Flow through anisotropic networks

This paper presents an analysis of Hagen-Poiseulle flow through plane random anisotropic networks of interconnected channels. Macroscopic permeability tensor of the network is expressed in terms of statistico-geometrical characteristics like the degree of anisotropy in channel orientations, average co-ordination number of the network and first two moments of channel length distribution. Analytical results are illustrated and verified using numerical analysis of flow in a simulated random network. The emphasis of the paper is on the effects of anisotropy on distributions of flow rates in channels. It is shown that, due to anisotropy the maximum flow rate generally occurs in channels that are not aligned along the direction of the macroscopic pressure gradient.     

Flow through anisotropic networks

This paper presents an analysis of Hagen-Poiseulle flow through plane random anisotropic networks of interconnected channels. Macroscopic permeability tensor of the network is expressed in terms of statistico-geometrical characteristics like the degree of anisotropy in channel orientations, average co-ordination number of the network and first two moments of channel length distribution. Analytical results are illustrated and verified using numerical analysis of flow in a simulated random network. The emphasis of the paper is on the effects of anisotropy on distributions of flow rates in channels. It is shown that, due to anisotropy the maximum flow rate generally occurs in channels that are not aligned along the direction of the macroscopic pressure gradient.     

Variable-density groundwater flow and solute transport in irregular 2D fracture networks

Numerical simulations of variable-density flow and solute transport have been conducted to investigate dense plume migration for various configurations of 2D fracture networks. For orthogonal fractures, simulations demonstrate that dispersive mixing in fractures with small aperture does not stabilize vertical plume migration in fractures with large aperture. Simulations in non-orthogonal 2D fracture networks indicate that convection cells form and that they overlap both the porous matrix and fractures. Thus, transport rates in convection cells depend on matrix and fracture flow properties. A series of simulations in statistically equivalent networks of fractures with irregular orientation show that the migration of a dense plume is highly sensitive to the geometry of the network. If fractures in a random network are connected equidistantly to the solute source, few equidistantly distributed fractures favor density-driven transport. On the other hand, numerous fractures have a stabilizing effect, especially if diffusive transport rates are high. A sensitivity analysis for a network with few equidistantly distributed fractures shows that low fracture aperture, low matrix permeability and high matrix porosity impede density-driven transport because these parameters reduce groundwater flow velocities in both the matrix and the fractures. Enhanced molecular diffusion slows down density-driven transport because it favors solute diffusion from the fractures into the low-permeability porous matrix where groundwater velocities are smaller. For the configurations tested, variable-density flow and solute transport are most sensitive to the permeability and porosity of the matrix, which are properties that can be determined more accurately than the geometry and hydraulic properties of the fracture network, which have a smaller impact on density-driven transport.     

The physical framework of the dependence between channel flow hydrographs and drainage network morphometry

The physical basis of the linkage between magnitude and timing of channel flow hydrographs and drainage network morphometry is reviewed. Small Hortonian and structurally Hortonian networks are analysed using numerical runoff simulation. For Hortonian networks the variability of the geometry of individual channels and subcatchments within each Strahler order has generally little effect upon the overall character of the hydrograph in channels of higher order. If the network is also structurally Hortonian, the analysis of the simultaneous formation, travel, and concentration of the hydrographs in all channels of the network can be simplified to a sequence of one representative hydrograph per channel order. This approach is used in this study. Three major runoff processes control the flow hydrograph characteristics: the overland flow process which determines the water supply to the drainage network; the channel flow process which translates the hydrograph in space and time; and the drainage network process which concentrates and magnifies the flow at the junctions of the drainage network. Functional relations for the hydrograph peak, timing, and flow velocity are presented. For a given uniform rainfall and infiltration rate, the peak of the channel flow hydrograph is shown to increase geometrically with channel order, and its magnitude is directly related to the bifurcation ratio. The travel time of the peak also increases geometrically with channel order, and it is directly related to the channel length ratio over velocity ratio. The flow velocity of the peak changes in a downstream direction as a function of the bifurcation and slope ratio. It was also found that for negligible channel storage the channel flow and drainage network processes do not contribute significantly to the observed nonlinear response of a watershed to precipitation.     

