Experimental Investigations of Fracture Deformation, Flow, and Transport Using a Pressure-Controlled Hele-Shaw Cell and Digital Fabrication

Abstract

In this paper we present a novel pressure-controlled Hele-Shaw cell to investigate different physical processes in rough fractures using 3D-printed rock analogs. Our system can measure high-resolution fracture aperture and tracer concentration maps under relevant field stress conditions. Using a series of hydraulic and visual measurements, combined with numerical simulations, we investigate the evolving fracture geometry characteristics, pressure-dependent hydraulic transmissivity, flow channeling, and the nature of mass transport as a function of normal stress. Our experimental results show that as the fracture closes and deforms under increasing normal loading: (a) the contact areas grow in number and size; (b) the flow paths become more focused and tortuous; and (c) the transport dynamics of conservative tracers evolve toward a higher dispersive regime. Moreover, under the applied experimental conditions, we observed excellent agreement between the simulated- and the experimentally measured-hydraulic behavior.

Key Points

• A novel pressure-controlled Hele-Shaw cell was introduced to study fracture deformation, flow, and mass transport in 3D-printed fractures

• In situ aperture maps and tracer concentration fields were measured under relevant field stress conditions

• We linked the changes in mass transport dynamics to the evolution of the aperture field and the channelization degree of the flow

1 Introduction

Flows affecting underground geological structures often occur through fractured rock. This is the case in enhanced geothermal systems, some hydrocarbon reservoirs and nuclear repositories, and in other environmental systems and civil engineering projects. Not surprisingly, in the past decades, significant efforts have been devoted to the study of flow and transport processes in these systems given their high economic, social, and environmental effects (e.g., National Academies of Sciences, Engineering, and Medicine, 2020).

Understanding the interplay between the different interacting physical processes in the fracture environment is essential to explain and predict the behavior of these systems. A classic example is the interaction between stress and flow. It is well recognized that the flow in stressed fractured systems tends to occur along a few fractures within the network (Berkowitz, 2002; C. Tsang & Neretnieks, 1998). When the far-field stresses evolve (e.g., during production, or upon stimulation treatments), some of the fractures are reactivated while others are likely to close, affecting the preferential flow paths within the network (P. K. Kang et al., 2019; Z. Zhao et al., 2010).

Similarly, at the single fracture scale, the flow tends to concentrate in a few preferential pathways (i.e., channels). The spatial structure of such channels is dictated by the combination of the aperture field resulting from the intrinsic fracture geometry, the stress level, and the fluid pressure gradient (Brown, 1987; C. Tsang & Neretnieks, 1998). Hence, fracture deformation may have a strong effect on flow channelization and the nature of the mass and heat transport in the fracture (e.g., P. K. Kang et al., 2016).

In this paper, we specifically investigate these fundamental physical processes using fracture analogs and Hele-Shaw cells. Fracture analogs are useful to simplify the mechanical and geochemical properties of the rock fracture while they enable and enhance visualization of fluid flow and mechanical deformation. In addition, these techniques facilitate experiment repeatability and allow one to systematically control the geometry and material properties of the specimens.

Hele-Shaw cells have been successfully used to study a great variety of processes such as (a) fluid flow including the validation of numerical models (Cheng et al., 2004; Nicholl et al., 1999), (b) the investigation of conservative (Detwiler et al., 2000; Nowamooz et al., 2013) and reactive transport (Detwiler, 2010; Jones & Detwiler, 2016), or (c) the settling and mobilization of proppants (Medina et al., 2018). While some of the experimental apparatuses in these studies can apply small normal loads to the fracture analogs (usually made of textured glass), they cannot produce stresses relevant to field conditions. Moreover, the aforementioned studies just applied loads to induce normal fracture closure without consideration of the mechanical properties of the fractures. In fact, to the best of our knowledge, the effect of normal stress on fracture deformation and its effect on flow, channeling and mass transport have not been explored so far using fracture analogs in Hele-Shaw cells. While this topic has been extensively investigated using real rocks and numerical simulations, there is still a lack of high-quality and high-resolution experimental data (Zou & Cvetkovic, 2020); a problem that can be overcome with the experimental methodology described in this paper.

In particular, new experimental efforts are needed to better understand the underlying physical processes related to (a) stress-dependent permeability relationships (e.g., Chen et al., 2017; Iwano & Einstein, 1995; Pyrak-Nolte & Nolte, 2016); (b) the evolution of the flow field with fracture deformation (e.g., C. Tsang & Neretnieks, 1998); and (c) the emergence of anomalous transport in rough fractures (e.g., Berkowitz et al., 2006; P. K. Kang et al., 2016; Nowamooz et al., 2013; C. Tsang & Neretnieks, 1998; Zou & Cvetkovic, 2020). For instance, in tracer tests, breakthrough curves (BTCs) frequently feature early arrival and long tails due to the complex advective processes in fracture flow (i.e., channeling). Other mechanisms such as adsorption onto the walls, matrix diffusion, diffusion into and out of stagnation zones, and eddy trapping may also be responsible for the occurrence of long tails (Hawkins et al., 2018; Lee et al., 2015). Moreover, on some occasions, BTCs may even show more than one peak (Moreno & Tsang, 1991).

In this work we use new techniques in digital fabrication combined with a novel pressure-controlled Hele-Shaw cell to replicate stress levels in the fracture environment relevant to field conditions. In Section 2, we detail our experimental methodology, including the design and fabrication of self-affine rough surfaces and a short discussion of scaling of material properties. Then through Sections 3, 4, 5, we obtain high-quality and high-resolution data of the aperture and tracer concentration fields needed to study fracture deformation and stiffness, the evolution of flow properties, and conservative mass transport. Finally, in Section 6 we summarize the work and outline potential applications for this novel setup.

2 Experimental Methodology

This section describes how the hydraulic and mechanical behavior of fractures can be explored using analog specimens. Analogs are powerful tools as they allow one to have a fine control of geometric and mechanical properties, as well as of test reproducibility. Specifically, we discuss the design and fabrication process of fracture analogs and describe a novel Pressure-Controlled Hele-Shaw Cell that was designed to measure fracture deformation, permeability, and transport under relevant in situ conditions.

2.1 Fracture Analogs and Digital Fabrication

The fracture specimens used in this study were designed and manufactured using a digital fabrication approach. Our workflow starts with the numerical generation of the specimen geometries following Brown (1995). This method, described in Text S1 in Supporting Information S1, generates 3D self-affine fractures based on three main components: the Hurst exponent, H, the mismatch wavelength, λ_m, and the root-mean-square roughness, h_rms. The generated self-affine surfaces have power spectral densities (PSD) of the form S^iso(ξ) ∝ ξ^−2−2H, where S^iso is the radial-averaged version of the 2-dimensional PSD, and ξ is the spatial frequency (i.e., ξ = 2π/λ, where λ is the wavelength), a.k.a. wavenumber (see, e.g., Jacobs et al., 2017, or Section 3.2.2).

Figure 1a shows a 3D self-affine mated fracture generated using the method described in Text S1 in Supporting Information S1. In 3-dimensions, the fracture aperture is defined as the void space structure limited by the upper and lower surfaces of the fracture. This aperture volume contains information regarding both the middle surface and the local separation between the upper and lower surfaces, , where is the position vector (Figure 1b). Note that in many applications the 2D distribution of the local separation (or apertures)—also known as the composite aperture (Figure 1c)—is assumed to describe the hydro-mechanical behavior of the fracture (Brown, 1995). For the sake of simplicity, the fractures in our experiments will be represented by a rigid flat surface and a deformable composite aperture (Figures 2c and 4a).

Figure 1

(a) Numerically generated 3D model of a fracture. The aperture volume corresponds to the 3-dimensional void space defined by the upper and lower fracture surfaces. (b) The 3D void space structure contains information about the local separation between the upper and lower surfaces (i.e., b(x) = z_up(x) − z_low(x)) in addition to the structure of the middle surface. (c) In many applications, this local separation b(x), known as the composite aperture, is assumed to describe the mechanical and hydraulic behavior of a fracture. The composite aperture describes how the fracture volume is connected in the XY-plane and does not include information regarding the undulation of the middle surface. Models using the composite aperture are often referred as 2.5D or depth averaged-descriptions.

Figure 2

Initial geometry of the two self-affine aperture distributions generated with the parameters listed in Table 1. (a) Aperture map of the two specimens, A and B. Since both aperture distributions were generated using the same H and λ_m they are basically scaled versions of each other. Hence the color scale is different for both specimens; they differ by a factor of 2 (recall that ). (b) Cross-section of the composite aperture of specimens A and B along the center line shown in panel (a). (c) 3D-printed specimen of the composite aperture using Clear™ resin. Our specimens have L × W dimensions of 76.2 × 38.1 mm, and thickness of 7.6 mm (which is more than 10 times the mean aperture).

