Flow In Fractured Rock Research Articles

A Nonlinear Version of the Reynolds Equation for Flow in Rock Fractures With Complex Void Geometries

AbstractThis study presents a nonlinear Reynolds equation (NRE) for single‐phase flow through rock fractures. The fracture void geometry is formed by connected wedge‐shaped cells at pore scale, based on the measured aperture field. An approximate analytical solution to two‐dimensional Navier‐Stokes equations is derived using the perturbation method to account for flow nonlinearity for wedge‐shaped geometries. The derived perturbation solution shows that the main contributors to the determination of general flow behaviors in local wedges are the degree of aperture variation relative to mean aperture, the ratio of aperture variation to wedge length, the Reynolds number, and the degree of wedge asymmetry. The transmissivity of the entire fracture is then solved with a field of local cell transmissivity that varies along both longitude and latitude directions on the fracture plane. The performance of the proposed NRE is tested against flow experiments and flow simulations by solving numerically the three‐dimensional Navier‐Stokes equations for three cases of rock fractures with different void geometries. Results from the proposed model are in close agreement with those obtained from simulations and experiments. As the Reynolds number increases, the pressure difference obtained from the NRE demonstrates the same nonlinear behavior as that obtained from the simulations. Overall, the mean discrepancy between the proposed model and flow simulations is 5.7% for Reynolds number ranging from 0.1 to 20, indicating that the proposed NRE can capture the flow nonlinearity in rock fractures.

Are Arid Regions Always that Appropriate for Waste Disposal? Examples of Complexity from Yucca Mountain, Nevada

The study of the hydrology of arid regions greatly expanded at the end of the 20th century as humans sought to reduce groundwater pollution from landfills, waste dumps and other forms of land disposal. Historically viewed as wastelands where little or no water percolated to the underlying water table, the discovery of large-scale contamination beneath arid disposal sites such as the Hanford nuclear complex in eastern Washington jumpstarted an industry in studying the hydrology of arid vadose zones and their transport behavior. These studies showed that, in spite of hyper aridity in many areas, precipitation often did infiltrate to deep water. The efforts at Yucca Mountain, Nevada to design a high-level nuclear repository stand out as one of the largest of such studies, and one that fundamentally changed our understanding of not only water flow in fractured rocks, but also of the range of our uncertainty of hydrologic processes in arid regions. In this review and commentary, we present some of the initial concepts of flow at Yucca Mountain, and the evolution in research to quantify the concepts. In light of continued stockpiling of high-level waste, and the renewed interest in opening Yucca Mountain for high-level waste, we then focus on the significant surprises and unanswered questions that remained after the end of the characterization and licensing period; questions that continue to demonstrate the challenges of a geologic repository and our uncertainty about critical processes for long-term, safe storage or disposal of some of our most toxic waste products.

Subdiffusive flow in a composite medium with a communicating (absorbing) interface

Two-dimensional subdiffusion in media separated by a partially communicating interface is considered. Starting with the appropriate Green’s functions, solutions are developed in terms of the Laplace transformation reflecting two circumstances at the interface: situations where there is perfect contact and situations where the interface offers a resistance. Asymptotic solutions are derived; limiting forms of the expressions reduce to known solutions for both classical diffusion and subdiffusion. Specifics are analyzed in depth with reference to flow in porous media with potential applications to the evaluation of the role of subsurface faults and flow in fractured rocks. Characteristics of the derivative responses are documented extensively as they are the linchpin for evaluation of pressure tests. Results given here may be used for evaluation at the Theis (1935; Eos Trans. AGU 2, 519–524) scale along with geological and geophysical properties, and production statistics. Yet a subdiffusive model does not imply a single value for properties. The method presented here may be extended to multiple contiguous media and to subdiffusive transport in many contexts (complex wellbores such as inclined, fractured and horizontal wells, situations such as sequestration, production of geothermal systems, etc.).