Assessment of critical path analyses of the relationship between permeability and electrical conductivity of pore networks

Critical path analysis (CPA) is a method for estimating macroscopic transport coefficients of heterogeneous materials that are highly disordered at the micro-scale. Developed originally to model conduction in semiconductors, numerous researchers have noted that CPA might also have relevance to flow and transport processes in porous media. However, the results of several numerical investigations of critical path analysis on pore network models raise questions about the applicability of CPA to porous media. Among other things, these studies found that (i) in well-connected 3D networks, CPA predictions were inaccurate and became worse when heterogeneity was increased; and (ii) CPA could not fully explain the transport properties of 2D networks. To better understand the applicability of CPA to porous media, we made numerical computations of permeability and electrical conductivity on 2D and 3D networks with differing pore-size distributions and geometries. A new CPA model for the relationship between the permeability and electrical conductivity was found to be in good agreement with numerical data, and to be a significant improvement over a classical CPA model. In sufficiently disordered 3D networks, the new CPA prediction was within ±20% of the true value, and was nearly optimal in terms of minimizing the squared prediction errors across differing network configurations. The agreement of CPA predictions with 2D network computations was similarly good, although 2D networks are in general not well-suited for evaluating CPA. Numerical transport coefficients derived for regular 3D networks of slit-shaped pores were found to be in better agreement with experimental data from rock samples than were coefficients derived for networks of cylindrical pores.     

Statistical characteristics of flow as indicators of channeling in heterogeneous porous and fractured media

We introduce two new channeling indicators D_ic and D_cc based on the Lagrangian distribution of flow rates. On the basis of the participation ratio, these indicators characterize the extremes of both the flow-tube width distribution and the flow rate variation along flow lines. The participation ratio is an indicator biased toward the larger values of a distribution and is equal to the normalized ratio of the square of the first-order moment to the second-order moment. Compared with other existing indicators, they advantageously provide additional information on the flow channel geometry, are consistently applicable to both porous and fractured media, and are generally less variable for media generated using the same parameters than other indicators. Based on their computation for a broad range of porous and fracture permeability fields, we show that they consistently characterize two different geometric properties of channels. D_ic gives a characteristic scale of low-flow zones in porous media and a characteristic distance between effectively flowing structures in fractured cases. D_cc gives a characteristic scale of the extension of high-flow zones in porous media and a characteristic channel length in fractured media. D_ic is mostly determined by channel density and permeability variability. D_cc is, however, more affected by the nature of the correlation structure like the presence of permeability channels or fractures in porous media and the length distribution in fracture networks.     

Applications of the random model of drainage basin composition

The random model of drainage basin composition is founded on the assumptions that (a) natural channels are topologically random in the absence of geological controls and (b) for channel networks developed in similar environments, the exterior and interior link lengths are independent random variables with a common distribution for each type. The effectiveness of this model in estimating the values of geomorphic variables and in explaining and predicting geomorphic relationships is illustrated by several examples. The data required for these examples were obtained from map studies of 30 channel networks, comprising a total of about 8700 links, in eastern Kentucky. A common factor in the success of all three applications of the model is the way in which the planimetric features of drainage basins are determined by their underlying topologic structure.     

