Seismic AVO statistical inversion incorporating poroelasticity

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

Finite element implementation of poroelasticity theory for swelling dynamics of hydrogels

Abstract Hydrogel can swell to many times of its dry volume, resulting in large deformation which is vital for its function. The swelling process is regulated by many physical and chemical mechanisms, and can, to some extent, be fairly described by the poroelasticity theory. Implementation of the poroelastieity theory in the framework of finite element method would aid the design and optimization of hydrogel-based soft devices. Choosing chemical potential and displacement as two field variables, we present the implementation of poroelastieity tailored for hydrogel swelling dynamics, detail the normalization of physical parameters and the treatment of boundary conditions. Several examples are presented to demonstrate the feasibility and correctness of the proposed strategy.

MULTIRATE TIME ITERATIVE SCHEME WITH MULTIPHYSICS FINITE ELEMENT METHOD FOR A NONLINEAR POROELASTICITY

In this paper,a multirate time iterative scheme with multiphysics finite element method is proposed and analyzed for the nonlinear poroelasticity model.The original problem is reformulated into a generalized nonlinear Stokes problem coupled with a diffusion problem of a pseudo pressure field by a new multiphysics approach.A multiphysics finite element method is adopted for the spatial discretization,and the generalized nonlinear Stokes problem is solved in a coarse time step and the diffusion problem is solved in a finer time step.The proposed algorithm is a decoupled algorithm,which is easily implemented in computation and reduces greatly computation cost.The stability analysis and the convergence analysis for the multirate iterative scheme with multiphysics finite element method are given.Some numerical tests are shown to demonstrate and validate the analysis results.

VISCOELASTICITY AND POROELASTICITY IN ELASTOMERIC GELS

An elastomeric gel is a mixture of a polymer network and a solvent. In response to changes in mechanical forces and in the chemical potential of the solvent in the environment, the gel evolves by two concurrent molecular processes： the conformational change of the network, and the migration of the solvent. The two processes result in viscoelasticity and poroelasticity, and are characterized by two material-specific properties： the time of viscoelastic relaxation and the effective diffusivity of the solvent through the network. The two properties define a material- specific length. The material-specific time and length enable us to discuss macroscopic observations made over different lengths and times, and identify limiting conditions in which viscoelastic and poroelastic relaxations have either completed or yet started. We formulate a model of homogeneous deformation, and use several examples to illustrate viscoelasticity-limited solvent migration, where the migration of the solvent is pronounced, but the size of the gel is so small that the rate of change is limited by viscoelasticity. We further describe a theory that evolves a gel through inhomogeneous states. Both infinitesimal and finite deformation are considered.

Multigrid Method for Poroelasticity Problem by Finite Element Method

In this paper,we will investigate a multigrid algorithm for poroelasticity problem by a new finite element method with homogeneous boundary conditions in two dimensional space.We choose N´ed´elec edge element for the displacement variable and piecewise continuous polynomials for the pressure variable in the model problem.In constructing multigrid algorithm,a distributive Gauss-Seidel iteration method is applied.Numerical experiments shows that the finite element method achieves optimal convergence order and the multigrid algorithm is almost uniformly convergent to mesh size h and parameter dt on regular meshes.

COMPUTATIONAL MULTISCALE METHODS FOR LINEAR HETEROGENEOUS POROELASTICITY

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling.This poroelasticity problem suffers from rapidly oscillating material parameters,which calls for a thorough numerical treatment.In this paper,we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity.Therein,local corrector problems are constructed in line with the static equations,whereas we propose to consider the full system.This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure.We prove the optimal first-order convergence of this method and verify the result by numerical experiments.

