Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation

A nonlocal study of the vibration responses of functionally graded(FG)beams supported by a viscoelastic Winkler-Pasternak foundation is presented.The damping responses of both the Winkler and Pasternak layers of the foundation are considered in the formulation,which were not considered in most literature on this subject,and the bending deformation of the beams and the elastic and damping responses of the foundation as nonlocal by uniting the equivalently differential formulation of well-posed strain-driven(ε-D)and stress-driven(σ-D)two-phase local/nonlocal integral models with constitutive constraints are comprehensively considered,which can address both the stiffness softening and toughing effects due to scale reduction.The generalized differential quadrature method(GDQM)is used to solve the complex eigenvalue problem.After verifying the solution procedure,a series of benchmark results for the vibration frequency of different bounded FG beams supported by the foundation are obtained.Subsequently,the effects of the nonlocality of the foundation on the undamped/damping vibration frequency of the beams are examined.

AVO forwarding modeling in two-phase media: multiconstrained matrix mineral modulus inversion

AVO forward modeling is based on two-phase medium theory and is considered an effective method for describing reservoir rocks and fluids. However, the method depends on the input matrix mineral bulk modulus and the rationality of the two-phase medium model. We used the matrix mineral bulk modulus inversion method and multiple constraints to obtain a two-phase medium model with physical meaning. The proposed method guarantees the reliability of the obtained AVO characteristicsin two-phase media. By the comparative analysis of different lithology of the core sample, the advantages and accuracy of the inversion method can be illustrated. Also, the inversion method can be applied in LH area, and the AVO characteristics can be obtained when the porosity, fluid saturation, and other important lithology parameters are changed. In particular, the reflection coefficient amplitude difference between the fast P wave and S wave as a function of porosity at the same incidence angle, and the difference in the incidence angle threshold can be used to decipher porosity.

Reflection and transmission of bottom simulating reflectors in gas hydrate-bearing sediments: Two-phase media models

The bottom simulating reflector （BSR） in gas hydrate-bearing sediments is a physical interface which is composed of solid, gas, and liquid and is influenced by temperature and pressure. Deep sea floor sediment is a porous, unconsolidated, fluid saturated media. Therefore, the reflection and transmission coefficients computed by the Zoeppritz equation based on elastic media do not match reality. In this paper, a two-phase media model is applied to study the reflection and transmission at the bottom simulating reflector in order to find an accurate wave propagation energy distribution and the relationship between reflection and transmission and fluid saturation on the BSR. The numerical experiments show that the type I compressional （fast） and shear waves are not sensitive to frequency variation and the velocities change slowly over the whole frequency range. However, type II compressional （slow） waves are more sensitive to frequency variation and the velocities change over a large range. We find that reflection and transmission coefficients change with the amount of hydrate and free gas. Frequency, pore fluid saturation, and incident angle have different impacts on the reflection and transmission coefficients. We can use these characteristics to estimate gas hydrate saturation or detect lithological variations in the gas hydrate-bearing sediments.

Numerical simulation of two-phase flow in fractured porous media using streamline simulation and IMPES methods and comparing results with a commercial software

Streamline simulation is developed to simulate waterflooding in fractured reservoirs. Conventional reservoir simulation methods for fluid flow simulation in large and complex reservoirs are very costly and time consuming. In streamline method, transport equations are solved on one-dimensional streamlines to reduce the computation time with less memory for simulation. First, pressure equation is solved on an Eulerian grid and streamlines are traced. Defining the "time of flight", saturation equations are mapped and solved on streamlines. Finally, the results are mapped back on Eulerian grid and the process is repeated until the simulation end time. The waterflooding process is considered in a fractured reservoir using the dual porosity model. Afterwards, a computational code is developed to solve the same problem by the IMPES method and the results of streamline simulation are compared to those of the IMPES and a commercial software. Finally, the accuracy and efficiency of streamline simulator for simulation of two-phase flow in fractured reservoirs has been proved.