Incorporation of conceptual and parametric uncertainty into radionuclide flux estimates from a fractured granite rock mass

Detailed numerical flow and radionuclide simulations are used to predict the flux of radionuclides from three underground nuclear tests located in the Climax granite stock on the Nevada Test Site. The numerical modeling approach consists of both a regional-scale and local-scale flow model. The regional-scale model incorporates conceptual model uncertainty through the inclusion of five models of hydrostratigraphy and five models describing recharge processes for a total of 25 hydrostratigraphic–recharge combinations. Uncertainty from each of the 25 models is propagated to the local-scale model through constant head boundary conditions that transfer hydraulic gradients and flow patterns from each of the model alternatives in the vicinity of the Climax stock, a fluid flux calibration target, and model weights that describe the plausibility of each conceptual model. The local-scale model utilizes an upscaled discrete fracture network methodology where fluid flow and radionuclides are restricted to an interconnected network of fracture zones mapped onto a continuum grid. Standard Monte Carlo techniques are used to generate 200 random fracture zone networks for each of the 25 conceptual models for a total of 5,000 local-scale flow and transport realizations. Parameters of the fracture zone networks are based on statistical analysis of site-specific fracture data, with the exclusion of fracture density, which was calibrated to match the amount of fluid flux simulated through the Climax stock by the regional-scale models. Radionuclide transport is simulated according to a random walk particle method that tracks particle trajectories through the fracture continuum flow fields according to advection, dispersion and diffusional mass exchange between fractures and matrix. The breakthrough of a conservative radionuclide with a long half-life is used to evaluate the influence of conceptual and parametric uncertainty on radionuclide mass flux estimates. The fluid flux calibration target was found to correlate with fracture density, and particle breakthroughs were generally found to increase with increases in fracture density. Boundary conditions extrapolated from the regional-scale model exerted a secondary influence on radionuclide breakthrough for models with equal fracture density. The incorporation of weights into radionuclide flux estimates resulted in both noise about the original (unweighted) mass flux curves and decreases in the variance and expected value of radionuclide mass flux.     

A Neumann expansion approach to flow through heterogeneous formations

A stochastic approach is used for the study of flow through highly heterogeneous aquifers. The mathematical model is represented by a random partial differential equation in which the permeability and the porosity are considered to be random functions of position, defined by the average value, constant standard deviation and autocorrelation function characterized by the integral scale. The Laplace transform of the solution of the random partial differential equation is first written as a solution of a stochastic integral equation. This integral equation is solved using a Neumann series expansion. Conditions of convergence of this series are investigated and compared with the convergence of the perturbation series. For mean square convergence, the Neumann expansion method may converge for a larger range of variability in permeability and porosity than the classic perturbation method. Formal expressions for the average and for the correlation moments of the pressure are obtained. The influence of the variability of the permeability and porosity on pressure is analyzed for radial flow. The solutions presented for the pressure at the well, as function of the permeability coefficient of variation, may be of practical interest for evaluating the efficiency of well stimulation operations, such as hydraulic fracturing or acidizing methods, aimed at increasing the permeability around the well.     

Impact of fracture network geometry on free convective flow patterns

The effect of fracture network geometry on free convection in fractured rock is studied using numerical simulations. We examine the structural properties of fracture networks that control the onset and strength of free convection and the patterns of density-dependent flow. Applicability of the equivalent porous medium approach (EPM) is also tested, and recommendations are given, for which situations the EPM approach is valid. To date, the structural properties of fracture networks that determine free convective flow are examined only in few, predominantly simplified regular fracture networks. We consider fracture networks containing continuous, discontinuous, orthogonal and/or inclined discrete fractures embedded in a low-permeability rock matrix. The results indicate that bulk permeability is not adequate to infer the occurrence and magnitude of free convection in fractured rock. Fracture networks can inhibit or promote convection depending on the fracture network geometry. Continuous fracture circuits are the crucial geometrical feature of fracture networks, because large continuous fracture circuits with a large vertical extent promote convection. The likelihood of continuous fracture circuits and thus of free convection increases with increasing fracture density and fracture length, but individual fracture locations may result in great deviances in strength of convection between statistically equivalent fracture networks such that prediction remains subject to large uncertainty.     