Permeability distribution and scaling in multi-stages carbonate damage zones: Insight from strike-slip fault zones in the Tarim Basin, NW China

Abstract Architecture and permeability of fault damage zones are important for understanding faulting mechanisms and fluid flow in fractured rocks. Nevertheless, this understanding is often hindered by limited availability of data, especially in the subsurface. A comprehensive suite of cores, logs and production data is here presented to unravel petrophysical properties and fracture characteristics of deep (>6000 m) Ordovician limestones within strike-slip fault zones of the Tarim intracratonic basin (NW China). The results show that (1) the carbonate porosity is mainly secondary dissolution porosity and comprises low porosity ( 5 mD) and high porosity (>8%); (2) fault damage zones are generally tight with fracture aperture

Effect of fluid slippage on eddy growth and non-Darcian flow in rock fractures

Fluid flow in rock fractures is usually theoretically and numerically investigated on the premise of no-slip boundary condition (BC). However, fluid slippage at rock surface is naturally present in some circumstances that might lead to a violation of no-slip BC. How and to what extent slip BC affects non-Darcian fracture flow remains poorly constrained. This study systematically investigated the slippery flow behaviors in rock fractures under sequentially-increasing pressure gradients. Two competing mechanisms impacting the apparent permeability (k) due to fluid slippage were revealed: k was not only enhanced by fluid slippage at fracture walls but also was reduced by the accelerated eddy growth due to enhanced velocity field. The increment in k caused by slip velocity initially dominated over the decrement caused by eddy growth during this competing process, but this dominant situation could reverse given a sufficiently large pressure gradient with fully-developed eddies. Moreover, the slippery fluid flow behavior was tested to be very sensitive to the slip length based on our sensitivity analysis. Therefore, a reasonable estimation of the slip length would be crucial to accurately determine the effects of fluid slippage. This study deepens our understanding of fluid flow in rock fractures when slip BC is present, and might provide theoretical guidance for organic pollution remediation, oil recovery, and geological carbon sequestration in fractured reservoirs.

Evaluate the Behavior of Fluid Flow in Rock Mass According to the Result of Water Pressure Tests in the Bakhtiari Dam Project

This paper aims to evaluate the behavior of fluid flow in rock fractures in a scale larger than the laboratory scale and to study the results of a number of water pressure tests in the Bakhtiari dam. The flow pattern of the fluid in the fractures existing in the tests or the relationship between the flow rate and fluid pressure has been divided into three groups comprising Darcy linear behavior, nonlinear behavior induced by inertial effect and nonlinear behavior under the influence of the fracture dilatation. The nonlinearity induced by inertial effect has been studied using nonlinear fitting of the water pressure test results. This analysis depicts that the Forchheimer equation properly describes the nonlinear behavior of fluid flow in the rock mass. Besides, the linear and nonlinear coefficients of the Forchheimer equation have been determined using a nonlinear fitting. By use of the quantitative criterion for flow nonlinearity in laboratory tests, a method has been presented for estimating critical Reynolds number in field tests. This quantification helps to estimate the critical Reynolds number and to determine the deviation from linear to nonlinear flow to better understand the fluid behavior in field tests such as water pressure test.

Effect of shear-induced aperture evolution on fluid flow in rock fractures

Understanding the shear-induced evolution of aperture size is critical for controlling fracture permeability in rock engineering projects. This study reports a robust procedure to evaluate the change of aperture size and its effect on fluid flow. A Finite Element Model is first calibrated based on the direct shear tests on a rough fracture to reproduce the uneven deformation of the fracture. A Fluent Computational Fluid Dynamics analysis is subsequently used to illuminate flow characteristics in the fracture without and with asperity deformation. The results indicate that the aperture size of the fracture increases dramatically with increasing shear displacement. However, the aperture size can be reduced slightly due to asperity deformation. Both the shear displacement and asperity deformation jointly influence flow characteristics in the sheared fracture (e.g., the development of eddy flow regions) and the accurate estimation of fracture permeability.

