A parallelizable method for two‐phase flows in naturally‐fractured reservoirs

A parallelizable, semi‐implicit numerical method is proposed for the study of naturally‐fractured reservoir systems. It has proved to be computationally efficient in producing accurate numerical solutions for the dual‐porosity model for immiscible, two‐phase flow in such reservoirs. The method combines hybridized mixed finite elements, a new version of the modified method of characteristics, a sophisticated operator‐splitting procedure for separating the pressure calculation in the fractures from that of the saturation, another operator splitting to handle the interaction of the matrix blocks and the fractures, and domain decomposition iterative procedures for both the pressure and the saturation. It permits moderately long time steps for the pressure and the saturation in the fractures and matrix blocks by using short, inexpensive microsteps to treat the transport portion of the saturation equation in the fractures. This paper is devoted to the formulation of the method and a discussion of numerical results for five‐spot and vertical cross‐section examples.     

Influence of flow velocity on conservative solute transport in sets of parallel fractures with fracture skin

Analysis of contaminant transport through fractured crystalline rocks has received considerable attention, particularly with regard to subsurface nuclear waste repositories. Most of the studies have employed the dual continuum approach, with the fractures and the rock matrix as the two continuums, assuming that fractures control the overall conductivity of the rock and the porous matrix just provides storage. However, field observations of rock fractures have shown that the real situation can be very complex. Based on some recent investigations, it has been reported that the portion of the rock matrix adjacent to many open fractures is physically and chemically altered. These alterations, referred to as the fracture skin, can have different sorption and diffusion properties compared to those of the undisturbed rock matrix and this may influence the transport of solutes through such formations. In the present study, a numerical model is developed to simulate conservative solute transport in a fractured crystalline rock formation using the triple continuum approach ?? with the fracture, fracture skin and the rock matrix as the three continuums. The model is solved using a fully implicit finite difference scheme. Contaminant migration in the fractured formation with and without skin has been simulated. It is observed that contaminant penetration along the fracture is enhanced at large flow velocities. The effect of flow velocity on conservative solute transport is investigated for different fracture apertures and fracture skin thicknesses. The influence of flow velocity on contaminant transport is demonstrated to be more with change in fracture aperture than with change in skin thickness.     

A Comparison of Methods for Calculating the Matrix Block Source Term in a Double Porosity Model for Contaminant Transport

Contaminant transport in a fractured porous medium can be modeled, under appropriate conditions, with a double porosity model. Such a model consists of a parabolic equation with a coupling term describing contaminant exchange between the fractures, which have high permeability, and the matrix block, which has low permeability. A locally conservative method based on mixed finite elements is used to solve the parabolic problem, and the calculation of the coupling term, which involves the solution of diffusion equations in the matrix blocks, is based on an analytic expression. Numerical experiments show that this semi-analytic method for the coupling term is accurate and faster than several other methods but at a small expense of computer memory.     

Boundary element analysis of contaminant transport in fractured porous media

A two-dimensional boundary integral method to analyse the flow of contaminant in fractured media having a two- or three-dimensional orthogonal fracture network is presented. The method assumes that the fractures provide the paths of least resistance for transport of contaminants while the matrix, because of its low permeability, acts as ‘storage blocks’ into which the contaminant diffuses. Laplace transform is used to eliminate the time variable in the governing equation in order to facilitate the formulation of a boundary integral equation in the Laplace transform space. Conventional boundary element techniques are applied to solve for the contaminant concentrations at specified locations in the spatial domain. The concentration in the time domain is then obtained by using an efficient inversion technique developed by Talbot. The method is able to analyse the behaviour of waste repositories which have diminishing concentration due to the mass transport of the contaminant into the surrounding fractured media.     

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.     

