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.     

The modified Reynolds equation for non-wetting fluid flow through a rough-walled rock fracture

The no-slip boundary condition has usually been assumed to hold for the Reynolds equations (local cubic law) for fluid flow through rough-walled fractures. However, its validity for non-wetting fluid flow, such as prevails in fractured oil reservoirs, has been questioned. A series of experiments with a rough-walled fracture with mean aperture of 760 μm finds a higher flow rate for non-wetting fluid than wetting fluid. A modified Reynolds equation with a slip boundary condition is derived for non-wetting fluid flow through rough-walled fractures. Comparison of the modified Reynolds equation predictions with experimental results confirms that slip was a plausible explanation for a higher flow rate. The amount by which the flow rate for non-wetting fluids exceeds that for wetting fluids is found to depend highly on, and increase with, the degree to which the flowing fluid was non-wetted to thin immobile films on the surfaces. Numerical studies using the modified Reynolds equation indicate that the flow rate of non-wetting fluid became higher than that of wetting fluid as the roughness of the fracture increases. As the aperture becomes smaller, the flow rate ratio of non-wetting fluid to wetting fluid becomes large, leading to the endpoint relative permeability for the non-wetting fluid to exceed 1. The experimental and numerical studies clearly show that as the aperture of the fracture became less than a few hundred microns, the modified Reynolds equation with slip boundary conditions provides a better model for flow of a non-wetting fluid through rough-walled fractures.     

Dependency of flow and transport properties on aperture distributions and compression states

Fluid conductivity and elastic properties in fractures depend on the aperture geometry – in particular, the roughness of fracture surfaces. In this study, we have characterized the surface roughness with a log-normal distribution and investigated the transport and flow behaviour of the fractures with varying roughness characteristics. Numerical flow and transport simulations have been performed on a single two-dimensional fracture surface, whose aperture geometry changes with different variances and correlation lengths in each realization. We have found that conventional measurement of hydraulic conductivity alone is insufficient to determine these two parameters. Transient transport measurements, such as the particle breakthrough time, provide additional constraints to the aperture distribution. Nonetheless, a unique solution to the fracture aperture distribution is still under-determined with both hydraulic conductivity and transport measurements. From numerical simulations at different compression states, we have found that the flow and transport measurements exhibit different rates of changes with respect to changes in compression. Therefore, the fracture aperture distribution could be further constrained by considering the flow and transport properties under various compression states.     

A smoothed particle hydrodynamics model for droplet and film flow on smooth and rough fracture surfaces

Flow on fracture surfaces has been identified by many authors as an important flow process in unsaturated fractured rock formations. Given the complexity of flow dynamics on such small scales, robust numerical methods have to be employed in order to capture the highly dynamic interfaces and flow intermittency. In this work we use a three-dimensional multiphase Smoothed Particle Hydrodynamics (SPH) model to simulate surface tension dominated flow on smooth fracture surfaces. We model droplet and film flow over a wide range of contact angles and Reynolds numbers encountered in such flows on rock surfaces. We validate our model via comparison with existing empirical and semi-analytical solutions for droplet flow. We use the SPH model to investigate the occurrence of adsorbed trailing films left behind droplets under various flow conditions and its importance for the flow dynamics when films and droplets coexist. It is shown that flow velocities are higher on prewetted surfaces covered by a thin film which is qualitatively attributed to the enhanced dynamic wetting and dewetting at the trailing and advancing contact lines. Finally, we demonstrate that the SPH model can be used to study flow on rough surfaces.     

