On the cubic law and variably saturated flow through discrete open rough-walled discontinuities

Dippenaar, Matthys Alois; Rooy, J.L. Van

doi:10.1016/j.ijrmms.2016.09.011

Cited by 58 publications

(30 citation statements)

References 51 publications

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“…Furthermore, they locally develop in talus and waste rock deposits (Tokunaga et al, 2005; Trinchero et al, 2011) and along fault zones (Bodvarsson et al, 1997; Liu et al, 2004), where they can strongly contribute to the overall flux of a system. However, classical volume‐effective modeling approaches (e.g., the Richards equation) and associated relationships between system geometry and flow properties might not be suitable to recover the nonlinear flow and mass partitioning processes that control preferential flow dynamics and are still under investigation (e.g., Dippenaar and Van Rooy, 2016). Different flow regimes, e.g., droplet, rivulet, and film flow, can coexist even in controlled settings and are difficult to cast into unified conceptual frameworks (Ghezzehei, 2004).…”

mentioning

confidence: 99%

Analogue Fracture Experiments and Analytical Modeling of Unsaturated Percolation Dynamics in Fracture Cascades

Noffz¹,

Dentz

Kordilla³

2019

Vadose Zone Journal

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Core Ideas Mass redistribution at unsaturated fracture intersections depends on free‐surface flow modes. A Washburn‐type flow regime is recovered via an analytical solution for capillary fracture inflow. A Gaussian transfer function predicts mass partitioning for arbitrary‐sized fracture cascades. Infiltration and recharge dynamics in fractured aquifer systems often strongly deviate from diffuse Darcy–Buckingham type flows due to the existence of a complex gravity‐driven flow component along fractures, fracture networks, and fault zones. The formation of preferential flow paths in the unsaturated or vadose zone can trigger rapid mass fluxes, which are difficult to recover by volume‐effective modeling approaches (e.g., the Richards equation) due to the nonlinear nature of free‐surface flows and mass partitioning processes at unsaturated fracture intersections. In this study, well‐controlled laboratory experiments enabled the isolation of single aspects of the mass redistribution process that ultimately affect travel time distributions across scales. We used custom‐made acrylic cubes (20 by 20 by 20 cm) in analog percolation experiments to create simple wide‐aperture fracture networks intersected by one or multiple horizontal fractures. A high‐precision multichannel dispenser produced gravity‐driven free‐surface flow (droplets or rivulets) at flow rates ranging from 1 to 5 mL min−1. Total inflow rates were kept constant while the fluid was injected via 15 (droplet flow) or three inlets (rivulet flow) to reduce the impact of erratic flow dynamics. Normalized fracture inflow rates were calculated and compared for aperture widths of 1 and 2.5 mm. A higher efficiency in filling an unsaturated fracture by rivulet flow observed in former studies was confirmed. The onset of a capillary‐driven Washburn‐type flow was determined and recovered by an analytical solution. To upscale the dynamics and enable the prediction of mass partitioning for arbitrary‐sized fracture cascades, a Gaussian transfer function was derived that reproduces the repetitive filling of fractures, where rivulet flow is the prevailing regime. Results show good agreement with experimental data for all tested aperture widths.

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confidence: 99%

Analogue Fracture Experiments and Analytical Modeling of Unsaturated Percolation Dynamics in Fracture Cascades

Noffz¹,

Dentz

Kordilla³

2019

Vadose Zone Journal

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“…The Brinkman model was also used for modeling two phase flows in a Hele-Shaw cell with cylindrical obstacles (Ferrari et al, 2015;Horgue et al, 2013). Although the local cubic law model has been often used to study the influence of contacts on the flow in rough-walled fractures (see for a review, e.g., Dippenaar & Rooy, 2016), already at low obstacle fractions, the contact regions may influence flow paths more significantly than predicted in the LCL models (Berkowitz, 2002;Durham & Bonner, 1994;Oron & Berkowitz, 1998) In the present work, we focus on the idealized fracture bounded by parallel planar walls and filled with circular obstacles. Laleian et al (2015) used 2-D and 3-D lattice Boltzmann method (LBM) simulations to study the accuracy of the depth-averaged flow model for variable aperture and cylindrical obstacles of different size, reporting an accuracy of 10% in terms of the effective permeability.…”

