Experimental study on non-Darcian flow through a single artificial fracture for different fracture apertures and surface roughness

Symbol Description

c

Aperture size

2B

Fracture spacing

μ

Dynamic viscosity of the fluid

ρ

Fluid density

g

Acceleration due to gravity

Q

Flow rate

J

Hydraulic gradient

R_e

Reynolds number

ν

Kinematic viscosity

V

Average flow velocity

K

Hydraulic conductivity

A, B

Forchheimer model coefficients (represent inertial and viscous forces)

a, b

Izbash model coefficients

h₁, h₂

Water levels in recharge and discharge flumes

l

The effective length of the flume

q

Unit width flux

Q_lcl

Discharge from local cubic law

Q_exp

Experimental discharge

H₁, H₂

Piezometric heads

L

Flow length between piezometers

∇P

Pressure gradients along the flow direction

β

Forchheimer coefficient (depends on the medium geometry)

F₀

Forchheimer number

Fractured rock aquifers play a crucial role in various fields, such as water resources management, disposal of radioactive waste, oil and gas production, geothermal utilization, and water production (Cook 2003). These aquifers consist of a network of fractures that traverse the rock matrix, serving as a storage medium for water due to their relatively high permeability compared to the surrounding impervious blocks. However, their permeability poses a significant challenge, as pollutants can readily travel through the fracture network and contaminate adjacent aquifers. Understanding the subsurface flow through fractured rocks is essential for effectively managing water resources, preventing contamination, and optimizing various industrial processes.

In the study of fractured aquifers, fluid flow behavior is strongly influenced by fracture characteristics such as tortuosity and connectivity. The tortuosity of the fracture path, in particular, plays a significant role in determining the flow rate, especially when the aperture size is small (Tsang 1984). To accurately simulate solute transport in fractures, it is necessary to consider the variation of hydraulic conductivity (K) with scale (Carrera 1993; Neuman 1994). While K is independent of the measurement scale for a homogeneous medium, it becomes a function of heterogeneity and fluid flow for a heterogeneous medium (Schulze-Makuch et al. 1999). Researchers have observed a correlation between hydraulic conductivity (K) values and the measurement scale in different granitic rocks, indicating that K increases until a particular rock volume is reached, beyond which it remains constant (Clauser 1992). Several methods, such as slug tests, flow meter tests, permeameter tests, pumping tests, grain size analyses, and geophysical tests, can determine the hydraulic conductivity of saturated groundwater flow (Neuman & Di Federico 2003). Anand et al. (2021) investigated several field and laboratory studies on the mobility of cohesionless sediments through open channels.

Studies have shown that the aperture of natural fractures can be larger than the hydraulic aperture. The distinction between hydraulic and mechanical aperture is essential, with the latter often being larger due to surface roughness and tortuosity (Hakami & Larsson 1996; Chen et al. 2000). Cornet et al. (2003) concluded that for a mechanical aperture greater than 15 μm, both mechanical and hydraulic aperture become equivalent for flow calculations in a single fracture, and the surface roughness no longer alters the flow. Fractured aquifers also exhibit faster flow rates than porous media and non-Darcian flow has been observed in various experiments and numerical models (Choi et al. 1997).

The nonlinear flow behavior is encountered for groundwater flow through porous media when the Reynolds number exceeds 10 (Oron & Berkowitz 1998). Rough-surfaced fractures can further complicate the flow, as permeability can vary even over small distances in hard-rock terrains (Baker 2000; Berkowitz 2002). It is recommended to consider the average value of hydraulic parameters to account for this variability. Bear & Corapcioglu (2012) characterized a fracture as a set of parallel plates with a well-defined aperture for modeling flow within the fracture. Local cubic law (LCL) was applied to calculate the flow rates. However, LCL for fracture flow has been inadequate in many case studies, as flow in fractured rocks often exhibits non-Darcian behavior (Field & Nash 1997; Kohl et al. 1997). LCL, Stokes, and Navier–Stokes equations have been further explored to model flow in rough-walled fractures for different aperture sizes (Brush & Thomson 2003).