Wave propagation and energy dissipation in viscoelastic granular media

In terms of viscoelasticity, the relevant theory of wave i n granular media is analyzed in this paper. Under the conditions of slight deforma tion of granules, wave equation, complex number expressions of propagation vecto r and attenuation vector, attenuation coefficient expressions of longitudinal wa ve and transverse wave, etc, are analyzed and deduced. The expressions of attenu ation coefficients of viscoelastic longitudinal wave and transverse wave show th a t the attenuation of wave is related to frequency. The higher the frequency is, the more the attenuation is, which is tested by the laboratory experiment. In a ddi tion, the energy dissipation is related to the higher frequency wave that is abs orbed by granular media. The friction amongst granular media also increase the e nergy dissipation. During the flowing situation the expression of transmission f actor of energy shows that the granular density difference is the key factor whi ch leads to the attenuation of vibrating energy. This has been proved by the exp eriment results.

Multiwave Amplitude Compensation and Its SensitivityAnalysis to AVO Inversion in Viscoelastic Media

We derive formulae of correction for multi-wave geometric spreading and absorption in layered viscoelastic media, this provides the theoretical foundation for true amplitude compensation of field data and for our sensitivity analysis. The imaging matrix at a plane reflector between viscoelastic media can be determined in the frequency domain using linearized reflection coefficients through Born approximation. We quantitatively analyze the sensitivity by studying eigenvalues and eigenvectors of the imaging matrix. The results show that two linear combinations of petrophysical parameters can be determined from the multi-wave AVO inversion in the case of amplitude compensation. Multi-wave AVO contains the information of attenuation in the media. However, the sensitivity of multi-wave AVO inversion to attenuation is small.

A control volume based finite element method for simulating incompressible two-phase flow in heterogeneous porous media and its application to reservoir engineering

Applying the standard Galerkin finite element method for solving flow problems in porous media encounters some difficulties such as numerical oscillation at the shock front and discontinuity of the velocity field on element faces.Discontinuity of velocity field leads this method not to conserve mass locally.Moreover,the accuracy and stability of a solution is highly affected by a non-conservative method.In this paper,a three dimensional control volume finite element method is developed for twophase fluid flow simulation which overcomes the deficiency of the standard finite element method,and attains high-orders of accuracy at a reasonable computational cost.Moreover,this method is capable of handling heterogeneity in a very rational way.A fully implicit scheme is applied to temporal discretization of the governing equations to achieve an unconditionally stable solution.The accuracy and efficiency of the method are verified by simulating some waterflooding experiments.Some representative examples are presented to illustrate the capability of the method to simulate two-phase fluid flow in heterogeneous porous media.

A lattice Boltzmann exploration of two-phase displacement in 2D porous media under various pressure boundary conditions

While experimental designs developed in recent decades have contributed to research on dynamic nonequilibrium effects in transient two-phase flow in porous media,this problem has been seldom investigated using direct numerical simulation(DNS).Only a few studies have sought to numerically solve Navier—Stokes equations with level-set(LS)or volume-of-fluid(VoF)methods,each of which has constraints in terms of meniscus dynamics for various flow velocities in the control volume(CV)domain.The Shan—Chen multiphase multicomponent lattice Boltzmann method(SC-LBM)has a fundamental mechanism to separate immiscible fluid phases in the density domain without these limitations.Therefore,this study applied it to explore two-phase displacement in a single representative elementary volume(REV)of two-dimensional(2D)porous media.As a continuation of a previous investigation into one-step inflow/outflow in 2D porous media,this work seeks to identify dynamic nonequilibrium effects on capillary pressure—saturation relationship(P_(c)—S)for quasi-steady-state flow and multistep inflow/outflow under various pressure boundary conditions.The simulation outcomes show that P_(c),S and specific interfacial area(a_(nw))had multistep-wise dynamic effects corresponding to the multistep-wise pressure boundary conditions.With finer adjustments to the increase in pressure over more steps,dynamic nonequilibrium effects were significantly alleviated and even finally disappeared to achieve quasisteady-state inflow/outflow conditions.Furthermore,triangular wave-formed pressure boundary conditions were applied in different periods to investigate dynamic nonequilibrium effects for hysteretical Pc—S.The results showed overshoot and undershoot of P_(c)to S in loops of the nonequilibrium hysteresis.In addition,the flow regimes of multistep-wise dynamic effects were analyzed in terms of Reynolds and capillary numbers(Re and Ca).The analysis of REV-scale flow regimes showed higher Re(1<Re<10)for more significant dynamic nonequilibrium effects.This indicates that inertia is critical for transient twophase flow in porous media under dynamic nonequilibrium conditions.