For the research reported in this paper, we generated two isotropic, self-affine fracture aperture distributions (Specimens A and B) using the same H and λ_m, and two different h_rms. Note that as H and λ_m are the same, these two distributions are basically scaled versions of each other. In fact, the scale is given by the ratio . Also, by definition, the spatial statistics of the aperture fields are independent of the direction of measurement (i.e., isotropic aperture fields). Our fracture analogs have a length to width ratio, L/W, of 2 with L = 76.2 mm. Figure 2 and Table 1 summarize the geometric characteristics of both aperture distributions. We selected H = 0.8 which is a reasonable value in real fractures (e.g., Brown & Scholz, 1985; Odling, 1994). To achieve a relatively homogeneous flow field, a low λ_m relative to the fracture dimensions L and W was chosen (λ_m = 0.05L = 0.10W). Recall that the mismatch wavelength is related to the correlation length of the aperture field, and, therefore, it controls the spatial structure of the permeability field.

Table 1. Specimen Geometric and Mechanical Properties

	L (mm)	W (mm)	H (–)	λm (mm)	hrms (mm)	E (GPa)	bm (mm)	bstd (mm)	θ (mL)
Specimen A	76.2	38.1	0.8	3.8	0.38	1.25	0.537	0.113	1.560
Specimen B	76.2	38.1	0.8	3.8	0.19	1.25	0.269	0.056	0.780

• Note. L: specimen length, W: specimen width, H: Hurst exponent, λ_m: mismatch wavelength, h_rms: root-mean-square roughness, E: Young's modulus, b_m: mean aperture, b_std: standard deviation of apertures, and θ: fracture volume.

The topographies of the rough face of both specimens are shown in Figure 2a. For the fabrication of the fracture analogs we used a Formlabs® Form3® printer. This printer uses a low force stereolithography technology, and features a maximum theoretical resolution of 25 μm in XYZ directions and a laser spot size of 85 μm (Formlabs, 2022). The Form3® can print with a wide variety of materials. We used Clear® resin due to its high optical transparency, which is advantageous for flow visualization in milifluidic applications (Figure 2c). The specimens were tested as printed, with no further post-processing treatment (i.e., no additional UV/temperature curing to enhance the material strength).

2.2 Scaling and Mechanical Properties

The mechanical properties of the printing material (Clear® resin) were determined in uniaxial compression tests, both parallel and perpendicularly to the printing direction. The cured resin has a Young's modulus, E, between 1 and 2 GPa, and a Poisson's ratio, ν, of ∼0.37. Since these Young's modulus values are significantly lower than those of real rocks in the field, E_rock/E_analog ∼ 10–60, the laboratory conditions (e.g., applied stress) must be scaled accordingly. To a first approximation, one can set the stress-stiffness ratio such that σ_lab/E_analog ∼ σ_field/E_rock. Figure 3 shows how this stress-stiffness ratio varies as a function of the real material properties (E_rock) and the underground depth.

Figure 3

Laboratory-Field scaling of mechanical properties. Variation of the stress-stiffness ratio as a function of depth and material properties (E_rock). In the field, the stress conditions are a function of multiple factors including the far field tectonic stresses and the local geology. To a first approximation the vertical stress (σ_v) at a given depth can be estimated by integrating the weight of the rock column at that point (i.e., σ_v = ∫ρgdz). The labels in the ordinate represent the depth of some relevant infrastructure (e.g., Eurotunnel, the Waste Isolation Pilot Plant) and natural resources.

For instance, let's assume one wants to scale down the field conditions to the lab for a sandstone (E_rock ∼ 20 GPa) at 1,000 m depth. From Figure 3 one can see that these conditions correspond to a stress-stiffness ratio ∼10⁻³. Therefore, the appropriate lab conditions will be given by σ_lab ∼ 10⁻³E_analog, and for our specimens σ_lab ∼ (10⁻³) (1 GPa) = 1 MPa.

Note that this first-order scaling is appropriate for simple physical processes as long as the material follows linear elastic deformation. More complex physics, for example, plastic and viscoelastic behaviors, may require additional temporal and spatial analysis. For instance, Hubbert (1945) suggests that different geological phenomena such as orogenic and gravitational processes may be investigated by means of the principles of physical similarity using plastic materials similar to the ones used in this study. Nevertheless, for the purpose of this paper the linear scaling shown above is sufficient.

Moreover, bear in mind that 3D-printed materials may exhibit anisotropic properties with respect to the printing direction (e.g., Jiang et al., 2020). Our preliminary mechanical characterization of the Clear® resin showed variations on the other of ∼20% in both elastic and strength properties. For the sake of simplicity, the effect of anisotropy on the mechanical behavior is not addressed in this paper (the printing direction for both Specimens A and B coincides with the fracture length (Figure 2c)).

Finally, note that the Poisson's ratio of this material (ν ∼ 0.37) is in the upper range for natural rocks (Gercek, 2007), and therefore, these analogs will deform laterally more than most rocks. Although this will affect the lateral spreading of the contact points, one can neglect this second-order effect for the sake of simplicity.

2.3 Pressure-Controlled Hele-Shaw Cell

In our experiments, the fractures are represented by a deformable rough surface (fracture analog) pressed against a rigid flat one (borosilicate glass window). This rough versus flat surface configuration is similar to the ones used in traditional Hele-Shaw cells setups (Figure 4). The novelty here is that the rough body representing the composite aperture is deformable, and that a uniform confining pressure is applied on the back and the side faces of the specimen.

Figure 4

(a) Cross section of the pressure-controlled Hele-Shaw cell with a rough surface (specimen) against a flat surface (window). Our cell consists of a rigid flat borosilicate window and a deformable fracture analog (Figure 2c). Uniform confining pressure is applied on the specimen using water as a confining fluid. (b) The specimens are sealed using customized polydimethylsiloxane membranes and a sealing ring. Fluid distributors inspired by microfluidic chips are used to assure uniform pressure distribution at the inlet and outlet of the specimen.

This pressurized Hele-Shaw cell can accommodate 38.1 × 76.2 mm prismatic specimens (Figure 2c), which are sealed using custom silicone membranes fixed with a sealing ring (Figures 4a and 4b). These membranes are fabricated with injection molding using polydimethylsiloxane (PDMS), a silicone-based polymer, which is completely transparent to visible light and does not react with the confining and fracture fluids (see Text S2 in Supporting Information S1 for details).

The experiments presented in this paper feature single-phase flow using water as the fracture fluid. Three-port coaxial needles located at the inlet and outlet of the cell are used for independent injection of water and a dye-tracer (Figure 4 and Figure S2 in Supporting Information S1). These needles enable direct fluid pressure measurements while reducing the fluid dead volume when switching between water and tracer injection. Uniform flow distribution at the inlet and outlet of the fracture is achieved thanks to a pair of fluid distributors with engraved multi-step channels inspired by microfluidic chips (Figure 4b). These fluid distributors are fabricated with transparent polymethyl methacrylate and have a relatively low dead volume (<0.1 mL), which alleviates issues related with tracer mixing and leakage. Please refer to Text S2 in Supporting Information S1 for an extended description of the experimental setup.

Very importantly, the setup includes a diffuse white-light source (Autograph® 930 LX LightPad®) and a scientific monochrome camera (Ximea® CB500MG-CM) for continuous imaging of fracture deformation and fluid flow under experimental conditions. The aperture and tracer concentration fields are determined from changes in the transmitted light intensity using the Beer-Lambert law as detailed in the next section. The camera is computer-operated and synchronized with the rest of electronic components using the control code mentioned above.

2.4 Aperture and Tracer Concentration Measurements. Light Transmission Techniques

Light transmission techniques based on the Beer-Lambert law are commonly used to determine the aperture and concentration fields in micro- and mili-fluidic investigations. Initially proposed by Glass et al. (1991), these techniques allow one to relate changes in the absorbed light intensity within the Hele-Shaw cell to specific properties of the light-transmitting media (Detwiler et al., 2000), in our case, the fracture fluid. Under experimental conditions, the radiance from the cell, I, is measured with the fracture filled with clear water (I₀) and during the injection of a dye tracer (I_C). For the case of a monochromatic light source, the light absorbance, A, can be related to the fracture aperture, b, and concentration, C, fields by the Beer-Lambert law:

(1)

where ɛ is the dye absorptivity (a solute property), and represents the equivalent aperture . In Equation 1, denotes local values at a given point. Further references to the Beer-Lambert law and its use for the determination of aperture and concentration maps can be found elsewhere (e.g., Detwiler et al., 1999; Nowamooz et al., 2013; Villamor-Lora et al., 2019).

