We investigate both laminar and turbulent flow regimes in a 3D rock fracture via numerical CFD simulations. We construct our realistic fracture model from 3D scan data of a fractured rock sample and implement changes in both shear displacement and contact ratio, examining their effect on fracture permeability and friction factor. While previous studies were investigating either fully viscous Darcy or inertial Forchheimer laminar flow regimes, we chose to cover the wide Reynolds number range of 0.1–\(10^6\). We introduce CFD simulation of a turbulent flow for realistic 3D fractures, implementing the RANS approach to turbulence modeling. We focus on 3D fracture geometries and implement changes in both shear displacement and contact ratio, systematically examining their effect on fracture permeability and friction factor in a manner similar to the fundamental studies of the flow in rough-walled pipes. Growing Re leads to first stationary–non-stationary and then laminar–turbulent transitions. The presence of contact spots severely disrupts the flow pattern and adversely affects the overall permeability of the fracture. Regardless of shear displacement, ‘no contact’ 3D models can be reasonably approximated by the 2D profiles. Depending on the fracture geometry, Forchheimer \(\beta\) fitting for both laminar and turbulent regimes can be performed either by a single or by two parameter values.
Notes
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
1 Introduction
Fluid flow through fractured media remain the more complex and less-studied counterpart to the flow through porous media. Networks of interconnected fractures grow in importance as the permeability of the rock matrix decreases, becoming dominant pathways for fluid transport in low-permeable rock formations. Related geoscientific applications include geothermal energy extraction, radioactive waste disposal and carbon capture and storage.
Two seminal experimental studies covering both laminar and turbulent flow regimes dealt with complex channel geometries: straight pipe with variable wall roughness (Nikuradse 1933), and planar ducts of different shape (including curvilinear constant-aperture fracture models) by Lomize (1951). While the former is well-known in hydraulic engineering, the latter is much more obscure due to its very specific focus on flow in rock fractures. Over time, multiple experimental studies have significantly increased the complexity of the implemented fracture models and expanded the range of Reynolds numbers of the flow; still, many issues and specifics of the fracture flow in a severely confined volume of the fracture can only be investigated via numerical modeling.
Advertisement
Main issues encountered in a study of the fracture flow stem from inherently complex geometry of the naturally-occurring rock fracture. On a macroscale, it may be either represented by a single plane or constitute a complex network of nearly randomly orientated segments; on a microscale, it exhibits both large-wavelength curvature and short-wavelength roughness. Inevitable formation of contact spots due to stresses present in the jointed rock mass adds another level of complexity.
The issue of fracture aperture has been extensively studied over time, both for scanned fractured samples and synthetic self-affine (fractal) surfaces. Apart from the traditional vertical aperture used by e.g. Brown (1987) or Zimmerman and Bodvarsson (1996), various other aperture metrics were developed, such as ball aperture (Mourzenko et al. 1995), true aperture normal to the fracture centerline (Ge 1997), segment-averaged aperture (Oron and Berkowitz 1998), or 3D Hausdorff distance (Finenko and Konietzky 2021). The main reason for this lies in the extreme sensitivity of the flowrate to the variations in the fracture aperture (\(Q \sim h^3\)), also known as the ‘cubic law’ (Witherspoon et al. 1980).
Geometry of the problem largely dictates the choice of the fluid flow modeling technique. Analytical solution exists only for a fully parallel plate model as a special case of the Poisseulle capillary flow, with flowrate Q following the ‘cubic law’ \(Q = - (w h^3 / 12 \mu ) \nabla p\), where w is the fracture width, h the distance between the plates (fracture aperture), \(\mu\) the fluid viscosity and \(\nabla p\) the applied pressure gradient. For models with variable aperture, a simpler approach assumes the validity of local cubic law (LCL) for every segment of the fracture plane, so that a solution is obtained via Reynolds lubrication equation \(\nabla \cdot [(h^3/12 \mu ) \nabla p] = 0\), with both aperture h(x, y) and pressure p(x, y) varying across the fracture plane. This approach essentially reduces a 3D fracture to a 2D grid of cells with variable aperture, and was first implemented by Brown (1987) for fractal surfaces.
Main deficiencies of LCL approximation are i) requirement of smooth and gradual aperture variations and ii) limitation to fully viscous flow regime where viscous forces dominate over inertial forces and \(Re \! \ll \! 1\). First geometrical limitation can be lifted altogether by turning to the numerical solution of Stokes equations which describe a creeping flow through arbitrary geometry: \(\mu \nabla ^2 {\textbf{u}} = \nabla p, \, \nabla \cdot {\textbf{u}} = 0\). In return, the volume of the fracture has to be discretised by a proper 3D mesh. This approach was first realised and compared against Reynolds LCL approximation by Mourzenko et al. (1995).
Advertisement
Second limitation of the fully viscous flow can only be remedied by solving full Navier–Stokes equations, which was first implemented for the 2D case by Skjetne et al. (1999) and for 3D case by Zimmerman et al. (2004). Subsequent studies modeled laminar high-velocity flow by solving NSEs for the 2D case (Koyama et al. 2008; Crandall et al. 2010; Zou et al. 2015; Briggs et al. 2017) and for the 3D case (Wang et al. 2016; Zou et al. 2017), employing finite element (FEM), finite volume (FVM) of even lattice Boltzmann (LBM) codes. Two main limitations can still be discerned: i) mesh dimensions for the 3D case and ii) upper limit of the Re number, which is kept \(<Re_{\textrm{cr}}\) to remain in the laminar domain. For a 3D fracture model, both Zimmerman et al. (2004) and Zou et al. (2017) solved NSEs using FEM with mesh dimensions of \(25 \times 25\) mm, whereas Wang et al. (2016) employed LBM for a \(50 \times 25\) mm model. As opposed to the Reynolds or Stokes approximations, for NSEs the mesh has to be sufficiently resolved in z-direction; thin and stretched-out geometry of the fracture with its high aspect ratio (\(L_{x,y} \gg L_z\)), small-scale roughness and/or contact spots thus is in itself a quite difficult problem for the meshing. Reported Reynolds numbers vary from 50 (Skjetne et al. 1999; Wang et al. 2016) and 70 (Zimmerman et al. 2004) to 500 (Briggs et al. 2017) and 1000 (Zou et al. 2015), with 2D cases keeping clearly ahead of 3D due to far lower computational cost.
Analysing the 2D data for the inertial laminar regime at \(Re \! \sim \! 50\), Skjetne et al. (1999) pointed out the decay of the symmetric laminar flow pattern into high-velocity flow tubes and low-velocity recirculation zones, as well as the frequent separation and reattachment of the flow upon encounter with fracture asperities. Zou et al. (2017) implemented pointwise contacts for a 3D geometry in laminar inertial flow regime and showed the buildup of the complex 3D streamline pattern around the asperities and contact spots.