Macroscale surface roughness and frictional resistance in overland flow

The hydraulics of overland flow on rough granular surfaces can be modelled and evaluated using the inundation ratio rather than the flow Reynolds number, as the primary dimensionless group determining the flow behaviour. The inundation ratio describes the average degree of submergence of the surface roughness and is used to distinguish three flow regimes representing partially inundated, marginally inundated and well-inundated surfaces. A heuristic physical model for the flow hydraulics in each regime demonstrates that the three states of flow are characterized by very different functional dependencies of frictional resistance on the scaled depth of flow. At partial inundation, flow resistance is associated with the drag force derived from individual roughness and therefore increases with depth and percentage cover. At marginal inundation, the size of the roughness elements relative to the depth of flow controls the degree of vertical mixing in the flow so that frictional resistance tends to decrease very rapidly with increasing depth of flow. Well-inundated flows are described using rough turbulent flow hydraulics previously developed for open channel flows. These flows exhibit a much more gradual decrease in frictional resistance with increasing depth than that observed during marginal inundation. A data set compiled from previously published studies of overland flow hydraulics is used to assess the functional dependence of frictional resistance on inundation ratio over a wide range of flow conditions. The data confirm the non-monotonic dependence predicted by the model and support the differentiation of three flow regimes based on the inundation ratio. Although the percentage cover and the surface slope may be of importance in addition to the inundation ratio in the partially and marginally inundated regimes, the Reynolds number appears to be of significance only in describing well-inundated flows at low to moderate Reynolds numbers. As these latter conditions are quite rare in natural environments, the inundation ratio rather than the Reynolds number should be used as the primary dimensionless group when evaluating the hydraulics of overland flow on rough surfaces. © 1997 by John Wiley & Sons, Ltd.     

Double-averaged velocity and stress distributions for hydraulically-smooth and transitionally-rough turbulent flows

We analyse experimental measurements of turbulent open-channel flow over hydraulically-smooth and transitionally-rough beds using the double-averaging methodology. Oil with a viscosity of 15×10^?6 m2/s is used instead of water so that transitional-range roughness Reynolds numbers can be achieved with large (11.1 mm) roughness elements, allowing spatial variations in the mean velocity field to more easily be measured. Distributions of double-averaged velocities, turbulence intensities, form-induced intensities, and viscous, Reynolds, form-induced and total shear stresses are studied with comparisons made between distributions for hydraulically-smooth, transitionally-rough, and fully-rough boundaries. Measured streamwise turbulence intensities for all experiments peaked at a constant distance from the bed (z ⁺+d ⁺ = 15) when elevation scale is adjusted using the zero-plane displacement d for the logarithmic velocity distribution. This collapse suggests that turbulence intensity distributions may be useful in assessing appropriate values of d for transitionally-rough and fully-rough boundaries. Form-induced normal and shear stresses above the roughness tops were found to collapse towards a common curve independent of roughness Reynolds number.     

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.     

Segmentation of soil roughness profiles

A robust and flexible algorithm to study spatial series like soil roughness profile has been introduced. It avoids using classic spectral analysis, considers the profile first and foremost as non-stationary and makes it possible to identify the separate domains inside the profile where chosen statistical parameters and roughness indexes have their own value. The method analyses a roughness profile considering it as an assemblage of several entities that may differ in terms of statistical properties and length, without establishing constraints as to number and extension. The method derives the variability of statistical and roughness properties along the profile and extracts the possible components - random and oriented - detectable inside the sample. Some examples of application illustrate the possibility offered by the method to study real roughness profiles recorded in the field by a portable laser microprofilometer. The procedure proposed allows the investigation of local roughness properties with varying degrees of accuracy and should be useful to monitor the differential evolution of roughness on patterned soil surface, increasing the overall information content. A general definition of ‘ordered roughness’ is introduced. The definition proposed seems more suited to current techniques for the numerical treatment of digital profiles and for the existing physical relationships between the scale of observation of roughness and the scale of the process investigated (hydraulic resistance, water storage in depressions). Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical simulation of three‐dimensional flow hydraulics in a braided channel

Results are presented from a numerical simulation of three‐dimensional flow hydraulics around a mid‐channel bar carried out using the FLUENT/UNS computational fluid dynamics (CFD) software package. FLUENT/UNS solves the three‐dimensional Reynolds‐averaged form of the Navier–Stokes equations. Turbulence closure is achieved using a RNG k–ϵ model. Simulated flow velocities are compared with measured two‐dimensional velocities (downstream and cross‐stream) obtained using an electromagnetic current meter (ECM). The results of the simulation are qualitatively consistent with the flow structures observed in the field. Quantitative comparison of the simulated and measured velocity magnitudes indicates a strong positive correlation between the two (r=0·88) and a mean difference of 0·09 m s⁻¹. Deviations between simulated and measured velocities may be identified that are both random and systematic. The former may reflect a number of factors including subgrid‐scale natural spatial variability in flow velocities associated with local bed structures and measurement uncertainty resulting from problems of ECM orientation. Model mesh configuration, roughness parameterization and inlet boundary condition uncertainty may each contribute to systematic differences between simulated and measured flow velocities. These results illustrate the potential for using CFD software to simulate flow hydraulics in natural channels with complex configurations. They also highlight the need for detailed spatially distributed datasets of three‐dimensional flow variables to establish the accuracy and applicability of CFD software. Copyright © 1999 John Wiley & Sons, Ltd.     