Note that the Beer-Lambert law is an empirical relationship that relies on the assumption of monochromatic light source. When a polychromatic light source is used, Equation 1 no longer holds as the dye absorptivity is a function of the irradiance wavelength, that is, ɛ = ɛ(λ)—see Text S3 in Supporting Information S1 for details.

As a dye tracer, we used Coomassie Brilliant Blue G-250 (Alfa Aesar® Brilliant Blue G, ultrapure) at a concentration C_in = 0.25 g/L. This dye is commonly used in analytical biochemistry for staining proteins, and under experimental conditions (pH ∼ 7) it presents a bright blue with an absorption maximum at ∼590 nm (Chial et al., 1993).

The light intensities (I₀, I_C) are measured with a high-resolution (47.5 MPx) monochrome camera equipped with a 100 mm f/2.8 macro lens. This camera has a 12bit CMOS sensor with a dynamic range of 64 dB, which translates to light intensity noise on the order of 1%–4%. To further reduce this noise, we apply a pixel binning to the original image using a 4-pixel neighborhood. The resulting images of the specimen have a resolution of 950 × 1,900 pixels (each pixel representing 40 × 40 μm in the fracture space domain). Also, under static conditions (i.e., before and after the tracer injection), we take 20 pictures of the same scene and then average them. When this averaging is done, the resulting precision error in the aperture estimation is about 2–3 μm. Further details describing the experimental methodology and the error analysis are included in Texts S4 and S5 in Supporting Information S1.

2.5 Experimental Protocol

For each of the two self-affine fracture analogs described in Section 2.1, we conducted a series of fracture deformation, flow, and transport measurements at different confining pressure levels. Each test starts with a back-pressure (BP), saturation phase followed by a series of confining pressure (CP), steps from 100 to 1,400 kPa, as illustrated in Figure 5. Recall that these specimens have a E ∼ 1.25 GPa, and therefore, the testing conditions correspond to σ/E ratios ranging from 10⁻⁵ to 10⁻³. Please, refer to Text S6 in Supporting Information S1 for a detailed description of the testing protocol.

Figure 5

Schematic of the experimental protocol. After a saturation phase to ensure that any residual bubbles are dissolved in the fracture fluid, the confining pressure is increased in a stepwise fashion. Each CP step is divided into three phases: equilibrium, permeability measurements, and dye injection. These phases are designed such that the specimen reaches equilibrium with the newly applied pressure, to then determine fracture permeability, and geometric and mass transport properties. More details are given in Text S6 in Supporting Information S1 and Sections 3, 4, 5.

In the following sections we will illustrate how our pressure-controlled Hele-Shaw cell setup can be used to study the evolution of fracture deformation, flow, and mass transport under different stress conditions. We will discuss why the investigation of coupled processes (i.e., fracture deformation, flow, and mass transport) require such independent measurements, and how our approach can fill the gaps in previous experimental studies.

3 Fracture Deformation and Stiffness

The evolution of the fracture geometric characteristics under stress conditions is key to understanding the pressure-dependence of flow and transport in the fracture environment. In numerical studies, the deformed fracture aperture can be determined using different models that account for elastic-, plastic-, viscoelastic-, and even brittle behaviors. However, in experimental investigations, the measurement of the 3D aperture space presents certain spatial-temporal challenges. Although modern computer tomography techniques allow one to reconstruct the fracture and pore space, such measurements are usually limited by resolution and/or time constraints. Moreover, in traditional laboratory testing, the measurement of the 3D fracture space is usually limited to global measurements (e.g., changes in the fracture -volume or -closure), or to a single snapshot of the aperture space (e.g., when pressure films or resins are used).

In contrast to the aforementioned limitations, the experimental setup presented in this paper allows one to conduct multiple 3D aperture measurements under different stress conditions in real time.

3.1 Determination of the Aperture Maps

In this section, we show the evolution of the aperture maps and the fracture geometric characteristics of the two specimens described earlier (Section 2.1). First, the aperture maps were determined using light transmission techniques at different effective stress-stiffness, σ_eff/E, ratios ranging from 10⁻⁵ to 10⁻³ as detailed in Sections 2.4 and 2.5. Figure 6 and Figure S10 in Supporting Information S1 illustrate the evolution of Specimens A and B with increasing confining pressure. Note how, as the fractures close, the contacting asperities—shown as white patches-grow in number and size. Recall that Specimens A and B feature aperture fields generated with the same Hurst exponent and mismatch wavelength, but two different root-mean-square of heights (i.e., they are scaled versions of each other).

Figure 6

Evolution of the aperture map for Specimen A with increasing stress levels. (a) The original aperture field was generated numerically with a self-affine surface model and used for the specimen 3D printing, see Section 2.1. (b, c) Measured aperture fields under low and high stress levels, respectively. As the confining pressure increases, the contacting asperities—shown as white patches-grow in number and size. (d) Comparison of the aperture profiles along the centerline (dashed line in panel (a)) for maps (a–c). Note: the original aperture was generated with a resolution of 75 μm in the XY directions (i.e., within the fracture plane), which is larger than the pixel size of the measured aperture maps (40 × 40 μm).

Figure 6 compares the original (design) aperture field to the measured maps under low (σ_eff/E ∼ 10⁻⁵) and high (σ_eff/E ∼ 10⁻³) stress levels for Specimen A. The original aperture (Figure 6a) refers to the numerically generated field that was used for 3D printing. Since we did not measure the specimen geometry under zero stress after the fabrication process, we cannot assure that our initial conditions (i.e., the aperture field under σ_eff = 0 kPa) perfectly correspond to Figure 6a. However, we are confident in the fabrication process as the measured aperture fields under stress conditions closely preserve the original (design) geometric structure as discussed below. Figure 6d compares the aperture profile along the centerline under low (σ_eff/E ∼ 10⁻⁵) and high (σ_eff/E ∼ 10⁻³) stress levels to the initial (design) aperture conditions. This clearly illustrates how, as the confining pressure increases, the contacting asperities deform. Finally, note while the measured profiles retain the overall design structure (e.g., the spatial location of peaks and valleys) they are smoother than expected from the original design. This difference in the roughness level may be explained by the limitations of the fabrication process (e.g., due to the finite laser spot size and XYZ resolutions), or due to the dispersion of the light that tends to smooth out our aperture measurements.

3.2 Fracture Geometric Characteristics

The aperture maps (Figure 6 and Figure S10 in Supporting Information S1) can be used to quantify different fracture characteristics such as aperture distributions, contact areas, or fracture -volumes and -closure (Figure 7, Figure S11 and Text S7 in Supporting Information S1). While in traditional laboratory testing the measurements are usually limited to the change of some of these properties (e.g., one can measure the change in aperture using strain gauges or linear variable differential transformers), with our methodology we can determine their absolute value and not just variations. Furthermore, through the analysis of the evolving aperture maps we can determine other characteristics that are very challenging (or virtually impossible) to measure through external sensors and transducers, for example, the univariate distribution of the aperture heights.

Figure 7

(a) Evolution of the aperture distribution for Specimen A with increasing stress levels. These three probability density functions correspond to three aperture fields under zero, low, and high stress levels shown in Figure 6. The insert at the top right of the figure reproduces the same data in log-log scale (the axes are the same as the parent figure's). (b, c) Evolution of Contact area and Fracture closure, respectively, for Specimens A and B with increasing stress. Here, the fracture closure of is normalized with the initial mean aperture, b_m0.

3.2.1 Univariate Description, Contacting Asperities, and Fracture Closure

Univariate descriptions are useful for a first-order description of the void structure. Besides the determination of the mean aperture value (a.k.a. mechanical aperture), b_m, and the standard deviation of apertures, b_std; it may be important, in certain scenarios, to characterize the overall “shape” of such distributions as a function of stress (see, e.g., Nowamooz et al., 2013).

