Hydro‐mechanical modeling of cohesive crack propagation in multiphase porous media using the extended finite element method

In this paper, a numerical model is developed for the fully coupled hydro‐mechanical analysis of deformable, progressively fracturing porous media interacting with the flow of two immiscible, compressible wetting and non‐wetting pore fluids, in which the coupling between various processes is taken into account. The governing equations involving the coupled solid skeleton deformation and two‐phase fluid flow in partially saturated porous media including cohesive cracks are derived within the framework of the generalized Biot theory. The fluid flow within the crack is simulated using the Darcy law in which the permeability variation with porosity because of the cracking of the solid skeleton is accounted. The cohesive crack model is integrated into the numerical modeling by means of which the nonlinear fracture processes occurring along the fracture process zone are simulated. The solid phase displacement, the wetting phase pressure and the capillary pressure are taken as the primary variables of the three‐phase formulation. The other variables are incorporated into the model via the experimentally determined functions, which specify the relationship between the hydraulic properties of the fracturing porous medium, that is saturation, permeability and capillary pressure. The spatial discretization is implemented by employing the extended finite element method, and the time domain discretization is performed using the generalized Newmark scheme to derive the final system of fully coupled nonlinear equations of the hydro‐mechanical problem. It is illustrated that by allowing for the interaction between various processes, that is the solid skeleton deformation, the wetting and the non‐wetting pore fluid flow and the cohesive crack propagation, the effect of the presence of the geomechanical discontinuity can be completely captured. Copyright © 2012 John Wiley & Sons, Ltd.     

A coupled chemo‐thermo‐hygro‐mechanical model of concrete at high temperature and failure analysis

A hierarchical mathematical model for analyses of coupled chemo‐thermo‐hygro‐mechanical behaviour in concretes at high temperature is presented. The concretes are modelled as unsaturated deforming reactive porous media filled with two immiscible pore fluids, i.e. the gas mixture and the liquid mixture, in immiscible–miscible levels. The thermo‐induced desalination process is particularly integrated into the model. The chemical effects of both the desalination and the dehydration processes on the material damage and the degradation of the material strength are taken into account. The mathematical model consists of a set of coupled, partial differential equations governing the mass balance of the dry air, the mass balance of the water species, the mass balance of the matrix components dissolved in the liquid phases, the enthalpy (energy) balance and momentum balance of the whole medium mixture. The governing equations, the state equations for the model and the constitutive laws used in the model are given. A mixed weak form for the finite element solution procedure is formulated for the numerical simulation of chemo‐thermo‐hygro‐mechanical behaviours. Special considerations are given to spatial discretization of hyperbolic equation with non‐self‐adjoint operator nature. Numerical results demonstrate the performance and the effectiveness of the proposed model and its numerical procedure in reproducing coupled chemo‐thermo‐hygro‐mechanical behaviour in concretes subjected to fire and thermal radiation. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite element modelling of three-phase flow in deforming saturated oil reservoirs

A numerical simulation is presented for three-dimensional three-phase fluid flow in a deforming saturated oil reservoir. The mathematical formulation describes a fully coupled governing equation systen which consists of the equilibrium and continuity equations for three immiscible fluids flowing in a porous medium. An elastoplastic soil model, based on a Mohr–Coulomb yield surface, is used. The finite element method is applied to obtain simultaneous solutions to the governing equations where displacement and fluid pressures are the primary unknowns. The final discretized equations are solved by a direct solver using fully implicit procedures. The developed model is used to investigate the problems of three-phase fluid flow in a deforming saturated oil reservoir.     

A virtual element method for the two-phase flow of immiscible fluids in porous media

A primal C⁰-conforming virtual element discretization for the approximation of the bidimensional two-phase flow of immiscible fluids in porous media using general polygonal meshes is discussed. This work investigates the potentialities of the Virtual Element Method (VEM) in solving this specific problem of immiscible fluids in porous media involving a time-dependent coupled system of non-linear partial differential equations. The performance of the fully discrete scheme is thoroughly analysed testing it on general meshes considering both a regular problem and more realistic benchmark problems that are of interest for physical and engineering applications.

