fractures in the rock under fluid–solid coupling is further illustrated. The aforementioned studies demonstrate that the novel hybrid PD-FEM-FVM approach not only ensures computational accuracy and effectiveness but also significantly enhances computational efficiency.

Measuring audible acoustic frequencies parameters of rigid porous media via an innovative impedance tube method - Solving the inverse problem

This study introduces an experimental approach to measuring audible acoustic frequency parameters of a rigid porous medium using an impedance tube. Grounded in the equivalent fluid model, a derivative of Biot's theory, our investigation delves into the propagation of waves within porous media, emphasizing the importance of effective density and dynamic compressibility of the saturated fluid. Our primary focus is on resolving the inverse problem by minimizing both experimental and theoretical absorption coefficient expressions within the audible frequency range. Concurrently, we determine four pivotal parameters: viscous and thermal permeability, the inertia factor introduced by Norris, and the thermal tortuosity introduced by Lafarge. The outcomes include a comparative analysis between experimental and simulated absorption coefficients, leveraging optimized parameters, across two distinct polyurethane foam samples.

The impact of boundary conditions on the acoustical behavior of lightweight granular particle stacks

The acoustical behavior of air-saturated granular stacks can differ significantly from that of more traditional sound absorbing materials such as fibrous webs and foams when tested in a standing wave tube. The latter materials can often be modeled as equivalent fluids by using one-dimensional theory, with the further assumption that their behavior does not depend on the input signal level. In contrast, level dependence of the response of lightweight granular materials such as glass bubbles has been consistently observed in previous studies. At low input levels, the absorption coefficient of glass bubble stacks shows solid-like behavior with multiple peaks associated with radial modes of the solid phase of the edge-constrained stack: i.e., the response is two-dimensional. In contrast, at high input levels, glass bubble stacks show one-dimensional fluid-like behavior, with the quarter wave depth resonance dominating the response. It is hypothesized here that at high input levels the particles slip axially at the circumferential boundary, thus showing a one-dimensional response, while at low levels, the particles are constrained to have zero axial velocity, thus creating a two-dimensional response. It is shown here that a two-dimensional finite difference implementation of the poro-elastic Biot theory can accurately reproduce the observed level-dependent behavior.

Dynamic Response Analysis and Liquefaction Potential Evaluation of Riverbed Induced by Tidal Bore

Tidal bores, defined by sudden upstream surges of tidal water in estuaries, exert significant hydrodynamic forces on riverbeds, leading to complex sedimentary responses. This study examines the dynamic response and liquefaction potential of riverbeds subjected to tidal bores in macro-tidal estuaries. An analytical model, developed using the generalized Biot theory and integral transform methods, evaluates the dynamic behavior of riverbed sediments. Key factors such as permeability, saturation, and sediment properties are analyzed for their influence on momentary liquefaction. The results indicate that fine sand reduces liquefaction risk by facilitating pore water discharge, while silt soil increases sediment instability. Additionally, the study reveals that pressure gradients induced by tidal bores can trigger momentary liquefaction, with the maximum liquefaction depth predicted based on horizontal pressure gradients being five times that predicted based on vertical pressure gradients. This research highlights the critical role of sediment characteristics in riverbed stability, providing a comprehensive understanding of the interactions between tidal bores and riverbed dynamics. The findings contribute to the development of predictive models and guidelines for managing the risks of tidal bore-induced liquefaction in coastal and estuarine environments.

Dynamic analysis of fractional poroviscoelastic reinforced subgrade under moving loading

This paper conducts the dynamic analysis of fractional poroviscoelastic reinforced subgrade under moving loading. Based on the Biot theory and transversely isotropic (TI) parameter expression of the geogrid reinforced subgrade, the governing equations of the poroelastic reinforced subgrade are established in the wavenumber domain by the double Fourier transform. Considering the viscosity of the soil skeleton and the flow-dependent viscosity between the soil skeleton and pore water, the governing equations are extended to the fractional poroviscoelastic medium by introducing the Zener viscoelastic model, fractional calculus theory and the dynamic elastic-viscoelastic correspondence principle. Combining boundary conditions and interlayer continuity conditions, the extended precise integration method (PIM) and double Fourier integral transform are employed to obtain the solution of fractional poroviscoelastic reinforced subgrade in the spatial domain. After the numerical validation, a sensitivity analysis of the relaxation time, permeability, reinforcement ratio and the load velocity are conducted.

