Effects of confining pressure and pore pressure on multipole borehole acoustic field in fluid-saturated porous media

In-situ stress is a common stress in the exploration and development of oil reservoirs. Therefore, it is of great significance to study the propagation characteristics of borehole acoustic waves in fluid-saturated porous media under stress.Based on the acoustoelastic theory of fluid-saturated porous media, the field equation of fluid-saturated porous media under the conditions of confining pressure and pore pressure and the acoustic field formula of multipole source excitation in open hole are given. The influences of pore pressure and confining pressure on guided waves of multipole borehole acoustic field in fluid-saturated porous media are investigated. The numerical results show that the phase velocity and excitation intensity of guided wave increase significantly under the confining pressure. For a given confining pressure, the phase velocity of the guided wave decreases with pore pressure increasing. The excitation intensity of guided wave increases at low frequency and then decreases at high frequency with pore pressure increasing, except for that of Stoneley wave which decreases in the whole frequency range. These results will help us get an insight into the influences of confining pressure and pore pressure on the acoustic field of multipole source in borehole around fluid-saturated porous media.

WAVE PROPAGATION WITH MASS-COUPLING EFFECT IN FLUID-SATURATED POROUS MEDIA

This article utilizes the theory of mixtures to formulate a general theory of wave propagation with mass-coupling effect in fluid-saturated porous media. An attempt is made to discuss the physical interpretation and the thermodynamic restriction of the coefficients appearing in the equations obtained, by the comparison it is shown that Biot's classical theory and the present one are essentially consistent. Also wave velocities in some special cases are calculated, from which it is concluded that mass-coupling and permeability of media greatly affect wave propagation behavior.

PARAMETERS INVERSION OF FLUID-SATURATED POROUS MEDIA BASED ON NEURAL NETWORKS

The multi- layers feedforward neural network is used for inversion ofmaterial constants of fluid-saturated porous media. The direct analysis of fluid-saturated porousmedia is carried out with the boundary element method. The dynamic displacement responses obtainedfrom direct analysis for prescribed material parameters constitute the sample sets training neuralnetwork. By virtue of the effective L-M training algorithm and the Tikhonov regularization method aswell as the GCV method for an appropriate selection of regu-larization parameter, the inversemapping from dynamic displacement responses to material constants is performed. Numerical examplesdemonstrate the validity of the neural network method.

GURTIN VARIATIONAL PRINCIPLE AND FINITE ELEMENT SIMULATION FOR DYNAMICAL PROBLEMS OF FLUID-SATURATED POROUS MEDIA

Based on the theory of porous media, a general Gurtin variational principle for the initial boundary value problem of dynamical response of fluid-saturated elastic porous media is developed by assuming infinitesimal deformation and incompressible constituents of the solid and fluid phase. The finite element formulation based on this variational principle is also derived. As the functional of the variational principle is a spatial integral of the convolution formulation, the general finite element discretization in space results in symmetrical differential-integral equations in the time domain. In some situations, the differential-integral equations can be reduced to symmetrical differential equations and, as a numerical example, it is employed to analyze the reflection of one-dimensional longitudinal wave in a fluid-saturated porous solid. The numerical results can provide further understanding of the wave propagation in porous media.

A wavelet finite-difference method for numerical simulation of wave propagation in fluid-saturated porous media

In this paper, we consider numerical simulation of wave propagation in fluidsaturated porous media. A wavelet finite-difference method is proposed to solve the 2-D elastic wave equation. The algorithm combines flexibility and computational efficiency of wavelet multi-resolution method with easy implementation of the finite-difference method. The orthogonal wavelet basis provides a natural framework, which adapt spatial grids to local wavefield properties. Numerical results show usefulness of the approach as an accurate and stable tool for simulation of wave propagation in fluid-saturated porous media.

FINITE ELEMENT ANALYSIS OF WAVE PROPAGATION IN FLUID-SATURATED POROUS MEDIA

With the porous media model based on mixture theory, a finite element formulation for dynamic transient analysis of fluid_saturated two_phase porous media is presented. Time integration of the equation, deduced with penalty method, can be performed by using implicit or explicit method. One_dimensional wave propagation in column under step loading and impulsive loading are analyzed with the developed finite element program. The obtained curves of displacements, velocities, effective stresses and pore pressures against time demonstrate the existence of wave propagation phenomena, which coincide with the theoretical results.

