Numerical Modeling and Spatial Moment Analysis of Solute Mobility and Spreading in a Coupled Fracture-Skin-Matrix System

This study deals with transport of solutes through a saturated sub-surface rock formation with well-defined horizontal parallel fractures. For this purpose, a simplified conceptual model consisting of a single fracture and its associated rock-matrix is considered in the presence of a fracture-skin in order to study the mobility and mixing of solutes along the fracture. In this paper, a coupled fracture-skin-matrix system is modeled numerically using finite difference method in a pseudo two-dimensional domain with a constant continuous source at fracture inlet. Flow and transport processes are considered parallel to the fracture axis, while the transport processes in fracture-skin as well as in rock-matrix are considered perpendicular to the fracture axis. Having obtained the concentration distribution along the fracture, method of spatial moments is employed to study the mobility and spreading of solutes. Sensitivity analyses have been done to understand the effect of various fracture-skin parameters like porosity, thickness, and diffusion coefficient. Further, the influence of non-linear sorption and radioactive decaying of solutes are carried out for different sorption intensities and decay constants. Results suggest that the presence of fracture-skin significantly influences the mobility and spreading of solutes along the fracture in comparison with a coupled fracture-matrix system without fracture-skin.     

http://www.sciencedirect.com/science/article/pii/S1674987111001277

A numerical model is developed for investigating the evolution of fracture permeability in a coupled fracture-matrix system in the presence of fracture-skin with simultaneous colloidal and bacterial tr...     

Numerical Modeling and Spatial Moment Analysis of Thermal Fronts in a Coupled Fracture-Skin-Matrix System

In this study, the behavior of thermal fronts along the fracture is studied in the presence of fracture-skin in a coupled fracture-matrix system. Cold water is injected into the fracture, which advances gradually towards production well, while extracting heat from the surrounding reservoir matrix. The heat conduction into the fracture-skin and the rock-matrix from the high permeability fracture is assumed to be one dimensional perpendicular to the axis of fluid flow along the fracture. Constant temperature cold water is injected through an injection well at the fracture inlet. The fluid flow takes place along the horizontal fracture which ensures connectivity between the injection and production wells. Since the rock-matrix is assumed to be tight, the permeability of fracture-skin as well as the rock-matrix is neglected. The present study focuses on the heat flux transfer at the fracture-skin interface as against the earlier studies on fracture-matrix interface, and the sensitivity of additional heterogeneity in the form of fracture skin in a conventional fracture-matrix coupled system is studied. The behavior of thermal fronts for various thermal conductivity values of the fracture-skin and rock-matrix is analyzed. Spatial moment analysis is performed on the thermal distribution profiles resulting from numerical studies in order to investigate the impact on mobility and dispersion behavior of the fluid in the presence of fracture-skin. The presence of fracture skin affects the heat transfer significantly in the coupled fracture-matrix system. The lower order spatial moments indicate that the effective thermal velocity increases with increase in skin thermal conductivity and a significant thermal dispersion is observed at the inlet of the fracture owing to the high thermal conductivity of the fracture-skin at the early stages. Furthermore the higher spatial moments indicate that the asymmetricity increases with decrease in skin thermal conductivity unlike the case with half fracture aperture and fluid velocity and the kurtosis is maximum with higher skin thermal conductivity which implies enhanced heat extraction from the fracture-skin into the fracture. Results suggest that the amount of heat extraction by the circulating fluid within the fracture from the reservoir not only depends on the rock-matrix module of the reservoir characteristics but also the fracture-skin characteristics of the system and subsequently influence the reservoir efficiency.     

http://dx.doi.org/10.1016/j.gsf.2015.11.001

A new method, the characteristic finite element method(CFEM), was developed to simulate solute transport in a cross-fracture. The solution of this mathematical model for solute transport considered that the contribution of convection and dispersion terms was deduced using the single-step, trace-back method and routine finite element method(FEM). Also, experimental models were designed to verify the reliability and validity of the CFEM. Results showed that experimental data from a single fracture model agreed with numerical simulations obtained from the use of the CFEM. However, routine FEM caused numerical oscillation and dispersion during the calculation of solute concentration. Furthermore, in this cross-fracture model, CFEM simulation results predicted that the arrival time of concentration peak values decreased with increasing flux. Also, the second concentration peak value was obvious with the decrease of flux, which may have resulted from the convergence of solute concentrations from main, and branch, fractures.     

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.     