Dynamic response in a transversely isotropic and layered nonlocal poroelastic seabed subjected to vertical P-waves

A novel model is put forward to characterize the seismic response excited by vertical P-waves in a transversely isotropic and layered nonlocal poroelastic seabed. The proposed model integrates nonlocal parameter, anisotropy, and stratification to accurately examine wave propagation behavior. The fundamental equations for the underlying seabed are formulated using Biot's poroelastodynamic theory and Eringen's nonlocal theory. The wave equation for the overlying water is expressed in terms of the velocity potential. General solutions in both the poroelastic seabed and water layer are derived by solving the involved ordinary differential equations. Employing the newly devised and unconditionally stable propagator matrix scheme, semi-analytical solutions are derived for the time-harmonic response in a layered poroelastic seabed subjected to vertical P-wave excitation within the frequency domain. The validity of the proposed solutions is confirmed through rigorous comparison with the previously established analytical solutions. The influence of key material properties of seabed on the velocities of P-waves and free-field response in the poroelastic seabed is estimated in detail, along with the free-field response in a nonhomogeneous poroelastic seabed.

Dispersion of thermoelastic guided waves in multi-layered porous media

The mechanical properties of the graphite electrode will dynamically change during the charge-discharge cycle, which will affect the propagation characteristics of guided waves in lithium-ion batteries. Meanwhile, lithium-ion batteries experience fluctuations in operating temperature during cycling, which will also influence the propagation of guided waves. This research aims to investigate the guided wave propagation problem of porous electrode material with electrolyte versus temperature. The problem is under the framework of Green-Naghdi theory and Biot theory, and a numerical approach is presented to solve the thermoelastic wave propagation characteristics in such a multi-layered porous electrode. A frequency domain simulation for a multi-layered porous anode will be established. The simulation results confirm the feasibility and accuracy of the proposed approach. The porosity of the anode material shows obvious dynamic variation during cycling, which illustrates a strong correlation with the elastic modulus of the solid skeleton. Then, the effects of porosity and temperature variations on the propagation characteristics of thermoelastic guided waves in multi-layered porous graphite electrodes will be analyzed in detail. The phase velocity of fundamental modes appears regularly shift in the low frequency range. This may provide theoretical support for the acoustic nondestructive characterization of the operating states of the lithium-ion batteries.

Influence of ice properties on wave propagation characteristics in partially frozen soils and rocks: A temperature-dependent rock-physics model

A better understanding of the temperature effects on the propagation characteristics of elastic waves in frozen soils and rocks is imperative for accurately quantifying their freezing degrees. Although the existing rock-physics models based on the three-phase Biot (TPB) theory adeptly interpret observed velocity variation with temperature (VVT) curves, they often lack a comprehensive understanding of the mechanisms underlying attenuation variation with temperature (AVT) curves. In this study, we first extend the TPB theory to incorporate the temperature-dependent properties of ice, such as changes in volumetric fraction, morphology, and viscoelasticity, by integrating the relevant thermodynamic laws. Model parameters related to ice properties and interactions, such as rigidity, shear moduli, density, and friction, are redefined. Then, using a numerical rock-physics modeling approach, we examine influential factors and modes of the wave VVT and AVT responses. Our results indicate that the P- and S-wave velocities increase with source frequency, consolidation degree, and frame-supporting ice content, whereas they decrease with temperature and pore-floating ice content. The P- and S-wave attenuation factors increase with the frame-supporting ice content and decrease with the consolidation degree. Rising temperatures tend to amplify the peak magnitude of the P-wave attenuation factors and shift the central frequency of the S-wave attenuation factors. Finally, within a temperature-controlled laboratory environment, we conduct ultrasonic wave transmission testing on brine-saturated sediment and rock specimens. The results demonstrate that as the temperature increases from −15°C to −3°C, the P- and S-wave velocities decrease, whereas the P-wave attenuation factors decrease and the S-wave attenuation factors initially rise before declining. Our viscoelastic TPB theory outperforms the existing ones in interpreting the S-wave AVT observations. This temperature-dependent rock-physics model holds promise for interpreting the sonic logging data in the time-lapse monitoring of permafrost, glaciers, and Antarctica.

