Spatial Moment Analysis for Transport of Nonreactive Solutes in Fracture-Matrix System
01 May 2005-Journal of Hydrologic Engineering (American Society of Civil Engineers)-Vol. 10, Iss: 3, pp 192-199
TL;DR: In this article, the authors present an analysis using spatial moments for transport of nonreactive solutes in a single fracture-matrix system using a dual porosity framework, where the effect of fracture velocity, fracture dispersivity, fracture spacing, matrix diffusion coefficient, and matrix porosity on both regimes are analyzed.
Abstract: This paper presents an analysis using spatial moments for transport of nonreactive solutes in a single fracture-matrix system using a dual porosity framework. The velocity and dispersion obtained using the first and second spatial moments are found to have two regimes. The effect of fracture velocity, fracture dispersivity, fracture spacing, matrix diffusion coefficient, and matrix porosity on both regimes are analyzed. The first regime is characterized by a behavior wherein both velocity and dispersion are functions of time and all of the above parameters of the fracture-matrix system are found to have an influence. In the second regime, they are independent of time similar to the behavior of conservative solutes in an ideal porous media. This regime is characterized by the influence of a few parameters of the fracture-matrix system. The empirical relationships for solute velocity, macrodispersion coefficient, and dispersivity in the asymptotic stage are presented.
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TL;DR: In this article, a numerical model is developed to simulate the combined effect of thermal and reactive solute transport in a coupled fracture-matrix system using dual porosity concepts, which includes solute dispersion in the fracture, lateral diffusion-limited transport of solutes from the fracture into the reservoir matrix, lateral conduction-limited thermal flux from the reservoir into the fracture and thermal conduction and dispersion, as well as thermal convexity and porosity.
54 citations
TL;DR: In this paper, an improved mathematical model is suggested that better describes fluid flow through a coupled fracture-matrix system using a dual-porosity approach, which differs from a conventional model as the fracture flow equation contains a hyperbolic term in addition to the conventional dispersive term.
Abstract: The present paper addresses critical issues that describe the transient transfer of stored rock-matrix flow into high-permeable fractures and rate-limited diffusive solute flux into low-permeable rock matrix using a typical dual-porosity approach. An improved mathematical model is suggested that better describes fluid flow through a coupled fracture-matrix system using a dual-porosity approach. The suggested model differs from a conventional model as the fracture flow equation contains a hyperbolic term in addition to the conventional dispersive term. The matrix flow equation contains the coupling term that controls the transient nature of fluid exchange from the stored rock matrix into the hydraulic conductors. The Langmuir sorption isotherm is suggested to describe the limited sorption sites available on fracture walls, while the Freundlich sorption isotherm is recommended to describe the unlimited sorption sites available within the rock matrix. The dispersion mechanism in a coupled fracture-ma...
34 citations
TL;DR: In this paper, the behavior of the solute velocity and effective macrodispersivity of solute front in a single fracture in the presence of rock matrix diffusion is analyzed using numerical modeling.
Abstract: The behavior of the solute velocity and effective macrodispersivity of solute front in the fracture for the transport in a single fracture in the presence of rock matrix diffusion is analyzed using numerical modeling. The study is limited to a constant continuous solute source boundary condition in a single fracture with constant aperture. Analysis is made for linear and nonlinear sorption cases. Expressions are provided for solute velocity and effective macrodispersivity of the solute fronts during asymptotic and preasymptotic stages using spatial moment analyses. The effective retardation factor and dispersivity of solutes in the fracture are found to follow an exponential function with an exponent related to fracture-matrix parameters.
32 citations
TL;DR: In this article, the authors analyzed the time-dependent dispersivity of the solute front in a single fracture with matrix diffusion and the expression for the time required to attain the asymptotic behavior.
Abstract: Field studies show that the variance of travel distance often increases nonlinearly with time elapsed after release of solute tracers. The nonlinear relationship between variance of travel distance and time is attributed to the heterogeneity of the porous media. To describe the transport in such a heterogeneous system, a time-dependent dispersivity is necessary. Though more attention has been devoted toward the study of non-Fickian dispersion at early time, there are no known studies that explicitly describe the dispersivity behavior in a fracture–matrix-coupled system. The observation from numerical results suggests that dispersivity has a time-dependent behavior and it reaches asymptotic values after a long time. The preasymptotic behavior of a solute front in fracture is characterized by increasing effective dispersivity with time. The role of fracture and matrix transport parameters on this behavior is analyzed for linearly sorbing solutes. Approximate expression is provided for the time-dependent dispersivity of the solute front in a single fracture with matrix diffusion and the expression for the time required to attain the asymptotic behavior is also obtained. A comparison of the front dispersivity behavior between parallel multiple fractures with a constant aperture width model and smooth parallel multiple fractures with a varying aperture width model is done.
