Effects of the geometry of two‐dimensional fractures on their hydraulic aperture and on the validity of the local cubic law

Flow through rough fractures is investigated numerically in order to assess the validity of the local cubic law for different fracture geometries. Two‐dimensional channels with sinusoidal walls having different geometrical properties defined by the aperture, the amplitude, and the wavelength of the walls' corrugations, the corrugations asymmetry, and the phase shift between the two walls are considered to represent different fracture geometries. First, it is analytically shown that the hydraulic aperture clearly deviates from the mean aperture when the walls' roughness, the phase shift, and/or the asymmetry between the fracture walls are relatively high. The continuity and the Navier–Stokes equations are then solved by means of the finite element method and the numerical solutions compared to the theoretical predictions of the local cubic law. Reynolds numbers ranging from 0.066 to 66.66 are investigated so as to focus more particularly on the effect of flow inertial effects on the validity of the local cubic law. For low Reynolds number, typically less than 15, the local cubic law properly describes the fracture flow, especially when the fracture walls have small corrugation amplitudes. For Reynolds numbers higher than 15, the local cubic law is valid under the conditions that the fracture presents a low aspect ratio, small corrugation amplitudes, and a moderate phase lag between its walls.     

A constitutive model for water flow in unsaturated fractured rocks

A conceptual model for describing effective saturation in fractured hard rock is presented. The fracture network and the rock matrix are considered as an equivalent continuum medium where each fracture is conceptualized as a porous medium of granular structure and the rock matrix is assumed to be impermeable. The proposed model is based on the representation of a rough‐walled fracture by an equivalent porous medium, which is described using classical constitutive models. A simple closed‐form equation for the effective saturation is obtained when the van Genuchten model is used to describe saturation inside fractures and fractal laws are assumed for both aperture and number of fractures. The relative hydraulic conductivity for the fractured rock is predicted from a simple relation derived by Liu and Bodvarsson. The proposed constitutive model contains three independent parameters, which may be obtained by fitting the proposed effective saturation curve to experimental data. Two of the model parameters have physical meaning and can be identified with the reciprocal of the air entry pressure values in the fractures of minimum and maximum apertures. Effective saturation and relative hydraulic conductivity curves match fairly well the simulated constitutive relations obtained by Liu and Bodvarsson. Copyright © 2008 John Wiley & Sons, Ltd.     

Fluid flow through rough fractures in rocks. II: A new matching model for rough rock fractures

An understanding of fluid flow through natural fractures in rocks is important in many areas, such as in the hydrocarbon and water industries, and in the safe design of disposal sites for domestic, industrial and nuclear waste. It is often impractical to obtain this information by field or laboratory scale measurements, so numerical modelling of fluid flow must be carried out using synthetic fractures with rough fracture surfaces that are representative of the natural rock fractures. Clearly there are two practical requirements; (i) the development of a method for analysing natural rock fractures to obtain their characteristic parameters, and (ii) the development of techniques for creating high quality synthetic fractures using these parameters. We have implemented these practical requirements in two new software packages. The first, ParaFrac allows the analysis and parameterisation of fracture surfaces and apertures. The second, SynFrac, enables the numerical synthesis of fracture surfaces and apertures with basic prescribed parameters. Synthetic fractures are created using, (i) a new model, which takes full account of the complex matching properties of fracture surfaces using two new parameters, a minimum matching fraction and a transition length and (ii) an improved method of partially correlated random number generation. This model more closely captures the often complex matching properties of real rock fractures than previous more simplified models.     

Experimental study of the effect of roughness and Reynolds number on fluid flow in rough‐walled single fractures: a check of local cubic law

Local cubic law (LCL) is one of the most commonly applied physical laws for flow in single fractures (SF) and fractured media. The foundation of LCL is Darcian flow. This experimental study examines if LCL is valid for flow in a single rough fracture and how the fracture roughness and Reynolds number (Re) affect flow. Similar to the Moody diagram for flow in pipes, a diagram for flow in a single rough fracture has been generated to relate the friction coefficient with Re and the roughness. Under the experimental condition of this study, flow appears to be substantially different from Darcian flow. The flow law of q ∝ eⁿJ^m appears to be valid for describing the flow scheme where q, e, and J are the unit width flux, the average aperture, and the hydraulic gradient. The value of the power index m is found to be around 0·83 ～ 0·98, less than what has been used in Darcian flow (m = 1). The power index n is around 11·2 and 13·0, much greater than the n value used in the LCL (n = 3), and it increases with the average velocity. The Moody type of diagram shows that the friction factor for flow in SFs is influenced by Re and the roughness. It decreases with Re when Re is small, and becomes less sensitive to Re when Re is large enough. It also increases with the roughness. Copyright © 2010 John Wiley & Sons, Ltd.     