Experimental studies have been conducted to investigate flow characteristics through natural fractures and laboratory-prepared fracture samples. These studies have focussed on characterizing nonlinear flow behavior, evaluating the applicability of flow equations such as Izbash and Forchheimer's equations, and examining the influence of hydraulic gradient, scale effect, and surface roughness (Qian et al. 2005, 2007, 2011a, 2011b; Zhang & Nemcik 2013; Yan et al. 2018). Chen et al. (2015) conducted laboratory experiments on rough-walled granite fracture samples and studied the variation of Forchheimer's coefficient with confining stress. Outflow experiments on smooth and artificial rough fractures were conducted, where both Izbash and Forchheimer's equations were found capable of describing the transition flow conditions (Tzelepis et al. 2015). Li et al. (2016) studied the influence of hydraulic gradient, scale effect, and surface roughness on single-fracture intersections having nonlinear flow. The determination of geometric parameters and the development of empirical models have also been explored to understand better and predict fluid flow in a single fracture (Rong et al. 2020; Xing et al. 2021; Chen et al. 2022). Ikhsan et al. (2022) analyzed the impact of grain size uniformity and shear stress on riverbed stability for steady and uniform flow, aiming to ascertain the minimum thickness required for a non-cohesive Armor layer. Qasim et al. (2022) further examined how the discordance of bed flumes can influence the flow characteristics of hydraulic structures. Energy dissipation by porous structure was studied, which in turn can help in mitigating scour around bridge abutments (Widyastuti et al. 2022)

The present study mainly focuses on the experimental investigation of groundwater flow in an artificially built single fracture for different surface roughness (fine, medium, and coarse), fracture apertures, and hydraulic gradients. To avoid the geometrical constraints and uncertainty in data interpretation and to precisely regulate the experimental parameters (aperture, surface roughness), an artificially built fracture plane with sand glued on one surface to simulate fracture roughness has been used. A few researchers, such as Qian et al. (2005, 2011a, 2011b), Chen et al. (2009), and Tzelepis et al. (2015), have also done a similar study using a single artificial fracture.

However, a distinct departure from previous research is observed in several aspects. Firstly, the fractures used in the present study are significantly larger, approximately twice the size of those examined in previous studies (Qian et al. 2005, 2007, 2011a, 2011b; Tzelepis et al. 2015; Yan et al. 2018). Secondly, the fracture surfaces are coated with sand of different particle sizes, conforming to the Indian Standards classification of fine, medium, and coarse roughness, presenting a novel approach to surface characterization. Additionally, the present study incorporates a diverse range of aperture sizes (3, 9, and 12 mm), in contrast to the limited aperture configurations examined in earlier research. Notably, plywood sheets were utilized to simulate the fracture planes, representing a departure from the plexiglass sheets employed in previous investigations. Furthermore, a comprehensive network of piezometers was installed along the fracture length to facilitate accurate and detailed monitoring of the flow behavior, a significant improvement over the limited number of piezometers typically used in earlier studies.

The present study emphasizes the experimental investigation of flow through a single fracture, which is essentially non-Darcian. The gradient of Reynolds number with different average flow velocities for different surface roughness and aperture size is studied. The parameters for different flow-characterizing models have been calculated, finding the best-suited model for the prevailing conditions. Further, hydraulic conductivity variation with flow length is studied for different apertures, surface roughness, and hydraulic gradient. Also, the variation of the Forchheimer coefficient with hydraulic gradient is discussed.

Through these novel experimental setups and measurements, we gained valuable insights into the influence of fracture size, aperture, and surface roughness on unconfined groundwater flow characteristics. The findings contribute to a better understanding of flow behavior in large-scale fractured rock aquifers and offer important implications for groundwater management and modeling in similar geological settings.

In this equation, Q [L³T⁻¹] represents discharge, and J (dimensionless) is the hydraulic gradient in the fracture plane. This approach is valid for laminar flow only. Zimmerman & Bodvarsson (1996) suggested that the aperture should be replaced by the hydraulic aperture in the case of rough-wall fracture for the applicability of cubic law.

For calculating the Reynolds number for different flow rates through the fracture, the above equation is used. It has been observed that at R_e> 10, flow no longer remains laminar (Zimmerman & Yeo 2000). LCL becomes invalid, and Darcy's law becomes inapplicable in such flow conditions.

In this equation, a [TⁿLⁿ] is a model parameter, and b (dimensionless) is the power index whose value varies between 1 and 2. After plotting V and J on a log-log scale, if the trend comes to be linear, then the slope corresponds to the power index. However, if a nonlinear trend is observed, the Forchheimer model should be applied. Compared to the Forchheimer model, this model offers an advantage as it involves a single pseudo hydraulic conductivity parameter instead of two, as in the Forchheimer equation.