The issue of closed recirculation zones (eddies) in the laminar flow regime has to be addressed separately. Presence of such structures, whether stationary or non-stationary, can often be erroneously linked to the turbulence; nevertheless, in the related CFD application areas such as pipe or duct flows those structures are quite common (e.g. elbow and backward-facing step), where separation bubble with closed recirculation is also present; a classical example of non-stationary vortices in a laminar regime is the von Kármán vortex street behind the cylinder. ‘True’ turbulence at \(Re > Re_{\textrm{cr}}\) is characterized by the energy cascade involving eddies of different length scales from the large scale, which is roughly proportional to the model length, down to the Kolmogorov scale \(\eta\), where turbulent kinetic energy k is dissipated into heat by molecular viscosity.
High-velocity laminar flow is per se a transient phenomenon with constant flow separation and reattachment and vortex shedding, which was mentioned by the previous research on this subject; turbulent flow is transient per definition. Transient phenomenon should preferably be modeled by a transient simulation; however, that implies a significant cost in terms of computational power and data storage space. Following an established practice in CFD modeling, we chose to cut the unnecessary computational cost by making two major decisions: i) employing Reynolds-averaged Navier–Stokes (RANS) technique where all turbulence length scales are modeled instead of resolved as in LES or DNS approaches, and ii) running a steady-state simulation, where large-scale transient effects not accounted for by the RANS turbulence model are also smoothed out, obtaining a fully time-averaged flow field. The following argument can be made in favor of steady-state assumption: random and rough geometry of the fracture results in a random number of bottlenecks, with each one of them acting as a vortex generator (turbulator). Vortices shed upstream are superimposed onto the downstream vortices; their cumulative effect should, therefore, be also averaged out in time, as opposed to a single vortex-shedding edge/step, where sharp oscillations are present.
In our 2D study (Finenko and Konietzky 2023), we have implemented both laminar and turbulent flow in rough-walled 2D fracture models, aiming for the widest possible Reynolds number range of 0.1–\(10^6\) to close all remaining gaps between fracture and pipe studies. To the best of our knowledge, no numerical simulation of turbulent flow in rock fracture (\(Re > Re_{\textrm{cr}}\)), whether 2D or 3D, has been reported to date. We now extend our developed 2D workflow to a 3D problem, performing a steady-state simulation of a turbulent incompressible flow for a 3D case, solving NSEs with a FVM code OpenFOAM. Having tested several popular RANS turbulence models for accuracy and efficiency in 2D case, we employ two-equation standard \(k{-}\omega\) (SKW) model in turbulent regime. Principal novelty of this study lies in the numerical simulation of both laminar and turbulent flow regimes in a 3D realistic (scan-based) fracture geometry, solving full Navier–Stokes equations and implementing turbulence modeling. We compare our numerical results with the available empirical solutions (Forchheimer equation), and with results of the fundamental experiments by Nikuradse (1933) and Lomize (1951). The findings of this study would help to better estimate fracture permeability in various geoscientific applications such as geothermal energy extraction, CO\(_2\) sequestration or petroleum engineering, where expected Re numbers may be significantly higher than those typical for naturally occurring groundwater systems.
2 Fracture Models
Fig. 1
Plots of Hausdorff distance d. Narrow \(1 {\textsf{x}} 10\) cm 3D models: \(\Delta x \!=\! 0\) mm, \(\Delta z \!=\! 0.3895\) mm (upper), \(\Delta x \!=\! 1\) mm, \(\Delta z \!=\! 1\) mm (lower). Full-size \(5 {\textsf{x}} 10\) cm 3D models: \(\Delta x \!=\! 0\) mm, \(\Delta z \!=\! 0.3895\) mm (top), \(\Delta x \!=\! 0\) mm, \(\Delta z \!=\! 0.3\) mm (bottom)
Fig. 2
3D fracture models, \(\Delta x = 0.5\) mm: plots of Hausdorff distance d for \(\Delta z = 1.0386\) mm (top), 0.75 mm (middle), 0.5 mm (bottom)
Fig. 3
3D fracture models, \(\Delta x = 1\) mm: plots of Hausdorff distance d for \(\Delta z = 1.712\) mm (top), 1 mm (middle), 0.75 mm (bottom)
×
×
×
Two most common approaches to the generation of the fracture model are i) synthetic self-affine models and ii) realistic models built from the 3D scans of the fracture surface. We follow the scan-based approach and generate our models from 3D surface scan data of a fractured basalt sample.
Given the computational limitations, the model size was limited at \(5 {\textsf{x}} 10\) cm. With its spatial resolution of \(\sim \! 0.35\) mm our scan data is on the smoother side, since geometry features under \(\sim \! 0.7\) mm are filtered out following the sampling theorem. However, we chose not to downsample the data during meshing in order to reduce the mesh cell count. For example, Zimmerman et al. (2004), given the remarkable spatial resolution of 20 µm, limit their model dimensions to \(20 {\textsf{x}} 20\) mm. Even then, they proceed to generate a coarser grid with a spatial resolution of 200 µm, citing RMS discrepancy between the original profiles and the resulting grid of \(< 1 \%\) of the mean aperture. Koyama et al. (2008a), having scanning their \(200 {\textsf{x}} 100\) mm sample with spatial resolution of 0.2 mm, also decimate the sampled points to form a grid of \(200 {\textsf{x}} 100\) points, reducing spatial resolution to 1 mm.
Roughness of the initial fractured sample surface was estimated via i) joint roughness coefficient (JRC), using expression by Tatone and Grasselli (2010) for the sampling interval of 0.5 mm, calculated for each profile in x-direction, and ii) Grasselli 3D roughness parameter \(\mathrm {G_{3D}} \,{=}\, \theta _{\textrm{max}}^* / (C+1)\) (Tatone and Grasselli 2009), which is calculated for analysis direction 0–\(360^{\circ }\). Following mean and \(\sigma\) values were obtained for the upper (A) and lower (B) halves:
Here \(\sigma _{\textrm{JRC}}\) denotes the standard deviation of JRC across all constitutive 2D profiles of the 3D surface, while \(\sigma _{\textrm{G3D}}\) stands for standard deviation of 3D Grasselli parameter across all analysis azimuths of the 3D surface. Calculated \(\mathrm {G_{3D}}\) anisotropy for top and bottom surfaces is 1.158 and 1.147, respectively, while surface roughness coefficient \(R_s\) is \({\sim }1.135\) for both. Note that mean values for JRC are \({\sim }2\) points above those of \(\mathrm {G_{3D}}\). Mean 2D tortuosity in xz-plane \(\tau _{xz}\) is \({\sim }1.064\) for both surfaces.