Study on effect of surface roughness on overland flow from different geometric surfaces through numerical simulation

Effect of variability in surface roughness on overland flow from different geometric surfaces is investigated using numerical solution of diffusion wave equation. Three geometric surfaces rectangular plane, converging and diverging plane at slopes 1 to 3% are used. Overland flow is generated by applying rainfall at constant intensity of 10 mm/h for period 30 min and 100 min. Three scenarios of spatial roughness conditions viz. roughness increasing in downstream direction, roughness decreasing in downstream direction and roughness distributed at random are considered. Effect of variability of roughness on overland flow in terms of depth, velocity of flow and discharge along the distance from upstream to downstream for different geometric surfaces are discussed in detail. Results from the study indicate that roughness distribution has significant effect on peak, time to peak and overall shape of the overland flow hydrograph. The peak occurs earlier for the scenario when roughness increases in downstream direction as compared to scenario when roughness is decreasing in downstream for all three geometric surfaces due to very low friction factor and more velocity at the top of the domain. The converging plane attains equilibrium state early as compared to rectangular and diverging plane. Different set of random values result in different time to peak and shape of hydrograph for rectangular and diverging plane. However, in case of converging plane, the shape of computed hydrographs remains almost similar for different sets of random roughness values indicating stronger influence of converging geometry than effect due to variation of roughness sequence on computed runoff hydrograph. Hierarchically, the influence of geometry on overland flow is stronger than the influence of slope and the influence of slope is stronger than the influence of roughness. Copyright © 2013 John Wiley & Sons, Ltd.     

Two-phase flow properties of a horizontal fracture: The effect of aperture distribution

A systematic numerical method has been presented to investigate the constitutive relationships between two-phase flow properties of horizontal fractures and aperture distributions. Based on fractal geometry, single rough-walled fractures are generated numerically by modified successive random addition (SRA) method and then aperture distributions with truncated Gaussian distribution are formed by shear displacement between lower and upper surfaces. (The truncated Gaussian distribution is used to describe aperture evolution under different normal stresses.) According to the assumption of two-dimensional porous media and local parallel plate model, invasion percolation approach is employed to model the two-phase flow displacement (imbibition) in generated horizontal fractures, in which capillary forces are dominant over viscous and gravity forces. For truncated Gaussian distributions, constitutive relationships from numerical simulation are compared to closed-form relationships and a good agreement is obtained. The simulation results indicate strong phase interference with the sum of two phase relative permeability values being less than one in the intermediate saturations. It is found that fracture properties related to residual saturations depend on spatial correlation of aperture distributions. Based on the simulation results, we proposed an empirical relationship between the fracture residual-saturation-rated parameters and the corresponding aperture distributions.     