For comparison, Figure 7a presents the probability density function of the aperture heights of three maps under zero, low (σ_eff/E ∼ 10⁻⁵) and high (σ_eff/E ∼ 10⁻³) stress levels. The apertures in the original field (σ_eff ∼ 0) follow a normal distribution as expected from topographic surfaces generated with PSD functions of the form of S^iso(ξ) ∝ ξ^−2−2H (see, e.g., Jacobs et al., 2017; Persson et al., 2005; Yastrebov et al., 2015). As the fracture deforms and closes with increasing normal stress, the aperture distributions move to the left. Note, however, that these distributions are not simply shifted and truncated. This becomes evident upon the inspection of the left tails of the distributions, especially around aperture values close to zero (i.e., contacts). Here, high probability density values around zero represent large contact areas, which rapidly increase with the applied stress.

To this regard, our experimental results show linear increase in the contact area with the stress level (Figure 7b), in good agreement with previous studies on real rocks and contact mechanics (Hanaor et al., 2015; Kling et al., 2018; Persson et al., 2005; Yastrebov et al., 2015; Zou et al., 2020). Moreover, we observed that contact areas and fracture closure follow very different trends with increasing normal stress (Figures 7b and 7c). In fact, for our fracture analogs, the normal stress versus fracture closure curves, σ versus δ, can be nicely described by simple power-law relationships (Brown & Scholz, 1986).

Upon closure, the asperities of the opposite fracture surfaces enter in contact and start to deform. At first, only few asperities are in contact and the fracture deforms rapidly. As the normal load increases, more and more asperities become in contact and deform, resulting in more and larger contact areas (Figure S10 in Supporting Information S1). The larger the contact area, the stiffer the fracture becomes. This stiffening is usually quantified from the slope of the σ versus δ curves. This slope is referred as the fracture stiffness, κ, a concept first introduced by Goodman et al. (1968). Note that the specific stiffness is especially interesting because it can be quantified remotely using geophysical methods (Pyrak-Nolte & Nolte, 2016). Therefore, identifying the relationship between fluid flow and fracture-specific stiffness may enable he quantification of fracture permeability through geophysical monitoring.

3.2.2 Geostatistical and Spectral Analysis

Although univariate descriptions and global geometric characteristics are informative and often sufficient to describe the overall mechanical behavior of fractures, in certain cases, more exhaustive geometrical analyses are required. In particular, in this section, we discuss geostatistical and spectral methods (see Text S9 in Supporting Information S1). Figure 8 compares the normalized omni-directional variogram and the power spectral density (PSD) of the three aperture maps of Specimen A shown in Figure 6 (i.e., the aperture fields under zero, low (σ_eff/E ∼ 10⁻⁵) and high (σ_eff/E ∼ 10⁻³) stress levels).

Figure 8

Evolution of the omni-directional variogram and the power spectral density (PSD) of the three aperture maps shown in Figure 6. (a) Experimental variogram. The position of the first valley reveals the largest scale where the repetitive nature of the aperture becomes evident, that is, the mismatch wavelength. (b) PSD. Here, the PSD corresponds to the radial-averaged version of the 2-dimensional PSD. The PSD curve of the original aperture (black) follows a decreasing power-law, which is one the characteristic signatures of self-affine surfaces. This curve has a slope equal to −2 − 2H, where H is the Hurt exponent (see Section 2.1). The corner spatial frequency (i.e., the largest wavenumber with significant amplitude) determines the long-distance correlation length, and in our case is coincident with the mismatch wavelength, λ_m. The PSD curves of the aperture maps under low (blue) and high (red) stress levels show smaller slopes than the original aperture (i.e., they have Hurst exponents >0.8, and therefore are characteristic of smoother surfaces than the original).

The variograms in Figure 8a show the typical characteristics of stationary, repetitive fields, including negative correlation at λ_m/2 < lag < λ_m (Detwiler et al., 2000; Ma & Jones, 2001). They reach global maxima at λ_m/2, and local minima at λ_m. Recall that the mismatch wavelength, λ_m, is the dominant wavelength in the aperture field. Therefore, two points separated by λ_m/2 (e.g., peak-to-valley) tend to be more dissimilar than two other points separated by λ_m (e.g., peak-to-peak or valley-to-valley). Here, this minimum is located at distance of ∼3.6 mm in close agreement with the design λ_m of 3.8 mm. The slight difference can be explained by the discretization approximations during the surface generation and during the determination of the variogram.

Comparing the three variogram curves in Figure 8a, one can notice how the positions of the crests and valleys are coincident, which suggests that correlation distances remain unchanged as the fracture deforms. Moreover, variogram curves can provide other insights such as the presence of non-zero slopes and the occurrence of rigid body rotations. Note that, by definition, the normalized variogram of a stationary, repetitive field should approach ∼1 for large lags (Bachmaier & Backes, 2011). However, if the surface has a non-zero overall slope, the variance will keep increasing. In our case, one can observe how the low stress curve steadily increases at large lags while the original aperture and high stress level curves approach ∼1. The presence of a non-zero overall slope under low stress level is further confirmed by inspection of the spatial distributions of the asperity contact in Figures 6b and 6c. Under low stress, most of contacting asperities are located within the upper half of the fracture, in contrast with the more homogeneous distribution under high stress, suggesting rigid body rotations with increasing confining pressure.

Finally, Figure 8b presents the power spectral density, PSD, of the same three apertures maps shown in Figure 6. Note that there are different ways that PSDs can be defined, here we report the radial-averaged two-dimensional PSD, S^iso, which has units of (L⁴) (Jacobs et al., 2017).

Recall that for self-affine fractures, the PSD follows a decreasing power-law across most scales (e.g., Persson et al., 2005). In fact, in this paper, the upper and lower surfaces of the fracture were generated with a PSD of the form S^iso(ξ) ∝ ξ^−2−2H, where ξ is the spatial frequency (ξ = 2π/λ) and H is the Hurst exponent equal to 0.8 (see Section 2.1). Brown (1995) showed that the resulting aperture also follows the same power law scaling with a corner spatial frequency equal to the mismatch wavenumber. This is confirmed in Figure 8b, where the PSD of the original aperture map follows a decreasing power law with slope −2 − 2H (with H = 0.8) for all frequencies greater than ξ_m = 2π/λ_m. While the curves of the aperture maps under low and high stress also show a decreasing PSD, they do not follow a strict power-law. For small frequencies (ξ < 10⁴ m⁻¹), the slope of the PSD corresponds to H ∼ 0.9 (slope ∼−3.8 = −2 − 2H), which is in accordance with the smoother profiles observed in Figure 6d (recall 0 < H < 1, with larger values being characteristic of smoother surfaces). Moreover, for 10⁴ < ξ < 2.5 × 10⁴ m⁻¹, the PSD curves are no longer characteristic of self-affine surfaces as their slopes would correspond to H greater than one (slope ∼−4.6 = −2 − 2H). Note that at these large ξ (i.e., small wavelengths), the PSD curves become affected by the increasing contact areas, which are essentially flat regions and do not follow a self-affine scaling. In fact, the increasing contact area with stress may explain why the PSD curve of the aperture map under high stress deviates from the self-affine scaling earlier than the low stress one.

The analyses above illustrate the importance of in situ fracture geometry measurements to quantify the changes in relation to the initial conditions. Moreover, the determination of the aperture maps under stress conditions is key for the validation of new numerical models (H. Kang et al., 2020; Kling et al., 2018), and to understand the changes in flow and transport regimes as we will illustrate in the following sections.

4 Evolution of Flow Properties. Experimental Results and Simulations

In this section we explore how normal stress-induced closure affects flow properties, and in particular, the fracture hydraulic impedance and the flow field. We first investigate the stress-dependent permeability of the fracture analogs using a series of direct hydraulic measurements from the permeability phase in our tests (i.e., fluid pressures measured at the specimen inlet and outlet, p_in and p_out, and flow rates, Q) (see the experimental protocol in Section 2.5). Then, we relate the observed hydraulic behavior to the evolution of fracture geometric characteristics, which were determined from the evolving aperture maps in Section 3. And finally, we conduct a series of fluid flow simulations, that take the measured aperture maps under different stress levels as inputs, to determine the evolving flow fields and study the evolution of the flow channelization. Moreover, we use this set of numerical simulations to re-evaluate the stress-dependent permeability of the fracture and compare to the hydraulic measurements.