Numerical modeling of stress effects on solute transport in fractured rocks

The effects of stress/deformation on fluid flow and contaminant transport in fractured rocks is one of the major concerns for performance and safety assessments of many subsurface engineering problems, especially radioactive waste disposal and oil/gas reservoir fields. However, very little progress has been made to study this issue due to difficulties in both experiments and numerical modeling. The objective of this study is to systematically investigate the influence of stress on solute transport in fractured rocks for the first time, considering different stress and hydraulic pressure conditions. A hybrid approach combining discrete element method (DEM) for stress-flow simulations and a particle tracking algorithm is developed. The impact of matrix diffusion (diffusion of molecular size solutes in and out of the rock matrix, and sorption onto the surface of micropores in rock matrix) is also included. The numerical results show that stress not only significantly changes the solute residence time through the fracture networks, but also changes the solute travel paths. Matrix diffusion plays a dominant role in solute transport when the hydraulic gradient is small, which is often encountered in practice.     

Investigations on Diffusion Characteristics of Granite and Chalk Rock Mass

Contaminant transport through fractured rock mass is predominated by diffusion. This is due to the continuous interaction of the mobile water present in the fracture network and relatively immobile pore water, which is adsorbed on the surface and in the rock matrix itself. Even though the advective flow through the fracture network is high, besides sorption of rock mass, the diffusive exchange into the rock mass leads to significant retardation of contaminant transport. Hence, for describing contaminant transport in fractured rock mass, more precisely, the effect of retardation attributed to the matrix diffusion must be taken in account. With this in view, a methodology, which can be employed for determination of the diffusion characteristics of the rock mass, has been developed and its details are presented in this paper. Validation of the methodology has been demonstrated with the help of Archie’s law.     

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。     

Predicting contaminant fate and transport in sediment caps: Mathematical modelling approaches

Sediment capping is a remedial option for managing contaminated sediments that involves the artificial placement of a layer of material over a contaminated area. Sorbent materials such as activated C and coke can be used to amend sand caps to improve cap performance. In this study, analytical and numerical modelling approaches were compared for predicting contaminant fate and transport in sediment caps using several diffusion-controlled and advection-dominated contaminant transport scenarios. An analytical tool was used to predict cap performance at steady-state. These results were compared with the results from the numerical CoReTranS model in which the effective diffusivity and degradation rates were modelled as discontinuous functions at a prescribed bioturbation depth. The numerical approach was also applied to modelling a sorptive cap. It was shown that, while the analytical approach can be used to predict steady-state contaminant transport, the numerical approach is needed to evaluate multiple sediment layers with different transport and sorption characteristics and to examine the transient performance between the time that the single layer transient is applicable (i.e., before penetration of the cap containment layer) and until steady-state in the upper layer. For the 30 cm thick sand cap that was considered in this study, the predicted time to reach steady-state conditions for a diffusion-controlled scenario is 1 ka. For an advection-dominated transport, the time to reach steady-state conditions is reduced to 100 a. The activated C-amended sand cap was more effective in isolating the contaminant within the sorbent layer for a sustained period of time (∼100 a). Results from both modelling approaches showed that capping can effectively reduce contaminant flux to the overlying water with critical variables being cap thickness, groundwater velocity, and sediment sorptivity.     

A TIME-STEPPING FINITE ELEMENT METHOD FOR ANALYSIS OF CONTAMINANT TRANSPORT IN FRACTURED POROUS MEDIA

This paper describes the development of a finite element method for analysing contaminant transport in double-porosity geomaterials using a time-stepping approach. In many cases, double-porosity models may be used to represent fractured rock formations and fissured soils. A distinctive feature of utilizing this kind of model is that it is not necessary to have an intimate knowledge of the nature, distribution and properties of individual fractures and fracture arrangement since the fracture geometry and details are considered only in an averaged or equivalent continuum sense. The flux exchange that occurs between the fluid in the fractures and in the solid matrix is represented by a linear heriditary process. This has the consequence that in order to carry the solution forward from time t to t+Δt, it is necessary to know and to store the complete contaminant history up to time t. This paper shows that all the hereditary information necessary to carry the solution forward is contained in the values of certain hereditary variables at time t so that it is not necessary to store the complete time history and consequently a more efficient numerical process can be developed.     