Scale Effects Related to Flow in Rough Fractures

— A numerical fracture flow simulation based on the lubrication approximation is used to investigate the influence of roughness on the flow inside a rough fracture, at low Reynolds number. Facing surfaces are described as self-affine topographies with identical roughness magnitude. Resolution of the Reynolds equation is achieved using two distinct numerical schemes, with consistency. Fracture closure is studied assuming perfect plastic contact between facing surfaces. Long-range correlations are shown to exist in the local aperture field due to the fracture geometry and subsequently in the local fluxes inside the fracture. Flow channeling is the result of these correlations in terms of spatial distribution of the flow, and is responsible for either flow-enhancing or flow-inhibiting behavior of the fracture. Matching between the two surfaces at scales larger than a mismatch scale is studied. The mismatch scale is the upper limit scale for the local apertures scale invariance. It appears to control flow channeling and the related dispersion of the possible behaviors over a large statistics of fractures with identical statistical features. Hydraulic anisotropy of a given fracture is investigated: the dependence of the fracture transmittivity on the pressure drop orientation is proved to be sinusoidal, with an amplitude that is controlled by the mismatch scale.     

Non-Darcian groundwater flow in leaky aquifers

Abstract

A simplified method has been developed for solving leaky aquifer non-Darcian flow hydraulics. The principle of volumetric approach is combined with the confined-aquifer, time-dependent drawdown equation in an observation well. The groundwater flow in the leaky aquifer is assumed to obey a non-Darcian flow law of exponential type. The results are obtained in the form of type-curve expressions from which the necessary bundles of curves are drawn for a set of selective non-Darcian flow aquifer parameters. Although application of the methodology appears as rather limited but it provides a scientific contribution and extension of leaky aquifer theory towards nonlinear flow conditions. The methodology developed herein is applied to some actual field data from the eastern sedimentary basin in the Kingdom of Saudi Arabia.     

Factors that influence fluid flow in a single fracture

To investigate the influence of compression, Poisson effect and turbulence on the fluid flow process and the inversion for fracture surface geometries, we simulate two sets of fractures: one with a defined fracture height standard deviation σ constant and a varying autocorrelation length λ and another with a fixed λ and a changing σ. Under compression, the normal stress closes fractures with a large aperture and thus reduces the effective permeability. However, the Poisson effect, which is induced by the compression, has little influence on the fluid flow properties and does not affect the inversion for fracture height standard deviation or the autocorrelation length. When introducing turbulence, we observe a significant difference between the performance of the Navier–Stokes equation and the local cubic law; compared with the Navier–Stokes equation, the local cubic law overestimates the peak value of the breakthrough time curve and effective permeability, thereby underestimating the mean fracture aperture.     

Non-Darcian flow in a single confined vertical fracture toward a well

Non-Darcian flow towards a well which fully penetrates a confined single vertical fracture is presented in this paper on the basis of the Izbash equation. We have obtained semi-analytical solutions for non-Darcian flow by using the Boltzmann transform and developed the non-Darcian flow well functions for cases with and without the effect of wellbore storage. The results show that the non-Darcian flow type curves are more or less deviated from the Darcian flow type curve. The non-linear effect is mainly attributable to the turbulent factor, v, a dimensionless parameter related to the pumping rate, the fracture aperture, the fracture thickness, and two constants k′ and n used in the Izbash power-law. The non-linear effect appears to be less sensitive to the power-law index, n. When excluding wellbore storage, the well function at early times is proportional to v^−1/(n−1)u^−n/(n−1), where u is a dimensionless term inversely proportional to time; whereas the well function at late times is approximated as , where A₀(n) is a finite term depending on n. When considering wellbore storage, drawdowns inside the well with different v values approach the same asymptotic value at small times, and the effect of wellbore storage is only found at the early stage of pumping.     