30 citations
TL;DR: In this article, the authors studied the behavior of thermal fronts along the fracture in the presence of fracture-skin in a coupled fracture-matrix system, where cold water is injected into the fracture, which advances gradually toward production well, while extracting heat from the surrounding reservoir matrix.
Abstract: 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.
20 citations
Cites methods from "Spatial Moment Analysis for Transpo..."
...Method of spatial moments has already been used for assessing the dispersive transport model formulations (Freyberg 1986), and in general, this method has been applied for analyzing the concentration profiles obtained along the fracture by many researchers (Taylor 1953; Aris 1956; Horn and Kipp 1971; Guven et al. 1984; Srivastava et al. 2002; Suresh Kumar and Sekhar 2005; Suresh Kumar and Ghassemi 2005)....
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TL;DR: Enginsera et al. as discussed by the authors proposed an idealized model for the purpose of studying the characteristic behavior of a permeable medium which contains regions which contribute significantly to the pore volume of the system but contribute negligibly to the flow capacity.
Abstract: An idealized model has been developed for the purpose of studying the characteristic behavioroja permeable medium which contains regions which contribute sigizificantly to tbe pore volume O! the system but contribute negligibly to the flow capacity; e.g., a naturally fractured or vugular reservoir, Vnsteady-state flow in this model reservoir has been investigated analytically. The pressure buiid-up performance has been examined insomedetait; and, a technique foranalyzing tbebuild.up data to evaluate the desired parameters has been suggested. The use of this ap$roacb in the interpretation of field data has been discussed. As a result of this study, the following general conclusions can be drawn: 1. Two parameters are sufficient to characterize the deviation of the behavior of a medium with “double porosity ”from that of a homogeneously porous medium. 2. These Parameters can be evaluated by the proper analy~is of pressure buildup data ob~ained from adequately designed tests. 3. Since the build-up curve associated with this type of porous system is similar to that obtained from a stratified reservoir, an unambiguous interpretation is not possible without additional information. 4, Dif@rencing methods which utilize pressure data from the /inal stages of a buik-kp test should be used with extreme caution. INTRODUCTION In order to plan a sound exploitation program or a successful secondary-recovery pro ject, sufficient reliable information concerning the nature of the reservoir-fluid system must be available. Sincef it is evident chat an adequate description of the reservoir rock is necessary if this condition is to be fulfilled, the present investigation was undertaken for the purpose of improving the fluid-flow characterization, based on normally available data, ofs particular porous medium. DISCUSSION OF THE PROBLEM For many years it was widely assumed that, for the purpose of making engineering studies, two psram. . -. . Origlml manuscriptreceived fn eociaty of Petroleum Ertatneere offiae AUS. 17, 1962.Revieed manuscriptreceived.March21, 1963. P eper pr+$eented at the Fetl Meeting of the %ciot Y of. Petreleum Enginsera In Lo= Ar@Ies on Oct. 7-10, 1962. ‘ . GULF RESEARCH d DEVELOPMENT CO. PITTSBURGH, PA, eters were sufficient to desckibe the single-phase flow properties of a prodttcing formation, i.e., the absolute permeability and the effective porosity. It : later became evident that the concept of directional permeability was of more thin academic interest; consequently, the de$ee of permeability anisotropy and the orientation of the principal axes of permeability were accepted as basic parameters governing reservoir performance. 1,2 More recently, 3“6 it was recognized that at least one additional parameter was required to depict the behavior of a porous system containing region,s which contributed significantly to the pore volume but contributed negligibly to the flow capacity. Microscopically, these regions could be “dead-end” or “storage” pores or, microscopically, they could be discrete volumes of lowpermeability inatrix rock combined with natural fissures in a reservoir. It is obvious thst some provision for the ;.ncIusion of all the indicated parameters, as weIl as their spatial vstiations$ must be made if a truly useful, conceptual model of a reaetvoir is to be developed. A dichotomy Qf the internaI voids of reservoir rocks has been suggested, r~s These two classes of porosity can be described as follows: a. Primary porosity is intergranular and controlled by deposition and Iithification. It ie highly intercoririected arid “usually can be correlated with permeability since it is largely dependent on the geometry, size distribution and spatial distribution of the grains. The void systems of sands, sandstones and oolitic limestones are typical of this type. b. Secondary porosity is foramenular and is controlled by fracturing, jointing and/or solution in circulating water although it may be modified by infilling as a result of precipitation. It is not highly interconnected and usually cannot be correlated with permeability. Solution channels or vugular voids developed during weathering or buriaI in sedimentary basins are indigenous to carbonate rocks such as limestones or dolomites. Joints or fissures which occur in massive, extensive formations composed of shale, siltstone, schist, limestone or dolomite are generally vertical, and they are ascribed to tensional failure, during mechanical deformation (the permeability associated with this type of void system is often anisotropic). Shrinkage cracks are the result 1 ~ef&ence. aiven atendof p@er. ‘-
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TL;DR: In this paper, a multirate model is proposed to model small-scale variation in rates and types of mass transfer by using a series of first-order equations to represent each of the mass transfer processes.