     

Finite element modelling for simulating the surface subsidence above a compacting hydrocarbon reservoir

The surface subsidence above a compacting saturated oil reservoir is the main topic of this paper. From a literature review, it is obvious that extensive efforts have been conducted for investigating this phenomenon in various situations. Herein, a numerical model, based on the finite element method, was used for simulating three-dimensional three-phase fluid flow in a deforming saturated oil reservoir. The mathematical formulation describes a fully coupled governing equation system which consists of the equilibrium and continuity equations for three immiscible fluids flowing in a porous media. An elastoplastic soil model, based on a Mohr Coulomb yield surface, was utilized. The finite element method was applied to obtain simultaneous solutions to the governing equations where the displacements and the fluid pressures are the primary unknowns. The final discretized equations are solved by a direct solver using fully implicit procedures. A linear analysis was used to study the stability conditions of the present model. Finally, a series of simulations were conducted to indicate the validity and the utility of the developed model.     

Coupled formulation and algorithms for the simulation of non‐planar three‐dimensional hydraulic fractures using the generalized finite element method

This paper presents a coupled hydro‐mechanical formulation for the simulation of non‐planar three‐dimensional hydraulic fractures. Deformation in the rock is modeled using linear elasticity, and the lubrication theory is adopted for the fluid flow in the fracture. The governing equations of the fluid flow and elasticity and the subsequent discretization are fully coupled. A Generalized/eXtended Finite Element Method (G/XFEM) is adopted for the discretization of the coupled system of equations. A Newton–Raphson method is used to solve the resulting system of nonlinear equations. A discretization strategy for the fluid flow problem on non‐planar three‐dimensional surfaces and a computationally efficient strategy for handling time integration combined with mesh adaptivity are also presented. Several three‐dimensional numerical verification examples are solved. The examples illustrate the generality and accuracy of the proposed coupled formulation and discretization strategies. Copyright © 2015 John Wiley & Sons, Ltd.     

Two‐scale modeling of solute dispersion in unsaturated double‐porosity media: Homogenization and experimental validation

A two‐scale modeling of solute transport in double‐porosity (DP) media under unsaturated water flow conditions is presented. The macroscopic model was developed by applying the asymptotic homogenization method. It is based on theoretical and empirical considerations dealing with the orders of magnitude of characteristic quantities involved in the process. For this purpose a physical model that mimics the behavior of DP medium was built. The resulting two‐equation model relies on a coupling exchange term between micro‐ and macro‐porosity subdomains associated with local non‐equilibrium solute concentrations. The model was numerically implemented (Comsol Multiphysics^®) to simulate the macroscopic one‐dimensional physical process taking place into the porous medium of 3D periodic microstructure. A series of dispersion experiments of NaCl solution under unsaturated steady‐state flow conditions were performed. The experimental results were used first to calibrate the dispersion coefficient of the model, and second to validate it through two other independent experiments. The excellent agreement between the numerical simulations and the measurements of the time evolution of the non‐symmetrical breakthrough curves provides a proof of predictive capacity of the developed model. Copyright © 2010 John Wiley & Sons, Ltd.     

A finite element solution of a fully coupled implicit formulation for reservoir simulation

An iterative method is presented for solving a fully coupled and implicit formulation of fluid flow in a porous medium. The mathematical model describes a set of fully coupled three-phase flow of compressible and immiscible fluids in a saturated oil reservoir. The finite element method is applied to obtain the simultaneous solution (SS) for the resulting highly non-linear partial differential equations where fluid pressures are the primary unknowns. The final discretized equations are solved iteratively by using a fully implicit numerical scheme. Several examples, illustrating the use of the present model, are described. The increased stability achieved with this scheme has permitted the use of larger time steps with smaller material balance errors.     

