Quantification of longitudinal dispersion by upscaling Brownian motion of tracer displacement in a 3D pore-scale network model

We present a 3D network model with particle tracking to upscale 3D Brownian motion of non-reactive tracer particles subjected to a velocity field in the network bonds, representing both local diffusion and convection. At the intersections of the bonds (nodes) various jump conditions are implemented. Within the bonds, two different velocity profiles are used. At the network scale the longitudinal dispersion of the particles is quantified through the coefficient D_L, for which we evaluate a number of methods already known in the literature. Additionally, we introduce a new method for derivation of D_L based on the first-arrival times distribution (FTD). To validate our particle tracking method, we simulate Taylor’s classical experiments in a single tube. Subsequently, we carry out network simulations for a wide range of the characteristic Péclet number Pe_? to assess the various methods for obtaining D_L. Using the new method, additional simulations have been carried out to evaluate the choice of nodal jump conditions and velocity profile, in combination with varying network heterogeneity. In general, we conclude that the presented network model with particle tracking is a robust tool to obtain the macroscopic longitudinal dispersion coefficient. The new method to determine D_L from the FTD statistics works for the full range of Pe_?, provided that for large Pe_? a sufficiently large number of particles is used. Nodal jump conditions should include molecular diffusion and allow jumps in the upstream direction, and a parabolic velocity profile in the tubes must be implemented. Then, good agreement with experimental evidence is found for the full range of Pe_?, including increased D_L for increased porous medium heterogeneity.     

Controls on at‐a‐station hydraulic geometry in steep headwater streams,Colorado, USA

Detailed hydraulic measurements were made in nine step‐pool, five cascade and one plane‐bed reach in Fraser Experimental Forest, Colorado to better understand at‐a‐station hydraulic geometry (AHG) relations in these channel types. Average values for AHG exponents, m (0·49), f (0·39), and b (0·16), were well within the range found by other researchers working in steep gradient channels. A principal component analysis (PCA) was used to compare the combined variations in all three exponents against five potential control variables: wood, D₈₄, grain‐size distribution (σ), coefficient of variation of pool volume, average roughness‐area (projected wetted area) and bed gradient. The gradient and average roughness‐area were found to be significantly related to the PCA axis scores, indicating that both driving and resisting forces influence the rates of change of velocity, depth and width with discharge. Further analysis of the exponents showed that reaches with m > b + f are most likely dominated by grain resistance and reaches below this value (m < b + f) are dominated by form resistance. Copyright © 2010 John Wiley & Sons, Ltd.     

甘肃三个台地磁日变赫斯特指数的时序特征

选用甘肃嘉峪关、兰州和天水地磁台2011—2013年的观测数据,重点研究地磁垂直向日变化波形下行段的赫斯特指数变化的时序特征。结果表明,在此3年内,嘉峪关台和兰州台地磁垂直向日变化波形下行段的赫斯特指数变化范围分别是0.08和0.06,天水台在2012年存在一个赫斯特指数幅度超过正常变化范围的过程,在此过程完成后的2013年7月22日,在距离天水130km的漳县和岷县交界发生M6.6地震。结合其他学者的研究结果,认为这一赫斯特指数的异常变化过程可能反映了区域性地球深部(下地壳与上地幔)热物质运移变化而引起地球局部居里面变化,也可能是地震孕育过程在地磁日变中的表征。这一发现有益于进一步研究地磁日变异常的机理、震磁关系以及地震预报的探索。     

Diapirism in the earth's mantle: Experiments on the motion of a hot sphere in a fluid with temperature-dependent viscosity

The ascent of magma diapirs through the earth's mantle is modelled experimentally by the motion of a hot metal sphere through a fluid whose viscosity varies strongly with temperature. The dimensionless drag on the sphere (drag number D) and the heat transfer from it (Nusselt number Nu) are found as functions of the dimensionless velocity of the sphere (Peclet number Pe) and the viscosity contrast μ_∞/μ₀ = 10^γ, where μ_∞ and μ₀ are the viscosities of the fluid far from the sphere and at its surface. The drag D = D(Pe, γ) has two limits. For large Pe and small γ (“Stokes” limit), the drag approaches the Stokes' Law result. For small Pe and large γ (“lubrication” limit), the drag is orders of magnitude less than that predicted by Stokes' Law. Nu is a function of Pe alone. For reasonable values of the diapir radius and the viscosity contrast, the dimensionless scale height Pe/3 Nu may exceed a critical value, resulting in progressive melting during ascent. This suggests that diapirs may ascend great distances through the mantle while remaining largely molten. Lamont-Doherty Geological Observatory Contribution No. 3414.     