A coupled compressible flow and geomechanics model for dynamic fracture aperture during carbon sequestration in coal

This paper presents the development of a discrete fracture model of fully coupled compressible fluid flow, adsorption and geomechanics to investigate the dynamic behaviour of fractures in coal. The model is applied in the study of geological carbon dioxide sequestration and differs from the dual porosity model developed in our previous work, with fractures now represented explicitly using lower-dimensional interface elements. The model consists of the fracture-matrix fluid transport model, the matrix deformation model and the stress-strain model for fracture deformation. A sequential implicit numerical method based on Galerkin finite element is employed to numerically solve the coupled governing equations, and verification is completed using published solutions as benchmarks. To explore the dynamic behaviour of fractures for understanding the process of carbon sequestration in coal, the model is used to investigate the effects of gas injection pressure and composition, adsorption and matrix permeability on the dynamic behaviour of fractures. The numerical results indicate that injecting nonadsorbing gas causes a monotonic increase in fracture aperture; however, the evolution of fracture aperture due to gas adsorption is complex due to the swelling-induced transition from local swelling to macro swelling. The change of fracture aperture is mainly controlled by the normal stress acting on the fracture surface. The fracture aperture initially increases for smaller matrix permeability and then declines after reaching a maximum value. When the local swelling becomes global, fracture aperture starts to rebound. However, when the matrix permeability is larger, the fracture aperture decreases before recovering to a higher value and remaining constant. Gas mixtures containing more carbon dioxide lead to larger closure of fracture aperture compared with those containing more nitrogen.     

非饱和土中镉离子传输模型参数反演

使用自行设计的的真空实验装置,采用一维实验室土柱实验方法在不同实验条件下进行了15组实验,并基于实验所得的穿透曲线,使用梯度正则化方法反演得到了镉离子在8种葡萄牙土样和1种比利时土样中相应的传输模型参数。同时通过数值模拟实验验证了梯度正则化方法的有效性和可靠性,进而证明所得结果是有效的。     

Thermo-hydro-mechanical modeling of fracturing porous media with two-phase fluid flow using X-FEM technique

In this paper, a fully coupled thermo-hydro-mechanical model is presented for two-phase fluid flow and heat transfer in fractured/fracturing porous media using the extended finite element method. In the fractured porous medium, the traction, heat, and mass transfer between the fracture space and the surrounding media are coupled. The wetting and nonwetting fluid phases are water and gas, which are assumed to be immiscible, and no phase-change is considered. The system of coupled equations consists of the linear momentum balance of solid phase, wetting and nonwetting fluid continuities, and thermal energy conservation. The main variables used to solve the system of equations are solid phase displacement, wetting fluid pressure, capillary pressure, and temperature. The fracture is assumed to impose the strong discontinuity in the displacement field and weak discontinuities in the fluid pressure, capillary pressure, and temperature fields. The mode I fracture propagation is employed using a cohesive fracture model. Finally, several numerical examples are solved to illustrate the capability of the proposed computational algorithm. It is shown that the effect of thermal expansion on the effective stress can influence the rate of fracture propagation and the injection pressure in hydraulic fracturing process. Moreover, the effect of thermal loading is investigated properly on fracture opening and fluids flow in unsaturated porous media, and the convective heat transfer within the fracture is captured successfully. It is shown how the proposed computational model is capable of modeling the fully coupled thermal fracture propagation in unsaturated porous media.     

华北地区碳酸盐岩热储层酸化压裂模拟方法与应用

碳酸盐岩地层是我国华北地区最重要的热储岩层之一,不同次级构造单元的碳酸盐岩地热井产量和回灌量存在较大差异,严重制约了碳酸盐岩热储的规模化开发。酸化压裂是碳酸盐岩热储增产稳产的有效手段,其涉及溶质运移、酸岩反应、渗流传热、反应放热等一系列复杂过程。本文提出了一种模拟工程尺度碳酸盐岩热储酸化压裂过程的数值方法,可考虑碳酸盐岩热储酸化压裂过程中热、水、力、化四场之间的耦合作用。通过与前人试验和模拟结果的对比,验证了该方法的正确性,并建立了碳酸盐岩热储非均质裂隙模型,对其酸化压裂效果进行了模拟。     