Almost similar values for both A and B halves indicate that surface alignment step was performed correctly and surfaces nearly match. Differences between vertical and Hausdorff apertures, as well as surface alignment specifics and fracture aperture distribution are discussed at length in Finenko and Konietzky (2021). For now, we recall that fracture aperture metrics are distributed normally and two factors with the strongest influence on aperture distribution are shear displacement \(\Delta x\) and vertical displacement \(\Delta z\), both of them jointly governing the contact ratio c.
In contrast to 2D fracture models, 3D geometry allows for contact spots, which are created by gradually reducing \(\Delta z\). Intersecting surfaces lead to negative apertures which are truncated to zero following Koyama et al. (2008a), creating contact spots. Depending on the shear displacement \(\Delta x\), contact spots appear at different absolute \(\Delta z\) values and exhibit distinctive contact patterns (small scattered archipelagoes vs large continuous ridges). Complexity of the contact spot distribution is largely limited by the subsequent meshing step where a CFD-viable mesh is to be created. In our experience two most problematic features for the commonly used CFD meshing algorithms are thin wedges and archipelagoes of tiny contact spots typical for non-sheared geometries. Resulting mesh should thus be fine enough to capture the relevant geometry features, at the same time not exceeding the total cell count limits imposed by the available computational resources.
Our starting reference model is a ‘no shear, no contact’ case with \(\Delta x \!=\! 0\) and \(\Delta z\) chosen so that \(h_{\textrm{min}} \!=\! 0\), meaning that both fracture halves are barely touching each other. Starting with the initial model, we implement the following geometry modifications:
Increasing shear displacement: we implement increasing shear displacement of \(\Delta x \,{=}\, 0, 0.5, 1\) mm while maintaining contact ratio \(c \!=\! 0\) by simultaneously increasing \(\Delta z\) so that \(h_{\textrm{min}} \!=\! 0\). Due to above-mentioned approach of determining \(\Delta z\) from the shifted full-size 3D model, aperture increase with shear fully reflects the actual behaviour of the realistic fracture model as long as there are no deformations of the initial geometry. For brevity’s sake, resulting ‘no contact’ models are denoted via respective shift values in mm as \(\Delta x\)-\(\Delta z\): 0\(-\)0.3895, 0.5\(-\)1.0386, and 1\(-\)1.712.
Increasing contact ratio: Next we compress the ‘no-contact’ models together, allowing the formation of the contact spots. Due to irregularity of the surfaces it is rather hard to match the desired c value across multiple shear displacements: we were aiming at creating two contact ratio levels (marked as A and B) at 0.5–\(1 \%\) and 3–\(5 \%\), which could be achieved only partially. Thin no-shear models with archipelagoes of scattered small contacts turned out to be particularly difficult to mesh, with even the \(c \! = \! 0.011\) getting close to the cell count limit of \(\sim \! 20\)–25M cells.
Narrow width: To facilitate testing routines we cut out \(1 {\textsf{x}} 10\) cm thin stripes from both no-contact 0\(-\)0.3895 and contact 1–1 models in our CAD software. Additionally, an extruded 3D model from 2D profile y20 (with no aperture changes in z-direction) was built to investigate flow tortuosity in z-direction not related to aperture variations.
Table 1
Aperture distribution for meshed 3D fracture models: h and d stand for vertical aperture and 3D Hausdorff distance, respectively; \({\bar{h}}\) denotes the mean and \({\tilde{h}}\) the median values, \(\sigma _h\) and \(\sigma _d\) denote the standard deviations. All values are in mm except for contact ratio c
\(\Delta x\)
\(\Delta z\)
\({\bar{h}}\)
\({\bar{d}}\)
\({\tilde{h}}\)
\({\tilde{d}}\)
\(\sigma _h\)
\(\sigma _d\)
c
1x10 cm
0
0.3895
0.3910
0.3547
0.3821
0.3488
0.0664
0.0600
0
1
1.0000
1.0002
0.8775
0.9796
0.9148
0.3596
0.2832
0.0281
5x10 cm
0
0.3895
0.3988
0.3571
0.3865
0.3495
0.0945
0.0813
0
0
0.3000
0.3267
0.2934
0.3147
0.2850
0.0966
0.0796
0.0011
0.5
1.0386
1.0135
0.9061
0.9976
0.9343
0.1922
0.1732
0
0.5
0.7500
0.7352
0.6573
0.7163
0.6720
0.1893
0.1567
0.0043
0.5
0.5000
0.5020
0.4497
0.4775
0.4491
0.1833
0.1420
0.0281
1
1.7120
1.6663
1.4910
1.6548
1.5488
0.3394
0.3057
0
1
1.0000
0.9764
0.8715
0.9581
0.8971
0.3277
0.2665
0.0113
1
0.7500
0.7470
0.6664
0.7186
0.6734
0.3110
0.2435
0.0425
Aperture distribution specifics for all meshed models is listed in Table 1; note that meshing introduces additional artifacts around contacts due to finite cell size, slightly increasing the overall contact ratio. Plots of Hausdorff aperture d for all 3D models are shown in Figs. 1, 2, 3; note the contact spot pattern changing from sparse spots for the no-shear 0\(-\)0.3 model to a system of elongated ridges running nearly transversally to the shear direction for the \(\Delta x \!=\! 0.5,1\) mm models. Round deep blue spots correspond to the sharp features torn off during the fracturing of the intact sample.
3 Flow Simulation
Following the approach tested and implemented in our 2D case study (Finenko and Konietzky 2023), we model the fluid flow numerically by solving full Navier–Stokes equations (NSE) with an open-source finite-volume (FVM) method-based CFD toolbox OpenFOAM; we choose a velocity-driven implementation for ease of setting specific Re values, with pressure gradient \(\nabla p\) as output parameter. We then calculate the fracture permeability k and the Darcy friction factor f, with latter being the standard dimensionless parameter for the pipe/duct flows that connects pressure drop with the kinetic energy of the fluid (cf. Darcy–Weisbach equation).
For both laminar and turbulent flow regimes we make a steady-state approximation and use a solver based on the SIMPLE algorithm. The viability of the steady-state approximation of the turbulent flow which is by definition transient stems was confirmed by comparing results of both steady-state and transient RANS solvers (cf. Finenko and Konietzky 2023). Strongest \(\nabla p\) fluctuations, in both pseudo-time iterations (steady-state) and in real-time steps (transient), were found in non-stationary laminar regime, with standard deviation of pressure gradient reaching \(7.4\%\) and \(9.4\%\), respectively. Both laminar stationary and turbulent regimes have shown no fluctuations in flowrate Q or pressure gradient \(\nabla p\) whatsoever.