The control mechanism of P-wave attenuation in unconsolidated porous media.

Unconsolidated porous media are distinct from consolidated porous rocks in the negligible bulk and shear moduli. This paper is motivated by resolving the control mechanism of P-wave attenuation in the media (represented by Toyoura sands and glass beads) saturated with water. The first model is Biot theory in which longitudinal friction (arising from velocity difference between the two phases) is quantified using dynamic permeability as a function of frequency. The first model simulates phase velocity (Vp) and the ultrasonically measured quality factor (Qp) well. A second model is the transverse squirt model in which squirt is induced by pressure differential between contact of grains (COG) and the main pore space. The second model outputs unrealistic Vp and Qp. The results reveal that P-wave attenuation in unconsolidated porous media (saturated with water) is governed by longitudinal friction rather than intrapore squirt. Remarkably, low-frequency dynamic permeability is much smaller than Darcy permeability, indicating that ultrasonic P-wave is surprisingly capable of indirectly detecting the very narrow gap at COG.

Acoustic response of patchy-saturated porous media: Coupling Biot's poroelasticity equations for mono- and biphasic pore fluids.

Patchy saturation is a term used in the seismic prospecting literature to describe the state of a geological formation in which two immiscible pore fluids prevail in mesoscopic-scale clusters. If the pore fluids have contrasting compressibilities, wave-induced fluid pressure diffusion (FPD) processes may induce significant attenuation and velocity dispersion on seismic waves. Biot's monophasic poroelasticity theory is widely used to model the seismic response of rocks containing binary patches of two immiscible pore fluids. Even though effective fluid approximations may help to represent more realistic partially saturated patches using Biot's monophasic equations, the so inferred dissipation may not be representative of actual biphasic FPD phenomena. In this work, Biot's equations for mono- and biphasic fluids are combined to model FPD processes in porous media, comprising fully and partially saturated patches. An analytical solution for one-dimensional layered patchy-saturated media is presented which permits to explain some, as of yet enigmatic, experimentally observed characteristics such as increased seismic attenuation and stiffening effects occurring at low saturations. The results show that the existence of an additional diffusive wave mode within partially saturated patches may render conventional binary and effective fluid approaches incorrect and error prone.

Rock-physics modeling and pre-stack seismic inversion for the Cambrian superdeep dolomite reservoirs in Tarim Basin, Northwest China

The Cambrian superdeep dolomite in Tarim Basin has great resource potentialities. However, the superdeep dolomite reservoirs have the seismogeological characteristics of deep burial, low porosity and permeability, complex fracture networks and low signal-to-noise, which cannot support the efficient exploration and development of superdeep dolomite reservoir due to restricting the precision of fracture prediction. The fractures and pores compose the main storage spaces of superdeep dolomite reservoir. To better characterize the superior reservoir of superdeep dolomite with complex fracture, we first describe the properties of dolomite using core data, which illustrates that the low-porosity and low-permeability dolomite reservoirs have three tendencies: relatively higher porosity and lower permeability, relatively lower porosity and lower permeability, and relatively lower porosity and higher permeability. The development of porosity is decided by the size of dolomite particle, and when the porosity is greater, the pore type controls the permeability of the reservoir. It is difficult to effectively characterize dolomite with multiple types fractures and pores with the traditional rock physical analysis. Considering the influence of complex pore structures on the elastic parameters of superdeep dolomites, we propose a fracture characterization indicator (FCI) based on Biot theory. We test the FCI using cores and well-logging data, which can quantitatively describe the change of pores in dolomite reservoirs, much better than conventional rock-physics modeling. Meanwhile, the pre-stack seismic gather optimization process is established to employ the combination of denoising and flattening. The density indicator (DI) is proposed due to the poor result of density inversion. The priority distributions of superdeep dolomite reservoirs are revealed through the FCI and DI applied to 3D pre-stack seismic inversions. The results are consistent with those by actual drilling, which confirms that the proposed method can serve as a new way to predict superdeep dolomite reservoirs.