The aperture of the fracture, which refers to the distance between the two plywood sheets, was adjusted to different sizes (3, 9, and 12 mm) for conducting the experiments. It was intended to keep the aperture size as minimal as possible while fabricating the fracture in the laboratory. Since plywood sheets were utilized to prepare the fracture, it was quite challenging to work on such small aperture sizes. The initial gap between fracture sheets was 12 mm. Thereafter, to create a 9 mm gap for creating an aperture size of 9 mm, the plywood sheets of 3 mm thickness were placed and stuck on one side of the fracture sheet. Similarly, to create a 3 mm aperture size, a 9 mm thick plywood sheet was inserted and stuck on one side of the fracture. The authors tried to create the aperture sizes in multiple of 3. However, due to the unavailability of a 6 mm thick plywood sheet, the experimental study was not done on a 6 mm aperture size. Therefore, the study focussed on 3, 9, and 12 mm aperture sizes.

The flow rate through the fracture could be altered by adjusting the hydraulic head in the recharge and discharge flumes. Nine piezometers were installed inside the fracture plane at equal distances to determine the hydraulic head. The calculations were made once the steady-state flow condition was obtained. To ensure an accurate representation of flow behavior within the fracture for turbulent flow through the media, the first piezometer was strategically positioned at a distance of 52 cm from the source. This location was chosen to minimize the impact of the boundary layer, keeping it at a distance greater than the entrance length (>10 c). As a result, the effective length under consideration extends for a total of 948 cm. The average flow velocity (V) determined the flow rate to the discharge flume.

Here, Q represents the total discharge rate through the fracture, and (h₁+h₂)/2 indicates the average saturated width of the fracture (Qian et al. 2011a, 2011b).

In this equation, V represents the average flow velocity, q is the unit width flux, and c denotes the average aperture. For the experimental runs conducted at an average temperature of 20 °C, the kinematic flow viscosity is assumed to be 1.005 × 10⁻⁶m²/s (White 1990).

For all the cases, the Reynolds number fell within the range of 12.34–474.22, suggesting the flow to be non-Darcian. The gradient of Reynolds number with flow velocity increased with larger aperture sizes, regardless of the fracture roughness.

The obtained experimental data were compared with three popular flow models, i.e., Darcy, Izbash, and Forchheimer, to identify the most suitable model for describing flow conditions in the single fracture model. The model parameters were computed using the SPSS statistical tool and are presented in Tables 2–4. These parameters were then used to fit the corresponding model equations, and the agreement between experimental and model data was assessed, as depicted in Figures 6 and 7.

Regression analysis revealed that non-laminar flow conditions prevailed throughout the experiments, rendering the Darcy equation unsuitable. However, for a 3 mm aperture, the Darcy equation provided a good fit, suggesting creeping flow conditions at smaller aperture values. The coefficients of determination for the Forchheimer and Izbash models were comparable, but the Forchheimer model exhibited greater precision. Due to the complexity involved in calculations with the Forchheimer equation and considering turbulent flow, the Izbash model was chosen for subsequent hydraulic conductivity calculations.

Table 5 represents the K–L relation approximation in the form of linear function K = aL+b, where ‘a’ is the slope and ‘b’ is the intercept.

The slope of the K–L regression lines was found to increase with wall roughness for 9 and 12 mm aperture sizes, and it exhibited an irregular trend for 3 mm wall roughness. Additionally, the slope increased with aperture size for medium and coarse-wall fractures, while a decreasing slope trend was observed for fine-wall roughness with increased aperture size. The correlation coefficient was highest for fine-wall roughness fracture and hence was considered more suitable for the present discussion. From Figure 8 and Table 5, it is evident that the slope of K–L regression lines decreases with aperture size, implying a higher scale dependency for smaller apertures. However, this trend was not observed in coarse surface fracture due to a weaker correlation between K and L.

The coefficient β in A depends on the geometry of the media, which needs to be calculated from experiments. With regard to B, k=c²/12 represents the intrinsic permeability of the fractured media. The main concern in the fracture is mainly coefficient A, in contrast to porous media, where both A and B need to be parameterized simultaneously.

This study aimed to investigate unconfined flow in a single fracture with uniform surface roughness and aperture sizes through well-controlled laboratory experiments. The experiments were designed to minimize the influence of the boundary layer by installing the first piezometer at a distance greater than the entrance length (>10 c) for turbulent flow through the media, ensuring that the flow behavior within the fracture was accurately captured.

The results indicated that the flow through the single fracture was non-Darcian for different surface roughnesses (fine, medium, and coarse) and aperture sizes (3, 9, and 12 mm). This conclusion was reached by calculating the Reynolds number and conducting regression analyses. The non-Darcian flow behavior observed in the study suggests that the Darcy equation is not suitable for describing groundwater flow at the laboratory scale, even within a smaller range of Reynolds numbers.