Fluvial seed dispersal of riparian trees: transport and depositional processes

Fluvial seed dispersal considers both the transport and deposition of seeds where channel geomorphic structures, hydrology and seed dispersal traits contribute to transport times and depositional locations. This study examines the influence of stream flow patterns on fluvial seed dispersal of buoyant white alder (Alnus rhombifolia) seeds by applying a one‐dimensional transport model. Conceptually, the model separates the stream into two components: (i) the main channel where the seeds are transported downstream; and (ii) the transient storage zone where seeds are temporarily detained or deposited on the river bank. Transport processes are characterized by an advection–dispersion equation which is coupled to a transient storage model using an exponential decay term. The model parameters: longitudinal dispersion (D_L), exchange coefficient (α), main channel area (A) and storage zone (A_s) are estimated based on field experiments conducted in a confined, bedrock‐gravel bed river with pool‐riffle morphology located in coastal northern California. The riparian zone is inhabited by Alnus rhombifolia that disperse buoyant seeds in mid‐spring coinciding with the end of the wet, Mediterranean season. Artificial seeds, with similar traits of buoyancy and density to alder seeds, were used to quantify transport times and depositional locations. Preferential deposition resulted in stream reaches with larger A_s, high A_s/A ratios, and faster exchange coefficients corresponding to divergent stream flow (back‐eddies, re‐circulating flow, flow expansions) caused by geomorphic structures such as the ends of bar/riffle features and bends in the stream. The results demonstrate the importance of transient storage for seed transport and depositional processes. Morphological features that increase a channel's complexity create complex flow structures that detain seeds and provide a greater opportunity for deposition to occur. The model provides a simplification of river hydraulics to represent dispersal dynamics and lends itself to further understanding of hydrochory processes and associated population structure. Copyright © 2015 John Wiley & Sons, Ltd.     

Flow resistance in gravel‐bed channels with large‐scale roughness

A previously published mixing length (ML) model for evaluating the Darcy–Weisbach friction factor for a large‐scale roughness condition (depth to sediment height ratio ranging from 1 to 4) is brie?y reviewed and modi?ed (MML). Then the MML model and a modi?ed drag (MD) model are experimentally tested using laboratory measurements carried out for gravel‐bed channels and large‐scale roughness condition. This analysis showed that the MML gives accurate estimates of the Darcy–Weisbach coef?cient and for Froude number values greater than 0·5 the MML model coincides with the ML one. Testing of the MD model shows limited accuracy in estimating ?ow resistance. Finally, the MML and MD models are compared with the performance of a quasi‐theoretical (QT) model deduced applying the P‐theorem of the dimensional analysis and the incomplete self‐similarity condition for the depth/sediment ratio and the Froude number. Using the experimental gravel‐bed data to calibrate the QT model, a constant value of the exponent of the Froude number is determined while two relationships are proposed for estimating the scale factor and the exponent of the depth/sediment ratio. This indirect estimate procedure of the coef?cients (b₀, b₁ and b₂) of the QT model can produce a negligible overestimation or underestimation of the friction factor. Copyright © 2003 John Wiley & Sons, Ltd.     

Temporal correlations and clustering of landslides

This paper examines temporal correlations and temporal clustering of a proxy historical landslide time series, 2255 reported landslides 1951–2002, for an area in the Emilia‐Romagna Region, Italy. Landslide intensity is measured by the number of reported landslides in a day (D_L) and in an ‘event’ (S_event) of consecutive days with landsliding. The non‐zero values in both time series D_L and S_event are unequally spaced in time, and have heavy‐tailed frequency‐size distributions. To examine temporal correlations, we use power‐spectral analysis (Lomb periodogram) and surrogate data analysis, confronting our original D_L and S_event time series with 1000 shuffled (uncorrelated) versions. We conclude that the landslide intensity series D_L has strong temporal correlations and S_event has likely temporal correlations. To examine temporal clustering in D_L and S_event, we consider extremes over different landslide intensity thresholds. We first examine the statistical distribution of interextreme occurrence times, τ, and find Weibull distributions with parameter γ << 1·0 [D_L] and γ < 1·0 [S_event]; thus D_L and S_event each have temporal correlations, but S_event to a lesser degree. We next examine correlations between successive interextreme occurrence times, τ. Using autocorrelation analysis applied to τ, combined with surrogate data analysis, we find for D_L linear correlations in τ, but for S_event inconclusive results. However, using Kendall's rank correlation analysis we find for both D_L and S_event the series of τ are strongly correlated. Finally, we apply Fano Factor analysis, finding for both D_L and S_event the timings of extremes over a given threshold exhibit a fractal structure and are clustered in time. In this paper, we provide a framework for examining time series where the non‐zero values are strongly unequally spaced and heavy‐tailed, particularly important in the Earth Sciences due to their common occurrence, and find that landslide intensity time series exhibit temporal correlations and clustering. Many landslide models currently are designed under the assumption that landslides are uncorrelated in time, which we show is false. Copyright © 2010 John Wiley & Sons, Ltd.     