4.1 Stress-Dependent Permeability Relationships: Hydraulic Impedance, Transmissivity, and Hydraulic Aperture

In permeability tests of fractured media, it is customary to evaluate the flow regime at different effective stress levels by measuring the fluid pressure drop between inlet and outlet at several flow rates. In a linear flow regime, the pressure drop versus flow rate curves, Δp vs. Q, follow a linear relationship. These data are usually normalized by the length and width of the specimen, and reported as pressure gradient versus normalized flow rate curves, ∇p vs. Q/W (Figure 9a), in which the slope, represents the hydraulic impedance at a given effective stress level. Note that under flow conditions, the mean effective stress is defined as σ_eff = CP − p_avg, with p_avg = (p_in + p_out)/2. When the hydraulic impedance of the fracture is high, increasing Q can result in large Δp relative to the confining pressure levels. In such cases, the effective stress in fracture specimen can no longer be considered uniform, and the specimen cannot be treated as representative (homogeneous) elementary volume. In other words, if the ratio is large, then is large, and there exist large variations in the effective stress relative to the mean value at the center of the fracture. To avoid this, we selected Q such that the ratio is always below 0.1 (i.e., variations in the effective stress within the fracture are less than 10% of the value of CP).

Figure 9

(a) Pressure gradient versus Normalized flow rate curves of Specimen B at different effective stress levels. At each stress level, the ∇p vs. Q/W data follow a linear relationship indicating linear flow, with the slope, , representing the hydraulic impedance at that stress level. The Specimen A shows a similar behavior but reaching lower ∇p ranges represented by the gray shaded area at the bottom of the plot. (b) Hydraulic behavior of Specimens A and B, expressed in terms of their fracture transmissivity (left ordinate) and hydraulic aperture (right ordinate), as a function of the normalized effective stress. As the fracture deforms with increasing normal stress, the aperture volume available for advective flow declines resulting in increasing hydraulic impedance (i.e., decreasing transmissivity). Note that the hydraulic impedance, T_f and b_h are three different ways to express similar information (see Text S9 in Supporting Information S1 for further details).

While many authors use the term permeability to describe the hydraulic behavior in fractured media, one should note that this term does not strictly refer to the traditional definition as used in Darcy's law for porous media. In fact, for fractured media, especially at the single fracture scale, other definitions are preferred (see Text S9 in Supporting Information S1).

Figure 9b illustrates the hydraulic behavior of Specimens A and B, expressed in terms of their fracture transmissivity and hydraulic aperture, as a function of the normalized effective stress. As the fracture deforms with increasing normal stress, the aperture volume available for advective flow decreases resulting in declining transmissivity with normal stress.

Previous investigations with real rocks (e.g., Iwano & Einstein, 1995; Raven & Gale, 1985) have found that the fracture transmissivity and the normal stress exhibit negative power-law relations of the form , where T₀ and n are fitting parameters. While we do observe such dependence at large effective stresses (Figure S12a in Supporting Information S1), our experimental results seem to be better explained by exponential relationships of the form , where a and b are fitting parameters likely related to the initial aperture and material properties (Figure S12b in Supporting Information S1). As a matter of fact, these results are in good agreement with recent numerical investigations (e.g., Pyrak-Nolte & Nolte, 2016; Wang & Cardenas, 2016) that found similar exponential relationships between flow and fracture stiffness.

4.2 Pressure-Dependent Permeability and the Evolution Fracture Geometric Characteristics

The observed trend in Figure 9b and Figure S12 in Supporting Information S1 is also followed by some fracture geometric characteristics such as the mechanical aperture (Figure S11a in Supporting Information S1). Moreover, note that how the mean mechanical aperture, b_m, is consistently greater than the hydraulic aperture, b_h, and that, in general, both decrease at a similar rate as shown in Figure 10. Chen et al. (2017) suggested that differences between b_m and b_h could be explained by the fracture roughness (fractal dimension) and the increasing contact area. Our results, however, do not show this dependence with increasing contact area but reveal constant absolute differences for Specimens A (b_m − b_h ≈ 30) and B (b_m − b_h ≈ 15 μm) resulting in mean relative deviations of ∼25%.

Figure 10

Mechanical aperture versus Hydraulic aperture. The hydraulic aperture is consistently smaller than the mechanical aperture in both experiments, and both b_m and b_h decrease at a similar rate for aperture below 180 μm (i.e., the datapoints are more or less parallel to the 1-to-1 dashed line). Note that hydraulic apertures greater than 180 μm may be subjected to large experimental error as they are close to the system's limit of quantification.

Although the mean mechanical aperture, b_m, displays a strong relation with the hydraulic properties of the fracture as suggested by the data shown in Figure 10, it is not sufficient to explain the complex behavior of flow and transport under evolving stress conditions. An important limitation is that b_m just represents the average value of the physical void space, and it does not contain information regarding the internal structure of the flow field (e.g., channelization), which is key to explain certain processes such as the onset of anomalous transport or multi-phase flow in fractures.

4.3 Flow Field: Geometric Characteristics and Spatial Relations

When the flow channelizes, the velocity field is no longer uniform, but it is characterized by a high-velocity zones (channels) and low-velocity (stagnation) areas, which control the transport of mass and heat within the fracture. To study the statistics of the evolving flow field and the channelization degree, we first determine the steady flow maps at different stress levels through a series of fluid flow simulations using the measured aperture fields (Figure 6) as inputs. Note that while there exist experimental techniques to directly measure the velocity field in microfluidic cells (e.g., particle image velocimetry), their use at our scale of interest (i.e., ∼40 × 80 mm) presents some technical challenges.

4.3.1 Flow Simulations

Here, we simulate 2D steady laminar flow of a Newtonian fluid with constant density and viscosity at each stress level using the Reynolds equation:

(2)

(3)

where represents the local volumetric fluxes, b is the local aperture, and is the two-dimensional depth-averaged local fluid velocity defined at each computational grid block (Figure 11). Our model solves the pressure equation using a finite-volume discretization in a 2D grid (n_x × n_y), with constant-pressure boundary conditions along the inlet and outlet of the fracture, and no-flow across the sides. Equations 2 and 3, also known as the Local Cubic Law (LCL), are a 2D depth-averaged approximation of the 3D Stokes equations. They form a linear system of equations that can approximate laminar fluid flow in thin, rough fractures when the variability of the aperture is gradual, that is, when 3D effects can be neglected (e.g., Brush & Thomson, 2003); see Text S10 in Supporting Information S1 for details.

Figure 11

The flow simulations solve the Reynolds equations in a 2D computational grid using the finite-volume method. (a) Domain discretization into (n_x × n_y) grid blocks using the same pixel resolution as the measured aperture fields (see Section 2.4). Boundary conditions: no-flux along left/right boundaries, and constant pressure along inlet/outlet boundaries. (b) Resulting pressure and flow field maps from the input aperture map shown in panel (a). See Text S10 in Supporting Information S1 for details.

In addition to the determination of the flow fields, we used this set of LCL simulations to obtain a second evaluation (after the hydraulic experiments) of the fracture transmissivity under different stress levels. Figure 12 compares the hydraulic apertures derived from the numerical simulations to the corresponding values determined from the experimental hydraulic measurements (Figure 9). We observed excellent agreement between both methods, with slight overestimations of the b_h from LCL simulations (5%–10%).

Figure 12

Hydraulic aperture: simulations versus permeability measurements. Blue dots = Specimen A; Red diamonds = Specimen B. Local Cubic Law, simulations tend to overestimate the hydraulic aperture, b_h, compared to the experimentally measured values. Note that measured hydraulic apertures greater than 180 μm may be subjected to large experimental error as they are close to the system's limit of quantification.

To date, a number of experimental, theoretical, and numerical studies have investigated the validity of the LCL to simulate flow in rough fractures (see, e.g., Berkowitz, 2002; Bodin et al., 2003; Oron & Berkowitz, 1998). While it remains an open question, the consensus is that, in general, the Reynolds equation tends to overestimate the fracture transmissivity. Previous experimental works on real rocks (e.g., Hakami & Larsson, 1996) and fracture analogs (e.g., Nicholl et al., 1999; Yeo et al., 1998) have reported discrepancies on the order of ∼10%–50% between the experimentally measured b_h and the simulation results. Such discrepancies are usually associated with the transgression of one or more of the underlying assumptions of the Reynolds equation (see Text S10 in Supporting Information S1). They may arise due to abrupt changes in the local aperture (Brown, 1987); large roughness and mean apertures compared to the correlation length (Brush & Thomson, 2003); the onset of nonlinear flow and eddy development (Briggs et al., 2017); or to increasing fracture closure and shear displacement (Brown et al., 1995; Zou et al., 2017).