It is important to note that in this study, the flow tortuosity, which can be present in natural fractures due to factors such as varied apertures, multiple fractures, and non-uniform roughness, was not considered. Therefore, future studies should take these factors into account to better simulate real-world conditions.

The influence of inertial terms on the flow behavior was found to increase with surface roughness, with the trend being coarse > medium > fine. The regression analysis indicated that the Forchheimer and Izbash models were suitable for describing the prevailing non-laminar flow conditions. However, considering the simplicity and accuracy of the Izbash model, it is recommended for estimating hydraulic conductivity.

This study also revealed that hydraulic conductivity exhibited a linear increase with flow length. This can be attributed to the 2-D torturous flow within the rough fracture surface and additional inertial effects, resulting in energy dissipation within the fracture. The slope of the hydraulic conductivity vs. flow length relationship was found to be more dependent on surface roughness than aperture size. Thus, hydraulic conductivity is more scale-dependent on the surface roughness than the aperture and hydraulic gradient.

Furthermore, the results showed that hydraulic conductivity decreased with an increase in hydraulic gradient, assuming other conditions remained constant. This can be attributed to the increased friction between the rough fracture surfaces at higher hydraulic gradients, as well as the higher turbulent flow velocity induced by the increased hydraulic gradient.

It is important to acknowledge that the present study had certain limitations that could be addressed in future research to enhance the understanding of flow behavior in real-world conditions. Firstly, the study utilized an idealized single-fracture model with parallel fracture plates and uniform surface roughness. While this provided valuable insights into the flow characteristics within the fracture, it may not fully represent the complexities observed in actual field conditions, where multiple fractures can exist with varying surface roughness. Therefore, future studies should focus on investigating flow characteristics in multiple fractures with different aperture sizes to better simulate the field conditions. Secondly, experimental runs were conducted for unconfined flow conditions, which may not fully represent the real-world scenario. Therefore, confined flow conditions must be incorporated in future studies to simulate better the dynamic interactions between fractures and surrounding rock formations.

Additionally, a comparative analysis between flow conditions in natural fractures and single artificial fractures, as utilized in this study, would provide a more comprehensive perspective on fluid migration through the geological formations. In summary, by addressing these limitations and incorporating more representative experimental setups, future studies can significantly contribute to the advancement of knowledge regarding flow behavior through the fracture. This would ultimately lead to a better understanding of fluid migration in subsurface reservoirs, improving our ability to manage natural resources and make informed decisions in various engineering and environmental applications.

Based on the study, the following conclusions were reached:

• (1)

  The flow through a single-fracture surface is essentially non-Darcian or fully turbulent under different surface roughness conditions. Reynolds number showed an increasing trend with average flow velocity, independent of the surface roughness. Decreasing the aperture size led to a decrease in Reynolds number for the same roughness, and the gradient of Reynolds number vs. average flow velocity increased with larger apertures.

• (2)

  Non-Darcian turbulent flow conditions were observed within the medium to extensive Reynolds number range (12.34–474.22), confirming the nonlinear relationship between average velocity and hydraulic gradient found in previous studies.

• (3)

  The ratio of calculated discharge (Q_lcl) to experimental discharge (Q_exp) was significantly greater than 1 for fractures with 9 and 12 mm apertures and different surface roughnesses. This suggests that no creeping flow occurred through the fracture, and the influence of inertial terms outweighed the viscous term. The impact of inertial terms was observed to increase with higher surface roughness. However, for a coarse surface fracture with a 12 mm aperture, a decreasing trend of Q_lcl vs.Q_exp indicated that a threshold value has been reached.

• (4)

  Both Forchheimer and Izbash models were suitable for describing the non-laminar flow conditions prevailing in unconfined fractures with different wall roughness and aperture sizes.

• (5)

  Hydraulic conductivity exhibited a linear increase in flow length for fractures with different apertures and surface roughnesses. The scale dependence of hydraulic conductivity on surface roughness was attributed to the torturous flow and energy dissipation within rough fractures.

• (6)

  Higher hydraulic gradient values corresponded to smaller hydraulic conductivity values, assuming other conditions remained constant. This can be attributed to the increased turbulent flow velocity induced by a higher hydraulic gradient and increased friction between rough surfaces.

• (7)

  The Forchheimer coefficient increased significantly with increasing confining stress for different β values of fine, medium, and coarse surface roughnesses until a critical value was reached.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.