Laboratory measurement of hydrodynamic saline dispersion within a micro-fracture network induced in granite

We report the first measurements of hydrodynamic dispersion in a microfractured granite using a combination of novel techniques. A fracture network was induced in a cylindrical plug of Ailsa Craig micro-granite by thermal stressing, to produce an isotropic network of fractures with an average aperture of  0.3 μm, a density of approximately 4 × 10⁴ fractures/mm³ and a permeability of 5.5 × 10^− 17 m². After saturating the cores with 0.01 M NaCl solution a step in the concentration profile to 1 M was advected into the plug at flow rates of 0.07 to 2.13 cm³ h^− 1. The longitudinal electrical impedance of the plug was measured continuously as the solute front advected through its length until the plug was saturated with the concentrated electrolyte. Analysis of the impedance versus time relationships allows the derivation of the longitudinal dispersion coefficient, D_L, and hydrodynamic retardation, R_H. The Peclet number–dispersion relationship for the micro-fracture network is very similar to that predicted for other, radically different, fracture networks. Thus dispersion may be more dependent on fracture connectivity and length than fracture density and display a relationship similar to that shown by particle beds and clastic sandstones. The high retardation values observed (2.2–4.9) reflect flow behaviour within a fracture network with a proportion of ‘blind’ sections, and demonstrates how such networks can slow the advance of conservative solute components.     

Network concepts to describe channel importance and change in multichannel systems: test results for the Jamuna River,Bangladesh

Most of the largest rivers on Earth have multiple active channels connected at bifurcations and confluences. At present a method to describe a channel network pattern and changes in the network beyond the simplistic braiding index is unavailable. Our objectives are to test a network approach to understand the character, stability and evolution of a multi‐channel river pattern under natural discharge conditions. We developed a semi‐automatic method to derive a chain‐like directional network from images that represent the multi‐channel river and to connect individual network elements through time. The Jamuna River was taken as an example with a series of Landsat TM and ETM+ images taken at irregular intervals between 1999 and 2004. We quantified the overall importance of individual channels in the entire network using a centrality property. Centrality showed that three reaches can be distinguished along the Jamuna with a different network character: the middle reach has dominantly one important channel, while upstream and downstream there are about two important channels. Temporally, relatively few channels changed dramatically in both low‐flow and high‐flow periods despite the increase of braiding index during a flood. Based on the centrality we calculated a weighted braiding index that represents the number of important channels in the network, which is about two in the Jamuna River and which is larger immediately after floods. We conclude that the network measure centrality provides a novel characterization of river channel networks, highlighting properties and tendencies that have morphological significance. Copyright © 2013 John Wiley & Sons, Ltd.     

Multichannel rivers: their definition and classification

The etymology and historic usage of such terms as ‘anabranch’, ‘anastamose’ and ‘braided’ within river science are reviewed. Despite several decades of modern research to define river channel typologies inclusive of single channels and multiple channel networks, typologies remain ill‐conditioned and consequently ill‐defined. Conventionally employed quantitative planform characteristics of river networks possibly cannot be used alone to define channel types, yet the planform remains a central part of all modern classification schemes, supplemented by sedimentological and other qualitative channel characteristics. Planform characteristics largely have been defined using non‐standardized metrics describing individual network components, such as link lengths, braiding intensity and bifurcation angles, which often fail to separate visually‐different networks of channels. We find that existing typologies remain pragmatically utilitarian rather than fundamentally physics‐based and too often fail to discriminate between two distinctive and important processes integral to new channel initiation and flow‐splitting: (i) in‐channel bar accretion, and (ii) channel avulsion and floodplain excision. It is suggested that, first, if channel planform is to remain central to river typologies, then more rigorous quantitative approaches to the analysis of extended integral channel networks at extended reach scales (rather than network components) are required to correctly determine whether ‘visually‐different’ channel patterns can be discriminated consistently; and, second, if such visually‐different styles do in fact differ in their governing processes of formation and maintenance. A significant question is why do so many seemingly equilibrium network geometries possess a large number of anabranches in excess of predictions from theoretical considerations? The key research frontier with respect to initiating and maintaining multichannel networks remains the understanding and discrimination of accretionary‐bar flow splitting versus avulsive processes. Existing and new knowledge on flow splitting processes needs to be better integrated into channel typologies. Copyright © 2013 John Wiley & Sons, Ltd.     