For the laminar flow regime, boundary conditions for all four walls of the fracture model are set to no-slip, which is itself a Dirichlet BC setting all velocity vector components at the wall to zero. Turbulent flow simulations follow the RANS approach, with all turbulence length scales being modeled via turbulence models; laminar no-slip boundary conditions are then replaced by respective wall functions. For the 3D case, we have selected the standard \(k{-}\omega\) (SKW) turbulence model which was compared against other popular models on our 2D data, proving itself to be both sufficiently accurate and robust.
4 Results and Discussion
4.1 Comparison with 2D Data
Fig. 4
Comparison of the 3D \(1 {\textsf{x}} 10\) and \(5 {\textsf{x}} 10\) cm models with the 2D profile data: friction factor (left) and permeability (right). Squares denote laminar, diamonds—turbulent (SKW model) solver regimes. Top row: no-shear no-contact model, \(\Delta x = 0, \Delta z = 0.3895\) mm. Bottom row: shear and contact model, \(\Delta x = 1, \Delta z = 1\) mm
Fig. 5
Comparison of the 3D \(5 {\textsf{x}} 10\) cm models with the 2D profile data with increasing shear displacement \(\Delta x\). Top row: ‘no-contact’ models, friction factor (left) and permeability (right). Bottom row: ‘contact’ models, permeability only. Squares denote laminar, diamonds—turbulent (SKW model) solver regimes
×
×
We compare our ‘narrow’ \(1 {\textsf{x}} 10\) cm and ‘wide’ \(5 {\textsf{x}} 10\) cm models with 2D data for 3 profiles cut out from the same surface which were examined in (Finenko and Konietzky 2023), having different aperture, waviness and roughness. Obtained results are shown in Fig. 4. Similar to our 2D case study, arbitrary curvilinear geometry causes both i) nonlinear bending and ii) vertical shift of the f(Re) curves away from analytical solutions such as e.g. laminar 96/Re or turbulent Blasius scalings, which are fully applicable to parallel plate flow. Scaling factor \(\eta\) introduced by Lomize (1951) and later Witherspoon et al. (1980) to account for the vertical shift in k, f produces a good fit for data from rough-walled straight channels (after minor adjustments), but clearly falls short for curvilinear geometries with both variable and constant aperture.
Extruded 3D model built from y20 profile stayed within \(2\%\) deviation from its parent 2D y20 profile throughout both laminar and turbulent regimes. It follows that any flow tortuosity and turbulent mixing in z-direction not caused by aperture variations in that direction is negligible even in high-Re turbulent flow regime.
Next we examine the 0\(-\)0.3895 \(1 {\textsf{x}} 10\) cm 3D model cut out between \(\Delta x \! = \! 20\)–30 mm (Fig. 4, top row). Due to complex aperture variations in z-direction, it is not a mere extension of a single 2D profile, with both permeability and friction factor curves rather staying in a corridor between smoother y20 and rougher y35 and y63 profiles. Degree of matching in the laminar regime between 2D and 3D models is remarkable. Lower turbulent regime above \(Re \! \sim \! 10^3\)–\(10^4\) shows a discrepancy due to different characters of laminar–turbulent transition: while all 2D profiles feature a clear gap, 3D models notably show a gapless transition. In upper turbulent regime above \(Re \!=\! 10^5\) 3D data again matches 2D profiles y35 and y63. Smoother y20 profile, being essentially the side wall of the 3D model, behaves as more hydraulically smooth compared with its 3D counterpart, even though its mean aperture \({\bar{h}} \!=\! 0.379\) mm is more narrow than 0.391 mm of the 3D model.
We then add a ‘wide’ \(5 {\textsf{x}} 10\) cm 0\(-\)0.3895 model. Although full-width model includes previously ‘unexplored’ parts of the fracture with major aperture variations (and a vast increase in potential flow-spoiling features), is still fits well with both ‘narrow’ 3D and 2D data, behaving slightly more hydraulically rough in uppermost Re region. Based on this comparison we can assume that in the absence of shear and/or contact spots both friction factor f and permeability k curves for a 3D case could be to a certain degree estimated directly from the 2D profile data.
Second, we analyse the 1–1 \(1 {\textsf{x}} 10\) cm 3D model cut out between \(\Delta x \! = \! 30\)–40 mm (Fig. 4, bottom row); this model is both sheared and compressed so that large contact spots are present with contact ratio of \(2.81 \%\). In the middle section of the model contacts form a ‘barrier ridge’ blocking nearly \(75 \%\) of the overall width; together with the next ridge at \(x \! \sim \! 55\) mm it forms an S-shaped chicane which is a major obstacle for the fluid flow. For comparison we use non-sheared 2D profile y20 with three different degrees of added roughness, as well as the version sheared by \(\Delta x \!=\! 1\) mm.
Regarding the f, k plots of this ‘shear and contact’ case we see that 2D and 3D data are far from each other; moreover, ‘narrow’ \(1 {\textsf{x}} 10\) cm model is \(\sim \! 3\) times less permeable than its full-width \(5 {\textsf{x}} 10\) cm counterpart where the chicane can be quite easily bypassed by the fluid. Fully viscous \(k_0\) of a sheared 2D profile is nearly an order of magnitude higher than the 3D models despite equal \(\Delta x\); we may safely say that the negative effect of contact spots on the fracture permeability easily outweighs any benefits brought by aperture dilation with shear. 3D models are in fact much closer to the non-sheared y20 profile and the less rough noise A and B variations; however, the roughest noise C model is yet another order of magnitude less permeable, suggesting the extreme negative effect of short-wavelength roughness on the overall fracture permeability. As Re increases, S-shaped chicane of the \(1 {\textsf{x}} 10\) cm model causes a considerably earlier onset of non-linearity at \(Re \! \sim \! 10\)–100, further widening the gap between ‘wide’ and ‘narrow’ 3D models from \(\sim \! 3\) at fully viscous flow to at least one order of magnitude throughout the turbulent regime at \(Re \! \sim \! 10^3\)–\(10^6\).
Another small point worth mentioning is the discrepancy in \(k_0\) (permeability for fully viscous flow) between flat and curved models. Both for 2D and 3D cases curved model will always show a lower \(k_0\) against a parallel plate flat model with equal \({\bar{h}}\), even if the inertial effects are practically nonexistent (\(Re \! < \! 1\)), whereas the flat model always matches the cubic law value of \(k \! = \! h^2/12\). For 2D y20 model and its flat counterpart the factor was 0.756. For 3D models, this factor amounted to 0.682 and 0.664 for \(1 {\textsf{x}} 10\) cm and \(5 {\textsf{x}} 10\) cm cases, respectively. We can thus conclude that for 3D case with its added curvature and roughness in z-direction even the ‘viscous’ permeability is pushed even further away from the ideal cubic law values; with growing Re it is further diminished by inertial effects and turbulence.