CHARACTERISTIC ANALYSIS FOR STRESS WAVE PROPAGATION IN TRANSVERSELY ISOTROPIC FLUID-SATURATED POROUS MEDIA

According to generalized characteristic theory, a characteristic analysis for stress wave propagation in transversely isotropic fluid-saturated porous media was performed. The characteristic differential equations and compatibility relations along bicharacteristics were deduced and the analytical expressions for wave surfaces were obtained. The characteristic and shapes of the velocity surfaces and wave surfaces in the transversely isotropic fluid-saturated porous media were discussed in detail. The results also show that the characteristic equations for stress waves in pure solids are particular cases of the characteristic equations for fluid-saturated porous media.

Near-field performance of chemical dissolution-front instability around a circular acid-injection-well in fluid-saturated porous media

This paper investigates the near-field performance of chemical dissolution-front instability(CDFI) around a circular acidinjection-well in fluid-saturated porous media(FSPM) through using purely mathematical deductions.After the mathematical governing equations of the CDFI problem involving radially divergent flow are briefly described,both analytical base solutions and perturbation solutions for the considered problem are mathematically derived.These analytical solutions lead to the theoretical expression of the perturbation induced dimensionless growth-rate and the following two new findings.The first new finding is that the critical Peclet number of a chemical dissolution system(CDS) associated with radially divergent flow in FSPM is not only a function of the permeability ratio between the undissolved and dissolved regions as well as the dimensionless wavenumber,but also a function of the circular chemical dissolution-front location relative to the circular acid-injection-well in FSPM.The second new finding is that as the direct result of considering a nonzero radius of the circular acid-injection-well,there exits a critical closeness number,which may be used to assess where the circular chemical dissolution-front starts becoming unstable in the CDS associated with radially divergent flow.Based on these two new findings,a theoretical criterion of two parts has been established.The first part of the established theoretical criterion answers the scientific question when a circular chemical dissolution-front can become unstable,while the second part of the established theoretical criterion answers the scientific question where a circular chemical dissolution-front can become unstable.Through applying the established theoretical criterion,a long-term existing mystery why the wormhole pattern of fractal nature and the compact pattern of fingering nature are formed at different locations away from the circular acid-injection-well circumference in fluid-saturated carbonate rocks has been successfully revealed.

ACOUSTOELASTIC THEORY FOR FLUID-SATURATED POROUS MEDIA

Based on the finite deformation theory of the continuum and poroelastic theory, the aeoustoelastic theory for fluid-saturated porous media （FSPM） in natural and initial coordi- nates is developed to investigate the influence of effective stresses and fluid pore pressure on wave velocities. Firstly, the assumption of a small dynamic motion superimposed on a largely static pre- deformation of the FSPM yields natural, initial, and final configurations, whose displacements, strains, and stresses of the solid-skeleton and the fluid in an FSPM particle could be described in natural and initial coordinates, respectively. Secondly, the subtraction of initial-state equations of equilibrium from the final-state equations of motion and the introduction of non-linear constitu- rive relations of the FSPM lead to equations of motion for the small dynamic motion. Thirdly, the consideration of homogeneous pre-deformation and the plane harmonic form of the small dynamic motion gives an acoustoelastic equation, which provides analytical formulations for the relation of the fast longitudinal wave, the fast shear wave, the slow shear wave, and the slow longitudinal wave with solid-skeleton stresses and fluid pore-pressure. Lastly, an isotropic FSPM under the close-pore jacketed condition, open-pore jacketed condition, traditional unjacketed condition, and triaxial condition is taken as an example to discuss the velocities of the fast and slow shear waves propagating along the direction of one of the initial principal solid-skeleton strains. The detailed discussion shows that the wave velocities of the FSPM are usually influenced by the effective stresses and the fluid pore pressure. The fluid pore-pressure has little effect on the wave velocities of the FSPM only when the components of the applied initial principal solid-skeleton stresses or strains are equal, which is consistent with the previous experimental results.

