How to predict flow through a network of discrete fractures in a three-dimensional domain is investigated. Fractures are modeled as circular discs of arbitrary size, orientation, transmissivity, and location. A fracture network is characterized by the statistical distributions of these quantities. Fracture traces observed on a wall form the basis for estimates of mean fracture radius, fracture orientation parameters, and fracture density. Fracture trace lengths are estimated with the scanline method and from areal sampling on circular regions. The traces observed on the wall can also be used to condition the network. This trace conditioning is achieved by forcing the network generator to always reproduce the observed traces. Conditioning might be a means of decreasing the variability of the fracture networks. A numerical simulation model has been developed which is capable of generating a fracture network of desired statistical properties and solving for the steady state flow. On each fracture disc the flow is discretized with the boundary element method. A series of hypothetical examples are analyzed. These examples consist of sets of Monte-Carlo simulations of flow through a series of networks generated from the same statistical distributions. The examples lead to the following conclusions. Large fractures and high fracture density implies good connectivity in the networks. A high fracture density implies a small variance in the flow through the network. Trace conditioning decreases estimation variance only when the fracture network consists of large fractures. Fracture statistics can be estimated reasonably well from fracture traces observed on a wall. INTRODUCTION Field investigations of flow in fractured rock have clearly demonstrated that modeling the rock as a homogeneous continuum might be a serious simplification. For example, the three-dimensional flow and tracer test by Neretnieks et al.[1985] in the Stripa test site in Sweden shows that the flow in the rock can be very unevenly distributed. In crystalline rock, flow is confined to the fractures, whereas the rock matrix is almost impervious. Furthermore, there is growing evidence that the hydraulic properties in a single fracture vary strongly in the fracture plane "channeling" (see, for example, Abelin [1986]). The uneven flow encountered in crystalline rock is probably caused both by the discrete fracture network and the channeling within a fracture. This paper only considers the discrete network; the fracture properties are assumed constant in each fracture. Channeling, which is to be investigated in future studies, may be viewed as a submodel to a discrete network model. Interestin discrete fracture network flow and transport models has grown in the past few years, although until recently these models have only been investigated in two dimensions [Long et. al., 1982; Schwartz et al., 1983; Andersson et al., 1984; Robinson, 1984; Smith and Schwartz, 1984; Long and Witherspoon, 1985; Andersson and Thunvik, 1986]. From these studies one concludes that...
A three‐dimensional variable aperture fracture network model for flow and transport in fractured rocks was developed. The model generates both the network of fractures and the variable aperture distribution of individual fractures in the network. Before solving for the flow and transport of the whole network, a library of single‐fracture permeabilities and particle transport residence time spectra is first established. The spatially varying aperture field within an individual fracture plane is constructed by geostatistical methods. Then the flow pattern, the fracture transmissivity, and the residence times for transport of particles through each fracture are calculated. The library of transmissivities and frequency distributions of residence times is used for all fractures in the network by a random selection procedure. The solution of flow through the fracture network and the particle‐tracking calculation of solute transport for the whole network are derived from one side of the network to the other. The model thus developed can handle flow and transport from the single‐fracture scale to the multiple‐fracture scale. The single‐fracture part of the model is consistent with earlier laboratory tests and field observations. The multiple‐fracture aspect of the model was verified in the constant aperture fracture limit with an earlier code. The simulated breakthrough curves obtained from the model display dispersion on two different scales as has been reported from field experiments.
We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an s-sparse signal from linear measurements (with high probability) is known to be m s(ln s) 3 ln N . We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of m × N -matrices, by considering what we call low entropy. We also present an improved condition on the so-called restricted isometry constants, δs, ensuring sparse recovery via 1 -minimization. We show that δ2s < 4/ √ 41 is sufficient and that this can be improved further to almost allow for a sufficient condition of the type δ2s < 2/3.
The uneven flow distribution observed in a migration experiment in the Stripa research mine is analyzed with a discrete fracture network model. Data on the measured inflow distribution and trace geometry in the experimental drift are the basic sources for estimation of the model parameters. Direct estimates cannot provide a unique definition of input data to the model; thus different combinations of input data need to be analyzed in the network model. Detailed measurements of the inflow distribution in the experimental drift provided an opportunity to determine the model parameters through calibration in the network model. Calibration was performed on the mean inflow, distribution of flow, and trace geometry observed in the roof of the experimental drift. The simulation results show that the model can be calibrated to produce an areal flow distribution that is consistent with the measured one. The uncertainties in the input parameters are thus reduced, but different combinations of input data are still possible. The calibration simulations show that the length of conductive fractures per area might be used as a calibration parameter. Simulations based on different combinations of fracture size and density but with the same length of conductive fracture traces produced similar flow distributions. The validity of the calibrated model is explored by predicting the flow into the boreholes at the experimental site. The resulting inflow distributions accord well with those measured in two of the three boreholes. The properties of the third borehole, which differ substantially from the other two, could not be explained by the simulation results.
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