Finally, we present the comparison between the 3D ‘no contact’ models and the 2D profile data with increasing shear displacement (Fig. 5). As we have mentioned earlier, ‘no contact’ 3D models are at the very least well compatible with the 2D data in the fully viscous regime, albeit they do tend to diverge in the direction of lower k and higher f as soon as we enter laminar inertial regime. Even more surprising is the second near match between 2D and 3D data at the laminar–turbulent transition and in lower turbulent range for \(Re \le 2 \cdot 10^4\): 3D models exhibit a gapless transition which again matches them with the turbulent regime data of the 2D profiles (which themselves always show a clear laminar–turbulent gap). With further increase in Re 3D data again drops under its 2D counterparts, indicating an inherently higher friction; the latter is clearly caused by the presence of fully three-dimensional roughness or waviness. Bottom line of this comparison is that 2D profile data can, in fact, be successfully employed to obtain a permeability estimate of the 3D models which are inherently much more computationally intensive. With respect to ‘contact’ models (Fig. 5 bottom), behaviour of 2D and 3D data is clearly different and no similar approximation would be possible.
Compression of the fracture model strongly influences aperture and contact spot distribution, so that each time we are essentially dealing with a completely new geometry. In Fig. 6 we group all 8 3D models together, with fully viscous range of \(Re \!=\! 0.1\)–\(10^1\) omitted for better visibility.
Firstly, all full-size 3D models exhibit a smooth gapless laminar–turbulent transition due to abundance of roughness and curvature features in three dimensions. Friction factor curves are less cluttered, with the expected increase in f coming both with shear and compression of the model. Nearly all models behave as hydraulically rough in the turbulent range with \(f(Re) \! \rightarrow \! const\); it is even more pronounced for the four models with both shear and contact, even though terminal f is reached at different Re ranging between \({\sim }10^4\)–\(10^5\).
Secondly, similar to 2D cases the ‘no contact’ models (dotted lines) show a superlinear increase of k with growing \(\Delta x\) between 0 and 1 mm; taking into account our 2D data, we can assume that with continuing shear permeability k would tend to some ‘saturation level’.
Finally, vertical compression and subsequent buildup of contact spots result in a drastic drop in permeability. The most compressed 0.5\(-\)0.5 and 1\(-\)0.75 models both show a viscous \(k_0\) less or equal to the non-sheared 0\(-\)0.3895 and 0\(-\)0.3 models; however, abundant contact spots strongly augment the inertial effects, so that both models show the earliest onset of non-linearity; in the upper turbulent flow regime (\(Re > 10^5\)) their permeability drops nearly one order of magnitude under that of the non-sheared compressed model.
The last finding brings in itself a quite bold statement that all beneficial effects of shearing connected with the increase of mean fracture aperture \({\bar{h}}\) can be effectively neutralised and even reverted by the formation of the contact spots and subsequent fluid flow channeling through narrow bottlenecks. Our experience with the chosen fracture surface, which in the no-shear state exhibits both longitudinal and transversal troughs and ridges, shows that introduction of shear will form predominantly ridges transversal (perpendicular) to the shear direction — which is quite logical since we are dealing with nearly matching surfaces. Of course, orientation of the contact spots relative to flow direction (ranging from parallel to perpendicular) would also play a major role — one can safely assume that contact spots parallel to the flow direction would be much less of a hindrance. By setting flow direction parallel to the shear direction (and, therefore, perpendicular to the forming ridges) we thus deliberately choose the worst-case scenario. In the next section we analyse the obtained flow fields and inspect the issue of contact spots and bottlenecks in more detail.
4.3 Increasing Shear Displacement
Fig. 7
Influence of increasing shear for scan-based 3D models. Mean Hausdorff aperture \({\bar{d}}\) vs shear displacement \(\Delta x\) (1st row left). Relative permeability \(k/k_{\textrm{CL}}\) for laminar regime and \(Re=1\) (1st row right). Plots of permeability k vs shear displacement \(\Delta x\) for fixed Re. Laminar regime: \(Re = 10^0, 10^1\) (2nd row), \(Re = 10^2, 10^3\) (3rd row). Turbulent regime: \(Re = 10^4, 10^6\) (4th row)
×
In our 3D models. we implement shear displacement of \(\Delta x \!=\! 0, 0.5, 1\) mm; subsequent shearing steps were omitted since they result only in gradual changes of the aperture and contact spot distribution. Strictly speaking, only the ‘no contact’ 3D models retain fully unchanged fracture surface geometry during shear; ‘contact’ models invariably alter the geometry via contact spots for each \(\Delta x\) step and compression level (A or B), which is reflected by the contact ratio c (cf. Table 1). We arrange our 3D models into 3 following groups: i) ‘no contact’ 0\(-\)0.3895, 0.5\(-\)1.0385, and 1\(-\)1.712 models, ii) ‘contact A’ 0\(-\)0.3, 0.5\(-\)0.75, and 1–1 models, and iii) ‘contact B’ 0.5\(-\)0.5 and 1\(-\)0.75 models. Figure 7 presents \(k(\Delta x)\) plots of permeability vs shear displacement for different Re values, supplemented with mean Hausdorff distance \({\bar{d}}\) plot (1st row left) and \(k/k_{\textrm{CL}}\) plot comparing obtained k with cubic law values calculated from mean fracture aperture \({\bar{d}}\) for \(Re \!=\! 1\) (1st row right).
First, we analyse the ‘no contact’ models. As expected, highest k values are reached in a fully viscous regime at \(Re \!<\! 10\); it is also in this regime that the shear-induced aperture dilation of \(\sim \! 4{\textsf{x}}\) brings the largest benefits, yielding a \(\sim \! 14{\textsf{x}}\) increase in k. In contrast, at the opposite fully turbulent end of Re range at \(Re \!=\! 10^6\) same dilation brings only a \(\sim \! 4{\textsf{x}}\) increase in permeability.
Performance of ‘contact’ models is considerably worse: for group A, shear-induced dilation of \(\sim \! 3{\textsf{x}}\) brings only an \(\sim \! 6{\textsf{x}}\) increase in k at \(Re \!=\! 1\); this increase is nearly zero at \(Re \!=\! 10^2\) and is effectively reverted at \(Re \!=\! 10^3\), with turbulence onset furthering the drop. Of course, much of this should be attributed to the marked increase in contact ratio from \(0.11\%\) to \(1.13\%\). For group B, \({\bar{h}}\) remains nearly constant with shear while c increases slightly from \(2.81\%\) to \(4.25\%\), which leads to the same pattern of stagnating/reverting permeability with growing shear displacement and Re.