帷幕灌浆扩散半径及数值模拟的研究

从连续性方程出发,考虑浆液压力对土体孔隙率的影响,分析推导了浆液在多孔介质中的渗流规律,并给出了扩散半径的简单近似计算公式;然后将二相流理论应用到注浆研究中,假设孔隙由水和浆液完全充填,且二者不相混溶,建立了浆液驱水的非稳定渗流模型。根据某大坝的实际情况,用FLAC软件中的二相流模块对大坝灌浆过程进行了模拟分析,并与推导公式相比较。由于二相流理论考虑了毛细压力的作用,模拟得到的浆液扩散速率递减得更快。模拟结果表明,浆液和水之间存在着一个过渡带,浆液的饱和度在不同时间和位置上是变化的。随着时间的延长,浆液扩散得越远,但其速率逐渐减小。扩散半径不仅与渗透系数有关,还与孔隙度有关,而且孔隙度较渗透系数对扩散半径有着更大的影响。二相流理论可以更好地模拟帷幕灌浆的浆液扩散情况。     

Fluid inclusion evidence for deep burial of the Tertiary accretionary wedge of the Carpathians

Hydrothermal quartz from mineralized joints of the Carpathian accretionary wedge contains immiscible aqueous, oil‐condensate, methane and carbon dioxide‐rich fluid inclusions. Distribution patterns of the inclusion trapping PT parameters point to a crack‐seal mechanism during upward and lateral migration of hot methane‐rich fluids from overpressured sediments at the base of the accretionary wedge. A simple equation is proposed to calculate depths from densities and trapping pressures of the buoyant inclusion fluids. In the Carpathian accretionary wedge, the paleofluid pressures of 52–306 MPa correspond to a 5‐ to 11‐km‐thick overburden. Prior to exhumation, thickness of the wedge must have attained 10–25 km, of which only c. 50% was preserved until recently. Anomalously high methane densities (up to 0.43 g cm^?3) recorded in the lowermost nappe sheets are provisionally interpreted as a result of supralithostatic overpressure due to thermal cracking of oil and kerogen to methane and pyrobitumen at temperatures above 200 °C.     

Diagenesis and fracturing of Paleocene–Eocene carbonate turbidite systems in the Ionian Basin: The example of the Kelçyra area (Albania)

The Kelçyra area is emplaced in a foreland fold-and-thrust belt (FFTB), characterized by a westward thrusting with the Triassic evaporites as the major décollement level. Several secondary features related with this evolution, like backthrusting, folding, duplex structures, evaporite diapirism are present. During the FFTB evolution, the study area has been subjected to several fracturing events with associated stages of fluid migration. During the pre-deformational stage, complex textures such as crack-and-seal features most likely reflect expulsion of overpressured fluids. These fluids were dominantly host-rock buffered. Within the post-deformational stage, a meteoric fluid caused cementation and development of a karst network during a period of emergence after the thrust emplacement. Subsequently, Mg calcite reprecipitated in the more stable carbonate phase calcite and dolomite, which filled part of the karts network. The latter is finally dedolomitized and locally partially dissolved by a second meteoric fluid flow, which greatly increased the secondary porosity.     

Fluid inclusion constraints on the uplift history of the metamorphosed massive sulphide deposits at Ducktown, Tennessee

Abstract Standard petrographic, microthermometric and Raman spectroscopic analyses of fluid inclusions from the metamorphosed massive sulphide deposits at Ducktown, Tennessee, indicate that fluids with a wide range of compositions in the C–O–H–N–S–salt system were involved in the syn- to post-metamorphic history of these deposits. Primary fluid inclusions from peak metamorphic clinopyroxene contain low-salinity, H₂O–CH₄ fluids and calcite, quartz and pyrrhotite daughter crystals. Many of these inclusions exhibit morphologies resembling those produced in laboratory experiments in which confining pressures significantly exceed the internal pressures of the inclusions. Secondary inclusions in metamorphic quartz from veins, pods, and host matrix record a complex uplift history involving a variety of fluids in the C–O–H–N–salt system. Early fluids were generated by local devolatilization reactions while later fluids were derived externally. Isochores calculated for secondary inclusions in addition to the chronology of trapping and morphological features of primary and secondary fluid inclusions suggest an uplift path which was concave toward the temperature axis over the P–T range 6–3 kbar and 550–225° C. Immiscible H₂O–CH₄–N₂–NaCl fluids were trapped under lithostatic to hydrostatic pressure conditions at 3–0.5 kbar and 215 ± 20° C. Entrapment occurred during Alleghanian thrusting, and the fluids may have been derived by tectonically driven expulsion of pore fluids and thermal maturation of organic material in lower-plate sedimentary rocks which are thought to underlie the deposits. Episodic fracturing and concomitant pressure decreases in upper-plate rocks, which host the ore bodies, would have allowed these fluids to move upward and become immiscible. Post-Alleghanian uplift appears to have been temperature-convex. Uplift rates of 0.10–0.05 mm year^?1 from middle Ordovician to middle Silurian – late Devonian, and 0.07–0.12 mm year^?1 from middle Silurian – late Devonian to late Permian are suggested by our uplift path and available geochronological data.     