Analysis of historical landslide time series in the Emilia‐Romagna region,northern Italy

A catalogue of historical landslides, 1951–2002, for three provinces in the Emilia‐Romagna region of northern Italy is presented and its statistical properties studied. The catalogue consists of 2255 reported landslides and is based on historical archives and chronicles. We use two measures for the intensity of landsliding over time: (i) the number of reported landslides in a day (D_L) and (ii) the number of reported landslides in an event (S_event), where an event is one or more consecutive days with landsliding. From 1951–2002 in our study area there were 1057 days with 1 ≤ D_L ≤?45 landslides per day, and 596 events with 1 ≤ S_event ≤ 129 landslides per event. In the first set of analyses, we find that the probability density of landslide intensities in the time series are power‐law distributed over at least two‐orders of magnitude, with exponent of about ?2·0. Although our data is a proxy for landsliding built from newspaper reports, it is the first tentative evidence that the frequency‐size of triggered landslide events over time (not just the landslides in a given triggered event), like earthquakes, scale as a power‐law or other heavy‐tailed distributions. If confirmed, this could have important implications for risk assessment and erosion modelling in a given area. In our second set of analyses, we find that for short antecedent rainfall periods, the minimum amount of rainfall necessary to trigger landslides varies considerably with the intensity of the landsliding (D_L and S_event); whereas for long antecedent periods the magnitude is largely independent of the cumulative amount of rainfall, and the largest values of landslide intensity are always preceded by abundant rainfall. Further, the analysis of the rainfall trend suggests that the trigger of landslides in the study area is related to seasonal rainfall. Copyright © 2010 John Wiley & Sons, Ltd.     

Conditions when anisotropy is negligible for solute transfer in sediment beds of lakes or streams

To reduce the complexity and save computation time, an isotropic and a scalar dispersion model are explored and compared to the anisotropic advection/dispersion model to study the interstitial flow in a stream and lake sediment induced by a periodic pressure wave. In these systems, the solute transport is controlled by the ratio (R = a/(LS)) of the pressure wave steepness (a/L) to the stream slope (S), and the dispersivity ratio (λ = α_L/L) that measures the longitudinal dispersivity (α_L) relative to the pressure wave length (L). Through a series of numerical experiments, the conclusion is reached that a scalar dispersion model can be applied with satisfactory results for advection-dominated transport, i.e. when R ?  0.1 and λ ? 0.01, or λ ? 0.0001, i.e. Peclet number (Pe) ? 10000; an isotropic dispersion model can be applied when R ? 10 or λ ? 0.001, and the full anisotropic advection/dispersion model has to be applied when R > 10 and λ > 0.001.     

The causes of bedload pulses in a gravel channel: the implications of bedload grain‐size distributions

Unsteady bedload transport was measured in two c. 5 m wide anabranches of a gravel‐bed braided stream draining the Haut Glacier d'Arolla, Switzerland, during the 1998 and 1999 melt seasons. Bedload was directly sampled using 152 mm square Helley–Smith type samplers deployed from a portable measuring bridge, and independent transport rate estimates for the coarser size fractions were obtained from the dispersion of magnetically tagged tracer pebbles. Bedload transport time series show pulsing behaviour under both marginal (1998) and partial (1999) transport regimes. There are generally weak correlations between transport rates and shear stresses determined from velocity data recorded at the measuring bridge. Characteristic parameters of the bedload grain‐size distributions (D₅₀, D₈₄) are weakly correlated with transport rates. Analysis of full bedload grain‐size distributions reveals greater structure, with a tendency for transport to become less size selective at higher transport rates. The bedload time series show autoregressive behaviour but are dif?cult to distinguish by this method. State–space plots, and associated measures of time‐series separation, reveal the structure of the time series more clearly. The measured pulses have distinctly different time‐series characteristics from those modelled using a one‐dimensional sediment routing model in which bed shear stress and grain size are varied randomly. These results suggest a mechanism of pulse generation based on irregular low‐amplitude bedforms, that may be generated in‐channel or may represent the advection of material supplied by bank erosion events. Copyright © 2003 John Wiley & Sons, Ltd.     