In our study, the good agreement between the experimentally measured and numerically determined hydraulic apertures (5%–10% error) may be explained by the geometric characteristics of our fractures - and their evolution under the applied stress—(Figure 7 and Figure S11 in Supporting Information S1); and the observed linear flow regime during the tests (Figure 9a). Brush and Thomson (2003) compared simulation results from the full Navier-Stokes Equations to the simplified Stokes and Reynolds equations, and found that the LCL, can indeed approximate flow in rough fractures when a series of geometric (b_std/b_m < 1, b_std/λ_m < 0.2, and b_m/λ_m < 0.5) and kinematic (Re < 1, Re ⋅ b_m/λ_m < 1, and Re ⋅ b_std/b_m < 1, where Re is the Reynolds number) conditions are met.

Note that in or tests, the fluid film thickness is small compared to the lateral extent of the fractures, and that the variability in the aperture fields is gradual (Figure 7, Figures S10 and S11 in Supporting Information S1). Such conditions are in good agreement with Brush and Thomson's geometric requisites and with the assumptions of the Reynolds equation (see Text S10 in Supporting Information S1).

Moreover, our experimental results suggest that the observed fluid flow is within the linear regime (Figure 9a). The first of Brush and Thomson's kinematic conditions (Re < 1), refers to the threshold beyond which the inertial forces significantly influence the bulk flow, violating one of the key assumptions of the Reynolds equations. Under such circumstances, the flow starts transitioning from laminar to turbulent, and the pressure drop versus flow rate curves become nonlinear. Previous experimental, numerical, and theoretical investigations have proposed a wide range of values for this critical Re, ranging from as little as 0.001 to more than 2,300 (e.g., Javadi et al., 2014; Louis, 1969; Oron & Berkowitz, 1998; Zimmerman et al., 2004). A significant variation that may be explained, among the other several potential reasons and sources of error, by how the Reynolds number itself is defined in rough fractures as we discuss in Appendix A.

4.3.2 Spatial Structure of the Flow Field and Channeling

As discussed in Section 4.3.1, we use the experimentally measured aperture fields (Figure 6 and Figure S10 in Supporting Information S1) as inputs for LCL flow simulations to determine the evolving flow field. Our experiments indicate that increasing fracture deformation has a profound effect on the flow field as shown in Figure 6 and Figure S15 in Supporting Information S1. At low stress levels, only few asperities are in contact, and the flow is relatively homogeneous. However, as the fracture deforms and these contacts areas begin to grow in number and size, the flow field starts to channelize.

The simulated flow fields shown here present clear evidence of boundary effects due to the more significant deformation along the edges of the fractures (Figures S10 and S15 in Supporting Information S1). Note that the development of such effects is an inherent challenge with small-scale laboratory experiments, however, in our experiments their influence can be directly quantified and accounted for.

To evaluate the evolution of the spatial structure of the flow field, we first compute the univariate distribution of the flow rate magnitudes and directions at different effective stress levels (Figure 14). We observe that at low stress the flow rate distribution has a truncated bell-shape, which then transitions toward a power-law-like distribution under increasing normal stress (Figure 14a). As the flow channelizes, pathways get increasingly dominated by large flow rates, and stagnation (low velocity) and flow-back (negative velocity) zones start to develop (Figure 14b). In addition, these new flow paths become more tortuous and less dominated by the macroscopic (i.e., fracture-scale) hydraulic gradient as shown in Figure 14c (note how the absolute value of the local flux component perpendicular to macroscopic flow—i.e., q_y—increases as the fracture deforms under increasing normal stress).

Figure 13

Evolution of the simulated flow field map for Specimen A with increasing stress levels from left to right: σ_eff/E = (a) 0.4 × 10⁻⁴, (b) 10⁻⁴, (c) 4 × 10⁻⁴, and (d) 11 × 10⁻⁴. The colormaps represent the normalized flow rates: values above 1 represent channelization, and those close to 0 indicate low flow (stagnation) zones. The local flow rates computed at each grid block, q_ij (see Text S10 in Supporting Information S1), are normalized by average flow rate, which is computed as the inlet flow rate divided by the number of grid blocks along the fracture width, Q_in/n_y (i.e., direction perpendicular to flow).

Figure 14

Evolution of the local flow rate distributions with effective stress (Specimen A). Absolute values above 1 represent channelization, while values below 1 indicate stagnation zones. We normalize the local flow rates by the average flow rate computed as the inlet flow rate divided by the number of grid blocks along the fracture width, Q_in/n_y (i.e., direction perpendicular to flow). (a) Absolute magnitudes of the flow rate. As the flow channelizes under increasing normal stress, the distribution transitions from a bell-shape to a power-law distribution. Channelization results in increasing number of stagnation zones around the contacting asperities and preferential pathways dominated by large flow rates, as indicated by the increasing probability density at large flow rates. The insert reproduces the same data in log-log scale. The axes are the same as the parent's plot. (b) Normalized component of the local flux along the flow direction (i.e., macroscopic hydraulic gradient) (x-direction, see Figure 11). Negative values represent local velocities pointing toward the fracture inlet, that is, flow-back zones. (c) Normalized component of the local flux perpendicular to macroscopic flow (y-direction, see Figure 11). When the flow paths follow the macroscopic hydraulic gradient (along x-direction), q_y tend to zero. Therefore, more tortuous channels are characterized by large q_y.

To further quantify the channelization degree within the fracture, we plot the channelized area-flow portion for different normal stress. Each of the curves in Figure 15a represents the minimum fracture area portion required to carry a given fraction of the flow (see Text S11 in Supporting Information S1). Under uniform flow (parallel-plate) conditions, this corresponds to the 1-to-1 line, for example, 80% of the fluid flows through 80% of the fracture. However, when channelization occurs in stressed rough fractures this percentage decreases. For instance, in Specimen A, 80% of the fracture fluid flows in a little bit more than 50% of the fracture area under low normal stress (σ_eff/E ∼ 4 × 10⁻⁵), and in less than 30% under high normal stress (σ_eff/E ∼ 10⁻³).

Figure 15

(a) Channelized area-flow portion of Specimen A under different normal stress levels. The curves represent the minimum portion of fracture area required to carry a given portion of the flow (e.g., under uniform conditions 80% of the fluid flows within 80% of the fracture area, but as the flow channelizes with increasing fracture deformation, this percentage decreases). The intersection of the curves with the right ordinate represents “1—contact area fraction,” as marked by the solid dots, that is, at the limit, 100% of the flow is carried within the entire fracture area excluding the area at the contacting asperities. (b) Evolution of the participation number with increasing normal stress revealing increasing channeling with normal deformation. The participation number statistically quantifies the channeling effect in groundwater flows. It takes values between 0 (strong channeling degree) and 1 (homogeneous flow).

Alternatives to quantify the spatial structure of the flow field may also include velocity field correlations (Le Borgne et al., 2007), the determination of the channeling area, fractal analysis, or the definition of statistical parameters that summarize the global channelization degree. Regarding the latter, Andrade et al. (1999) proposed a participation number, π, to assess the univariate distribution of the kinetic energy of the flow field,

(4)

where n is the total number of grid blocks in the 2D flow velocity field, is the kinetic energy in each grid block, and q_x and q_y are the local components of the volumetric flow. The participation number, π, is a dimensionless parameter that takes values between 0 (strong channeling) and 1 (homogeneous flow), and quantifies the degree to which the flow is concentrated in a small portion of the fracture without considering spatial correlations. Figure 15b shows the evolution of the participation number in Specimens A and B as a function of the normalized effective stress, indicating increasing channelization (i.e., decreasing π) with fracture deformation. In the next section, we will make use of π to link the structure of the flow field to the evolution mass transport processes.

Finally, note that in this section we quantify the flow topologies shown in Figure 13 and Figure S15 in Supporting Information S1. These flow fields resulted from the application of a macroscopic fluid pressure gradient between the inlet and the outlet of the fracture, and that a different flow field would result if the gradient directionality changed. However, Y. Tsang and Tsang (1989) show that the different flow fields in fractures with isotropic aperture fields have the same statistical properties regardless the directionality of the macroscopic pressure gradient (C. Tsang & Neretnieks, 1998). Therefore, our analysis remains valid.

5 Conservative Mass Transport

In this last section we explore the effect of normal stress-induced closure on the evolution of transport properties in rough fractures. For the sake of simplicity, in these experiments, we limit ourselves to the study of non-reactive transport using a conservative tracer (i.e., non-reactive, non-sorbing tracer). We have seen in previous sections how increasing normal stress affects the aperture field and the spatial structure of the flow, specifically regarding the channelization degree and the tortuosity of the flow paths. These two are specific features of flow in rough fractures that have a major effect on mass transport.

5.1 Determination of the 2D Concentration Maps and 1D Profiles

In field tests (and laboratory experiments with real rocks) mass transport measurements are usually limited to the observation of the BTC at a single point in space, for example, at the production well (or at the outlet of the specimen). In these scenarios, high-spatial resolution measurements of the concentration field are very challenging, or virtually impossible.