Modeling patchy saturation of fluids in nanoporous media probed by ultrasound and optics.

Nanoporous materials provide high surface area per unit mass and are capable of fluids adsorption. While the measurements of the overall amount of fluid adsorbed by a nanoporous sample are straightforward, probing the spatial distribution of fluids is nontrivial. We consider literature data on adsorption and desorption of fluids in nanoporous glasses reported along with the measurements of ultrasonic wave propagation. We analyze these using the so-called dynamic equivalent medium approach, which is based on Biot's theory of dynamic poroelasticity with coefficients that are continuous random functions of position. When further constrained by optical scattering data for similar systems, the calculations show that on adsorption the characteristic patch size is of the order of 100 pore diameters, while on desorption the patch size is comparable to the sample size. Our analysis suggests that one can employ ultrasound to probe the uniformity of the spatial distribution of fluids in nanoporous materials.

Analytical solution for horizontal vibration of partially embedded pipe pile group in layered saturated soil

AbstractBased on Biot's dynamic wave theory and Novak's plane strain model (PSM), an analytical model for the horizontal vibration of a partially embedded pipe pile group (PEPPG) in layered saturated soil (LSS) that considers the soil‐plugging effect was established: First, the frequency‐domain analytical expressions of the saturated surrounding soil (SSS) and soil plug were obtained by introducing operator decomposition theory and the variable separation method. Second, the analytical solution of the horizontal impedance of a partially embedded pipe pile in the LSS is derived using the coupled condition of the pile–soil interface and transfer matrix method. Finally, an analytical solution for the horizontal impedance of PEPPG in LSS was derived by introducing the horizontal dynamic interaction factor of piles and the superposition method. The derived solution is then reduced and compared with existing theoretical solutions to verify its rationality. In addition, further parametric analysis was conducted to investigate the influences of pile spacing, embedment ratio, properties of layered SSS, and soil‐plugging on the horizontal vibration characteristics (HVCs) of PEPPG.

A mixed nonlocal finite element model for thermo‐poro‐elasto‐plastic simulation of porous media with multiphase fluid flow

SummaryA new mixed nonlocal finite element framework is developed for nonlinear thermo‐poro‐elasto‐plastic simulation of porous media with multiphase pore fluid flow and thermal coupling. The solid‐fluid interaction is accounted for using the mixture theory of Biot based on the volume fractions concept. Different sources of nolinearities arising from the multiphase fluid flow effects, advective‐diffusive heat transfer, inelastic deformation, fluid flux injection induced mechanical tractions, solid skeleton deformation permeability dependence, and temperature dependent viscosity are included in developing a robust numerical solver for the targeted coupled multiphysics problem. To address the effect of microstructure in inelastic localized deformation behaviour with dilational softening, a nonlocal plasticity model is proposed based on a characteristic length scale which rectifies the non‐physical pathological mesh dependence problem encountered in conventional plasticity. The accuracy and strength of the developed model is shown with comparing the obtained numerical results of a benchmark bilateral compression test with existing published data in the literature. To show the versatility and robustness of the developed computational framework in modelling the geomechanics of real‐case engineering practices, large scale thermo‐hydro‐mechanical (THM) subsurface stimulation processes with applications in enhanced oil recovery (EOR) are effectively simulated and the targeted enhanced recovery and performances are demonstrated. The current formulation does not include phase transformation modelling capability, and therefore, the developed models may not be applicable for simulation of the engineering processes that involve phase change behaviour (e.g., steam injection).