Corrigendum to “A Review of Critical Conditions for the Onset of Nonlinear Fluid Flow in Rock Fractures”

Corrigendum to “A Review of Critical Conditions for the Onset of Nonlinear Fluid Flow in Rock Fractures”

An Efficient Computational Model for Simulating Stress-dependent Flow in Three-dimensional Discrete Fracture Networks

In-situ modelling of stress-dependent fluid flow in fractured rocks is important for various applications in rock engineering. However, precise determination of the hydraulic aperture of subsurface fractures, particularly at great depths, is often quite difficult. One of the most important parameters affecting the aperture is the stress field. Therefore, in this study, a new FEM model is proposed to study the effect of the in-situ stresses on the flow rate in fractured rocks with limited fracture lengths using a one-way hydromechanical coupling scheme and various non-linear joint constitutive models. The model is computationally efficient and of low-cost for various applications, and it provides results that are consistent with those from time-consuming two-way coupling methods. A series of sensitivity analyses have also been carried out to investigate key parameters in the model and to demonstrate how the fracture aperture and fluid flow change with variation of the in-situ stress field.

Numerical analysis of fluid–rock interactions in hydraulic fracturing

Hydraulic fracturing is a process of fluid injection, which creates tensile stresses in rock in order to overcome the tensile strength of the rock. In this study, a three-phase hydro-mechanical model is developed for simulating hydraulic fracturing. The three phases include: porous solid, fracturing fluid and reservoir fluid. Two numerical simulators are coupled together to model multiphase fluid flow in fractured rock undergoing deformations, ranging from linear elastic to large, non-linear inelastic deformations. The fluid flow model involves solving the Navier–Stokes equations using the finite-volume method. The flow model is coupled with the geomechanics model to simulate the interaction between fluid flow inside the fractures with rock deformations. For any time step, the pore pressures from the flow model are used as input for the geomechanics model for the determination of stresses, strains and displacements. The strains derived from the geomechanics model are in turn used to calculate changes to the reservoir parameters that are fed as input to the flow model. This iterative process continues until both models are converged. A parametric study is conducted and the results show that changes in rock mechanical properties as well as fluid parameters could lead to significant changes in the hydraulic fracture propagation.

FDEM‐TH3D: A three‐dimensional coupled hydrothermal model for fractured rock

SummaryThis paper proposes a three‐dimensional coupled hydrothermal model for fractured rock based on the finite‐discrete element method to simulate fluid flow and heat transport. The 3D coupled hydrothermal model is composed of three main parts: a heat conduction model for the rock matrix, a heat transfer model for the fluid in the fractures (including heat conduction and heat convection), and a heat exchange model between the rock matrix and the fluid in the fractures. Four examples with analytical solutions are provided to verify the model. A heat exchange experiment of circulating water in a cylindrical granite sample with one fracture is simulated. The simulation results agree well with the experimental results. The effects of the fracture aperture, fluid viscosity, and pressure difference on the heat exchange between the fluid and rock are studied. Finally, an application concerned with heat transport and fluid flow in fractured rock is presented. The simulation results indicate that the 3D fully coupled hydrothermal model can capture the fluid flow and temperature evolution of rocks and fluids.

Simulation of Fluid Flow in Fractured Rocks Based on the Discrete Fracture Network Model Optimized by Measured Information

Simulation of Fluid Flow in Fractured Rocks Based on the Discrete Fracture Network Model Optimized by Measured Information

Numerical investigation on permeability evolution behavior of rock by an improved flow-coupling algorithm in particle flow code