Stochastic analysis of immiscible displacement of the fluids with arbitrary viscosities and its dependence on support scale of hydrological data

Stochastic analysis is commonly used to address uncertainty in the modeling of flow and transport in porous media. In the stochastic approach, the properties of porous media are treated as random functions with statistics obtained from field measurements. Several studies indicate that hydrological properties depend on the scale of measurements or support scales, but most stochastic analysis does not address the effects of support scale on stochastic predictions of subsurface processes. In this work we propose a new approach to study the scale dependence of stochastic predictions. We present a stochastic analysis of immiscible fluid–fluid displacement in randomly heterogeneous porous media. While existing solutions are applicable only to systems in which the viscosity of one phase is negligible compare with the viscosity of the other (water–air systems for example), our solutions can be applied to the immiscible displacement of fluids having arbitrarily viscosities such as NAPL–water and water–oil. Treating intrinsic permeability as a random field with statistics dependant on the permeability support scale (scale of measurements) we obtained, for one-dimensional systems, analytical solutions for the first moments characterizing unbiased predictions (estimates) of system variables, such as the pressure and fluid–fluid interface position, and we also obtained second moments, which characterize the uncertainties associated with such predictions. Next we obtained empirically scale dependent exponential correlation function of the intrinsic permeability that allowed us to study solutions of stochastic equations as a function of the support scale. We found that the first and second moments converge to asymptotic values as the support scale decreases. In our examples, the statistical moments reached asymptotic values for support scale that were approximately 1/10000 of the flow domain size. We show that analytical moment solutions compare well with the results of Monte Carlo simulations for moderately heterogeneous porous media, and that they can be used to study the effects of heterogeneity on the dynamics and stability of immiscible flow.     

Supraglacial melt channel networks in the Jakobshavn Isbræ region during the 2007 melt season

Supraglacial channels are an important mechanism for surface water transport over the ablation zone of western Greenland. The first assessment of the spatio‐temporal distribution of surface melt channels and their relationship to supraglacial lakes over the Jakobshavn Isbræ region of Western Greenland was analysed using Landsat Enhanced Thematic Mapper Plus panchromatic images during the 2007 melt season. A total of 1188 melt channels were delineated and show an increase in the number of melt channels throughout the season, reaching a peak on 9 August. Water‐filled melt channels advanced to a maximum elevation of 1647 m on 9 August and attained a minimum average slope of 0.009 on 8 July. The ablation zone demonstrates two hydrologic modes, where crevasse and moulin terminating channels dominate at elevations <800 m and higher‐order channel networks >800 m. Development of higher‐order networks is interrupted by flow divergence due to partitioning of melt water into vertical infiltration through moulins and crevasse fields prevalent at lower elevations. Tributary and connector networks between 800 and 1200 m in elevation are correlated with fewer lake occurrences, lower surface velocities (~50 m a^?1), and ice flow dominated by internal deformation over basal sliding. High‐order channels are associated with lake basins that exceed melt water storage capacity. Evolution of channel networks is coupled to changes in melt water production, runoff, and ice dynamics with implication for the englacial and subglacial environments. © 2013 The Authors. Hydrological Processes Published by John Wiley & Sons, Ltd.     

Regional watershed characterization and classification with river network analyses

In order to understand and manage a hydrological region, one usually needs to comprehensively characterize the watersheds (basins) and their river networks. This usually and primarily involves analysis of hydrological and geomorphological properties of the watershed derived from the digital terrain model (DTM), but this approach neglects the information content of the associated river networks. In this study, we used a combination of traditional DTM and original river network‐related indices to the watersheds of an understudied region, Haiti. We also used Monte Carlo simulations to estimate index confidence levels of these indices. Compared to commonly used indices, the network indices provided valuable information that could then be used in statistical analyses as a way to identify the dominant features of the country's watershed morphology. On this basis, we identified four watershed groups in Haiti: (i) high, elongated watersheds, (ii) lowlands, with sinuous networks and relatively slow runoff, (iii) high watersheds with dendritic networks, and (iv) lowlands with high downstream–upstream contrast in properties and rapid runoffs. We argue that river network features provide complementary information in terms of flow topology, highly relevant to characterize contrasting relief countries, such as Haiti. Hence, more exhaustive characterization of watersheds would predictably benefit from the approach outlined in this paper. Copyright © 2017 John Wiley & Sons, Ltd.     