A plot of relative permeability \(k/k_{\textrm{CL}}\) presents the ratio of the actual fracture permeability k to the cubic law value given by \(k_{\textrm{CL}} \!=\! {\bar{h}}^2/12\), where \({\bar{h}}\) is the mean fracture aperture. Just as for the 2D case, both curvature and aperture variations of the model geometry lead to \(k/k_{\textrm{CL}}\) always being \(<\! 1\); here we skip the redundant calculation of the hydraulic aperture \(h_H \!<\! h\). Note the particularly poor performance of the more ‘compressed’ group B models, although for \(Re \!=\! 1\) all inertial effects are taken out of the equation. Since we cannot effectively ‘freeze’ the contact ratio and cancel its influence, more general conclusion would be that with the increase of c the related flow pattern degradation largely overrules the beneficial effect of shear-induced aperture dilation and leads first to stagnation and then to decrease of the overall fracture permeability.
Finally, we compare sheared 3D models with the sheared 2D profiles from our 2D study. Performance of the ‘no contact’ models is more similar to the ‘normal’ 2D profiles without closing bottlenecks. For the viscous flow we observe the same superlinear increase in k, which is flattened out with growing Re. We may also assume that k will reach ‘saturation level’ with further shear beyond \(\Delta x \!=\! 1\) mm, quite similar to the 2D models. The ‘contact’ 3D models behave more akin to the 2D profile with the closing bottleneck, showing both much lower viscous \(k_0\) and k decrease with shear — however, an important distinction is that for 3D ‘contact’ models this decrease only occurs for laminar inertial and turbulent regimes, while y35 profile shows k decrease with \(\Delta x\) even for a fully viscous flow. Our explanation of this discrepancy is that although both cases exhibit an increasing \({\bar{d}}\) with growing \(\Delta x\), 2D geometry is entirely constricted by a single closing bottleneck (which is reflected by the decrease in \(d_{\textrm{min}}\)), whereas the 3D geometry provides the fluid with far more bypasses around any contact spots appearing with shear.
4.4 Streamline Tortuosity
Fig. 8
Full 3D tortuosity \(\tau _{xyz}\) and its 2D components \(\tau _{xy}, \tau _{xz}\) for scan-based 3D models: 0\(-\)0.3895 and 0\(-\)0.3 models (1st row), 0.5\(-\)1.0386 and 0.5\(-\)0.75 models (2nd row), 0.5\(-\)0.5 and 1\(-\)1.712 models (3rd row), 1–1 and 1\(-\)0.75 models (4th row). Squares denote laminar, diamonds — turbulent (SKW model) solver regimes
Fig. 9
Full 3D tortuosity \(\tau _{xyz}\) and its 2D components \(\tau _{xy}, \tau _{xz}\) for scan-based 3D models. General view with all models (1st row and 2nd row left); zoomed-in view view no-contact 0\(-\)0.3895, 0.5\(-\)1.0386 and 1\(-\)1.712 models only (2nd row right and 3rd row). Squares denote laminar, diamonds — turbulent (SKW model) solver regimes
×
×
Having obtained the velocity field \({\bar{u}}\), we generate flow streamlines and estimate their tortuosity \(\tau\). Unlike the 2D case where tortuosity in horizontal plane \(\tau _{xy} \!=\! 0\) and \(\tau _{xyz} \!=\! \tau _{xz}\), for a 3D model streamlines are free to bend and undulate across the xy-plane and \(\tau _{xy}\) gains a clear physical meaning. Full tortuosity \(\tau _{xyz}\) can be roughly approximated by a product \(\tau _{xy} \tau _{xz}\), with the error being in the order of \(1\%\).
We first examine the tortuosity components for each individual fracture geometry (Fig. 8). Recall that our 3D model geometry is curved in xz-plane with mean \(\tau _{xz} \! \sim \! 1.064\) for both upper and lower fracture surfaces. For no-contact cases (e.g. 0\(-\)0.3895 or 1\(-\)1.712 models) \(\tau _{xy}\) is typically much smaller than \(\tau _{xz}\); both components diminish with growing Re, although decrease in \(\tau _{xy}\) is \({\sim }3\) times larger (\({\sim }0.015\) vs \({\sim }0.005\)), meaning that lateral streamline undulations caused by aperture variations in xy-plane are ‘straightened out’ more easily than the more immutable undulations is xz-plane due to the large-scale fracture curvature. As soon as sizeable contact spots appear causing strong lateral streamline undulations, two components trade places as in e.g. 1–1 and 1\(-\)0.75 models. An exception to this rule is the 0\(-\)0.3 model with its rather negligible contact ratio of \({\sim }0.1\%\); for the 0.5\(-\)0.75 model with \(c = 0.43\%\) both components are quite close, especially at the opposite ends of the Re range. All other models have \(c > 1\%\) so that \(\tau _{xy}\) clearly dominates over \(\tau _{xz}\) since lateral undulations are hard to avoid in the presence of contact spots. Additionally, note that while for the viscous laminar regime \(\tau _{xz}\) never exceeds the wall tortuosity \(\tau _{xz}=1.064\) for any of the models (as the laminar flow tends to be the most energy-efficient), nearly all ‘shear and contact’ models overstep this limit for \(Re > 10^2\), as the flow becomes non-stationary and undulating in xz-plane similar to the 2D cases analysed in the previous chapter. In a way, tortuosity falls from the fully viscous plateau quite similar to the Forchheimer permeability drop in k(Re) plots; contrary to k this fall in \(\tau\) is stabilised or even reverted at the onset of non-stationary and turbulent flow regimes.
While analysing the tortuosity in no-contact models with increasing shear displacement (Fig. 9, zoomed-in plots), we notice that as \(\Delta x\) increases, \(\tau _{xy}\) grows while \(\tau _{xz}\) decreases. Regarding the \(\tau _{xy}\) first and recalling the corresponding streamline plots and aperture distributions, we see that increasing shear causes the formation of wider channels which present a much more appealing pathway for the fluid, while the non-sheared 0\(-\)0.3895 model is rather a ‘curved flat’ case in comparison, despite the noticeable streamline convergence around the hollows. On the other hand, the decrease in \(\tau _{xz}\) is most probably due to the fact that the increased overall aperture allows the fluid to ‘cut corners’ instead of meticulously following every bend of the geometry in the xz-plane. Finally, we have to note that both effects are present throughout the whole simulated Re range. On the other hand, all ‘shear and contact’ models show a steady increase in both \(\tau _{xy}\) and \(\tau _{xz}\) with increasing shear displacement, with the nearly-contactless 0\(-\)0.3 model being no exception.
