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LINA NEUNHÄUSERER(1),
ANNETTE HEMMINGER(1), RAINER HELMIG(1)
(1) University of Technology Braunschweig, Institute for Computer Applications in Civil Engineering, Germany
Pockelsstraße
3, 38106 Braunschweig
Phone: +49
(0)531 391 7595
Fax: +49 (0)531 391 7599
Fractured porous formations show vastly different properties concerning permeability and storage capacity, thus giving rise to mass exchange processes between fractures and surrounding matrix. This interaction between fracture and matrix has a certain impact on the flow and transport processes in the fractured subsurface which can be observed on each scale considered for investigation purposes. The influence of fracture - matrix - interaction has to be examined carefully when dealing with safety investigations or remediation possibilities.
This paper shows some of the effects of fracture - matrix - interaction on groundwater flow and solute transport in a saturated fractured aquifer and on the parameters describing those processes with respect to different scales. A discrete model concept has been employed containing fracture network generation, mesh generation and appropriate discretization techniques. The influence of a matrix of finite porosity on the effective hydraulic conductivity tensor of a fractured system is illustrated by an example.
Keywords: Fracture - matrix - interaction, upscaling concept, effective parameters
The question of groundwater flow and solute transport behaviour in fractured porous media is of major importance when considering the long term safety of waste disposal sites, the employment of remediation techniques or the safety of fractured aquifers used as drinking water reservoirs. Flow and transport processes in fractured porous rock are strongly influenced by the heterogeneous properties of such geological systems. Under saturated conditions fractures are usually characterized by a comparatively high permeability and a low storativity and therefore might represent preferential pathways to a rapid migration of solutes. In contrast the surrounding rock matrix shows a rather low permeability and a high storage capacity. Contaminants entering the matrix by diffusive processes may cause a long term contamination of the system by advancing slowly in the low-mobility matrix or by re-entering the fractures. This interaction between fracture and matrix plays a significant role in solute transport processes.
Field tests like tracer tests or pumping tests and laboratory investigations provide some of the data necessary for the examination of flow and transport conditions. However, while the data received from field tests are usually not detailed enough to assess the behaviour of a fractured aquifer the data achieved from laboratory experiments are not likely to be representative for larger scales regarding the heterogeneity of the system. Within the research project "Aquifer Analogy", laboratory as well as field experiments with a fractured sandstone and numerical simulations with different model concepts are performed in order to formulate upscaling possibilities which allow to include the information gained from small scale investigations into regional scale models [Neunhäuserer7].
The work presented in this paper describes the influence of fracture-matrix-interaction on flow and transport conditions in the fractured subsurface and the resulting effective parameters. The concept of scales and effective parameters will be explained and the modelling set-up used for numerical investigations with a discrete model concept will be outlined. A simple example designed to illustrate the effects of fracture-matrix-interaction on groundwater flow and system conductivity will be presented. Further examples can be found in [Neunhäuserer7].
Prior to the investigation and the understanding of the fracture - matrix - interactions influencing the flow and transport processes and the resulting effective parameters of different scales the terms "scales" and "effective parameters" have to be explained. The complexity of a fractured rock system, i.e. the disparity of the hydraulic properties of fractures and matrix combined with the irregular geometry of a fracture
Fig.1. Upscaling by applying effective parameters using different model concepts
network, requires investigations on different scales in space and time. The definition of scales helps to characterize the system behaviour and to assess the flow and transport conditions to be expected. The influence of geological properties and physical processes differs from scale to scale. Flow and transport phenomena and geological heterogeneities that can be observed on smaller scales influence the system behaviour on larger scales. But due to computer limitations and with respect to computing time it is not advisable to include all the small scale information on larger scales.
To reduce the complexity of the system on each scale the governing processes have to be identified, analysed and formulated in terms of effective parameters. Effective parameters allow to consider the effects of smaller scale processes on larger scales while simplifying the system. Usually effective parameters are obtained by an averaging process over the scale volume which is typically a representative elementary volume (REV, [Bear1]) with respect to the parameter of interest. In order to determine the governing processes and the resulting effective parameters of each scale and to include the effective parameters of smaller scales the appropriate numerical model concept has to be applied to each scale (Fig.1).
In the context of the work presented in this paper the following scales will be considered: single fractures without and with matrix and fracture networks without and with matrix.