A non‐coaxial critical state soil model and its application to simple shear simulations

The yield vertex non‐coaxial theory is implemented into a critical state soil model, CASM (Int. J. Numer. Anal. Meth. Geomech. 1998; 22 :621–653) to investigate the non‐coaxial influences on the stress–strain simulations of real soil behaviour in the presence of principal stress rotations. The CASM is a unified clay and sand model, developed based on the soil critical state concept and the state parameter concept. Without loss of simplicity, it is capable of simulating the behaviour of sands and clays within a wide range of densities. The non‐coaxial CASM is employed to simulate the simple shear responses of Erksak sand and Weald clay under different densities and initial stress states. Dependence of the soil behaviour on the Lode angle and different plastic flow rules in the deviatoric plane are also considered in the study of non‐coaxial influences. All the predictions indicate that the use of the non‐coaxial model makes the orientations of the principal stress and the principal strain rate different during the early stage of shearing, and they approach the same ultimate values with an increase in loading. These ultimate orientations are dependent on the density of soils, and independent of their initial stress states. The use of the non‐coaxial model also softens the shear stress evolutions, compared with the coaxial model. It is also found that the ultimate shear strengths by using the coaxial and non‐coaxial models are dependent on the plastic flow rules in the deviatoric plane. Copyright © 2006 John Wiley & Sons, Ltd.     

Fluid Inclusions and Metallization of the Kendekeke Polymetallic Deposit in Qinghai Province,China

The Kendekeke polymetallic deposit, located in the middle part of the magmatic arc belt of Qimantag on the southwestern margin of the Qaidam Basin, is a polygenetic compound deposit in the Qimantag metallogenic belt of Qinghai Province. Multi-periodic ore-forming processes occurred in this deposit, including early-stage iron mineralization and lead-zinc-gold-polymetallic mineralization which was controlled by later hydrothermal process. The characteristics of the ore-forming fluids and mineralization were discussed by using the fluid inclusion petrography, Laser Raman Spectrum and micro-thermometry methods. Three stages, namely, S1-stage(copper-iron-sulfide stage), S2-stage(lead-zinc-sulfide stage) and C-stage(carbonate stage) were included in the hydrothermal process as indicated by the results of this study. The fluid inclusions are in three types: aqueous inclusion(type I), CO2-aqueous inclusion(type II) and pure CO2 inclusion(type III). Type I inclusions were observed in the S1-stage, having homogenization temperature at 240–320oC, and salinities ranging from 19.8% to 25.0%(wt % NaCl equiv.). All three types of inclusions, existing as immiscible inclusion assemblages, were presented in the S2-stage, with the lowest homogenization temperature ranging from 175 oC to 295oC, which represents the metallogenic temperature of the S2-stage. The salinities of these inclusions are in the range of 1.5% to 16%. The fluid inclusions in the C-stage belong to types I, II and III, having homogenization temperatures at 120–210oC, and salinities ranging from 0.9% to 14.5%. These observations indicate that the ore-forming fluids evolved from high-temperature to lowtemperature, from high-salinity to low-salinity, from homogenization to immiscible separation. Results of Laser Raman Spectroscopy show that high density of CO2 and CH4 were found as gas compositions in the inclusions. CO2, worked as the pH buffer of ore-forming fluids, together with reduction of organic gases(i.e. CH4, etc), affected the transport and sediment of the minerals. The fluid system alternated between open and close systems, namely, between lithostatic pressure and hydrostatic pressure systems. The calculated metallogenic pressures are in the range of 30 to 87 Mpa corresponding to 3 km mineralization depth. Under the influence of tectonic movements, immiscible separation occurred in the original ore-forming fluids, which were derived from the previous highsalinity, high-temperature magmatic fluids. The separation of CO2 changed the physicochemical properties and composition of the original fluids, and then diluted by mixing with extraneous fluids such as meteoric water and groundwater, and metallogenic materials in the fluids such as lead, zinc and gold were precipitated.     