Porous-DeepONet:Learning the Solution Operators of Parametric Reactive Transport Equations in Porous Media

Reactive transport equations in porous media are critical in various scientific and engineering disciplines,but solving these equations can be computationally expensive when exploring different scenarios,such as varying porous structures and initial or boundary conditions.The deep operator network(DeepONet)has emerged as a popular deep learning framework for solving parametric partial differential equations.However,applying the DeepONet to porous media presents significant challenges due to its limited capability to extract representative features from intricate structures.To address this issue,we propose the Porous-DeepONet,a simple yet highly effective extension of the DeepONet framework that leverages convolutional neural networks(CNNs)to learn the solution operators of parametric reactive transport equations in porous media.By incorporating CNNs,we can effectively capture the intricate features of porous media,enabling accurate and efficient learning of the solution operators.We demonstrate the effectiveness of the Porous-DeepONet in accurately and rapidly learning the solution operators of parametric reactive transport equations with various boundary conditions,multiple phases,and multiphysical fields through five examples.This approach offers significant computational savings,potentially reducing the computation time by 50–1000 times compared with the finite-element method.Our work may provide a robust alternative for solving parametric reactive transport equations in porous media,paving the way for exploring complex phenomena in porous media.

The impact of heterogeneity and pore network characteristics on single and multi-phase fluid propagation in complex porous media:An X-ray computed tomography study

This study investigates the impact of pore network characteristics on fluid flow through complex and heterogeneous porous media,providing insights into the factors affecting fluid propagation in such systems.Specifically,high-resolution or micro X-ray computed tomography(CT)imaging techniques were utilized to examine outcrop stromatolite samples of the Lagoa Salgada,considered flow analogous to the Brazilian Pre-salt carbonate reservoirs.The petrophysical results comprised two distinct stromatolite depositional facies,the columnar and the fine-grained facies.By generating pore network model(PNM),the study quantified the relationship between key features of the porous system,including pore and throat radius,throat length,coordination number,shape factor,and pore volume.The study found that the less dense pore network of the columnar sample is typically characterized by larger pores and wider and longer throats but with a weaker connection of throats to pores.Both facies exhibited less variability in the radius of the pores and throats in comparison to throat length.Additionally,a series of core flooding experiments coupled with medical CT scanning was designed and conducted in the plug samples to assess flow propagation and saturation fields.The study revealed that the heterogeneity and presence of disconnected or dead-end pores significantly impacted the flow patterns and saturation.Two-phase flow patterns and oil saturation distribution reveal a preferential and heterogeneous displacement that mainly swept displaced fluid in some regions of plugs and bypassed it in others.The relation between saturation profiles,porosity profiles,and the number of fluid flow patterns for the samples was evident.Only for the columnar plug sample was the enhancement in recovery factor after shifting to lower salinity water injection(SB)observed.

A study on the temperature sensitivity of NMR porosity in porous media based on the intensity of magnetization Dedicated to the special issue “Magnetic Resonance in Porous Media”

The measurement of nuclear magnetic resonance(NMR)porosity is affected by temperature.Without considering the impact of NMR logging tools,this phenomenon is mainly caused by variations in magnetization intensity of the measured system due to temperature fluctuations and difference between the temperature of the porous medium and calibration sample.In this study,the effect of temperature was explained based on the thermodynamic theory,and the rules of NMR porosity responses to temperature changes were identified through core physics experiments.In addition,a method for correcting the influence of temperature on NMR porosity measurement was proposed,and the possible factors that may affect its application were also discussed.