The influence of fracture-matrix-interaction on the effective parameters used for upscaling purposes will be set out in the sequel with two examples, namely the effective hydraulic conductivity tensor and the effective hydrodynamic dispersion. The effective hydraulic conductivity tensor describes the permeability of a porous medium, respectively of a fractured formation which has been subjected to an averaging process over an REV with respect to fluid properties and gravitation. To determine the hydraulic conductivity tensor of a fracture network the system is rotated with respect to the pressure gradient and for each rotation angle the kf-value is computed by means of the corresponding in- and outflow [Long5]. The better the computed kf-values fit into an ideal balancing ellipse the better the considered fracture network is represented by a porous medium with the corresponding conductivity tensor.
By the same procedure an effective hydraulic conductivity tensor can be determined for fracture-matrix-networks. However, considering a porous matrix in a fracture network affects the resulting hydraulic conductivity values in several ways. First, the porous matrix allows groundwater flow not only in the fractures but throughout the domain of interest. Consequently while rotating the domain with respect to the pressure gradient the resulting kf-values fit more to a balancing ellipse than the kf-values of the corresponding pure fracture network would do. Second, the connectivity of the fracture network plays a less important role when a matrix is involved. In contrast to pure fracture networks in which a connective fracture pathway from one edge of the domain to the other is a prerequisite for finding an effective conductivity value, fracture-matrix-systems can balance the missing fracture connection by establishing flow pathways in the matrix. Third, any anisotropy of the matrix may influence the effective conductivity tensor of the fracture-matrix-system. The described effects depend primarily on the permeability difference between fracture and matrix and on the fracture density of the considered networks.
The quantity of effective dispersion has been introduced to describe the spreading of a solute due to velocity fluctuations which arise from the heterogeneities of the underlying medium. Regarding the scale of the single fracture, dispersive processes occur due to channeling effects in the rough fracture plane itself. If the single fracture is embedded in a matrix of finite porosity, the mass exchange between fracture and matrix has to be taken into account. Depending on the porosity and the permeability of the matrix as well as on the pressure gradient between fracture and matrix a considerable quantity of solute will enter the matrix leading to a strong tailing effect in the resulting breakthrough curve as well as a fast reduction of the concentration peak. Fig. 2 illustrates this effect vividly. It shows the analytic solution of [Gillham3] describing advective transport in a one-dimensional fracture and diffusive transport in the adjacent two-dimensional matrix.
Fig. 2. Breakthrough curves in a single fracture with matrix at three different time steps, analytical solution after [Gillham3]
A similar behaviour can be observed on the fracture network scale. Contaminants migrating through a fracture network will use different possible pathways of connected fractures in the network leading to a macroscopic dispersion effect. If the mass exchange between the fracture network and a surrounding porous matrix is considered additionally, the resulting breakthrough curve will show a pronounced tailing while the corresponding concentration peak diminishes rapidly. The contrast between fracture and matrix properties may lead to completely different time scales for the rate of solute migration in both components of the system. A description of the strongly asymmetric behaviour of the breakthrough curves resulting from transport processes in fracture-matrix-systems by the Scheidegger tensor can only provide a first approximation. Alternatives are the employment of a multi-layer model [Wollrath8] or the analysis of the contaminant plume behaviour by evaluating spatial moments [Dagan2].
To analyse the influence of fracture-matrix-interaction on the flow and transport processes on different scales a discrete model concept has been applied. This implies that the single fracture and the adjacent matrix respectively the fracture network and the matrix in between are described discretely in space. The modelling set-up includes several steps which will be briefly outlined in the following. The first step in the discrete modelling of fracture-matrix-systems is the generation of a heterogeneous permeability field when considering a single fracture with variable aperture respectively the generation of a fracture network. The necessary geometry and material property data base is gained from laboratory or field investigations and evaluated using stochastic methods and geostatistical optimization tools. If the fracture density in the domain of interest is low, a number of different realisations of fracture networks with the same statistic properties is usually generated to fulfil the requirements of the REV - concept [Hemminger4]. The next step is the geometrical description of fracture and matrix with different types of elements. For this purpose a highly flexible mesh generator based on an optimized Delaunay - Triangulation is used [Neunhäuserer6]. The governing equations for single phase flow and transport in a saturated aquifer are the continuity equation combined with Darcy's law:
and the advection-dispersion transport-equation:
where S0 represents the specific storage coefficient, h the
piezometric head, t the time, K the hydraulic conductivity tensor, c the
solute concentration, va the seepage velocity, D the
Scheidegger tensor and q respectively r externally applied source and sink
terms. To discretize the flow equation in space a Standard-Galerkin method is
applied. For the transport equation a Flux-Corrected-Transport algorithm is
implemented. This algorithm combines a monotonic low order method (e.g. Control
Volume Finite Element method) with a high order method (e.g. Taylor-Galerkin
method) thus taking into account the different physical properties of the
advective and the dispersive term of the transport equation with respect to the
heterogeneous flow regime in the domain. For time discretization a finite
difference method is employed [Neunhäuserer6].
