Fluid flow in rock fractures: From the Navier-Stokes equations to the cubic law

Zimmerman, Robert W.; Yeo, In Wook

doi:10.1029/gm122p0213

Cited by 143 publications

(136 citation statements)

References 31 publications

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“…However, for such realistic fracture geometries, the Navier-Stokes equations have not yet been solved (cf. Zimmerman and Yeo, 2000). Our field observations (Figure 12) and numerical model (Figure 14) indicate that, although shallow fractures may be more likely to be stratabound than fractures at deeper crustal levels (Odling et al, 1999;Gillespie et al, 2001), hydrofractures can become arrested at any crustal depth if there is a strong contrast in the mechanical properties of adjacent rock layers.…”

Section: Discussionmentioning

confidence: 99%

Effects of mechanical layering on hydrofracture emplacement and fluid transport in reservoirs

2013

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Fractures generated by internal fluid pressure, for example, dykes, mineral veins, many joints and man-made hydraulic fractures, are referred to as hydrofractures. Together with shear fractures, they contribute significantly to the permeability of fluid reservoirs such as those of petroleum, geothermal water, and groundwater. Analytical and numerical models show that-in homogeneous host rocks-any significant overpressure in hydrofractures theoretically generates very high crack tip tensile stresses. Consequently, overpressured hydrofractures should propagate and help to form interconnected fracture systems that would then contribute to the permeability of fluid reservoirs. Field observations, however, show that in heterogeneous and anisotropic, e.g., layered, rocks many hydrofractures become arrested or offset at layer contacts and do not form vertically interconnected networks. The most important factors that contribute to hydrofracture arrest are discontinuities (including contacts), stiffness changes between layers, and stress barriers, where the local stress field is unfavorable to hydrofracture propagation. A necessary condition for a hydrofracture to propagate to the surface is that the stress field along its potential path is everywhere favorable to extension-fracture formation so that the probability of hydrofracture arrest is minimized. Mechanical layering and the resulting heterogeneous stress field largely control whether evolving hydrofractures become confined to single layers (stratabound fractures) or not (non-stratabound fractures) and, therefore, if a vertically interconnected fracture system forms. Non-stratabound hydrofractures may propagate through many layers and generate interconnected fracture systems. Such systems commonly reach the percolation threshold and largely control the overall permeability of the fluid reservoirs within which they develop.

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Section: Discussionmentioning

confidence: 99%

Effects of mechanical layering on hydrofracture emplacement and fluid transport in reservoirs

2013

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“…This relation, known as the cubic law, is derived by assuming Darcy flow in the rock matrix and Stokes flow inside the fracture. More accurate approximations can be found by accounting for fracture wall roughness (Zimmerman and Yeo, 2000) and viscous forces at the fracture-matrix interface (Vernerey, 2012). In the normal direction, the fluid experiences no flow resistance within the fracture.…”

Section: Tab 1: Parameters For Characterizing Ellipsoidal Fracturesmentioning

confidence: 99%

“…A 3D computational study of effective medium methods applied to fractured media Some numerical tests of effective medium theory have been conducted, with encouraging results (Zimmerman and Yeo, 2000;Pozdniakov and Tsang, 2004;Barthélémy, 2008;Tawerghi and Yi, 2009). These works are limited to spherical/near-spherical inclusions, or isotropic/transversely isotropic media, or two-dimensional geometries.…”