Anisotropic dynamic permeability model for porous media

Based on the tortuous capillary network model,the relationship between anisotropic permeability and rock normal strain,namely the anisotropic dynamic permeability model(ADPM),was derived and established.The model was verified using pore-scale flow simulation.The uniaxial strain process was calculated and the main factors affecting permeability changes in different directions in the deformation process were analyzed.In the process of uniaxial strain during the exploitation of layered oil and gas reservoirs,the effect of effective surface porosity on the permeability in all directions is consistent.With the decrease of effective surface porosity,the sensitivity of permeability to strain increases.The sensitivity of the permeability perpendicular to the direction of compression to the strain decreases with the increase of the tortuosity,while the sensitivity of the permeability in the direction of compression to the strain increases with the increase of the tortuosity.For layered reservoirs with the same initial tortuosity in all directions,the tortuosity plays a decisive role in the relative relationship between the variations of permeability in all directions during pressure drop.When the tortuosity is less than 1.6,the decrease rate of horizontal permeability is higher than that of vertical permeability,while the opposite is true when the tortuosity is greater than 1.6.This phenomenon cannot be represented by traditional dynamic permeability model.After the verification by experimental data of pore-scale simulation,the new model has high fitting accuracy and can effectively characterize the effects of deformation in different directions on the permeability in all directions.

Prediction of Porous Media Fluid Flow with Spatial Heterogeneity Using Criss-Cross Physics-Informed Convolutional Neural Networks

Recent advances in deep neural networks have shed new light on physics,engineering,and scientific computing.Reconciling the data-centered viewpoint with physical simulation is one of the research hotspots.The physicsinformedneural network(PINN)is currently the most general framework,which is more popular due to theconvenience of constructing NNs and excellent generalization ability.The automatic differentiation(AD)-basedPINN model is suitable for the homogeneous scientific problem;however,it is unclear how AD can enforce fluxcontinuity across boundaries between cells of different properties where spatial heterogeneity is represented bygrid cells with different physical properties.In this work,we propose a criss-cross physics-informed convolutionalneural network(CC-PINN)learning architecture,aiming to learn the solution of parametric PDEs with spatialheterogeneity of physical properties.To achieve the seamless enforcement of flux continuity and integration ofphysicalmeaning into CNN,a predefined 2D convolutional layer is proposed to accurately express transmissibilitybetween adjacent cells.The efficacy of the proposedmethodwas evaluated through predictions of several petroleumreservoir problems with spatial heterogeneity and compared against state-of-the-art(PINN)through numericalanalysis as a benchmark,which demonstrated the superiority of the proposed method over the PINN.

Modeling of multiphase flow in low permeability porous media:Effect of wettability and pore structure properties

Multiphase flow in low permeability porous media is involved in numerous energy and environmental applications.However,a complete description of this process is challenging due to the limited modeling scale and the effects of complex pore structures and wettability.To address this issue,based on the digital rock of low permeability sandstone,a direct numerical simulation is performed considering the interphase drag and boundary slip to clarify the microscopic water-oil displacement process.In addition,a dual-porosity pore network model(PNM)is constructed to obtain the water-oil relative permeability of the sample.The displacement efficiency as a recovery process is assessed under different wetting and pore structure properties.Results show that microscopic displacement mechanisms explain the corresponding macroscopic relative permeability.The injected water breaks through the outlet earlier with a large mass flow,while thick oil films exist in rough hydrophobic surfaces and poorly connected pores.The variation of water-oil relative permeability is significant,and residual oil saturation is high in the oil-wet system.The flooding is extensive,and the residual oil is trapped in complex pore networks for hydrophilic pore surfaces;thus,water relative permeability is lower in the water-wet system.While the displacement efficiency is the worst in mixed-wetting systems for poor water connectivity.Microporosity negatively correlates with invading oil volume fraction due to strong capillary resistance,and a large microporosity corresponds to low residual oil saturation.This work provides insights into the water-oil flow from different modeling perspectives and helps to optimize the development plan for enhanced recovery.

A Level Set Method for the Inverse Problem of Wave Equation in the Fluid-Saturated Porous Media

In this paper,a level set method is applied to the inverse problem of 2-D wave equation in the fluid-saturated media.We only consider the situation that the parameter to be recovered takes two different values,which leads to a shape reconstruction problem.A level set function is used to present the discontinuous parameter,and a regularization functional is applied to the level set function for the ill-posed problem.Then the resulting inverse problem with respect to the level set function is solved by using the damped Gauss-Newton method.Numerical experiments show that the method can recover parameter with complicated geometry and the noise in the observation data.