Theoretical and numerical study of the steady‐state flow through finite fractured porous media

This paper investigates the two‐dimensional flow problem through an anisotropic porous medium containing several intersecting curved fractures. First, the governing equations of steady‐state fluid flow in a fractured porous body are summarized. The flow follows Darcy's law in matrix and Poiseuille's law in fractures. An infinite transversal permeability is considered for the fractures. A multi‐region boundary element method is used to derive a general pressure solution as a function of discharge through the fractures and the pressure and the normal flux on the domain boundary. The obtained solution fully accounts for the interaction and the intersection between fractures. A numerical procedure based on collocation method is presented to compute the unknowns on the boundaries and on the fractures. The numerical solution is validated by comparing with finite element solution or the results obtained for an infinite matrix. Pressure fields in the matrix are illustrated for domains containing several interconnected fractures, and mass balance at the intersection points is also checked. Copyright © 2013 John Wiley & Sons, Ltd.     

Evaluation of a capture zone overlay method for designing groundwater remediation systems

 Simplified approaches are often used to model the removal of groundwater contamination. These approaches can yield poor remediation schemes because they incorrectly portray the effects of multiple pumping wells. In this study, a pumping configuration designed by graphically overlaying capture zones having an identical, quasi-elliptical shape was evaluated with a numerical mass transport model. After a 3-year period (within which the hypothetical aquifer was to be remediated) the contaminant mass had been reduced by 77%. Due to stagnation zones which developed between extraction wells, approximately 15 years of pumping was required to remediate the aquifer with the overlay configuration. An alternative design, consisting of an extraction well between two injection wells along the long axis of the plume, removed the contaminant within the 3-year design period. Received: 23 October 1995 · Accepted: 18 June 1996     

A new model for flow in shale-gas reservoirs including natural and hydraulic fractures

In this work, we construct a new coupled Multiscale/Discrete Fracture Model for compressible flow in a multiporosity shale gas reservoir containing networks of natural and hydraulic fractures. The geological formation is characterized by four distinct length scales and levels of porosity. The window of observation of the finest (nanoscale) portraits the nanopores within organic matter containing adsorbed gas. At the microscale, the medium is formed by two solid phases: organic, composed by kerogen aggregates, and inorganic (clay, quartz, calcite). Such phases are separated by the network of partially-saturated interparticle pores where microscopic free gas flow influenced by Knudsen effects along with gas diffusion in the immobile water phase occur simultaneously. The upscaling of the local flow to the mesoscale gives rise to a nonlinear homogenized pressure equation in the shale matrix which lies adjacent to the system of natural fractures. Homogenization of the coupled matrix/preexisting fractures to the macroscale leads to a microstructural model of dual porosity type. Such homogenized model is subsequently coupled with the hydrodynamics in the network of induced fractures which, in the context of the discrete fracture modeling, are treated as (n ? 1), (n = 2, 3) lower dimensional objects. In order to handle numerically the nonlinear interaction between the different flow equations, we adopt a superposition argument, firstly proposed by Arbogast (1996), in each iteration of a fixed-point algorithm. The resultant governing equations are discretized by the finite element method and numerical simulations of gas production in stratified arrangements of the fracture networks are presented to illustrate the potential of the multiscale approach.     

一种用于突发性场地污染模拟的解析模型

为了及时有效地应对各种突发性环境污染事故,有必要开发一种简单实用、适于各类型污染物的场地污染数学模型。通过污染事故发生后污染物在包气带、饱和带迁移转化的概化,建立了污染物运移的自由入渗模型以及降雨入渗模型并给出各自相应的解析解。无降雨时,考虑污染物在重力作用下随包气带向下渗透的作用,建立一维垂直入渗模型。有降雨时,考虑污染场地（包气带）中污染物迁移和转化的对流作用、扩散作用及挥发、生物降解、吸附、根系吸收等作用,建立包气带剖面二维溶质运移模型和饱和带平面二维溶质运移数学模型。建模过程中,假定降雨量的平均分布及土壤质地、水力参数以及有机物成分、种类均相同,同时假定污染物与多孔介质间的作用为线性吸附,植物根系对污染物的吸附遵循一级动力学。基于模型的解析解,实现案例的模拟计算。模拟结果表明：该模型具有适用范围广、模拟高效快捷等优点,能够较准确预测污染发生后污染物在土壤中的动向、到达饱和带的时间以及饱和带中污染物的迁移情况。     

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).     