Section: Transport In Porous Media Manuscript No (Will Be Inserted Bmentioning

confidence: 99%

A 3D Computational Study of Effective Medium Methods Applied to Fractured Media

et al. 2013

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This work evaluates and improves upon existing effective medium methods for permeability upscaling in fractured media. Specifically, we are concerned with the asymmetric self-consistent, symmetric self-consistent and differential methods. In effective medium theory, inhomogeneity is modeled as ellipsoidal inclusions embedded in the rock matrix. Fractured media correspond to the limiting case of flat ellipsoids, for which we derive a novel set of simplified formulas. The new formulas have improved numerical stability properties, and require a smaller number of input parameters. To assess their accuracy, we compare the analytical permeability predictions with accurate, three-dimensional finite-element simulations. We also compare the results with a semi-analytical method based on percolation theory and curve fitting, which represents an alternative upscaling approach. A large number of cases is considered, with varying fracture aperture, density, matrix/fracture permeability contrast, orientation, shape and number of fracture sets. The differential method is seen to be the best choice for sealed fractures and thin open fractures. For high-permeable, connected fractures, the semi-analytical method provide the best fit to the numerical data, whereas the differential method breaks down. The two self-consistent methods can be used for both unconnected and connected fractures, although the asymmetric method is somewhat unreliable for sealed fractures. For open fractures, the symmetric method is generally the more accurate for moderate fracture densities, but only the asymmetric method is seen to have correct asymptotic behaviour. The asymmetric method is also surprisingly accurate at predicting percolation thresholds.

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“…In addition to that, the finite-volume method has been increasingly and successfully used to simulate fracture flow in the last few years (Al-Yaarubi et al, 2005;Brush and Thomson, 2003;Crandall et al, 2010;Huber et al, 2012;Schwarz and Enzmann, 2013). However, in practice, solving three-dimensional (3-D) mathematical models such as the Navier-Stokes equation with these methods can be computationally elaborate so that many simulations prefer simplified flow models such as the "local cubic law" approach (Brush and Thomson, 2003;Konzuk and Kueper, 2004;Koyama et al, 2008;Oron and Berkowitz, 1998;Zimmerman and Yeo, 2013).…”

Section: Introductionmentioning

confidence: 99%

Simulating stress-dependent fluid flow in a fractured core sample using real-time X-ray CT data

et al. 2016

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Abstract. Various geoscientific applications require a fast prediction of fracture permeability for an optimal workflow. Hence, the objective of the current study is to introduce and validate a practical method to characterize and approximate single flow in fractures under different stress conditions by using a core-flooding apparatus, in situ X-ray computed tomography (CT) scans and a finite-volume method solving the Navier-Stokes-Brinkman equations. The permeability of the fractured sandstone sample was measured stepwise during a loading-unloading cycle (0.7 to 22.1 MPa and back) to validate the numerical results. Simultaneously, the pressurized core sample was imaged with a medical X-ray CT scanner with a voxel dimension of 0.5 × 0.5 × 1.0 mm 3 . Fracture geometries were obtained by CT images based on a modification of the simplified missing attenuation (MSMA) approach. Simulation results revealed both qualitative plausibility and a quantitative approximation of the experimentally derived permeabilities. The qualitative results indicate flow channeling along several preferential flow paths with less pronounced tortuosity. Significant changes in permeability can be assigned to temporal and permanent changes within the fracture due to applied stresses. The deviations of the quantitative results appear to be mainly caused by both local underestimation of hydraulic properties due to compositional matrix heterogeneities and the low CT resolution affecting the accurate capturing of sub-grid-scale features. Both affect the proper reproduction of the actual connectivity and therefore also the depiction of the expected permeability hysteresis. Furthermore, the threshold value CT mat (1862.6 HU) depicting the matrix material represents the most sensitive input parameter of the simulations. Small variations of CT mat can cause enormous changes in simulated permeability by up to a factor of 2.6 ± 0.1 and, thus, have to be defined with caution. Nevertheless, comparison with further CT-based flow simulations indicates that the proposed method represents a valuable method to approximate actual permeabilities, particularly for smooth fractures (< 35 µm). However, further systematic investigations concerning the applicability of the method are essential for future studies. Thus, some recommendations are compiled by also including suggestions of comparable studies.

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Fluid flow in rock fractures: From the Navier-Stokes equations to the cubic law

Cited by 143 publications

References 31 publications

Effects of mechanical layering on hydrofracture emplacement and fluid transport in reservoirs

Effects of mechanical layering on hydrofracture emplacement and fluid transport in reservoirs

A 3D Computational Study of Effective Medium Methods Applied to Fractured Media

Simulating stress-dependent fluid flow in a fractured core sample using real-time X-ray CT data

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