PML Absorbing Boundary Condition for Seismic Numerical Modeling by Convolutional Differentiator in Fluid-Saturated Porous Media

The perfectly matched layer（PML） was first introduced by Berenger as an absorbing boundary condition for electromagnetic wave propagation.In this article,a method is developed to ex-tend the PML to simulating seismic wave propagation in fluid-saturated porous medium.This non-physical boundary is used at the computational edge of a Forsyte polynomial convolutional differenti-ator（FPCD） algorithm as an absorbing boundary condition to truncate unbounded media.The incor-poration of PML in Biot＇s equations is given.Numerical results show that the PML absorbing bound-ary condition attenuates the outgoing waves effectively and eliminates the reflections adequately.

PENALTY FINITE ELEMENT METHOD FOR NONLINEAR DYNAMIC RESPONSE OF VISCOUS FLUID-SATURATED BIPHASE POROUS MEDIA

The governing equations as well as boundary land initial conditions for nonlinear dynamic response problems of viscous fluid-saturated biphase porous medium model, based on mixture theory, are presented. With Galerkin weighted residual method the corresponding nonlinear dynamic penalty finite element equation, in which the dependencies of volume fraction and permeation coefficients an deformation are included, is obtained. The iteration solution method of the nonlinear system equation is also discussed. As a numerical example, the dynamic response of a porous medium column under impulsive loading action is analyzed with the developed finite element program. The numerical results demonstrate the efficiency and correctness of the method.

Flow characteristics and regime transition of aqueous foams in porous media over a wide range of quality,velocity,and surfactant concentration

Aqueous foam is broadly applicable to enhanced oil recovery(EOR).The rheology of foam as a function of foam quality,gas and liquid velocities,and surfactant concentration constitute the foundation of its application.The great variations of the above factors can affect the effectiveness of N2 foam in EOR continuously in complex formations,which is rarely involved in previous relevant studies.This paper presents an experimental study of foam flow in porous media by injecting pre-generated N2 foam into a sand pack under the conditions of considering a wide range of gas and liquid velocities and surfactant concentrations.The results show that in a wide range of gas and liquid velocities,the pressure gradient contours are L-shaped near the coordinate axes,but V-shaped in other regions.And the surfactant concentration is a strong factor influencing the trend of pressure gradient contours.Foam flow resistance is very sensitive to the surfactant concentration in both the high-and low-foam quality regime,especially when the surfactant concentration is less than CMC.The foam quality is an important variable to the flow resistance obtained.There exists a transition point from low-to high-quality regime in a particular flow system,where has the maximum flow resistance,the corresponding foam quality is called transition foam quality,which increases as the surfactant concentration increases.The results can add to our knowledge base of foam rheology in porous media,and can provide a strong basis for the field application of foams.

Numerical Approach of a Coupled Pressure-Saturation Model Describing Oil-Water Flow in Porous Media

Two-phase flow in porous media is a very active field of research,due to its important applications in groundwater pollution,CO_(2)sequestration,or oil and gas production from petroleum reservoirs,just to name a few of them.Fractional flow equations,which make use of Darcy's law,for describing the movement of two immiscible fluids in a porous medium,are among the most relevant mathematical models in reservoir simulation.This work aims to solve a fractional flow model formed by an elliptic equation,representing the spatial distribution of the pressure,and a hyperbolic equation describing the space-time evolution of water saturation.The numerical solution of the elliptic part is obtained using a finite-element(FE)scheme,while the hyperbolic equation is solved by means of two dif-ferent numerical approaches,both in the finite-volume(FV)framework.One is based on a monotonic upstream-centered scheme for conservation laws(MUSCL)-Hancock scheme,whereas the other makes use of a weighted essentially non-oscillatory(ENO)reconstruc-tion.In both cases,a first-order centered(FORCE)-αnumerical scheme is applied for inter-cell flux reconstruction,which constitutes a new contribution in the field of fractional flow models describing oil-water movement.A relevant feature of this work is the study of the effect of the parameterαon the numerical solution of the models considered.We also show that,in the FORCE-αmethod,when the parameterαincreases,the errors diminish and the order of accuracy is more properly attained,as verified using a manufactured solution technique.