Finite element equations and numerical simulation of elastic wave propagation in two-phase anisotropic media

Based on Biot theory of two-phase anisotropic media and Hamilton theory about dynamic problem,finite element equations of elastic wave propagation in two-phase anisotropic media are derived in this paper.Numerical solution of finite element equations is given.Finally,Properties of elastic wave propagation are observed and analyzed through FEM modeling.

MATHEMATICAL MODEL OF TWO-PHASE FLUID NONLINEAR FLOW IN LOW-PERMEABILITY POROUS MEDIA WITH APPLICATIONS

A mathematical model of two-phase fluid nonlinear flow in the direction of normal of ellipse through low-permeability porous media was established according to a nonlinear flow law expressed in a continuous function with three parameters, a mass conservation law and a concept of turbulent ellipses. A solution to the model was obtained by using a finite difference method and an extrapolation method. Formulas of calculating development index not only before but also after water breaks through an oil well in the condition of two-phase fluid nonlinear flow in the media were derived. An example was discussed. Water saturation distribution was presented. The moving law of drainage front was found. Laws of change of pressure difference with time were recognized. Results show that there is much difference of water saturation distribution between nonlinear flow and linear flow; that drainage front by water moves faster, water breaks through sooner and the index gets worse because of the nonlinear flow; and that dimensionless pressure difference gets larger at the same dimensionless time and difficulty of oil development becomes bigger by the nonlinear flow. Thus, it is necessary that influence of nonlinear flow on development indexes of the oil fields be taken into account. The results provide water-flooding development of the oilfields with scientific basis.

Characterization for wave equations in viscoelastic media based on the constant Q property

The constant Q property in viscoelastic media assumes that the quality factor Q does not change with frequency(i.e.,the Q value is independent of the frequency).For seismic waves propagating in viscoelastic media,the wave equation is determined by the viscoelastic media model.Equivalence relations exist between various frequency domain mathematical models and physical rheological models for the constant Q property.Considering two elastic moduli and three attenuation variables,24 kinds of wave equations based on diff erent generalized rheological models are divided into six classes in this study,and the 12 kinds of specifi c representation for the wave equations in the time domain are derived.On the basis of the equivalence relations between the generalized rheological models,the diff erence and equivalence relation between diff erent wave equations are proven and clarifi ed.Results show that the high-order generalized rheological model can accurately characterize the attenuation characteristics of seismic waves and has advantages in characterizing the dispersion characteristics in viscoelastic media.Lastly,the seismic refl ection characteristics caused by the diff erence of Q value are verifi ed by the forward modeling of the constant Q wave equation in this study,thereby providing a theoretical basis for the analysis and inversion of the formation Q value from refl ection seismic data.

Wave field in two-phase media by the convolutional differentiator method

This paper applies the convolutional differentiator method, based on generalized Forsyte orthogonal polynomial （CFPD）, to simulate the seismic wave propagation in two-phase media. From the numerical results we can see that three types of waves, fast P-waves, S-waves and slow P-waves, can be observed in the seismic wave field. The experiments on anisotropic models demonstrate that the wavefront is elliptic instead of circular and S-wave splitting occurs in anisotropic two-phase media. The research has confirmed that the rules of elastic wave propagation in fluid-saturated porous media are controlled by Biot＇s theory. Experiment on a layered fault model shows the wavefield generated by the interface and the fault very well, indicating the effectiveness of CFPD method on the wavefield modeling for real layered media in the Earth. This research has potential applications to the investigation of Earth＇s deep structure and oil/gas exploration.