Data time-step dependency of conceptual rainfall—streamflow model parameters: an empirical study with implications for regionalisation

Abstract

Employing 1-, 2-, 4-, 6-, 12- and 24-hourly data sets for two catchments (10.6 and 298 km²) in Wales, the calibrated parameters of a unit hydrograph-based model are shown to change substantially over that range of data time steps. For the smaller basin, each model parameter reaches, or approaches, a stable value as the data time step decreases, providing a straightforward method of estimating time-step independent model parameter values. For the larger basin, the model parameters also reach, or approach, stable values using hourly data, but, for reasons given in the paper, interpretation of the results is more difficult. Model parameter sensitivity analyses are presented that give insights into the relative precision on the parameters for both catchments. The paper discusses the importance of accounting for model parameter data time-step dependency in pursuit of a reduction in the uncertainty associated with estimates of flow in ungauged basins, and suggests that further work along these lines be undertaken using different catchments and models.     

Modeling fracture porosity development using simple growth laws

A model of porosity development has been developed to investigate general relationships between simple fracture aperture growth laws and fracture porosity in evolved fracture arrays in aquifers. The growth of fracture apertures in two-dimensional orthogonal arrays with initially spatially uncorrelated lognormal aperture distributions has been studied, where aperture growth rate is proportional to an exponent of the flow rate through each fracture. The evolved arrays show geometrical phase changes as a function of the aperture growth rate exponent, e, and the standard deviation of the initial aperture distribution, sigma(z). Low values of e and sigma(z) lead to bimodal aperture distributions, where apertures parallel to flow are preferentially enlarged. At moderate values of e and sigma(z), there is a transition to a regime of more complex geometries consisting of networks of channel-like structures of preferentially enlarged apertures. At larger values of e, array-spanning channel-like paths of preferentially enlarged apertures develop, where the tortuosity of the channel-like paths is a linear function of sigma(z). Following an initial growth phase, during which dynamically stable aperture configurations develop, arrays undergo simple amplification. The geometry of the evolved aperture fields is diverse and they can be highly complex; consequently, parameterization and prediction of their evolution in terms of the initial aperture distributions and growth rate laws is not trivial.     

Evaluation of storage–discharge relationships and recession analysis-based distributed hourly runoff simulation in large-scale,mountainous and snow-influenced catchment

ABSTRACT

Evaluation of a recession-based “top-down” model for distributed hourly runoff simulation in macroscale mountainous catchments is rare in the literature. We evaluated such a model for a 3090 km² boreal catchment and its internal sub-catchments. The main research question is how the model performs when parameters are either estimated from streamflow recession or obtained by calibration. The model reproduced observed streamflow hydrographs (Nash-Sutcliffe efficiency up to 0.83) and flow duration curves. Transferability of parameters to the sub-catchments validates the performance of the model, and indicates an opportunity for prediction in ungauged sites. However, the cases of parameter estimation and calibration excluding the effects of runoff routing underestimate peak flows. The lower end of the recession and the minimum length of recession segments included are the main sources of uncertainty for parameter estimation. Despite the small number of calibrated parameters, the model is susceptible to parameter uncertainty and identifiability problems.

EDITOR D. Koutsoyiannis; ASSOCIATE EDITOR A. Carsteanu     

Erosion modelling for land management in the Tahoe basin,USA: scaling from plots to forest catchments

Abstract

Recent developments in hydrological modelling of river basins are focused on prediction in ungauged basins, which implies the need to improve relationships between model parameters and easily-obtainable information, such as satellite images, and to test the transferability of model parameters. A large-scale distributed hydrological model is described, which has been used in several large river basins in Brazil. The model parameters are related to classes of physical characteristics, such as soil type, land use, geology and vegetation. The model uses two basin space units: square grids for flow direction along the basin and GRU—group response units—which are hydrological classes of the basin physical characteristics for water balance. Expected ranges of parameter values are associated with each of these classes during the model calibration. Results are presented of the model fitting in the Taquari-Antas River basin in Brazil (26 000 km² and 11 flow gauges). Based on this fitting, the model was then applied to the Upper Uruguay River basin (52 000 km²), having similar physical conditions, without any further calibration, in order to test the transferability of the model. The results in the Uruguay basin were compared with recorded flow data and showed relatively small errors, although a tendency to underestimate mean flows was found.     

An asymptotic model for resistive instability in the Earth's outer core

Abstract

Numerical work indicates that resistive instability may be the dominant mode of instability in the Earth's outer core for realistic core parameter regimes. In this paper, we assume that the Elsasser number is large in order to obtain an asymptotic analysis of resistive instability in an electrically conducting fluid confined to a rotating cylindrical shell of infinite extent in the axial direction. The dimensionless equations of motion are linearized about an ambient magnetic field which is purely azimuthal and depends only on the cylindrical radial variable. Applying the theory of ordinary differential equations with a large parameter, we obtain an asymptotic approximation to the solution. Relatively simple analytic expressions for the complex frequencies are obtained by applying the boundary conditions for insulating boundaries at the cylindrical sidewalls and then assuming that the ambient magnetic field vanishes at one or both of those sidewalls. The results appear to be consistent with previous numerical work.     