Effect of stress on fracture development in the Asmari reservoir in the Zagros Thrust Belt

Development and production from fractured reservoirs require extensive knowledge about the reservoir structures and in situ stress regimes.For this,this paper investigates fractures and the parameters(aperture and density)through a combination of wellbore data and geomechanical laboratory testing in three separate wells in the Asmari reservoir,Zagros Belt,Iran.The Asmari reservoir(Oligo-Miocene)consists mainly of calcitic and dolomitic rocks in depths of 2000e3000 m.Based on the observation of features in several wellbores,the orientation and magnitude of the in situ stresses along with their influence on reservoir-scale geological structures and neotectonics were determined.The study identifies two regional tectonic fracture settings in the reservoir:one set associated with longitudinal and diagonal wrinkling,and the other related to faulting.The former,which is mainly of open fractures with a large aperture,is dominant and generally oriented in the N45°-90°W direction while the latter is obliquely oriented relative to the bedding and characterized by N45°-90°E.The largest aperture is found in open fractures that are longitudinal and developed in the dolomitic zones within a complex stress regime.Moreover,analysis of drilling-induced fractures(DIFs)and borehole breakouts(BBs)from the image logs revealed that the maximum horizontal stress(SHmax)orientation in these three wells is consistent with the NE-SW regional trend of the SHmax(maximum principal horizontal stress)in the Zagros Belt.Likewise,the stress magnitude obtained from geomechanical testing and poroelastic equations confirmed a variation in stress regime from normal to reverse,which changes in regard to active faults in the study area.Finally,a relationship between the development degree of open fractures and in situ stress regime was found.This means that in areas where the stress regime is complex and reverse,fractures would exhibit higher density,dip angle,and larger apertures.

Similarities and differences in inherent mechanism and characteristic frequency between the one-dimensional poroelastic model and the layered White model

The similarities and differences in inherent mechanism and characteristic frequency between the onedimensional(1D)poroelastic model and the layered White model were investigated.This investigation was conducted under the assumption that the rock was homogenous and isotropic at the mesoscopic scale.For the inherent mechanism,both models resulted from quasi-static flow in a slow P-wave diffusion mode,and the differences between them originated from saturated fluids and boundary conditions.On the other hand,for the characteristic frequencies of the models,the characteristic frequency of the 1D poroelastic model was first modified because the elastic constant and formula for calculating it were misused and then compared to that of the layered White model.Both of them moved towards higher frequencies with increasing permeability and decreasing viscosity and diffusion length.The differences between them were due to the diffusion length.The diffusion length for the 1D poroelastic model was determined by the sample length,whereas that for the layered White model was determined by the length of the representative elementary volume(REV).Subsequently,a numerical example was presented to demonstrate the similarities and differences between the models.Finally,published experimental data were interpreted using the 1D poroelastic model combined with the Cole-Cole model.The prediction of the combined model was in good agreement with the experimental data,thereby validating the effectiveness of the 1D poroelastic model.Furthermore,the modified characteristic frequency in our study was much closer to the experimental data than the previous prediction,validating the effectiveness of our modification of the characteristic frequency of the 1D poroelastic model.The investigation provided insight into the internal relationship between wave-induced fluid flow(WIFF)models at macroscopic and mesoscopic scales and can aid in a better understanding of the elastic modulus dispersion and attenuation caused by the WIFF at different scales.

Teleseismic waves reveal anisotropic poroelastic response of wastewater disposal reservoir

Connecting earthquake nucleation in basement rock to fluid injection in basal,sedimentary reservoirs,depends heavily on choices related to the poroelastic properties of the fluid-rock system,thermo-chemical effects notwithstanding.Direct constraints on these parameters outside of laboratory settings are rare,and it is commonly assumed that the rock layers are isotropic.With the Arbuckle wastewater disposal reservoir in Osage County,Oklahoma,high-frequency formation pressure changes and collocated broadband ground velocities measured during the passing of large teleseismic waves show a poroelastic response of the reservoir that is both azimuthally variable and anisotropic;this includes evidence of static shifts in pressure that presumably relate to changes in local permeability.The azimuthal dependence in both the static response and shear coupling appears related to tectonic stress and strain indicators such as the orientations of the maximum horizontal stress and faults and fractures.Using dynamic strains from a nearby borehole strainmeter,we show that the ratio of shear to volumetric strain coupling is~0.41 which implies a mean Skempton's coefficient of A=0.24 over the plausible range of the undrained Poisson's ratio.Since these observations are made at relatively low confining pressure and differential stress,we suggest that the hydraulically conductive fracture network is a primary control on the coupling between pore pressure diffusion and elastic stresses in response to natural or anthropogenic sources.