Digital image processing of saturation for two-phase flow in planar porous media model

In this paper, the accuracy of estimating stained non-wetting phase saturation using digital image processing is examined, and a novel post-processing approach for calculating threshold is presented. In order to remove the effect of the background noise of images and to enhance the high-frequency component of the original image, image smoothing and image sharpening methods are introduced. Depending on the correct threshold, the image binarization processing is particularly useful for estimating stained non-wetting phase saturation. Calculated saturation data are compared with the measured saturation data during the two-phase flow experiment in an artificial steel planar porous media model. The results show that the calculated saturation data agree with the measured ones. With the help of an artificial steel planar porous media model, digital image processing is an accurate and simple method for obtaining the stained non-wetting phase saturation.

NUMERICAL SIMULATION OF ATTENUATION OF LOVE-TYPE CHANNEL WAVES IN VISCOELASTIC MEDIA

Proceeding from wave equations, the paper strictly deduced the dispersion relation equation of Love-type channel waves in Standard Linear Solid model. The e quation is a complex one with its real part signifying the dispersion characteristics of the channel wave while the imaginary part, the attenuation characteristics. In calcu lating the attenuation value, the author has set up a mathematical model of a horizon tal symmetric sequence (a three layer sequence of rock-coal-rock), given out some physical parametersl and adopted the dichotomy method that is more of ten used in root resolving of an equatlon. The calculation indicates that the influence of non-elas tic absorption on the attenuation of the propagation of channel wave varies with the frequency. In the frequency band of the Airy phase, the attenuation increases steep like, which is unfavorable for the channel wave seismic prospecting. The study of channel wave attenuation has provided a theoretical basis for the compensation of at tenuation.

Study of the generalized mixture rule for determining effective conductivity of two-phase stochastic models

The generalized mixture rule （GMR） is usually applied in determining mechanical properties such as the rheological property and Young’s modulus of multi-phase rocks. However, it is rarely used to determine electrical conductivity of multi-phase rocks presently. In this paper, we calculate the effective conductivity using the 3D finite element method for a large number of two-phase medium stochastic models. The GMR is then employed as an effective conductivity model to fit the data. It shows a very close relationship between the parameter J of GMR and the ratio of conductivities of the two phases. We obtain the equations of the parameter J with the ratio of conductivity of two phases for the first time. On this basis, we can quickly predict （or calculate） the effective conductivity of any twophase medium stochastic model. The result is much more accurate than two other available effective conductivity models for the stochastic medium, which are the random model and effective medium theory model, laying a solid base for detailed evaluation of oil reservoirs.

Multi-fracture interactions during two-phase flow of oil and water in deformable tight sandstone oil reservoirs

Tight oil reservoirs are complex geological materials composed of solid matrix,pore structure,and mixed multiple phases of fluids,particularly for oil reservoirs suffering from high content of in situ pressurized water found in China.In this regard,a coupled model considering two-phase flow of oil and water,as well as deformation and damage evolution of porous media,is proposed and validated using associated results,including the oil depletion process,analytical solution of stress shadow effect,and physical experiments on multi-fracture interactions and fracture propagation in unsaturated seepage fields.Then,the proposed model is used to study the behavior of multi-fracture interactions in an unsaturated reservoir in presence of water and oil.The results show that conspicuous interactions exist among multiple induced fractures.Interaction behavior varies from extracted geological profiles of the reservoir due to in situ stress anisotropy.The differential pressures of water and that of oil in different regions of reservoir affect interactions and trajectories of multi-fractures to a considerable degree.The absolute value of reservoir average pressure is a dominant factor affecting fracture interactions and in favor of enhancing fracture network complexity.In addition,difference of reservoir average pressures in different regions of reservoir would promote the fracturing effectiveness.Factors affecting fracture interactions and reservoir treatment effectiveness are quantitatively estimated through stimulated reservoir area.This study confirms the significance of incorporating the two-phase flow process in analyses of multifracture interactions and fracture trajectory predictions during tight sandstone oil reservoir developments.