An index to assess hydromorphological quality of Estonian surface waters based on macroinvertebrate taxonomic composition

We developed an index (MESH – Macroinvertebrates in Estonia: Score of Hydromorphology) to assess hydromorphological quality of Estonian surface waters based on macroinvertebrate taxonomic composition. The MESH is an average score based on the affinities of selected indicator taxa to flow velocity and bottom type. As both parameters were highly correlated (r = 0.65) indicator response to both parameters were combined. The list of MESH indicators includes 394 freshwater macroinvertebrate taxa derived from 3282 samples collected from rivers and lakes during 1985–2009. The indicators were selected out of 690 taxa, by applying the information-theoretical Kullback–Leibler divergence. The individual scores of macroinvertebrates range from 0 to 3, the higher scores indicating faster flow and/or solid bottom substrate. For standing waters, flow velocity was always considered zero. Among the reference waterbodies, mean MESH was the highest for small streams followed by middle streams, large streams, and lakes. In lakes with medium water hardness (the prevailing type in Estonia), the MESH decreased gradually from stony to muddy bottom. The highest MESH values for standing waters were observed in the stony surf zone of very large lakes (area > 100 km²). The lowest values occurred for small lakes with exceptional hydrochemical characteristics (soft- and darkwater, and calcareous types). Similarly, MESH indicated stream degradation by damming. Mean MESH in reservoirs with a muddy bottom was significantly lower than in reservoirs with a hard bottom, or in unregulated stream sections.     

华北地区地震序列参数的分布特征

为系统地考察华北地区地震序列参数的分布特征,以期构建适合区域地震活动特征的短期概率预测模型和评估地震危险性,本文利用当前较为前沿的ETAS模型和R-J模型,采用连续滑动、多时段拟合的方式,对华北地区1970年以来的16个地震序列进行了参数拟合,并对参数的整体情况、参数之间的关系、参数与大地热流之间的关系等特征进行了分析研究.研究结果表明,主震发生后的早期阶段,地震序列参数变化相对较为剧烈,误差也较大;两种地震预测模型的序列参数呈现出一种优势分布特征,主要参数的平均值分别为α_ETAS=1.7404±0.3420,p_ETAS=0.9769±0.1396,a_OML=-1.6638±0.5284,b_OML=0.8312±0.1658,p_OML=0.9053±0.1527,这与国际上其他区域的研究具有很强的一致性;大多数情况下,ETAS模型参数p_ETAS高于R-J模型参数p_OML,平均偏高0.0716;序列参数α_ETAS与大地热流值整体上呈现一种负相关关系,激发次级余震的能力与大地热流具有一定的相关性.地震序列参数的分布特征以及与大地热流之间的关系对地震序列类型的判断,及基于地震序列参数构建定量化的短期概率预测模型,均有重要的应用价值.     

Unsteady inter-porosity flow modeling for a multiple media reservoir

The paper deals with unsteady inter-porosity flow modeling of underground fluid in a multiple media reservoir. Assuming spherical vugs, symmetrically distributed pressure, negligible inter-porosity flow between matrix and vug systems and centrifugal flow of the fluid from matrix blocks or vugs to fractures, and treating media directly connected with wellbore as the fracture system, we establish and solve a model of unsteady inter-porosity flow for dual and triple porosity media reservoirs. We provide simulated graphs of pressure and pressure derivative log-log type curves, and analyze the transient flow process and characteristics of type curves affected by different parameters. The new type curves of unsteady inter-porosity flow modeling are evidently different in shape and characteristics from those of pseudo-steady inter-porosity flow modeling. The location of dimensionless pressure of unsteady inter-porosity is lower than that of pseudo-steady inter-porosity, which indicates that unsteady inter-porosity flow accelerates an energy supplement during production. Qualitatively, the unsteady inter-porosity flow modeling reduces the classical V-shaped response. We also estimated parameters from well test data in real applications using this model.     

Dynamos on stream surfaces of a highly conducting fluid

Abstract

This paper discusses dynamo action in generalisations of the Ponomarenko dynamo at large magnetic Reynolds number. The original Ponomarenko dynamo consists of a spiralling flow in which the stream surfaces are concentric cylinders of circular cross section, and the flow depends only on distance from the axis in cylindrical polar coordinates.