Section: 1002/2017jb014509mentioning

confidence: 99%

“…However, due to a reduced-order approach, the Reynolds model cannot satisfy the no-slip condition at the contacts. Although the local cubic law model has been often used to study the influence of contacts on the flow in rough-walled fractures (see for a review, e.g., Dippenaar & Rooy, 2016), already at low obstacle fractions, the contact regions may influence flow paths more significantly than predicted in the LCL models (Berkowitz, 2002;Durham & Bonner, 1994;Oron & Berkowitz, 1998) In the present work, we focus on the idealized fracture bounded by parallel planar walls and filled with circular obstacles. The studied geometry can directly represent hydraulic fractures propped with large proppant grains (Alramahi & Sundberg, 2012;Brannon et al, 2004), or it can be a good approximation to fractures formed by reactive processes (Budek et al, 2017;Detwiler et al, 2003;Szymczak & Ladd, 2009).…”

Section: 1002/2017jb014509mentioning

confidence: 99%

The Effective Transmissivity of a Plane‐Walled Fracture With Circular Cylindrical Obstacles

Jasinski

Dąbrowski

2018

JGR Solid Earth

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Stimulated and propped fractures provide the conductive pathways in low matrix permeability rocks. We study the impact of fracture aperture and proppant size and fraction on the effective transmissivity of a stimulated fracture filled with a partial monolayer of proppant grains. The proppant grains are treated as circular cylindrical obstacles, and the fracture walls are planar. The key geometric parameters are the obstacle fraction f and the ratio between the fracture aperture and the obstacle diameter α. We use three‐dimensional Stokes flow numerical simulations to demonstrate that the fracture flow model given by the Reynolds equations may largely overestimate the flow rate. To circumvent the inability of the Reynolds model to fulfill the no‐slip boundary condition at the rims of the obstacles, we use the Brinkman flow model adapted to the case of fracture flow. The relative difference between the effective transmissivities computed with the Stokes and Brinkman models for systems with multiple obstacles is below 10% for fractions up to 0.5. We use the Brinkman model to study the effective transmissivity of a plane‐walled fracture with circular cylindrical obstacles. Systematic numerical simulations show that the normalized effective transmissivity is predominantly dependent on f and α, and the effects of obstacle ordering are minor. The presented numerical results can be used by petroleum engineers for estimating transmissivities of propped fractures under in situ conditions. The model can also be applied to microfluidic systems and for deriving first‐order estimates of the effective transmissivity of rough‐walled, natural fractures with load‐bearing contact areas.

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“…Thereafter, the flux or discharge (Q) occurring through a single discrete fracture can be determined using a basic analytical method derived from the Navier-Stokes equation, and is known as the cubic law due to the proportionality of the flux to the third power of aperture (e.g. Bear 1972;Dippenaar and Van Rooy 2016;Silberhorn-Hemminger et al 2005;Singhal and Gupta 2010). The validity of the cubic law, however, is often queried given the assumptions from this strong oversimplification of natural conditions (see for e.g.…”

Section: Empirical and Mathematical Relationships For Fracture Flowmentioning

confidence: 99%

Lugeon Tests at Partial Saturation: Experimental and Empirical Contributions

2018

Self Cite

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Implications of improved understanding of variably saturated flow are numerous, especially given the complexity, heterogeneity and anisotropy of the intermediate fractured vadose zone. One such an implication is the quantification of water movement for engineering purposes, as flow through unsaturated discontinuities cannot be quantified through commonly applied saturated approaches. This paper presents an experimental study using geotechnical centrifuge modelling to investigate flow behaviour during Lugeon tests, through the fundamental concept of a smooth, clean, open fracture, at partial saturation. The study comprised flow visualisation experiments conducted on transparent replicas of horizontal and inclined fractures, with water injected under a consecutive series of ascending and descending pressures. Findings from the research proved that preferential flow occupied the minority of cross-sectional area despite the hydraulic pressure. Furthermore, the observed behaviour of these preferential pathways indicated non-linear flow. Deviation from linearity occurred at small fluxes and is likely as a result of inertial effects due to fluid bending at the inlet source into the fracture. In order to assess these non-linear results, the Forchheimer relationship was used to predict the flow rate at the imposed hydraulic heads. As the width of the fracture could not be used as input into the equation, due to the lack of saturation across its width, the width of the flow path was used instead. This resulted in the predicted results comparing well with the measured flow rates, and indicates that the Forchheimer relationship can potentially be used to describe unsaturated flow in discrete, open fractures.

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On the cubic law and variably saturated flow through discrete open rough-walled discontinuities

Cited by 58 publications

References 51 publications

Analogue Fracture Experiments and Analytical Modeling of Unsaturated Percolation Dynamics in Fracture Cascades

Analogue Fracture Experiments and Analytical Modeling of Unsaturated Percolation Dynamics in Fracture Cascades

The Effective Transmissivity of a Plane‐Walled Fracture With Circular Cylindrical Obstacles

Lugeon Tests at Partial Saturation: Experimental and Empirical Contributions

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