Effects of non-submerged boulder on flow characteristics – A field investigation

The effect of fully submerged boulders on the flow structure in channels has been studied by some researchers. However, many natural streams have bed material with boulders that are not fully submerged under water. In many natural streams, boulders cover between 1% and 10% of the area of the stream reach. The effect of non-submerged boulders on the velocity profile and flow characteristics is very important for assessing riverbed deformation. The objectives of this paper are to find the pattern of velocity distribution around a non-submerged boulder and to compare it with the classical studies on flow resistance and Reynolds stress distribution in open channels. Also, by considering the variation in the Reynolds stress distribution at different locations around a non-submerged boulder, the effect of a non-submerged boulder on the estimation of shear velocity and resistance to flow has been investigated. Results indicates that inside the scour hole caused by a non-submerged boulder in a river velocity distributions are irregular. However, velocity distributions are regular outside the scour hole. The presence of the boulder causes a considerable deviation of the Reynolds shear stress from the classic distribution, showing a non-specific distribution with negative values. The classical methods for calculating shear velocity are not suitable because these methods do not give detailed velocity and Reynolds stress distributions in natural rivers with a lot of boulders. Thus, the effect of a non-submerged boulder on the estimation of the resistance to flow by considering the variations in velocity and Reynolds stress distributions at different locations around a non-submerged boulder is important and needs to be studied in a natural river instead of just in laboratory flumes. The negative values in Reynolds stress distribution around a boulder indicate that the classical methods are unable to predict resistance to flow, and also show strong turbulence inside the scour hole where the complex flow conditions present ambiguous Reynolds stress distributions. In the current study, to obtain a reasonable estimation of parameters in natural rivers, the classical method has been modified by considering velocity and Reynolds stress distributions through the boundary layer method.     

A double-porosity model for a fractured aquifer with non-Darcian flow in fractures

Abstract

Non-Darcian flow in a finite fractured confined aquifer is studied. A stream bounds the aquifer at one side and an impervious stratum at the other. The aquifer consists of fractures capable of transmitting water rapidly, and porous blocks which mainly store water. Unsteady flow in the aquifer due to a sudden rise in the stream level is analysed by the double-porosity conceptual model. Governing equations for the flow in fractures and blocks are developed using the continuity equation. The fluid velocity in fractures is often too high for the linear Darcian flow so that the governing equation for fracture flow is modified by Forcheimer's equation, which incorporates a nonlinear term. Governing equations are coupled by an interaction term that controls the quasi-steady-state fracture—block interflow. Governing equations are solved numerically by the Crank-Nicolson implicit scheme. The numerical results are compared to the analytical results for the same problem which assumes Darcian flow in both fractures and blocks. Numerical and analytical solutions give the same results when the Reynolds number is less than 0.1. The effect of nonlinearity on the flow appears when the Reynolds number is greater than 0.1. The higher the rate of flow from the stream to the aquifer, the higher the degree of nonlinearity. The effect of aquifer parameters on the flow is also investigated. The proposed model and its numerical solution provide a useful application of nonlinear flow models to fractured aquifers. It is possible to extend the model to different types of aquifer, as well as boundary conditions at the stream side. Time-dependent flow rates in the analysis of recession hydrographs could also be evaluated by this model.     

Hydrodynamic characteristics of rill flow on steep slopes

Rill erosion is a dominant sediment source on sloping lands. However, the amount of soil loss from rills on steep slopes is vastly more than that on gentle slopes because of differences in rill shape and hydraulic patterns. The aims of this paper are to determine the hydrodynamic characteristics of rills and the friction coefficients in steep slope conditions and to propose modifications of some hydraulic parameters used in soil loss prediction models. A series of inflow experiments was conducted on loess slopes. The results show that the geometric and hydraulic properties of rill on the steep loess slopes, which are characterized by the mean width of cross sections, mean velocity and mean depth of flow, are related to discharge and slope gradient in power functions. However, the related exponents to discharge are 0.26, 0.48 and 0.26, respectively, which are different from the exponents derived in previous studies, which were conducted on gentle slopes. The Manning roughness coefficient ranged from 0.035 to 0.071, with an average of 0.0536, and the Darcy–Weisbach friction coefficients varied from 0.4 to 1.9. The roughness coefficients are closely related to the Reynolds numbers and flow volumes; however, the correlations vary with slope gradient. The roughness coefficients are directly proportional to the Reynolds number and the flow volume on steep slopes, in contrast with the roughness coefficients found on gentle slopes, which decrease as the Reynolds number and flow volume increase. This difference is caused by the interactions among the hydraulics of the flow, the shape of the rills and the sediment concentrations on steep slopes. The results indicate that parameters used in models to predict rill erosion have to be modified according to slope gradient. Copyright © 2015 John Wiley & Sons, Ltd.     