For the sheared no-contact 0.5\(-\)1.0386 and 1\(-\)1.712 models, an interesting effect can be seen at the transition between stationary and non-stationary laminar flow at \(Re \sim 10^2\), where tortuosity seems to ‘blow up’, albeit not exceeding the highest level reached in a fully viscous laminar regime. This effect is even more pronounced for the sheared contact models, where large contact spots act as quite effective vortex generators even in the laminar regime — here the ‘blow-up’ considerably exceeds the maximum level set in viscous laminar regime. With respect to the turbulent regime, while the no-contact models clearly exhibit a gradual but steady ‘dampening’ of the streamline tortuosity (especially the no-shear 0\(-\)0.3895 model), all models with contact spots further increase the already ‘exploded’ tortuosity levels with growing Re, which corresponds to the chaotic pattern of superimposed vortex shedding and channeling through the maze seen in the respective streamline plots.
4.5 Forchheimer Beta Fitting
Fig. 10
Forchheimer \(\beta\) fitting. 1st row: \(\Delta x \!=\! 0\) mm (left) and \(\Delta x \!=\! 0.5\) mm (right) models. 2nd row: \(\Delta x \!=\! 1\) mm (left); ‘no contact’ models (right). 3rd row: ‘contact A’ models (left) and ‘contact B’ models (right). 4th row: laminar \(\beta\) for 3D models vs increasing shear (left) and \({\bar{d}}\) (right). Dashed lines denote fitting to laminar, dotted — to turbulent data
Fig. 11
Forchheimer \(\beta\) fitting, classical \(\rho \beta {\bar{u}}^2\) convention. 1st row: 0\(-\)0.3895 (left) and 0\(-\)0.3 (right) models. 2nd row: 0.5\(-\)1.0386 (left) and 0.5\(-\)0.75 (right) models. 3rd row: 0.5\(-\)0.5 (left) and 1\(-\)1.712 (right) models. 4th row: 1–1 (left) and 1\(-\)0.75 (right) models. Dashed lines denote fitting to laminar, dash-dotted — to turbulent data
×
×
Empirical Forchheimer coefficient \(\beta\) is commonly used to estimate the non-linearity of the fracture flow. Two different \(\beta\) conventions exist: i) the classical one with the second term expressed as \(\beta \rho {\bar{u}}^2\) so that \(\beta \!>\! 0\), and ii) one employed by Zimmerman et al. (2004) (their eq. 11) where \(\beta \!<\! 0\).
We start with the simpler second convention. Plots of normalised permeability \(k_n\) are shown in Fig. 10, with each model fitted by two \(\beta\) values for the laminar and turbulent regime, respectively. All models exhibit a good fit in the laminar regime. Despite ‘gapless’ laminar–turbulent transition, permeability curve in the turbulent range is underestimated by the laminar \(\beta\). This issue is more obvious for ‘no shear’ and ‘no contact’ models which feature S-shaped bends in the k curves; vice versa, if we try to fit the turbulent range we overestimate the k in the laminar regime between \(Re \!=\! 10^2\)–\(10^4\). At the same time, ‘shear and contact’ models (e.g. 0.5\(-\)0.5 and 1\(-\)0.75) can, in fact, be well matched with a single \(\beta\) value throughout the whole Re range.
In the next step, we fit our data with two separate \(k, \beta\) pairs for laminar and turbulent regime, respectively. Following Chauveteau and Thirriot (1967), we switch to the classical convention with a modified Forchheimer equation for turbulent flow given by
where \(k_t, \beta _t\) denote the permeability and Forchheimer coefficient for turbulent flow.
Results are presented in Fig. 11. In accordance with the findings of Chauveteau and Thirriot (1967) and Skjetne and Auriault (1999), our data obeys the \(k_t \!\le \! k\) and \(\beta _t \!\le \! \beta\) rule. For all models, both laminar and turbulent data are clearly fitted much better with two \(k, \beta\) pairs. Note the stark difference in k and \(k_t\), ranging from \({\sim }3.5{\textsf{x}}\) for ‘no shear’ models up to \({\sim } 9{\textsf{x}}\) for \(\Delta x \!=\! 1\) mm models. Unfortunately, both parameter pairs still have to be defined empirically for every single fracture geometry.
4.6 Flow Field Analysis
In this section, we demonstrate flow field specifics for the simulated 3D models. Parameters obtained from a CFD simulation can be roughly divided into ‘universal’ (\({\bar{u}}, p\)) and ‘turbulent only’ (\(\nu _t, k, \omega\)); we use resulting velocity field to generate streamlines through the model. Additionally, given both Hausdorff distance d and velocity \({\bar{u}}\), we calculate local Reynolds number \(Re_L = 2{\bar{u}}d/\nu\) over the entire fracture area. Note that here we use the hydraulic diameter formulation of Re so that \(Re_{\textrm{cr}} = 2300\) in full accordance with the convention of the governing Re set at the inlet. Resulting mean values of the local Reynolds number \(Re_L\) are slightly above the inlet Re (since we are dealing with a channel of highly variable aperture), but this difference gradually diminishes with growing Re — e.g. for the 0\(-\)0.3895 model we obtained \({\bar{Re}}_L \!=\! 1.57, 151, 1278, 10,974\) and 104632 for \(Re \!=\! 1, 10^2, 10^3, 10^4\) and \(10^5\), respectively.
For brevity’s sake we demonstrate only the most characteristic cases: streamline plots for \(\Delta x \!=\! 0, 0.5\) mm, and plots of \({\bar{u}},p\), \(\nu _t, k, \omega\) and \(Re_L\) for ‘shear and contact’ 0.5\(-\)0.5 model which features a quite complex distribution of contact spots. For 3D models, we do not delve into vertical cross-sections which were thoroughly examined in our 2D study, and show the horizontal fracture plane only; note that real geometry is much more bent and all data is projected onto a plane.