The following simple example illustrates some of the described effects of fracture-matrix-interaction on groundwater flow in a fractured porous system. Fig. 1a) and 2a) show two realisations of a fracture network with a comparatively low fracture density, each combined with a porous matrix to which a homogeneous anisotropic hydraulic conductivity tensor has been assigned. The hydraulic conductivity tensor of both fracture-matrix-systems is determined by the procedure outlined in the Chapter "Fracture-Matrix-Interaction". In Fig. 1b) and 2b) the resulting kf-values and the corresponding balancing ellipses are depicted for each system respectively. For this case the permeability between fracture and matrix differs by five orders of magnitude. The comparison of computed kf-values and balancing ellipse shows that realisation 1a) can be represented sufficiently well by a homogeneous medium with an effective anisotropic hydraulic conductivity tensor although no connected fracture pathways exist. As expected, the largest kf-values can be found in the direction of the largest number of fractures (mind that the depicted axes show 1/sqrt(kf)). Realisation 2a) includes two fractures connecting each two edges, thus leading to much larger kf-values. However, two exceptions appear with 30° and 45°. A close observation of the rotating fracture system reveals that one of the two fractures connective with 0° does not reach the opposite edge with 30° and 45° anymore. This leads to a distinct decrease in the kf- values of these directions. A representation of this fracture-matrix-system with an effective hydraulic conductivity tensor would be impossible. It has to be emphasized again that two of several fracture-matrix-systems with the same statistic properties have been used here to show the influence of the fracture-
Fig. 3. Two realisations of a fracture system (1a, 2a), computed values 1/sqrt(kf) with corresponding balancing ellipse for five (1b, 2b) and three (1c, 2c) orders of magnitude difference between fracture and matrix permeability
Matrix-interaction on groundwater flow processes. To determine the corresponding effective parameters correctly the results of a sufficiently large number of realisations have to be evaluated statistically. Fig. 1c) and 2c) show computed kf-values and corresponding balancing ellipses for both realisations with a difference of three orders of magnitude between the permeability of fracture and matrix. This choice has been confirmed by laboratory investigations in the context of the project "Aquifer Analogy" to be valid for a fractured sandstone. In both cases the computed kf-values fit very well to the corresponding balancing ellipses. Both fracture-matrix-systems could be represented by an effective hydraulic conductivity tensor. Since the fracture density of the system is rather low, the direction of the corresponding hydraulic conductivity tensor is strongly influenced by the direction of the hydraulic conductivity tensor of the matrix.
A discrete model concept containing fracture generation, mesh generation, and numerical discretization techniques has been employed to examine the influence of fracture-matrix-interaction on flow and transport processes and the corresponding effective parameters in fractured systems with respect to different scales. The results presented in this paper have shown that, depending on the system properties, the presence of a porous matrix in a fractured system leads to a considerable mass exchange between fracture and matrix, strongly affecting the flow and transport conditions in the domain of investigation. Future work will focus principally on the parameterization of solute exchange between fracture and matrix regarding single fracture and fracture networks and on the description of macro dispersion induced by fracture systems.
This work is supported by the Deutsche Forschungsgemeinschaft under the contract He2531/1-1. We would like to thank our colleagues of the Lehrstuhl für Angewandte Geologie, Karlsruhe and of the Lehrstuhl für Angewandte Geologie, Tübingen, who put the field and laboratory data at our disposal, and our colleagues of the Lehrstuhl und Institut für Wasserbau und Wasserwirtschaft, Aachen, who deal with the multi-continuum concept.
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