Fracture Toughness and Fracture Roughness Interrelationship in Thermally treated Westerly Granite

This paper presents an experimental work aimed at assessing the correlation between fracture toughness (K _IC ) and fracture roughness for a series of Westerly granite specimens thermally treated up to 850°C. Mode I fracture toughness as a function of thermal treatment is determined using Cracked Chevron Notched Brazilian Disc specimens. The degree of roughness of the resultant fracture surfaces is analyzed with the aid of a high accuracy, high precision stereo-topometric measurement system. Roughness and toughness values display a negative correlation as a function of temperature. Fracture toughness decreases with increasing temperature due to the gradual opening of grain-grain boundaries in response to thermal stresses. Mode I fractures preferentially follow these weakened grain-grain boundaries, which in addition to the thermal expansion of individual grains, result in rougher failure profiles with increasing temperature. At low temperature, a distinct anisotropy in roughness was observed in all fracture surfaces with higher roughness values perpendicular to the direction of fracture propagation. However, higher treatment temperatures resulted in the homogenization of fracture roughness in all directions. These results confirm the important link among petrofabric analysis, fracture toughness, and fracture roughness in response to thermal treatment.     

The differential impact of some soil loss factors on flow,runoff creep and rainwash

The Hortonian model of runoff flow which had been thought to be applicable in arid areas has previously been shown not to be valid, notably in Israel, where inverse relations have been observed between slope angle, and runoff discharge and slope erosion. The paper discusses laboratory experiments on simulated slope conditions in a rather arid environment. It is shown by rain simulation on granite grus that infiltration capacity is a function of rainfall intensity, slope angle and runoff discharge. The infiltration capacity f can equal the rainfall intensity beyond a critical distance x(m) so that discharge becomes constant. Debris covers affect runoff hydraulics, especially on poorly cohesive soils, and both slow downslope and upslope movements which correspond to the process of so-called runoff creep can occur. Coarse debris and grass covers, as roughness factors, induce hydraulic discontinuities and activate local turbulent flow and slope erosion. Instead of being merely protective elements these factors tend to catalyze the slope wash, in comparison with naked surfaces, if the Reynolds number of the flow exceeds a certain critical value.     

Quantifying flow resistance in mountain streams using computational fluid dynamics modeling over structure-from-motion photogrammetry-derived microtopography

Flow resistance in mountain streams is important for assessing flooding hazard and quantifying sediment transport and bedrock incision in upland landscapes. In such settings, flow resistance is sensitive to grain-scale roughness, which has traditionally been characterized by particle size distributions derived from laborious point counts of streambed sediment. Developing a general framework for rapid quantification of resistance in mountain streams is still a challenge. Here we present a semi-automated workflow that combines millimeter- to centimeter-scale structure-from-motion (SfM) photogrammetry surveys of bed topography and computational fluid dynamics (CFD) simulations to better evaluate surface roughness and rapidly quantify flow resistance in mountain streams. The workflow was applied to three field sites of gravel, cobble, and boulder-bedded channels with a wide range of grain size, sorting, and shape. Large-eddy simulations with body-fitted meshes generated from SfM photogrammetry-derived surfaces were performed to quantify flow resistance. The analysis of bed microtopography using a second-order structure function identified three scaling regimes that corresponded to important roughness length scales and surface complexity contributing to flow resistance. The standard deviation σ_z of detrended streambed elevation normalized by water depth, as a proxy for the vertical roughness length scale, emerges as the primary control on flow resistance and is furthermore tied to the characteristic length scale of rough surface-generated vortices. Horizontal length scales and surface complexity are secondary controls on flow resistance. A new resistance predictor linking water depth and vertical roughness scale, i.e.  H/σ_z, is proposed based on the comparison between σ_z and the characteristic length scale of vortex shedding. In addition, representing streambeds using digital elevation models (DEM) is appropriate for well-sorted streambeds, but not for poorly sorted ones under shallow and medium flow depth conditions due to the missing local overhanging features captured by fully 3D meshes which modulate local pressure gradient and thus bulk flow separation and pressure distribution. An appraisal of the mesh resolution effect on flow resistance shows that the SfM photogrammetry data resolution and the optimal CFD mesh size should be about 1/7 to 1/14 of the standard deviation of bed elevation. © 2019 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.