Simulating the effects of spatial configurations of agricultural ditch drainage networks on surface runoff from agricultural catchments

The study of runoff is a crucial issue because it is closely related to flooding, water quality and erosion. In cultivated catchments, agricultural ditch drainage networks are known to influence runoff. As anthropogenic elements, agricultural ditch drainage networks can therefore be altered to better manage surface runoff in cultivated catchments. However, the relationship between the spatial configuration, i.e. the density and the topology, of agricultural ditch drainage networks and surface runoff in cultivated catchments is not understood. We studied this relationship by using a random network simulator that was coupled to a distributed hydrological model. The simulations explored a large variety of spatial configurations corresponding to a thousand stochastic agricultural ditch drainage networks on a 6.4 km² Mediterranean cultivated catchment. Next, several distributed hydrological functions were used to compute water flow paths and runoff for each simulation. The results showed that (i) denser networks increased the drained volume and the peak discharge and decreased hillslopes runoff, (ii) greater network density did not affect the surface runoff any further above a given network density, (iii) the correlation between network density and runoff was weaker for small subcatchments (< 2 km²) where the variability in the drained area that resulted from changes in agricultural ditch drainage networks increased the variability of runoff and (iv) the actual agricultural ditch drainage network appeared to be well optimized for managing runoff as compared with the simulated networks. Finally, our results highlighted the role of agricultural ditch drainage networks in intercepting and decreasing overland flow on hillslopes and increasing runoff in drainage networks. Copyright © 2011 John Wiley & Sons, Ltd.     

Self‐Organized Permeability in Carbonate Aquifers

Advances over the past 40 years have resulted in a clear understanding of how dissolution processes in carbonate rocks enhance aquifer permeability. Laboratory experiments on dissolution rates of calcite and dolomite have established that there is a precipitous drop in dissolution rates as chemical equilibrium is approached. These results have been incorporated into numerical models, simulating the effects of dissolution over time and showing that it occurs along the entire length of pathways through carbonate aquifers. The pathways become enlarged and integrated over time, forming self‐organized networks of channels that typically have apertures in the millimeter to centimeter range. The networks discharge at point‐located springs. Recharge type is an important factor in determining channel size and distribution, resulting in a range of aquifer types, and this is well demonstrated by examples from England. Most carbonate aquifers have a large number of small channels, but in some cases large channels (i.e., enterable caves) can also develop. Rapid velocities found in ground water tracer tests, the high incidence of large‐magnitude springs, and frequent microbial contamination of wells all support the model of self‐organized channel development. A large majority of carbonate aquifers have such channel networks, where ground water velocities often exceed 100 m/d.     

A new upscaling method for fractured porous media

We present a method to determine equivalent permeability of fractured porous media. Inspired by the previous flow-based upscaling methods, we use a multi-boundary integration approach to compute flow rates within fractures. We apply a recently developed multi-point flux approximation Finite Volume method for discrete fracture model simulation. The method is verified by upscaling an arbitrarily oriented fracture which is crossing a Cartesian grid. We demonstrate the method by applying it to a long fracture, a fracture network and the fracture network with different matrix permeabilities. The equivalent permeability tensors of a long fracture crossing Cartesian grids are symmetric, and have identical values. The application to the fracture network case with increasing matrix permeabilities shows that the matrix permeability influences more the diagonal terms of the equivalent permeability tensor than the off-diagonal terms, but the off-diagonal terms remain important to correctly assess the flow field.