When transparent analogs are used, on the other hand, one can easily measure these concentration fields with high-spatial and -temporal resolution. Figures 16a–16d show the evolution of the normalized concentration, , field during continuous injection of a tracer into Specimen A under low stress levels (σ_eff/E = 0.4 × 10⁻⁴). Here, we present our results in terms of the normalized (dimensionless) concentration, , time, , and position, ; where C_in is inlet tracer concentration, Q is the inlet flow rate, θ is the fracture volume, and L is the fracture length.

Figure 16

Tracer injection into Specimen A under a low stress level (σ_eff/E = 0.4 × 10⁻⁴). (a–d) Snapshots of the 2D dimensionless concentration maps, , at different dimensionless times, . (e) Flow map showing the underlying flux field. At low stress levels, the flow field is relatively homogeneous, and the tracer front remains parallel to the fracture inlet, similar to transport in a parallel plate. However, the presence of contacting asperities (white patches) and variable aperture results in the development of a few preferential channels. After the injection of one fracture volume, , the fracture is close to full saturation. (f) Breakthrough curves at different positions from the fracture inlet, . These 1D dimensionless concentration profiles are determined using flux-averages from the 2D maps. Note that the complete saturation of the fracture will occur after 3+ fracture volumes (see Figure 17d).

Under relatively homogeneous flow conditions (Figure 16e), the tracer front remains relatively parallel to the fracture inlet during injection (Figures 16a–16d). Nevertheless, the dye flow reveals a few preferential paths due to the presence of the contacting asperities and the variable aperture. Note that this phenomenon is different from flow instabilities such as flow-fingering, which arise during multiphase flow and one-phase flows with variable density and viscosity. Here, dye transport is purely controlled by molecular diffusion and mechanical dispersion (i.e., macrodispersion in the plane of the fracture and Taylor dispersion across the fracture aperture) (Detwiler et al., 2000).

Although these 2D concentration maps are quite informative, to gain further insights we compute 1D concentration profiles, , at different positions using a flux-weighted approach:

(5)

where and q_i are the normalized local concentrations and local fluxes at position from the fracture inlet. Note that while the local concentrations, , come from direct experimental measurements, the local fluxes are the result from LCL simulations built using the measured aperture maps. In Section, 4.3.1 we validated the accuracy of the numerical predictions (Figure 12), and therefore, we are confident on the that the local fluxes, q_i are accurate and do not introduce significant errors.

These 1D profiles represent the average concentration as a function of time and position (Figure 16f). They will be used to quantify transport properties by fitting existing analytical models (Section 5.3). We also investigated the use of aperture-weighted 1D concentration profiles, but flux-averages are preferred as they better represent the mass transfer rate within the fracture. Recall that the individual local fluxes do not depend exclusively on the corresponding local apertures, but on the entire permeability field and macroscopic gradient.

5.2 The Role of Fracture Geometry and Normal Stress on Channeling

Figure 16d shows the resulting 2D concentration map after 1 fracture volume of dye injection (i.e., at ). If transport occurred under no molecular diffusion or mechanical dispersion, at , the 2D normalized concentration map should be uniform and equal to 1. However, due to some channelization of the flow field, it takes longer times to fully saturate the fracture (Figure 16f indicates that complete saturation occurs after 3+ fracture volumes). This phenomenon is exacerbated by increasing fracture closure and channelization as shown in Figure 17, Figures S16 and S17 in Supporting Information S1. As the flow concentrates in a few preferential paths, it takes relatively longer times to saturate the fracture with the inlet concentration C_in.

Figure 17

Effect of normal stress on tracer transport. As the fracture deforms and the flow channelizes into preferential paths, dye saturation takes relatively longer times (i.e., more fracture volumes) due to the increasing number of low velocity and stagnation zones. (a, b) Measured concentration maps after the injection of one fracture volume, , into Specimens A and B, respectively, under different stress levels increasing from left to right: σ_eff/E = 10⁻⁴, 3 × 10⁻⁴, 6 × 10⁻⁴, and 11 × 10⁻⁴. The white patches represent contacting asperities, which grow in number and size with increasing normal stress. (c, d) Breakthrough curves at position for low and high stress levels in Specimens A and B in linear and logarithmic scales, respectively.

For qualitative comparison, we show the 1D concentration profiles at in Figures 17c and 17d for low and high stress levels in Specimens A and B (recall that Specimens A and B are scaled versions of each other, i.e., B has the same spatial structure than A but smaller aperture—see Figure 2). The data are presented in both linear (Figure 17c) and logarithmic (Figure 17d) scale to better illustrate early and late transport behaviors. Note that 99% concentration is achieved after only four fracture volumes in Specimen A under low stress (low channelization), but it takes more than 100 fracture volumes for Specimen B under high normal stress levels (high channelization). Our analysis suggest that such difference in saturation times might related to the changes in the flow field structure as discussed in the next Section.

Recall that normal stress-induced fracture closure results not only in the channelization of the flow into preferential paths, but also in the development of stagnation and flow-back zones (Section 4.3.2). The dominant mechanisms responsible for the saturation of these structures that is, preferential advective channels and low-velocity areas-might be different and evolve with the flow field. In fact, recent studies in disorganized porous media have shown that such small-scale flow structures can significantly affect the macroscopic mass transport (e.g., Bordoloi et al., 2022). Moreover, some of the observed BTCs—specially under high channelized conditions-exhibit interesting features including asymptotic-like behaviors (Figure 17d). These might be explain by the interactions between main the flow channels and stagnant zones—as in Bordoloi et al. (2022), and/or due to sample size and boundary effects.

5.3 The Evolution of Transport Properties With Increasing Normal Stress and Channeling

To further quantify the evolution of transport properties with increasing fracture closure, it is customary to fit the 1D experimental data to known transport models (see, e.g., Nowamooz et al., 2013). Recall that in our case these 1D concentration profiles, (i.e., the experimental BTCs, are obtained from the measured 2D concentration maps using a flux-weighted approach (Equation 5). Also note that during this dimension-reduction process the information regarding the transversal spreading is lost, and the resulting 1D data only relate to the longitudinal component of the transport (i.e., along the macroscopic flow direction). In fact, although it is possible to observe the transversal spreading in our tests, our experimental boundary conditions (i.e., uniform dye injection along the fracture inlet) are better suited for the study of the 1D longitudinal mass transport.

While the non-Fickian (a.k.a. anomalous) nature of mass transport in rough fractures requires the use of advanced models such as the Continuous Time Random Walk framework (e.g., Berkowitz et al., 2006), in this paper we limit ourselves to one of the analytical solutions to the widely known Advection Diffusion Equation (ADE). The reasons are threefold: (a) the ADE is simple and easy to interpret as it only depends on two parameters (i.e., the dispersion coefficient, D, and the average velocity, U); (b) it can be used to illustrate the differences between Fickian and non-Fickian behaviors; and (c) the discussion and validation of complex transport models is beyond the scope of this paper.

In one-dimension, the ADE can be expressed as,

(6)

where t is the elapsed time, C and x are the one-dimensional concentration and position, D is the longitudinal dispersion coefficient (which accounts for both molecular diffusion and mechanical dispersion), and U is the average flow velocity. For the case of a continuous tracer injection into a semi-infinite medium [C(∞, t) = 0; t ≥ 0] with initial concentration equal to zero [C(x, 0) = 0; x ≥ 0], and a step change at the inlet [C(0, t) = C_in; t ≥ 0], the ADE can be approximated by Ogata and Banks (1961),

(7)

where erfc[*] is the error function of *. Figure 18a shows this analytical solution fitted to the experimental BTCs at for two extreme cases: Specimen A under low normal stress (Figure S16a in Supporting Information S1) and Specimen B under high normal stress (Figure S17b in Supporting Information S1). Since the ADE cannot model the anomalous transport behavior (e.g., the long tails), we did not attempt to fit the entire concentration data, but we limited the fit to concentrations . In general, the analytical solution can capture of the transport features caused by channelized flow, that is, early arrivals and the dispersive nature of the transport. However, it fails to capture the trends at late stages including the long tails.