Acoustic viscoelastic modeling by frequency-domain boundary element method

Earth medium is not completely elastic, with its viscosity resulting in attenuation and dispersion of seismic waves. Most viscoelastic numerical simulations are based on the finite-difference and finite-element methods. Targeted at viscoelastic numerical modeling for multilayered media, the constant-Q acoustic wave equation is transformed into the corresponding wave integral representation with its Green＇s function accounting for viscoelastic coefficients. An efficient alternative for full-waveform solution to the integral equation is proposed in this article by extending conventional frequency-domain boundary element methods to viscoelastic media. The viscoelastic boundary element method enjoys a distinct characteristic of the explicit use of boundary continuity conditions of displacement and traction, leading to a semi-analytical solution with sufficient accuracy for simulating the viscoelastic effect across irregular interfaces. Numerical experiments to study the viscoelastic absorption of different Q values demonstrate the accuracy and applicability of the method.

Gas Condensate Two Phase Flow Performance in Porous Media Considering Capillary Number and Non-Darcy Effects

Retrograde condensation frequently occurs during the development of gas condensate reservoirs. The loss of productivity is often observed due to the reduced relative permeability to gas as condensate accumulates near the well bore region. How to describe the condensate blockage effect exactly has been a continuous research topic. However, up to now, the present methods usually over-estimate or underestimate the productivity reduction due to an incorrect understanding of the mechanism of flow in porous medium, which inevitably results in an inaccurate prediction of production performance. It has been found in recent numerous theoretical and experimental studies that capillary number and non-Darcy flow have significant influence on relative permeability in regions near the well bore. The two effects impose opposite impacts on production performance, thus leading to gas condensate flow showing characteristics different from general understanding. It is significant for prediction of performance in gas condensate wells to understand the two effects exactly. The aim of the paper is to describe and analyze the flow dynamics in porous media accurately during the production of gas condensate reservoirs. Based on the description of three-zone flow mechanism, capillary number and non-Darcy effect are incorporated in the analysis of relative permeability, making it possible to describe the effect of condensate blockage. The effect of capillary number and inertial flow on gas and condensate relative permeability is analyzed in detail. Novel Inflow Performance Relation (IPR) models considering high velocity effects are formulated and the contrast analysis of different IPR models is conducted. The result shows that the proposed method can help predict the production performance and productivity more accurately than conventional methods.

Examination of two-phase behaviors in porous media during pool boiling

The enhancement of pool boiling heat transfer using porous media has been extensively studied.Although the two-phase distribution and evolution in porous media are crucial to the heat transfer performance,including the critical heat flux(CHF)and heat transfer coefficient(HTC),direct observation of the two-phase flow inside the media is limited owing to the blockage of the direct view from the porous structures.In this study,pool boiling visualization experiments were conducted on porous samples with different throat widths in deionized water.The results showed that the HTC increased with the throat width.Additionally,the growth-contraction cycle of the vapor region and the formation and drying of the wall liquid film inside the porous media were investigated.The vapor region,including the maximum and minimum areas in the boiling cycle,was quantitatively described.Furthermore,the relationship between the minimum gas-phase area and HTC peak was identified.A one-dimensional transient model was developed considering solid skeleton heat conduction,liquid film evaporation,and vapor region growth to quantitatively study the influence of heat flux on the internal two-phase flow.The model successfully captured the maximum gas-phase area,duration of boiling cycles,and HTC trends at specific heat fluxes.The results of this quantitative study provide insights into the internal two-phase distribution and evolution induced by pool boiling.

Empirical mode decomposition of multiphase flows in porous media:characteristic scales and speed of convergence

We apply a proper orthogonal decomposition(POD)to data stemming from numerical simulations of a fingering instability in a multiphase flow passing through obstacles in a porous medium,to study water injection processes in the production of hydrocarbon reservoirs.We show that the time evolution of a properly defined flow correlation length can be used to identify the onset of the fingering instability.Computation of characteristic lengths for each of the modes resulting from the POD provides further information on the dynamics of the system.Finally,using numerical simulations with different viscosity ratios,we show that the convergence of the POD depends non-trivially on whether the fingering instability develops or not.This result has implications on proposed methods to decrease the dimensionality of the problem by deriving reduced dynamical systems after truncating the system’s governing equations to a few POD modes.