米尺度裂隙岩体模型水流-传热试验的数值模拟分析

为了研究高放射性核废物地下处置库近场的水流-传热耦合问题,采用国内高放废物地下处置库预选场址--甘肃北山地区的花岗岩石块体,加工组合成米尺度的规则裂隙岩体模型,设置边界热源和裂隙水流,试验模拟裂隙水水流与传热之间的相互作用。作为该室内模型试验的前期理论研究,采用等效孔隙介质数值模型,着重分析了裂隙开度、裂隙流量和热源功率对流场和温度场的影响。在设定条件下,计算分析表明：热传导和裂隙水水流由热源作用初期的不耦合很快转化为耦合;不流动的裂隙水主要表现为热存储和热传导,而流动的裂隙水还引起流动传热和水与岩石之间的对流换热,使岩体温度场明显不同于单纯热传导的情况;如果保持裂隙水流量不变,则裂隙开度的变化对水流-传热影响不大;如果保持裂隙水流速不变,则裂隙开度的变化对水流-传热影响显著;热源功率越大,通过裂隙水的热流量越大,裂隙水压强越大,而当温度超过100 ℃时,裂隙水会因汽化而压强显著增大;加热7 d时,热量的输入和输出几乎相等,裂隙水流带走的热量接近热源供给的热量,模型系统基本达到了热平衡。     

Nonaqueous liquid pool dissolution in three-dimensional heterogeneous subsurface formations

A three-dimensional numerical flow and contaminant transport model is developed to investigate the effect of variable hydraulic conductivity on average mass transfer coefficients associated with the dissolution of dense nonaqueous phase liquid (DNAPL) pools in heterogeneous, water-saturated subsurface formations. Randomly generated, three-dimensional hydraulic conductivity fields are used to represent a heterogeneous confined aquifer. Model simulations indicate that the average mass transfer coefficient is inversely proportional to the variance of the log-transformed hydraulic conductivity. A power law correlation relating the Sherwood number to the variance of the log-transformed hydraulic conductivity and appropriate Peclet numbers is developed. A reasonable fit between predicted and numerically determined mass transfer coefficients is observed.     

Numerical analysis of thermal fracturing in subsurface cold water injection by finite element methods

Integration of poromechanics and fracture mechanics plays an important role in understanding a series of thermal fracturing phenomena in subsurface porous media such as cold water flooding for enhanced oil recovery, produced‐water reinjection for waste disposal, cold water injection for geothermal energy extraction, and CO₂ injection for geosequestration. Thermal fracturing modeling is important to prevent the potential risks when fractures propagate into undesired zones, and it involves the coupling of heat transfer, mass transport, and stress change as well as the fracture propagation. Analytical method, finite element method, and finite difference method as well as boundary element method have been used to perform the thermal fracturing modeling considering different degrees and combinations of coupling. In this paper, extended finite element method is employed for the thermal fracturing modeling in a fully coupled fashion with remeshing avoided, and the stabilized finite element method is employed to account for the convection‐dominated heat transfer in the fracturing process with numerical oscillation circumvented. With the thermal fracturing model, a hypothetical numerical experiment on cold water injection into a deep warm aquifer is conducted. Results show that parameters such as injection rate, injection temperature, aquifer stiffness, and permeability can affect the fracture development in different ways and extended finite element method and stabilized finite element method provide effective tools for thermal fracturing simulation. Copyright © 2012 John Wiley & Sons, Ltd.     

裂隙岩体流-热耦合传热的三维数值模拟分析

通过对潘西煤矿水文地质条件的分析,基于裂隙岩体的流-热耦合数学模型,描述了裂隙岩体渗流场分布和水流及岩体的温度场分布,并结合边界条件及计算参数对裂隙岩体的流-热耦合传热进行了数值模拟和分析。数值模拟结果表明,岩体内裂隙水流所引发的热量迁移,对裂隙岩体的温度场分布有重要影响。断裂带及地下水流的存在改变了岩体的原有温度场分布。在渗流初期,温度梯度矢量沿渗流方向向两侧岩体方向流动,由于两侧岩体的渗透性系数低于断裂带处的渗透性系数,右侧等温线及温度梯度矢量方向逐渐向渗流方向移动,改变了两侧岩体的温度场分布。通过对断裂带内裂隙水流渗透性系数的折减,分析渗透性系数发生变化时对岩体温度场分布的影响,渗透性系数越大,伴随的热量迁移增大,对岩体的温度场分布的影响也越大。     