Landslide inventories and their statistical properties

Landslides are generally associated with a trigger, such as an earthquake, a rapid snowmelt or a large storm. The landslide event can include a single landslide or many thousands. The frequency–area (or volume) distribution of a landslide event quanti?es the number of landslides that occur at different sizes. We examine three well‐documented landslide events, from Italy, Guatemala and the USA, each with a different triggering mechanism, and ?nd that the landslide areas for all three are well approximated by the same three‐parameter inverse‐gamma distribution. For small landslide areas this distribution has an exponential ‘roll‐over’ and for medium and large landslide areas decays as a power‐law with exponent ‐2·40. One implication of this landslide distribution is that the mean area of landslides in the distribution is independent of the size of the event. We also introduce a landslide‐event magnitude scale m_L = log(N_LT), with N_LT the total number of landslides associated with a trigger. If a landslide‐event inventory is incomplete (i.e. smaller landslides are not included), the partial inventory can be compared with our landslide probability distribution, and the corresponding landslide‐event magnitude inferred. This technique can be applied to inventories of historical landslides, inferring the total number of landslides that occurred over geologic time, and how many of these have been erased by erosion, vegetation, and human activity. We have also considered three rockfall‐dominated inventories, and ?nd that the frequency–size distributions differ substantially from those associated with other landslide types. We suggest that our proposed frequency–size distribution for landslides (excluding rockfalls) will be useful in quantifying the severity of landslide events and the contribution of landslides to erosion. Copyright © 2004 John Wiley & Sons, Ltd.     

Determination of critical shear stress of non‐cohesive soils using submerged jet test and turbulent kinetic energy

Laboratory tests using Jet Erosion Testing (JET) apparatus, impinging normally on a horizontal boundary, were conducted to determine the critical shear stress (τ_c) of non‐cohesive soil samples. A three‐dimensional (3D) SonTek/YSI 16 MHz Micro‐Acoustic Doppler Velocimeter (MicroADV) was used to measure turbulent kinetic energy (TKE) at a radial limit of entrainment in the wall jet zone and the measurements were used to calculate τ_c of the samples. The results showed that TKE increases exponentially with increasing particle size. The τ_c from this study were comparable (R² = 0.8) to the theoretical τ_c from Shields diagram after bed roughness scale ratio (D/k_s), due to the non‐uniform bed conditions, was accounted for. This study demonstrated that JET and TKE can be used to determine τ_c of non‐cohesive soils. The use of JET and TKE was found to be faster and easier when compared to the conventional approach of using flumes. A relationship of TKE at the onset of incipient motion (TKE_c) and samples’ D₅₀ developed in this study can be used to predict τ_c of non‐cohesive soils under similar non‐uniform conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Eddy correlations for water flow in a single fracture with abruptly changing aperture

Fluid flow in single fractures with non‐uniform apertures is an important research subject in many disciplines. The abruptly changing aperture is a special case of such non‐uniformity. This paper simulates water flow in a single fracture with abruptly changing aperture (SF‐ACA) using the Lattice Boltzmann Method (LBM) and the Finite Volume Method (FVM). The flow occurs with the Reynolds number (Re) ranging from 5 to 900 and a ratio of aperture change (E) of 3 (E = D/d, where D and d are the larger and smaller apertures, respectively). For Re values between 5 and 100, both LBM and FVM can successfully simulate the eddy development in the expansion regime of an SF‐ACA. Flow with high Re values (up to 900) is simulated by FVM, which appears to be numerically more stable than LBM for high‐Re flow problems studied here. The flow symmetry in the expansion regime breaks at the Re value between 400 and 500. Our simulation result shows a linear relationship between l₁/d and Re at low Re (5–100) or higher Re (110–900) values, where defined as the length from the location of abrupt expansion to the right edge of the first eddy along the flow direction. If considering the simulation results for the entire simulated range of Re (5–900), the l₁/d–Re relationship is better described by a non‐linear logarithmical function. The l₁/d approaches an asymptotic constant at large Re. Copyright © 2011 John Wiley & Sons, Ltd.     