Finite element simulations of seismic effects on retaining walls with liquefiable backfills

Finite element simulations of two centrifuge tests on the same cantilever retaining wall model holding liquefiable backfill were conducted using the Biot formulation‐based program DIANA–SWANDYNE II. To demonstrate the effects due to different pore fluids in seismic centrifuge experiments, water was used as the pore fluid in one experiment whereas a substitute pore fluid was used in the second experiment. The cantilever wall model parameters were determined by comparing simulations with measurements from free‐vibration tests performed on the model wall without backfill. The initial stress conditions for dynamic analysis for the soil backfill were obtained by simulating static loads on the retaining wall from the soil backfill. Level‐ground centrifuge model results were used to select the parameters of the Pastor–Zienkiewicz mark III constitutive model used in the dynamic simulations of the soil. The effects due to different pore fluids were captured well by the simulations. The magnitudes of excess pore pressures in the soil, lateral thrust and its line of action on the wall, and wall bending strains, deflections, and accelerations were predicted well. Predictions of settlements and accelerations in the backfill were less satisfactory. Relatively high levels of Rayleigh damping were needed to be used in the retaining wall simulations in order to obtain numerically stable results, which is one of the shortcomings of the model. The procedure may be used for engineering purpose dealing with seismic analysis of flexible retaining walls where lateral pressures, bending strains and deflections in the wall are typically of importance. Copyright © 2008 John Wiley & Sons, Ltd.     

Shallow-level metamorphic fluids in a high uplift rat metamorphic belt; Alpine Schist, New Zealand

Abstract Fluids, some of which are CO₂-rich (up to 40 mol.% CO₂) and some of which are highly saline (up to 18 wt% NaCl equivalent), are trapped as fluid inclusions in quartz-calcite (∼ metallic minerals) veins which cross-cut the pumpellyite-actinolite to amphibolite facies rocks of the Alpine Schist. Fluids were commonly trapped as immiscible liquid-vapour mixes in quartz and calcite showing open-space growth textures. Fluid entrapment occurred at fluid pressures near 500 bars (possibly as low as 150 bars) at temperatures ranging from 260 to 330° C. Saline fluids may have formed by partitioning of dissolved salts into an aqueous phase on segregation of immiscible fluids from a low-density CO₂-rich fluid. Calcite deposited by these fluids has δ¹³C ranging from – 8.4 to – 11.5 and δ¹⁸O from + 4 to + 13. Isotopic data, fluid compositions and mode of occurrence suggest that the fluids are derived from high-grade metamorphic rocks. Fluid interaction with wall-rock has caused biotite crystallization and/or recrystallization in some rocks and retrogression of biotite to chlorite in other rocks.
Fluid penetration through the rock is almost pervasive in many areas where permeability, probably related to Alpine Fault activity, has focussed fluids on a regional scale into fractured rocks. The fluid flow process is made possible by high uplift-rates (in excess of 10 mm/year) bringing hot rocks near to the surface.     