Analytical poro‐elastic solution of deep lined tunnels in anisotropic rock with consideration of tunnel face advance

AbstractThis paper aims at deriving a closed‐form solution for deep lined tunnels within a saturated anisotropic poro‐elastic medium. The derivation of the solution is carried out with the assumption of plane strain conditions along the tunnel axis, an isotropic elastic liner, a steady‐state flow, and perfect contact between the liner and rock. For this purpose, the complex potential function approach pioneered by Lekhnitskii for the anisotropic elastic body was used to develop the mechanical response of the rock mass. A complex hydraulic potential function is also introduced to solve the steady state fluid flow. Then the well‐known Biot theory is chosen to describe the hydro‐mechanical coupling of the anisotropic saturated rock. The stress and displacement solutions of both the liner and the surrounding rock, considering the tunnel face advance with respect to the considered section, are derived based on the principle of the convergence‐confining method. Comparisons with previous closed‐form solutions for isotropic case and finite element solution for anisotropic case were made to show the accuracy of the current closed form solution. Sensitivity analysis is performed based on this analytical solution to highlight the effect of some rock parameters on the stress and displacement of the liner.

Designing hybrid aerogel-3D printed absorbers for simultaneous low frequency and broadband noise control

Recent studies have demonstrated that granular aerogels with sub-50 μm particles surpass conventional acoustic materials like glass fibers and polyurethane foams in low-frequency sound absorption. However, incorporating such granular materials within practical structural solutions remains a challenge. In this study, we use additive manufacturing to overcome this challenge by designing a modular geometry that allows us to encapsulate such granular materials within a 3D printed scaffold. Using the 3D printed porous scaffold provides the added benefits of tunability and durability. Further, we introduce a design methodology and software tool to facilitate the efficient design of such hybrid sound absorbers. The proposed method uses the Johnson-Champoux-Allard model, Biot theory, and the transfer matrix method to model the acoustical behavior of hybrid absorber designs that use layered granular aerogels and 3D printed bulk absorbers. The tool can also be used to inversely characterize the acoustic bulk properties of such materials. We validate the design concept and tool by comparing the model predictions with experimental measurements. Finally, we outline strategies for designing hybrid absorbers that can provide application specific low-frequency and broadband absorption performance within specific frequency ranges, while considering engineering constraints on their total mass and depth.

Semi‐implicit material point method for simulating infiltration‐induced failure of unsaturated soil structures

AbstractThis study presents a semi‐implicit MPM to adequately characterize the mechanical behavior of unsaturated soil based on Biot's mixture theory. To represent the dependency of the degree of saturation on the suction, we employ the VG model along with a soil‐water characteristic curve, which determines a functional form of permeability called the Mualem model. Hencky's hyperelastic model and the Drucker‐Prager model assuming nonassociativity are adopted for elastic and plastic deformations, respectively. The novelty of this study is the incorporation of the fractional‐step method into the MPM framework so that the pore water pressure is obtained by implicitly solving the pressure Poisson's equation, which reduces numerical instability and improves computational efficiency. Also, because the drag force between solid and liquid phases is evaluated using the intermediate velocity of pore water relative to the intermediate velocity of solid skeleton, the time increment can be chosen without considering the magnitude of water permeability. In addition, to suppress “odd‐even” oscillation, we employ a sub‐grid method in which two grids with different spatial resolutions are used for the velocities and pore water pressure. Furthermore, considering the advantages and disadvantages of two different interpolation schemes for pore water pressure, we suggest switching the schemes depending on the model conditions. Several numerical examples are presented to demonstrate the performance of the proposed method. Specifically, unidirectional consolidation and leak flow analyses are performed for verification purposes, followed by validation analysis of a model experiment of infiltration‐induced landslides.