3D geomechanical modeling of the response of the Wilzetta Fault to saltwater disposal

From 2009 to 2017,parts of Central America experienced marked increase in the number of small to moderate-sized earthquakes.For example,three significant earthquakes(~Mw 5)occurred near Prague,Oklahoma,in the U.S.in 2011.On 6 Nov 2011,an Mw 5.7 earthquake occurred in Prague,central Oklahoma with a sequence of aftershocks.The seismic activity has been attributed to slip on the Wilzetta fault system.This study provides a 3 D fully coupled poroelastic analysis(using FLAC3 D)of the Wilzetta fault system and its response to saltwater injection in the underpressured subsurface layers,especially the Arbuckle group and the basement,to evaluate the conditions that might have led to the increased seismicity.Given the data-limited nature of the problem,we have considered multiple plausible scenarios,and use the available data to evaluate the hydromechanical response of the faults of interest in the study area.Numerical simulations show that the injection of large volumes of fluid into the Arbuckle group tends to bring the part of the Wilzetta faults in Arbuckle group and basement into near-critical conditions.

Propagation of plane P-waves at interface between elastic solid and unsaturated poroelastic medium

A linear viscoporoelastic model is developed to describe the problem of reflection and transmission of an obliquely incident plane P-wave at the interface between an elastic solid and an unsaturated poroelastic medium, in which the solid matrix is filled with two weakly coupled fluids （liquid and gas）. The expressions for the amplitude reflection coefficients and the amplitude transmission coefficients are derived by using the potential method. The present derivation is subsequently applied to study the energy conversions among the incident, reflected, and transmitted wave modes. It is found that the reflection and transmission coefficients in the forms of amplitude ratios and energy ratios are functions of the incident angle, the liquid saturation, the frequency of the incident wave, and the elastic constants of the upper and lower media. Numerical results are presented graphically. The effects of the incident angle, the frequency, and the liquid saturation on the amplitude and the energy reflection and transmission coefficients are discussed. It is verified that in the transmission process, there is no energy dissipation at the interface.

Normal compression wave scattering by a permeable crack in a fluid-saturated poroelastic solid

A mathematical formulation is presented for the dynamic stress intensity factor (mode I) of a finite permeable crack subjected to a time-harmonic propagating longitudinal wave in an infinite poroelastic solid. In particular, the effect of the wave-induced fluid flow due to the presence of a liquid-saturated crack on the dynamic stress intensity factor is analyzed. Fourier sine and cosine integral transforms in conjunction with Helmholtz potential theory are used to formulate the mixed boundary-value problem as dual integral equations in the frequency domain. The dual integral equations are reduced to a Fredholm integral equation of the second kind. It is found that the stress intensity factor monotonically decreases with increasing frequency, decreasing the fastest when the crack width and the slow wave wavelength are of the same order. The characteristic frequency at which the stress intensity factor decays the fastest shifts to higher frequency values when the crack width decreases.

Uncertainty Analysis of Seepage-Induced Consolidation in a Fractured Porous Medium

Numerical modeling of seepage-induced consolidation process usually encounters significant uncertainty in the properties of geotechnical materials.Assessing the effect of uncertain parameters on the performance variability of the seepage consolidation model is of critical importance to the simulation and tests of this process.To this end,the uncertainty and sensitivity analyses are performed on a seepage consolidation model in a fractured porous medium using the Bayesian sparse polynomial chaos expansion(SPCE)method.Five uncertain parameters including Young’s modulus,Poisson’s ratio,and the permeability of the porous matrix,the permeability within the fracture,and Biot’s constant are studied.Bayesian SPCE models for displacement,flow velocity magnitude,and fluid pressure at several reference points are constructed to represent the input-output relationship of the numerical model.Based on these SPCE models,the total and first-order Sobol’indices are computed to quantify the contribution of each uncertain input parameter to the uncertainty of model responses.The results show that at different locations of the porous domain,the uncertain parameters show different effects on the output quantities.At the beginning of the seepage consolidation process,the hydraulic parameters make major contributions to the uncertainty of the model responses.As the process progresses,the effect of hydraulic parameters decreases and is gradually surpassed by the mechanical parameters.This work demonstrates the feasibility to apply Bayesian SPCE approach to the uncertainty and sensitivity analyses of seepage-induced consolidation problems and provides guidelines to the numerical modelling and experimental testing of such problems.

