Numerical investigation of geometrical and hydraulic properties in a single rock fracture during shear displacement with the Navier–Stokes equations
08 Mar 2015-Environmental Earth Sciences (Springer Berlin Heidelberg)-Vol. 73, Iss: 11, pp 7061-7074
TL;DR: In this article, a more detailed 2D numerical model was developed using a laser scanner system with a spacing grid of 0.1mm, and the results of the procedure for shear displacement simulation illustrate the distribution of the absolute velocity and pressure drop under the constant pressure gradient.
Abstract: Extensive research has shown that fluid flow through rock fractures is greatly influenced by surface roughness. For a single rock fracture, the roughness of the upper and bottom surfaces is the same in the initial condition and then its deformation occurs with normal stress and shear stress imposed on the natural rock. Previous researchers have paid considerable attention to describing the roughness of the single fracture and its effects on fluid flow. However, few studies have explained the fluid flow with shear displacement and the direction of the fluid flow velocity field. In this work, a more detailed 2D numerical model was developed using a laser scanner system with a spacing grid of 0.1 mm. To investigate the influence of shear displacement accurately, the COMSOL multiphase codes were applied. By applying the Navier–Stokes equations, the results of the procedure for shear displacement simulation illustrate the distribution of the absolute velocity and pressure drop under the constant pressure gradient. The velocities predicted at the vertical profiles of the inlet were similar to the parabolic velocity curve defined by the cubic laws. The mean mechanical aperture was usually larger than the hydraulic aperture from the measured flow rates, and a devised empirical equation was proposed to describe the relationship between the mechanical aperture and the hydraulic aperture values. The recirculation zones observed in directional fluid flow during shear explain the anisotropy of roughness of a single fracture, and the phenomenon argues the applicability of local cubic laws which overestimate the total fluid flow rate.
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11 Jun 2010
Abstract: The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.
1,557 citations
TL;DR: In this article, the impact of surface roughness on the nonlinear fluid flow through 3D self-affine rock fractures is investigated, and the authors show that the primary roughness mostly controls the pressure distribution and fracture flow paths at a large scale, whereas the secondary roughness determines the non-linear properties of the fluid flow at a local scale.
182 citations
TL;DR: In this article, the roles of hydraulic gradient, surface roughness, intersecting angle, and scale effect in the nonlinear fluid flow behavior through single fracture intersections were investigated using a CCD (charged coupled device) camera.
102 citations
TL;DR: In this article, the authors analyzed the influence of shear processes on nonlinear flow behavior through 3D rough-walled rock fractures and found that the relationship between the volumetric flow rate and hydraulic gradient can be well fit using Forchheimer's law.
101 citations
TL;DR: In this paper, a numerical approach was established to assess the effects of shear on the hydraulic properties of 2-D discrete fracture networks (DFNs) in both linear and nonlinear regimes.
77 citations
References
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TL;DR: The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm.
Abstract: The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.
1,729 citations
11 Jun 2010
Abstract: The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.
1,557 citations
TL;DR: In this article, the authors derived the cubic law of the Navier-Stokes equations for flow between smooth, parallel plates and showed that the effective hydraulic aperture is less than the mean aperture, by a factor that depends on the ratio of the mean value of the aperture to its standard deviation.
Abstract: The flow of a single-phase fluid through a rough-walled rock fracture is discussed within the context of fluid mechanics. The derivation of the ‘cubic law’ is given as the solution to the Navier-Stokes equations for flow between smooth, parallel plates - the only fracture geometry that is amenable to exact treatment. The various geometric and kinematic conditions that are necessary in order for the Navier-Stokes equations to be replaced by the more tractable lubrication or Hele-Shaw equations are studied and quantified. In general, this requires a sufficiently low flow rate, and some restrictions on the spatial rate of change of the aperture profile. Various analytical and numerical results are reviewed pertaining to the problem of relating the effective hydraulic aperture to the statistics of the aperture distribution. These studies all lead to the conclusion that the effective hydraulic aperture is less than the mean aperture, by a factor that depends on the ratio of the mean value of the aperture to its standard deviation. The tortuosity effect caused by regions where the rock walls are in contact with each other is studied using the Hele-Shaw equations, leading to a simple correction factor that depends on the area fraction occupied by the contact regions. Finally, the predicted hydraulic apertures are compared to measured values for eight data sets from the literature for which aperture and conductivity data were available on the same fracture. It is found that reasonably accurate predictions of hydraulic conductivity can be made based solely on the first two moments of the aperture distribution function, and the proportion of contact area.
1,003 citations
TL;DR: In this article, a simulation of flow between rough surfaces was done using a fractal model of surface topography and the hydraulic aperture was compared to the mean separation of the surfaces.
Abstract: Fluid flow through rock joints is commonly described by the parallel plate model where the volume flow rate varies as the cube of the joint aperture. However, deviations from this model are expected because real joint surfaces are rough and contact each other at discrete points. To examine this problem further, a computer simulation of flow between rough surfaces was done. Realistic rough surfaces were generated numerically using a fractal model of surface topography. Pairs of these surfaces were placed together to form a “joint” with a random aperture distribution. Reynolds equation, which describes laminar flow between slightly nonplanar and nonparallel surfaces, was solved on the two-dimensional aperture mesh by the finite-difference method. The solution is the local volume flow rate through the joint. This solution was used directly in the cubic law to get the so-called “hydraulic aperture.” For various surface roughnesses (fractal dimensions) the hydraulic aperture was compared to the mean separation of the surfaces. At large separations the surface topography has little effect. At small separations the flow is tortuous, tending to be channeled through high-aperture regions. The parameter most affecting fluid flow through rough joints is the ratio of the mean separation between the surfaces to the root-mean-square surface height. This parameter describes the distance the surface asperities protrude into the fluid and accounts for most of the disagreement with the parallel plate model. Variations in the fractal dimension produce only a second-order effect on the fluid flow. For the range of joint closures expected during elastic deformation these results show that the actual flow rate between rough surfaces is about 70–90% of that predicted by the parallel plate model.
826 citations
"Numerical investigation of geometri..." refers background in this paper
...Because of the necessity of understanding the laws and mechanisms coupling seepage field and stress field in rock masses, the study of fluid flow in a single fracture is of great value and has extensively emerged by theoretical, experimental and numerical approaches (Brown 1987; Brown et al. 1995; Thompson and Brown 1991; Zimmerman et al. 2004)....
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...…and mechanisms coupling seepage field and stress field in rock masses, the study of fluid flow in a single fracture is of great value and has extensively emerged by theoretical, experimental and numerical approaches (Brown 1987; Brown et al. 1995; Thompson and Brown 1991; Zimmerman et al. 2004)....
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TL;DR: The results of experiments using radial and unidirectional flow in a carefully described single rough aperture are reported and compared with numerical predictions as discussed by the authors, showing that the aperture distribution became more closely correlated in the direction parallel to the roughness ridges than in the shear direction.
438 citations
"Numerical investigation of geometri..." refers background in this paper
...…filled fracture on the condition of variable length by the single-tube and double-tube fracture models (Huang et al. 2013), while others tend to use two kinds of artificial parallel plates, with different combinations of shear displacement to investigate the fluid flow behaviors (Yeo et al. 1998)....
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...kinds of artificial parallel plates, with different combinations of shear displacement to investigate the fluid flow behaviors (Yeo et al. 1998)....
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