Study of the free surface flow of water–kaolinite mixture by moving particle semi‐implicit (MPS) method

The flow of water–kaolinite mixtures exhibits a non‐Newtonian nature that differs from the flow of Newtonian fluid. The varying viscosities and shear history of non‐Newtonian fluid flows necessitate the use of a rheology model in moving particle semi‐implicit (MPS) for the numerical studies. On the other hand, the Lagrangian method has the advantage of handling free surface flows with large deformation and fragmentation. This study proposes a mesh‐free Lagrangian method, namely, the MPS method, together with a simple rheology model to investigate the non‐Newtonian free surface flows. The rheological parameters required in the rheology model are determined based upon experiments. The proposed method is applied to a water–kaolinite mixture collapse problem and is proved to be capable of reproducing the significant flow features observed in laboratory experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Identification of non‐linear stress–strain–time relationship of soils using genetic algorithm

Recognition of non‐linear constitutive rock/soil model from experimental results is often multi‐modal in the large parameter space. A genetic evolution algorithm is thus proposed for its recognition, including that of structure of the model and coefficients in the model. The structure of the model can be firstly determined according to mechanical mechanism if the mechanism is clearly understood or searched by using evolutionary algorithm. The coefficients to be determined are then searched in global optional space. With the new evolutionary algorithm, the non‐linear stress–strain–time constitutive law to describe strain softening behaviours of diatomaceous soil under consolidated and undrained state was recognized by learning stress–strain–time behaviour of an intact sample under consolidated pressure of σ_c=0.1 MPa and strain velocity of_a=0.175%/min. This model gave reasonable prediction for diatomaceous soils under varying consolidated pressures (0.1–3.5 MPa) and strain velocities (0.0044–1.75%/min). It indicates that the methodology proposed in this paper is robust enough and strongly attractive for recognition of non‐linear constitutive model of soil and rock materials. Copyright © 2002 John Wiley & Sons, Ltd.     

Free-surface Simulations of Newtonian and Non-Newtonian Fluids with the Lattice Boltzmann Method

This paper describes the application of a three-dimensional lattice Boltzmann method (LBM) to Newtonian and non-Newtonian (Bingham fluid in this work) flows with free surfaces. A mass tracking algorithm was incorporated to capture the free surface, whereas Papanastasiou’s modified model was used for Bingham fluids. The lattice Boltzmann method was first validated using two benchmarks: Newtonian flow through a square cross-section tube and Bingham flow through a circular cross-section tube. Afterward, the dam-break problem for the Newtonian fluid and the slump test for Bingham fluid were simulated to validate the free-surface-capturing algorithm. The numerical results were in good agreement with analytical results, as well as other simulations, thereby proving the validity and correctness of the current method. The proposed method is a promising substitute for time-consuming and costly physical experiments to solve problems encountered in geotechnical and geological engineering, such as the surge and debris flow induced by a landslide or earthquake.     

Finite element modelling of transport of organic chemicals through soils

Aquifer contamination by organic chemicals in subsurface flow through soils due to leaking underground storage tanks filled with organic fluids is an important groundwater pollution problem. The problem involves transport of a chemical pollutant through soils via flow of three immiscible fluid phases: namely air, water and an organic fluid. In this paper, assuming the air phase is under constant atmospheric pressure, the flow field is described by two coupled equations for the water and the organic fluid flow taking interphase mass transfer into account. The transport equations for the contaminant in all the three phases are derived and assuming partition equilibrium coefficients, a single convective – dispersive mass transport equation is obtained. A finite element formulation corresponding to the coupled differential equations governing flow and mass transport in the three fluid phase porous medium system with constant air phase pressure is presented. Relevant constitutive relationships for fluid conductivities and saturations as function of fluid pressures lead to non-linear material coefficients in the formulation. A general time-integration scheme and iteration by a modified Picard method to handle the non-linear properties are used to solve the resulting finite element equations. Laboratory tests were conducted on a soil column initially saturated with water and displaced by p-cymene (a benzene-derivative hydrocarbon) under constant pressure. The same experimental procedure is simulated by the finite element programme to observe the numerical model behaviour and compare the results with those obtained in the tests. The numerical data agreed well with the observed outflow data, and thus validating the formulation. A hypothetical field case involving leakage of organic fluid in a buried underground storage tank and the subsequent transport of an organic compound (benzene) is analysed and the nature of the plume spread is discussed.