The effect of the fracture distribution on CO_2 injection into a coal seam

The effect of the fracture distribution on CO 2 injection into coal seams was studied with a heterogeneous model having dual porosity to represent both the primary medium (the coal matrix) and the secondary medium (the fractures) under variable stress conditions. A numerical generation method and a digital image processing method were used to model the heterogeneous fracture distribution in the coal. The model solutions demonstrate that: (1) the fractures are the main channel for gas flow and their distribution has an important impact on the gas injection rate; (2) the fractures only affect the injection rate of CO 2 into the coal but not the final storage amount; (3) when gas is injected into coal the fractures will first expand and then close due to the changing effective stresses and the adsorption induced swelling of coal grains. This fully coupled dual-porosity model with a heterogeneous fracture distribution provides a way to predict the CO 2 injection into a coal seam.

Effects of CO₂Injection on the Seismic Velocity of Sandstone Saturated with Saline Water

Geological sequestration (GS) of carbon dioxide (CO2) is considered as one of the most promising technologies to reduce the amount of anthropogenic CO2 emission in the atmosphere. To ensure success of CO2 GS, monitoring is essential on ascertaining movement, volumes and locations of injected CO2 in the sequestration reservoir. One technique is to use time-lapsed seismic survey mapping to provide spatial distribution of seismic wave velocity as an indicator of CO2 migration and volumes in a storage reservoir with time. To examine the use of time-lapsed seismic survey mapping as a monitoring tool for CO2 sequestration, this paper presents mathematical and experimental studies of the effects of supercritical CO2 injection on the seismic velocity of sandstone initially saturated with saline water. The mathematical model is based on poroelasticity theory, particularly the application of the Biot-Gassmann substitution theory in the modeling of the acoustic velocity of porous rocks containing two-phase immiscible pore fluids. The experimental study uses a high pressure and high temperature triaxial cell to clarify the seismic response of a sample of Berea sandstone to supercritical CO2 injection under deep saline aquifer conditions. Measured ultrasonic wave velocity changes during CO2 injection in the sandstone sample show the effects of pore fluid distribution in the seismic velocity of porous rocks. CO2 injection was shown to decrease the P-wave velocity with increasing CO2 saturation whereas the S-wave velocity was almost constant. The results confirm that the Biot-Gassmann theory can be used to model the changes in the acoustic P-wave velocity of sandstone containing different mixtures of supercritical CO2 and saline water provided the distribution of the two fluids in the sandstone pore space is accounted for in the calculation of the pore fluid bulk modulus. The empirical relation of Brie et al. for the bulk modulus of mixtures of two-phase immiscible fluids, in combination with the Biot-Gassmann theory, was found to satisfactorily represent the pore-fluid dependent acoustic P-wave velocity of sandstone.

Mathematical osteon model for examining poroelastic behaviors

An extended and reasonable stress boundary condition at an osteon exte- rior wall is presented to solve the model proposed by Remond and Naili. The obtained pressure and fluid velocity solutions are used to investigate the osteonal poroelastic behaviors. The following results are obtained. （i） Both the fluid pressure and the velocity amplitudes are proportional to the strain amplitude and the loading frequency. （ii） In the physiological loading state, the key role governing the poroelastic behaviors of the osteon is the strain rate. （iii） At the osteon scale, the pressure is strongly affected by the permeability variations, whereas the fluid velocity is not.