Flow resistance in steep mountain streams

Resistance to flow at low to moderate stream discharge was examined in five small (12–77 km² drainage area) tributaries of Chilliwack River, British Columbia, more than half of which exhibit planar bed morphology. The resulting data set is composed of eight to 12 individual estimates of the total resistance to flow at 61 cross sections located in 13 separate reaches of five tributaries to the main river. This new data set includes 625 individual estimates of resistance to flow at low to moderate river stage. Resistance to flow in these conditions is high, highly variable and strongly dependent on stage. The Darcy–Weisbach resistance factor (ff) varies over six orders of magnitude (0·29–12 700) and Manning's n varies over three orders of magnitude (0·047–7·95). Despite this extreme range, both power equations at the individual cross sections and Keulegan equations for reach‐averaged values describe the hydraulic relations well. Roughness is divided into grain and form (considered as all non‐grain sources) components. Form roughness is the dominant component, accounting for about 90% of the total roughness of the system (i.e., form roughness is on average 8.6 times as great as grain roughness). Of the various quantitative and qualitative form‐roughness indicators observed, only the sorting coefficient (σ = D₈₄/D₅₀) correlates well with form roughness. Copyright © 2008 John Wiley & Sons, Ltd.     

Modelling spatial variations in dissolved oxygen in fine‐grained streams under uncertainty

This paper presents a novel triple‐layer model, called VART DO‐3L, for simulation of spatial variations in dissolved oxygen (DO) in fine‐grained streams, characterized by a fluid mud (fluff or flocculent) layer (an advection‐dominated storage zone) as the interface between overlying stream water and relatively consolidated streambed sediment (a diffusion‐dominated storage zone). A global sensitivity analysis is conducted to investigate the sensitivity of VART DO‐3L model input parameters. Results of the sensitivity analysis indicate that the most sensitive parameter is the relative size of the advection‐dominated storage zones (A_s/A), followed by a lumped reaction term (R) for the flocculent layer, biological reaction rate (μ_o) in diffusive layer and biochemical oxygen demand concentration (L) in water column. In order to address uncertainty in model input parameters, Monte Carlo simulations are performed to sample parameter values and to produce various parameter combinations or cases. The VART DO‐3L model is applied to the Lower Amite River in Louisiana, USA, to simulate vertical and longitudinal variations in DO under the cases. In terms of longitudinal variation, the DO level decreases from 7.9 mg l at the Denham Springs station to about 2.89 mg l^?1 at the Port Vincent station. In terms of vertical variation, the DO level drops rapidly from the overlying water column to the advection‐dominated storage zone and further to the diffusive layer. The DO level (C_F) in the advective layer (flocculent layer) can reach as high as 40% of DO concentration (C) in the water column. The VART DO‐3L model may be applied to similar rivers for simulation of spatial variations in DO level. Copyright © 2014 John Wiley & Sons, Ltd.     

Testing the Universal Soil Loss Equation‐MB equation in plots in Central and South Italy

Planning soil conservation strategies requires predictive techniques at event scale because a large percentage of soil loss over a long‐time period is due to relatively few large storms. Considering runoff is expected to improve soil loss predictions and allows relation of the process‐oriented approach with the empirical one, furthermore, the effects of detachment and transport on soil erosion processes can be distinguished by a runoff component. In this paper, the empirical model USLE‐MB (USLE‐M based), including a rainfall‐runoff erosivity factor in which the event rainfall erosivity index EI₃₀ of the Universal Soil Loss Equation (USLE) multiplies the runoff coefficient Q_R raised to an exponent b₁ > 1 is tested by the measurements carried out for the Masse (10 plots) and Sparacia (22 plots) experimental stations in Italy. For the Masse experimental station, an exponent b₁ > 1 was also estimated by tests carried out by a nozzle‐type rainfall simulator. For each experimental site in fallow conditions, the effect of the sample size of the plot soil loss measurements on the estimate of the b₁ coefficient was also studied by the extraction of a fixed number N of randomly obtained pairs of the normalized soil loss and runoff coefficient. The analysis showed that the variability of b₁ with N is low and that 350 pairs are sufficient to obtain a stable estimate of b₁. A total of 1,262 soil loss data were used to parameterize the model both locally and considering the two sites simultaneously. The b₁ exponent varied between the two sites (1.298–1.520), but using a common exponent (1.386) was possible. Using a common b₁ exponent for the two experimental areas increases the practical interest for the model and allows the estimation of a baseline component of the soil erodibility factor, which is representative of the at‐site soil intrinsic and quasi‐static properties. Development of a single USLE‐MB model appears possible, and sampling other sites is advisable to develop a single USLE‐MB model for general use.     