Abstract: Mass transfer between immobile and mobile zones is a consequence of simultaneous processes. We develop a “multirate” model that allows modeling of small-scale variation in rates and types of mass transfer by using a series of first-order equations to represent each of the mass transfer processes. The multirate model is incorporated into the advective-dispersive equation. First, we compare the multirate model to the standard first-order and diffusion models of mass transfer. The spherical, cylindrical, and layered diffusion models are all shown to be specific cases of the multirate model. Mixtures of diffusion from different geometries and first-order rate-limited mass transfer can be combined and represented exactly with the multirate model. Second, we develop solutions to the multirate equations under conditions of no flow, fast flow, and radial flow to a pumping well. Third, using the multirate model, it is possible to accurately predict rates of mass transfer in a bulk sample of the Borden sand containing a mixture of different grain sizes and diffusion rates. Fourth, we investigate the effects on aquifer remediation of having a heterogeneous mixture of types and rates of mass transfer. Under some circumstances, even in a relatively homogeneous aquifer such as at Borden, the mass transfer process is best modeled by a mixture of diffusion rates.
834 citations
TL;DR: In this paper, a general analytical solution is developed for the problem of contaminant transport along a discrete fracture in a porous rock matrix, which takes into account advective transport in the fracture, longitudinal mechanical dispersion in a fracture, molecular diffusion along the fracture axis, adsorption into the face of the matrix, adhesion within the matrix and radioactive decay.
Abstract: A general analytical solution is developed for the problem of contaminant transport along a discrete fracture in a porous rock matrix. The solution takes into account advective transport in the fracture, longitudinal mechanical dispersion in the fracture, molecular diffusion in the fracture fluid along the fracture axis, molecular diffusion from the fracture into the matrix, adsorption into the face of the matrix, adsorption within the matrix, and radioactive decay. Certain assumptions are made which allow the problem to be formulated as two coupled, one-dimensional partial differential equations: one for the fracture and one for the porous matrix in a direction perpendicular to the fracture. The solution takes the form of an integral which is evaluated by Gaussian quadrature for each point in space and time. The general solution is compared to a simpler solution which assumes negligible longitudinal dispersion in the fracture. The comparison shows that in the lower ranges of groundwater velocities this assumption may lead to considerable error. Another comparison between the general solution and a numerical solution show excellent agreement under conditions of large diffusive loss. Since these are also the conditions under which the formulation of the general solution in two orthogonal directions is most subjectmore » to question, the results are strongly supportive of the validity of the formulation.« less
772 citations
TL;DR: In this paper, an exact analytical solution for the problem of transient contaminant transport in discrete parallel fractures situated in a porous rock matrix is developed for the same problem, taking into account advective transport in the fractures, molecular diffusion and mechanical dispersion along the fracture axes.
Abstract: An exact analytical solution is developed for the problem of transient contaminant transport in discrete parallel fractures situated in a porous rock matrix. The solution takes into account advective transport in the fractures, molecular diffusion and mechanical dispersion along the fracture axes, molecular diffusion from the fracture to the porous matrix, adsorption onto the face of the matrix, adsorption within the matrix, and radioactive decay. The general transient solution is in the form of a double integral that is evaluated using Gauss-Legendre quadrature. A transient solution is also presented for the simpler problem that assumes negligible longitudinal dispersion along the fracture. This assumption is usually reasonable when the advective flux in a fracture is large. A comparison between two steady state solutions, one with dispersion and one without, permits a criterion to be developed that is useful for assessing the significance of longitudinal dispersion in terms of the overall system response. Examples of the solutions demonstrate that penetration distances along fractures can be substantially larger through multiple, closely spaced fractures than through a single fracture because of the limited capability of the finite matrix to store solute.
626 citations