Figures 12, 13, 14, 15, 16 show the streamline plots for different shear steps and Re numbers, where color denotes the Hausdorff distance d. In a fully viscous Darcy regime (\(Re \!=\! 1\)) streamlines tend to adjust themselves nearly seamlessly to all aperture variations, converging in the wider sections and diverging around narrows or contact spots. In the smoother sections of the fracture streamline density is nearly homogeneous, indicating the absence of channeling. As Re grows and inertial forces increase (\(Re \!=\! 10^2\)), streamlines begin to ‘straighten out’ and form dominant channels marked by high streamline density. As the flow becomes non-stationary, laminar pattern breaks down into dominant channels and large recirculation zones behind obstacles, some of them sporting clear undulating von Kármán vortex streets in their wake. This divided non-stationary pattern is barely changing from upper laminar to turbulent regime and is shown only for \(Re \!=\! 10^5\). Most complex flow pattern is exhibited by the ‘shear and contact’ 0.5\(-\)0.75 and 0.5\(-\)0.5 models, where abundant contact spots force the fluid to run through a complex maze, completely disintegrating the homogeneous flow pattern at the inlet and contracting it into a few high-velocity flow channels and large recirculation zones behind the obstacles which are filled by the shed vortices and are not contributing to fluid transport.
Fig. 12
3D 0–0.3895 model, streamline plots. Laminar flow regime: \(Re \!=\! 1\) (top), \(10^2\) (middle); turbulent flow regime: \(Re \!=\! 10^5\) (bottom). Color denotes the Hausdorff distance d
Fig. 13
3D 0−0.3 model, streamline plots. Laminar flow regime: \(Re \!=\! 1\) (top), \(10^2\) (middle); turbulent flow regime: \(Re \!=\! 10^5\) (bottom). Color denotes the Hausdorff distance d
Fig. 14
3D 0.5−1.0386 model, streamline plots. Laminar flow regime: \(Re \!=\! 1\) (top), \(10^2\) (middle); turbulent flow regime: \(Re \!=\! 10^5\) (bottom). Color denotes the Hausdorff distance d
Fig. 15
3D 0.5−0.75 model, streamline plots. Laminar flow regime: \(Re \!=\! 1\) (top), \(10^2\) (middle); turbulent flow regime: \(Re \!=\! 10^5\) (bottom). Color denotes the Hausdorff distance d
Fig. 16
3D 0.5−0.5 model, streamline plots. Laminar flow regime: \(Re \!=\! 1\) (top), \(10^2\) (middle); turbulent flow regime: \(Re \!=\! 10^5\) (bottom). Color denotes the Hausdorff distance d
×
×
×
×
×
Pressure, velocity and \(Re_L\) plots for 0.5\(-\)0.5 model and \(Re \!=\! 10^4\) are shown in Fig. 18. Pressure field is controlled by the falling gradient from inlet to outlet; transversal geometry features such as ridges or contact spots act as barriers or ‘steps’ in this otherwise smooth gradient field. Turbulent flow at \(Re \!=\! 10^4\) is marked by larger wind shadows behind the larger obstacles and local disturbances in the pressure field downwind from the bottlenecks, roughly coinciding with the centers of shed vortices. Velocity plot at \(Re \!=\! 10^4\) compliments the streamline plots, illustrating the complexity of the turbulent flow through complex fracture geometry: uniform flow pattern is effectively broken apart into a large number of jet-like high-velocity flow channels and darker recirculation zones. Note that aperture variations are clearly visible in the uniform velocity field before the first contact ridge. Local \(Re_L\) number compensates for aperture-related velocity changes and provides a more consistent and focused image of dominant flow paths and wind shadows; still, the effects of turbulent diffusion remain clearly visible.
Plots of turbulent parameters \(k, \omega , \nu _t\) for 0.5\(-\)0.5 model at \(Re \!=\! 10^4\) are shown in Fig. 17, showing the finest details of turbulence production and dissipation. We remind the reader that k denotes the turbulent kinetic energy, \(\omega\) stands for its specific dissipation rate, while \(\nu _t \!=\! k/ \omega\) is the turbulent viscosity. All turbulent quantities show a gradual increase over the length of the model; every geometry feature encountered by the fluid acts as a turbulator: most importantly, the narrow aperture bottlenecks and contact spots. Generated turbulent quantities are then transported downwind along the dominant flow paths. A noticeable distinction between two obstacle types can be made: smaller point-source contacts leave clear vortex streets in their wake; naturally, those are somewhat smeared and averaged out by the steady-state RANS simulation. Wind shadows behind the larger contact spots are delimited by the stretching jets with the highest values of \(\nu _t, k, \omega\); further downwind these tight jets are then transformed into pairs of counterrotating vortices resembling the wingtip vortices or the convective vortices in the mushroom clouds. These vortices are most noticeable in the \(\nu _t\) plots and diffuse into the recirculation zones which are otherwise denied areas not contributing to fluid transport.
Fig. 17
3D 0.5−0.5 model, turbulent flow regime, \(Re \!=\! 10^4\): plots of pressure (top), velocity (middle), local Reynolds number \(Re_L\) (bottom)
Fig. 18
3D 0.5−0.5 model, turbulence parameters for \(Re \!=\! 10^4\): \(\nu _t\) (top), k (middle), \(\omega\) (bottom)
×
×
5 Conclusions
We have performed CFD simulation by solving full Navier–Stokes equations for both laminar and turbulent flow regimes. We developed realistic 3D scan-based fracture models, implementing both shear displacement and dilation, as well as the contact spots produced by model compression. Obtained permeability and friction factor data supplements and extends the results of previous experimental and numerical studies, covering a wide range of \(Re \!=\! 0.1\)–\(10^6\). With growing Re flow undergoes first stationary–non-stationary and then laminar–turbulent transitions: the former occurs at \(Re \! \sim \! 10^2\)–\(10^3\), depending on the shear displacement and contact ratio. Nonlinearity in both laminar inertial and turbulent regimes can by fitted by Forchheimer \(\beta\) coeffcient. All of the 3D models exhibit a gapless laminar–turbulent transition and largely behave as hydraulically rough channels (\(f \! \sim \! const\)) throughout the turbulent range, which allows for a good fit with a single \(k, \beta\) pair. ‘No contact’ models behaving as ‘hydraulically smooth’ are best fitted with two \(k, \beta\) pairs for laminar and turbulent regimes, respectively. For ‘no contact’ cases, the permeability of a 3D model can be reasonably approximated from a selection of 2D profiles regardless of shear displacement; however, as soon as the contact spots appear any 2D matching is off the table. Although contact ratios were kept rather low at \(\le \! 4\%\), cumulative adverse influence of contact spots on the fracture permeability was strong enough to override any beneficial effects of shear-induced aperture dilation. Fully viscous laminar flow is rather diffusive, filling all available fracture volume. Contact spots severely disrupt the inertial and turbulent flow patterns into a very complex maze of narrow high-velocity channels and large recirculation zones, which can be neither estimated from the aperture distribution nor predicted without full CFD simulation. Implemented wide ranges of both shear displacement and contact ratio produce essentially singular fracture geometries with non-recurring flow patterns, limiting the degree of generalisation.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