Video‐based gravel transport measurements with a flume mounted light table

The study of bedload transport processes is constrained by an inability to monitor the mass, volume and grain size distribution of sediment in transport at high temporal frequencies. Building upon a previously published design, we have integrated a high‐resolution (1392 × 1024 pixels) video camera with a light table to continuously capture images of 2–181 mm material exiting a flume. The images are continuously recorded at a rate of 15 to 20 frames per second and are post‐processed using LabView^(?) software, yielding continuous grain‐size‐specific transport information on a per second basis. The video capture rate is sufficient to record multiple images of each grain leaving the flume so that particle velocities can be measured automatically. No manual image processing is required. After calibration the method is accurate and precise for sediment in the 2 mm through to 45 mm grain size classes compared with other means of measuring bedload. Based on a set of validation samples, no statistically significant difference existed between the D₁₀, D₁₆, D₂₅, D₅₀, D₇₅, D₈₄, D₉₀ and D₉₅ determined by sieving captured samples and the D_i values determined with the system. On average the system overpredicted transport by 4 per cent (n = 206, SD = 42%). This error can be corrected easily by simply weighing the mass of sediment that leaves the flume. The technology is relatively inexpensive and provides high‐resolution data on coarse sediment transport out of a flume. Copyright © 2008 John Wiley & Sons, Ltd.     

Dispersions for the pushover‐based risk assessment of reinforced concrete frames and cantilever walls

The paper presents the results of an investigation into the dispersion values, expressed in terms of limit‐state spectral accelerations, which could be used for the pushover‐based risk assessment of low‐height to mid‐height reinforced concrete frames and cantilever walls. The results of an extensive parametric study of a portfolio of test structures indicated that the dispersion values due to record‐to‐record variability and modelling uncertainty (β_LS,RU) are within the range from 0.3 to 0.55 for the near collapse limit state, and between 0.35 and 0.60 for the collapse limit state. The dispersions β_LS,RU proposed for the code‐conforming and the majority of old (non code‐conforming) frames are in between these values. On the other hand, the dispersions proposed for the old frames with a soft storey and an invariant plastic mechanism, and for the code‐conforming cantilever walls, are at the lower and upper bounds of the presented values, respectively. The structural parameters that influence these dispersions were identified, and the influence of different ground motion sets, and of the models used for the calculation of the rotation capacities of the columns, on the calculated fragility parameters was examined and quantified. The proposed dispersion values were employed in a practice‐oriented pushover‐based method for the estimation of failure probability for eight selected examples. The pushover‐based risk assessment method, although extremely simple and economical when compared with more rigorous probabilistic methods, was able to predict seismic risk with reasonable accuracy, thus showing it to be a practical tool for engineers. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical investigation of alternative fracture stiffness measures and their respective scaling behaviours

We study the mechanical deformation of fractures under normal stress, via tangent and specific fracture stiffnesses, for different length scales using numerical simulations and analytical insights. First, we revisit an equivalent elastic layer model that leads to two expressions: the tangent stiffness is the sum of an “intrinsic” stiffness and the normal stress, and the specific stiffness is the tangent stiffness divided by the fracture aperture at current stress. Second, we simulate the deformation of rough fractures using a boundary element method where fracture surfaces represented by elastic asperities on an elastic half‐space follow a self‐affine distribution. A large number of statistically identical “parent” fractures are generated, from which sub‐fractures of smaller dimensions are extracted. The self‐affine distribution implies that the stress‐free fracture aperture increases with fracture length with a power law in agreement with the chosen Hurst exponent. All simulated fractures exhibit an increase in the specific stiffness with stress and an average decrease with increase in length consistent with field observations. The simulated specific and tangent stiffnesses are well described by the equivalent layer model provided the “intrinsic” stiffness slightly decreases with fracture length following a power law. By combining numerical simulations and the analytical model, the effect of scale and stress on fracture stiffness measures can be easily separated using the concept of “intrinsic” stiffness. We learn that the primary reason for the variability in specific stiffness with length comes from the fact that the typical aperture of the self‐affine fractures itself scales with the length of the fractures.     

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.