基于DDA方法一种流-固耦合模型的建立及裂隙体渗流场分析和应用

以非连续变形分析方法（DDA）为基础并采用稳态流体计算方法将二者结合进行裂隙岩体流-固耦合分析。利用DDA方法生成裂隙岩体模型,在此基础上采用矩阵搜索等方法形成新的裂隙水通网络模型。采用稳态迭代算法和立方定律求得裂隙水压力,并把裂隙水压力作为线载荷施加到块体边界,在DDA算法中每个迭代步完成后更新裂隙开度和水压值,与DDA算法结合研究裂隙水与块体之间相互作用关系。利用以上裂隙岩体流-固耦合计算方法研究了某水封油库开挖和运行过程洞室围岩流量和密封性,为该工程预测水封效果提供了有益的主要依据,也是国内首次采用DDA方法做大型工程的流-固耦合模型分析。     

MT3D-通用的三维地下水污染物运移数值模型

定量研究污染物在地下水中的运移过程通常采用数值模拟方法。MT3D是一套基于有限差分方法的污染物运移模拟软件，近年来在国外水文地质和水环境模拟等领域的研究中已经得到较为广泛的认可。MT3D比较全面地考虑了污染物在地下水中的对流、弥散和化学反应等过程，可以灵活处理各种复杂的源汇项和边界条件，能够准确模拟承压、无压和越流含水层中的污染物运移过程。MT3D具有模块化的程序结构、灵活的求解方法以及全面的模拟功能，非常适合实际问题的研究，值得在国内推广使用。     

Measurement of pressure drop in drainage boreholes and its effects on the performance of coal seam gas extraction: a case study in the Jiulishan Mine with strong coal and gas outburst dangers

Although it is a well-accepted belief in the petroleum industry that horizontal well productivity can be limited by the pressure drop within the wellbore, little has been reported regarding how this pressure drop affects gas extraction from a coal seam and its further effects on mitigating coal and gas outburst dangers in coal. One of the major reasons for this scarcity is that the pressure-drop distribution in horizontal drainage boreholes is difficult to obtain. In this study, measurements of pressure drops in 54 drainage boreholes were performed in the No. 2₁ coal seam, which is the primary mining layer of Jiulishan Mine and poses a strong danger of coal and gas outbursts. Next, a coupled governing finite-element model, which includes the pressure drop in the borehole, Darcy flow in fractures, gas diffusion in the matrix blocks, and the dynamic evolution of the permeability of coal, was developed and implemented using a finite-element method to quantify the pressure-drop effects. Field tests of the pressure drops indicate that the pressure increases in a parabolic form with the increasing depth of the borehole, and lower outer end pressure is associated with larger pressure increments. The numerical results indicate that the pressure drop does affect the coal seam gas extraction, the pressure around the borehole increases with increasing borehole depth, and the increment of the pressure becomes larger when the borehole’s drainage effect is enhanced. However, the impact is small and can be ignored in engineering.     

Finite element simulation of fluid flow in fractured rock media

Fluid dynamics models are used by the petroleum industry to model single- and/or multi-phase flow within fractured rock formations, in order to facilitate extraction of fluids such as oil and natural gas, and in other areas of engineering to study groundwater flow, as well as to estimate contaminant seepage and transport. In this paper, the numerical modelling software Comsol is used to simulate air and water flow through a specimen of granite with a single vertical fracture subjected to triaxial loading conditions. The intent of the model is to simulate triaxial test findings on a rock specimen with a natural fracture. Fluid flow is simulated at various confining and inlet pressures using the cubic law. Model results were in good agreement with laboratory findings. Pressure distribution along the fracture and across the specimen are as expected with a near linear pressure distribution along the length of the fracture. A drawdown effect on pressure distribution across the specimen in the vicinity of the fracture is also observed. Pressure gradient was largely uniform; however, some localised zones of high gradient along the fracture are observed.