CCL吸附特性及孔隙率降低对污染物运移的影响

假定孔隙均匀地分布于土体的物质空间内和土骨架对污染物的吸附特性服从平衡线性,对基本体积质量关系进行分析,提出了由于土体对污染物的吸附而引起的孔隙率降低的估算公式。在考虑土体孔隙率变化的条件下,建立了污染物一维运移的控制方程,并考虑垃圾生物降解效应、压实黏土衬里（CCL）防渗层、下覆有限厚度含水层等实际情况,确定了初始条件和边界条件。对所建立的初边值问题进行了数值求解,且对某假想填埋场情况进行了变动参数与对比计算,结果表明,由于土颗粒对污染物的吸附所引起的孔隙率降低,显著地降低了污染物对压实黏土衬里的穿透能力。与常孔隙率情况相比,CCL中污染物的峰值浓度降低近10 %,含水层中污染物浓度降低更显著。当考虑土体孔隙率变化时,弥散对污染物运移具有控制作用,分布系数对污染物的运移具有重要影响。     

Hybrid analytical and finite element numerical modeling of mass and heat transport in fractured rocks with matrix diffusion

Quantification of mass and heat transport in fractured porous rocks is important to areas such as contaminant transport, storage and release in fractured rock aquifers, the migration and sorption of radioactive nuclides from waste depositories, and the characterization of engineered heat exchangers in the context of enhanced geothermal systems. The large difference between flow and transport characteristics in fractures and in the surrounding matrix rock means models of such systems are forced to make a number of simplifications. Analytical approaches assume a homogeneous system, numerical approaches address the scale at which a process is operating, but may lose individual important processes due to averaging considerations. Numerical stability criteria limit the contrasts possible in defining material properties. Here, a hybrid analytical–numerical method for transport modeling in fractured media is presented. This method combines a numerical model for flow and transport in a heterogeneous fracture and an analytical solution for matrix diffusion. By linking the two types of model, the advantages of both methods can be combined. The methodology as well as the mathematical background are developed, verified for simple geometries, and applied to fractures representing experimental field conditions in the Grimsel rock laboratory.     

饱和多孔介质中水力-力学-传质耦合过程的混合有限元法

对饱和多孔介质提出了一个含溶混污染物输运（传质）过程的混合元方法,其中污染物输运过程数学模型包含了对流、机械逸散、分子弥散和吸附等机制。固相位移、应变和有效应力,孔隙水压力、压力空间梯度和Darcy速度,污染物浓度、浓度空间梯度和浓度流量在单元内均为独立变量分别插值。基于胡海昌-Washizu三变量广义变分原理,结合可以滤掉虚假振荡的特征线方法,推导出饱和土中水力-力学-传质耦合问题控制方程的单元弱形式,并导出了混合元计算公式。数值模拟证明了所提出的方法可以提供与传统4点积分方案同样精度,同时能够提高计算效率。     

渗流-化学溶解耦合作用下岩石单裂隙渗透特性研究

为揭示在渗流-化学溶解耦合作用下单裂隙渗透特性的变化规律,建立了描述二维渗流-化学溶解耦合作用的偏微分方程组,并利用COMSOL Multiphysics软件成功地求解该方程组。首先,模拟了文献[1]中的盐岩渗流-溶解耦合渗流试验结果,数值模拟结果与试验结果较为吻合,验证了数学模型的正确性和有效性。然后,利用分形理论生成了一个粗糙的裂隙面数字模型,着重分析了二维石灰岩粗糙裂隙面在水流、矿物溶解和输运过程中其渗透特性的变化规律。数值分析显示,（1）溶质浓度对裂隙面的溶解具有非常重要的作用,从而水流进口端的溶解厚度比出口端大得多。（2）裂隙的整体渗透性在初始时刻增加较慢,随着裂隙开度的增大和贯通,溶解速度会逐渐增大,是一个加速的过程。     

Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D

A new method to determine semi-analytical solutions of one-dimensional contaminant transport problem with nonlinear sorption is described. This method is based on operator splitting approach where the convective transport is solved exactly and the diffusive transport by finite volume method. The exact solutions for all sorption isotherms of Freundlich and Langmuir type are presented for the case of piecewise constant initial profile and zero diffusion. Very precise numerical results for transport with small diffusion can be obtained even for larger time steps (e.g., when the Courant-Friedrichs-Lewy (CFL) condition failed).     

吸附对土壤水环境中有机污染物生物降解过程的影响研究

在土壤水环境系统中,吸附作用是影响有机污染物生物降解过程的主要制约因素。污染物的吸附性越强,则其存在于土壤水溶液的重量百分比就越小,生物降解可能性也就越低。本文建立了定量描述吸附作用对有机污染物在土壤颗粒内部传输过程影响的数学模型。通过模型计算发现,有机污染物的土壤-水吸附分配系数Kd越大,则其从土壤颗粒内部传输到外部水溶液的速率就越小,总的降解速率也就越低。