Fluid discrimination incorporating amplitude variation with angle inversion and squirt flow of the fluid

Pre-stack seismic inversion is an important method for fluid identification and reservoir characterization in exploration geophysics. In this study, an effective fluid factor is initially established based on Biot poroelastic theory, and a pre-stack seismic inversion method based on Bayesian framework is used to implement the fluid identification. Compared with conventional elastic parameters, fluid factors are more sensitive to oil and gas. However, the coupling effect between rock porosity and fluid content is not considered in conventional fluid factors, which may lead to fuzzy fluid identification results. In addition,existing fluid factors do not adequately consider the physical mechanisms of fluid content, such as squirt flow between cracks and pores. Therefore, we propose a squirt fluid factor(SFF) that minimizes the fluid and pore mixing effects and takes into account the squirt flow. On this basis, a novel P-wave reflection coefficient equation is derived, and the squirt fluid factor is estimated by amplitude variation with offset(AVO) inversion method. The new reflection coefficient equation has sufficient accuracy and can be utilized to estimate the parameters. The effectiveness and superiority of the proposed method in fluid identification are verified by the synthetic and field examples.

Poroelastic behaviors of the osteon: A comparison of two theoretical osteon models

In the paper, two theoretical poroelastic osteon models are presented to compare their poroelastic behaviors, one is the hollow osteon model （Haversian fluid is neglected） and the other is the osteon model with Haversian fluid con- sidered. They both have the same two types of imperme- able exterior boundary conditions, one is elastic restraint and the other is displacement constrained, which can be used for analyzing other experiments performed on similarly shaped poroelastic specimens. The obtained analytical pressure and velocity solutions demonstrate the effects of the loading fac- tors and the material parameters, which may have a signifi- cant stimulus to the mechanotransduction of bone remodel- ing signals. Model comparisons indicate： （1） The Haversian fluid can enhance the whole osteonal fluid pressure and ve- locity fields. （2） In the hollow model, the key loading fac- tor governing the poroelastic behavior of the osteon is strain rate, while in the model with Haversian fluid considered, the strain rate governs only the velocity. （3） The pressure ampli- tude is proportional to the loading frequency in the hollow model, while in the model with Haversian fluid considered, the loading frequency has little effect on the pressure ampli- tude.

An analytical poroelastic model for laboratorial mechanical testing of the articular cartilage (AC)

The articular cartilage （AC） can be seen as a biphasic poroelastic material. The cartilage deformation under compression mainly leads to an interstitial fluid flow in the porous solid phase. In this paper, an analytical poroelastic model for the AC under laboratorial mechanical testing is developed. The solutions of interstitial fluid pressure and velocity are obtained. The results show the following facts. （i） Both the pressure and fluid velocity amplitudes are proportional to the strain loading amplitude. （ii） Both the amplitudes of pore fluid pressure and velocity in the AC depend more on the loading amplitude than on the frequency. Thus, in order to obtain the considerable fluid stimulus for the AC cell responses, the most effective way is to increase the loading amplitude rather than the frequency. （iii） Both the interstitiM fluid pressure and velocity are strongly affected by permeability variations. This model can be used in experimental tests of the parameters of AC or other poroelastic materials, and in research of mechanotransduction and injury mechanism involved interstitial fluid flow.

Combined approach of poroelastic and earthquake nucleation applied to the reservoir-induced seismic activity in the Val d’Agri area,Italy

In this work,an approach is developed to study the seismicity associated with the impoundment and level changes of a water reservoir(reservoir induced seismicity e RIS).The proposed methodology features a combination of a semi-analytical poroelastic model with an earthquake nucleation approach based on rate-and-state frictional law.The combined approach was applied to the case of the Pertusillo Lake,located in the Val d’Agri area(Italy),whose large seasonal water level changes are believed to induce protracted micro-seismicity(local magnitude ML<3).Results show that the lake impoundment in 1962 could have produced up to 0.5 bar(1 bar=100 kPa)changes in Coulomb failure stress(DCFS),while the seasonal water level variation is responsible for variation up to 0.05 bar.Modeling results of the seismicity rates in 20012014 show that the observed earthquakes are well correlated with the modeled DCFS.Finally,the reason that the seismicity is only observed at southwest of the Pertusillo Lake is provided,which is likely attributed to different rock lithologies and depletion caused by significant hydrocarbon exploitation in the northeastern sector of the lake.