In this study, the stream surfaces are allowed to be cylinders of arbitrary cross section, and the flow is only required to be independent of the coordinate along the cylinder axes. For smooth flows alpha and eddy diffusion effects are identified, in terms of the geometry of the stream surfaces, and asymptotic formulae for growth rates in the limit of large magnetic Reynolds number are obtained. Numerical support for these results is presented using direct simulation of dynamo action in selected flows at high conductivity. Finally the case is considered when in spherical polar coordinates the flow is independent of the azimuthal coordinate and the stream surfaces, which are tori, have arbitrary cross sections.     

A comparison of the asymptotic and no-z approximations for galactic dynamos

Abstract

The asymptotic and the no-z approximation methods of solving the axisymmetric mean field αΩ dynamo equation in a galactic disc are compared. The behaviour of the solutions is explored in both the linear and nonlinear regimes for a variety of dynamo parameters and two different rotation curves. The solutions obtained from the two different approaches are found to be in good agreement.     

Unified fractional differential approach for transient interporosity flow in naturally fractured media

A unified approach to modeling flows of slightly compressible fluids through naturally fractured media is presented. The unified fractional differential model is derived by combining the flow at micro scale for matrix blocks and macro scale for fractures, using the transient interporosity flow behavior at the interface between matrix blocks and fractures. The derived model is able to unify existing transient interporosity flow models formulated for different shapes of matrix blocks in any medium dimensions. The model is formulated in the form of a fractional order partial differential equation that involves Caputo derivative of order 1/2 with respect to time. Explicit solutions for the unified model are derived for different axisymmetrical spatial domains using Hankel or Hankel–Weber finite or infinite transforms. Comparisons between the predictions of the unified model and those obtained from existing transient interporosity flow models for matrix blocks in the form of slabs, spheres and cylinders are presented. It is shown that the unified fractional derivative model leads to solutions that are very close to those of transient interporosity flow models for fracture-dominant and transitional fracture-to-matrix dominant flow regimes. An analysis of the results of the unified model reveals that the pressure varies linearly with the logarithm of time for different flow regimes, with half slope for the transitional fracture-to-matrix dominant flow regime vs. the fracture and matrix dominant flow regimes. In addition, a new re-scaling that involves the characteristic length in the form of matrix block volume to surface area ratio is derived for the transient interporosity flow models for matrix blocks of different shapes. It is shown that the re-scaled transient interporosity flow models are governed by two dimensionless parameters Θ and Λ compared to only one dimensionless parameter Θ for the unified model. It is shown that the solutions of the transient interporosity flow models for different shapes of matrix blocks are almost identical for the re-scaled variables. Furthermore, the driving parameters for solution behavior are identified based on asymptotic approximations for different flow regimes. It is found that the matrix diffusion and the matrix area-to-volume ratio affect the solution behavior only for the transitional fracture-to-matrix dominant flow regime, that the capacitance ratio affects the solution behavior only for transitional and matrix dominant flow regimes and that the fracture diffusion is involved in all three flow regimes. Similar identification of the driving parameters is also presented in the re-scaled case.     

Estimation of asymptotic variances of quantiles for the generalized logistic distribution

In this study, the parameter estimations for the 3-parameter generalized logistic (GL) distribution are presented based on the methods of moments (MOM), maximum likelihood (ML), and probability weighted moments (PWM). The asymptotic variances of the MOM, ML, and PWM quantile estimators for the GL distribution are expressed as functions of the sample size, return period, and parameters. A Monte Carlo simulation was performed to verify the derived expressions for variances and covariances between parameters and to evaluate the applicability of the derived asymptotic variances of quantiles for the MOM, ML and PWM methods. The simulation results generally show good agreement with the analytical results estimated from the asymptotic variances of parameters and quantiles when the shape parameter (β) of the GL distribution is between −0.10 and 0.10 for the MOM method and between −0.25 and 0.45 for the ML and PWM methods, respectively. In addition, the actual sample variances and the root mean square error (RMSE) of asymptotic variances of quantiles for various sample sizes, return periods, and shape parameters were presented. In order to evaluate the applicability of the estimation methods to real data and to compare the values of estimated parameter, quantiles, and confidence intervals based on each parameter estimation method, the GL distribution was fitted to the 24-h annual maximum rainfall data at Pohang, Korea.