Permeability is a vital property of rock mass, which is highly affected by tectonic stress and human engineering activities. A comprehensive monitoring of pore pressure and flow rate distributions inside the rock mass is very important to elucidate the permeability evolution mechanisms, which is difficult to realize in laboratory, but easy to be achieved in numerical simulations. Therefore, the particle flow code (PFC), a discrete element method, is used to simulate permeability behaviors of rock materials in this study. Owe to the limitation of the existed solid-fluid coupling algorithm in PFC, an improved flow-coupling algorithm is presented to better reflect the preferential flow in rock fractures. The comparative analysis is conducted between original and improved algorithm when simulating rock permeability evolution during triaxial compression, showing that the improved algorithm can better describe the experimental phenomenon. Furthermore, the evolution of pore pressure and flow rate distribution during the flow process are analyzed by using the improved algorithm. It is concluded that during the steady flow process in the fractured specimen, the pore pressure and flow rate both prefer transmitting through the fractures rather than rock matrix. Based on the results, fractures are divided into the following three types: I) fractures link to both the inlet and outlet, II) fractures only link to the inlet, and III) fractures only link to the outlet. The type I fracture is always the preferential propagating path for both the pore pressure and flow rate. For type II fractures, the pore pressure increases and then becomes steady. However, the flow rate increases first and begins to decrease after the flow reaches the stop end of the fracture and finally vanishes. There is no obvious pore pressure or flow rate concentration within type III fractures.

A Forchheimer Equation-Based Flow Model for Fluid Flow Through Rock Fracture During Shear

Shear deformation-induced hydraulic conductivity change in fracture has been studied for decades. However, the existing models to link shear deformation and hydraulic behaviors are less accurate due to complex flow in rock fractures. This study presents an improved flow model for calculating nonlinear flow behaviors in rock fractures during shear. In this model, the linear and nonlinear coefficients in the Forchheimer equation were determined using mechanical aperture and fracture roughness coefficients. The mechanical aperture was equal to the initial aperture plus the change of the aperture due to shear-induced dilation. The dilation curve was divided into three stages and analytical expressions for modeling the dilation curve were established by incorporating the joint roughness coefficient (JRC) and the mobilized roughness coefficient (JRCmob). In addition, shear-flow tests were conducted on marble and granite fractures with normal stress between 0.5 and 3.0 MPa. The experimental data were used to verify the proposed model. The results show that the proposed model predicts flow in rock fractures well.

Two-phase cement grout propagation in homogeneous water-saturated rock fractures

Modeling of cement grout flow in rock fractures is important for the design, monitoring and execution of rock grouting that is widely used in a variety of rock engineering applications. This study presents a mathematical model based on the Reynolds flow equation for cement grout flow in a homogeneous water-saturated rock fracture. The model is based on two-phase flow, i.e. grout as a Bingham fluid and groundwater as a Newtonian fluid, and is used for investigating the importance of the water phase in rock grouting. The modeling results for the two-phase flow generally show the importance of the water phase that can significantly affect the pressure distribution and grout penetration in the fracture, especially under the condition of grout hardening. Such effects depend on the viscosity ratio between the grout and groundwater, which becomes increasingly important for cases with smaller values of the viscosity ratio. The grout density also affects the grout penetration length. Applying an analytical solution based on single-phase flow, i.e. neglecting the impact of groundwater flow, for modeling grout injection, will generally overestimate the penetration length.

Evolution of permeability in sand injectite systems

Sand injectite complexes comprise kilometer-scale clastic intrusion networks that act as effective conduits for the migration, accumulation and then recovery of hydrocarbons and other fluids. An equivalent continuum model is constructed to represent a sand injectite reservoir, coupling stress and fluid flow in fractured rock using the continuum simulator TOUGHREACT coupled with FLAC3D to follow deformation and fluid flow. A permeability model, which uses staged percolation models, is proposed to improve permeability estimation of fracture networks by accommodating four different levels of fracture connectivity. This permeability model is confirmed against field and laboratory data, corresponding to the different connectivities of fracture networks. The new constitutive permeability model is incorporated into the coupled hydro-mechanical simulator framework and applied to sand injectites with the analysis of permeability evolution mechanisms and mechanical sensitivity. The results indicate that when the magnitudes of principal stresses increase in a constant ratio, normal closure is the dominant mechanism in reducing fracture aperture and thereby permeability. Conversely, the evolution of stress difference can accentuate aperture and permeability due to an increase in shear dilation for critically or near-critically oriented fractures. Also, the evolution of aperture and related permeability of fractured rock are more sensitive at lower stress states than at higher stress states due to the hyperbolic relationship between normal stress and normal closure of the fractures.

A Review of Critical Conditions for the Onset of Nonlinear Fluid Flow in Rock Fractures

Selecting appropriate governing equations for fluid flow in fractured rock masses is of special importance for estimating the permeability of rock fracture networks. When the flow velocity is small, the flow is in the linear regime and obeys the cubic law, whereas when the flow velocity is large, the flow is in the nonlinear regime and should be simulated by solving the complex Navier-Stokes equations. The critical conditions such as critical Reynolds number and critical hydraulic gradient are commonly defined in the previous works to quantify the onset of nonlinear fluid flow. This study reviews the simplifications of governing equations from the Navier-Stokes equations, Stokes equation, and Reynold equation to the cubic law and reviews the evolutions of critical Reynolds number and critical hydraulic gradient for fluid flow in rock fractures and fracture networks, considering the influences of shear displacement, normal stress and/or confining pressure, fracture surface roughness, aperture, and number of intersections. This review provides a reference for the engineers and hydrogeologists especially the beginners to thoroughly understand the nonlinear flow regimes/mechanisms within complex fractured rock masses.

Experimental study of flow characteristics in non-mated rock fractures considering 3D definition of fracture surfaces

Three-dimensional (3D) morphology of rock fracture has a great influence on its hydraulic behavior. To experimentally investigate the 3D roughness dependent non-linear flow characteristics in deformable rock fractures, water flow tests through non-mated fractures were conducted under normal stresses ranging from 1.0MPa to 5.0MPa. Three types of granite samples were selected to produce rock fracture with different roughness. A 3D morphology parameter was used in the experiments to better understand the influence of joint roughness on the flow characteristics. By incorporating the 3D roughness metric, we proposed a new formula to describe the non-linear flow in rock fracture. The non-linear flow characteristics including the critical Reynolds number, Forchheimer's linear coefficient and nonlinear coefficient are analyzed and discussed. In addition, comparison among the proposed equation, Lomize's equation, Forchheimer equation and Izbash's law are examined from the perspective of the rationality of the formula. The limitations of Lomize's equation, Forchheimer equation and Izbash's law are analyzed in detail. The results show that the proposed equation is easier in the structure and more meaningful from an engineering point of view.

Fractal model and Lattice Boltzmann Method for Characterization of Non-Darcy Flow in Rough Fractures

The irregular morphology of single rock fracture significantly influences subsurface fluid flow and gives rise to a complex and unsteady flow state that typically cannot be appropriately described using simple laws. Yet the fluid flow in rough fractures of underground rock is poorly understood. Here we present a numerical method and experimental measurements to probe the effect of fracture roughness on the properties of fluid flow in fractured rock. We develop a series of fracture models with various degrees of roughness characterized by fractal dimensions that are based on the Weierstrass–Mandelbrot fractal function. The Lattice Boltzmann Method (LBM), a discrete numerical algorithm, is employed for characterizing the complex unsteady non-Darcy flow through the single rough fractures and validated by experimental observations under the same conditions. Comparison indicates that the LBM effectively characterizes the unsteady non-Darcy flow in single rough fractures. Our LBM model predicts experimental measurements of unsteady fluid flow through single rough fractures with great satisfactory, but significant deviation is obtained from the conventional cubic law, showing the superiority of LBM models of single rough fractures.

Numerical Analysis of Unsaturated Seepage Flow in Two-Dimensional Fracture Networks

Abstract Fracture-dominated flow in fractured rocks is of interest in many areas of scientific and technical research. In contrast to the continuum approach, this study employed a discrete fracture model method to analyze water movement in unsaturated fractured rocks, in which the flow behavior for single fractures is governed by Darcy’s law and the Richards equation, and the partial differential equations defined on the whole fracture network domain were formulated. A systematic finite-element algorithm was developed along with a corresponding numerical procedure. The results for the porous medium indicate that the proposed method is valid. Good agreement was found between the numerical predictions and experimental observations, and thus the equivalent discrete fracture network model may be used in place of the continuum model. The infiltration seepage analysis in a fractured rock slope demonstrated that the distribution of fracture-dominated flow was heterogeneous and that the movement of variably satur...