Figure 18

Inversion of the Advection-Diffusion Equation (ADE) parameters from the experimental breakthrough curves (BTCs) at . (a) Comparison between the experimentally measured BTCs at and the analytical solution of the ADE, in logarithmic scale. Increasing channeling results in early arrival of the tracer and log tails, which is a common signature of anomalous transport that cannot be captured by the ADE. Evolution of (b) the dimensionless longitudinal dispersion coefficient and (c) the dimensionless average velocity as a function of normal stress. Increasing deformation of the fracture with normal stress results in more heterogeneous flow fields (Figure S15 in Supporting Information S1), which increase tracer dispersion. Note: the analytical solution of the ADE is fitted for concentrations using the nonlinear least squares method with a trust-region algorithm.

The inverted ADE parameters (i.e., the dispersion coefficient, and the average velocity) from the experimental BTCs at are shown in Figures 18b and 18c. These results reveal that the transport dynamics becomes more dispersive with increasing normal stress leading to fracture closure. This was already anticipated during the analysis of the flow field structure in Section 4.3.2: increasing deformation of the fracture with normal stress results in more heterogeneous (and tortuous) flow fields, which enhance tracer dispersion (also shown in Figure 18b).

In fact, the evolution of the dispersion coefficient correlates quite well with some fracture characteristics and flow statistics related to the channelization degree, specifically to the asperity contact area, the flux-averaged Reynolds number (Appendix A), and the participation number (Figure 19). Note, however, that some scattering in the data is observed for highly channelized flows. This could be the result of the ADE's inability to capture the nature of anomalous transport in highly channelized flows, and/or the need for a better indices to define the flow statistics in relation to mass transport.

Figure 19

Evolution of the dimensionless longitudinal dispersion coefficient, D (Equation 6) as a function of (a) asperity contact area; (b) flux-averaged Reynolds number, ; and (c) channeling degree (∼1/π). Note how the dispersion coefficient nicely relates to some fracture geometric characteristics (e.g., contact area) and flow statistics (e.g., Re and π), highlighting the link between the transport dynamics and the fracture geometry and flow field.

Moreover, one could repeat this fitting exercise at other positions and investigate the ability (or the lack of ability) of the ADE to fit the experimental data. However, this is beyond the scope of this paper, and it was already explored elsewhere (Nowamooz et al., 2013). Our objective here is to illustrate how simple models such as the ADE can be used to quantify the evolution of hydrodynamic dispersion in the fracture due to normal closure as a result of increasing normal stress (Figure 19a).

6 Final Remarks

In this paper we illustrate how the latest advances in digital fabrication of rocks can be combined with a novel pressure-controlled Hele-Shaw cell to investigate different fracture deformation, flow, and transport processes in rough fractures. We introduced our experimental methodology and discussed how through the appropriate scaling our system can be used to replicate relevant field conditions.

Through a series of examples, we have shown that our setup can successfully measure aperture, flow, and tracer concentration fields under different stress conditions. Using light-transmission techniques, we measured the in-situ aperture maps with increasing fracture closure, determined several fracture geometric characteristics as a function of normal stress, and performed geostatistical and spectral analyses on the evolving aperture field. Then, with a series of hydraulic tests and numerical simulations we studied the stress-dependent transmissivity of the fracture, validated the LCL, and analyzed the evolving flow structure and channelization degree under increasing fracture closure. We finally quantified the transport characteristics of the flow by fitting tracer concentration data to the analytical solution of ADE and linked these evolving characteristics to the channelization degree of the flow.

Fracture closure and hydraulic transmissivity experimental results show similar trends to those previously observed with real rocks, including (a) linear increase of the contact area with normal loading, and (b) power-law decay of permeability at high normal stress levels. Moreover, we demonstrate that this power-law decay is only a restricted view of a more general exponential behavior. Our analysis demonstrates that increasing fracture closure results in growing flow channelization, affecting the transport dynamics as shown by the increasing dispersion coefficient with normal stress.

Note that some of the findings and mechanisms observed in this paper have been previously described in earlier numerical and theoretical studies; and occasionally indirectly observed in experimental works with real rocks. However, here we provide direct experimental evidence on the interplay between these mechanisms with increasing normal stress. Moreover, by comparing well-documented processes to our experimental results (e.g., monotonic fracture closure, single-phase fluid flow, and conservative mass transport), we aim to judge, evaluate and validate the quality of our experimental methodology.

To the best of our knowledge, this is the first time a single experimental setup is used to generate high-resolution and high-quality datasets of fracture deformation, flow and transport under the influence of applied normal stresses. Therefore, the real potential of this new setup and the proposed methodology relates to the study of new physics and complex mechanisms, including the onset of nonlinear flow and its effect on anomalous transport, multi-phase flow, proppant transport and embedment, contact mechanics, plastic and time-dependent fracture deformation, reactive transport, etc.

Digital fabrication of fractures provide us an exceptional opportunity to systematically control the fracture geometry and mechanical properties. This may allow researchers to study individual, or coupled, physical processes by specifically choosing and engineering the mechanical and geometrical properties of the analogs.

Acronyms

ADE

Advection Diffusion Equation

BTC

Breakthrough Curve

CAD

Computer-Aided Design

CAM

Computer-Aided Manufacturing

LCL

Local Cubic Law

EGS

Enhanced Geothermal Systems

NSE

Navier-Stokes Equations

PDMS

Polydimethylsiloxane

PVA

Pressure Volume Actuator

STL

Stereolithography

Notation

Variogram (L²)

δ

Fracture closure (L)

ɛ

Dye absorptivity (M⁻¹ L²)

θ

Fracture/void volume (L³)

κ

Fracture stiffness (M L⁻² T⁻²)

λ

Wavelength (L)

λ_m

Mismatch wavelength (L)

μ

Fluid dynamic viscosity (M L⁻¹ T⁻¹)

ν

Poisson's ratio (–)

π

Participation number (–)

ρ

Fluid density (M L⁻³)

σ

Stress (M L⁻¹ T⁻²)

σ_eff

Effective stress (M L⁻¹ T⁻²)

ξ

Spatial frequency—a.k.a. wavenumber—(L⁻¹)

A

Light absorbance (–)

BP

Back Pressure (M L⁻¹ T⁻²)

Fracture local aperture (L)

b_h

Hydraulic aperture (L)

b_m

Mean (mechanical) aperture (L)

b_std

Standard deviation of apertures (L)

Equivalent aperture (M L⁻²)

C

Concentration (M L⁻³)

C_in

Inlet concentration (M L⁻³)

Normalized concentration (–)

1D concentration profile (–)

CP

Confining Pressure (M L⁻¹ T⁻²)

D

Dispersion coefficient (L² T⁻¹)

D_F

Fractal dimension (–)

E

Young's modulus (M L⁻¹ T⁻²)

g

Gravity acceleration (L T⁻²)

H

Hurst exponent (–)

h_rms

Root-mean-square roughness (L)

I

Light intensity (radiance) (–)

L

Fracture length (L)

n_x/y

Number of computational grid cells along the x/y directions (–)

p

Fluid pressure (fracture) (M L⁻¹ T⁻²)

p_avg

Average fluid pressure (M L⁻¹ T⁻²)

p_in

Inlet fluid pressure (M L⁻¹ T⁻²)

p_out

Outlet fluid pressure (M L⁻¹ T⁻²)

Δp

Pressure drop (M L⁻¹ T⁻²)

∇p

Pressure gradient (M L⁻¹ T⁻²)

Q

Flow rate (L³ T⁻¹)

Local fluxes (x, y) (L² T⁻¹)

Re

Reynolds number (–)

S^iso

Radial-averaged 2D PSD (L⁴)

t

Time (T)

Normalized time (–)

T_f

(Fracture) transmissivity (L² T⁻¹)

Fluid velocity (x, y) (L T⁻¹)

U

Average fluid velocity (L T⁻¹)

W

Fracture width (L)

Position vector (x, y, z) (L)

Normalized position (–)

Acknowledgments

This work was partially funded by the Abu Dhabi National Oil Company under the research Project 015-RCM-2015. The authors gratefully acknowledge the help from Stephen W. Rudolph (Department of Civil and Environmental Engineering, MIT) during the design and fabrication of the experimental setup. Special thanks to Dr. Jean E. Elkhoury (Schlumberger-Doll Research), and our colleague Dr. Wei Li for many helpful discussions. We would also like to thank Mr. Majed Almubarak for his assistance in the mechanical characterization of the 3D-printed material.

Appendix A: Definition of the Reynolds Number in Rough Fractures

In fluid mechanics, the Reynolds number, Re, is a common measure of the ratio of inertial forces to viscous forces, and in rough fractures it is usually defined as,

(A1)

where ρ and μ are the fluid density and dynamic viscosity; and l_c and u_c are the characteristic length scale and fluid velocity, which are often represented in terms of the hydraulic aperture (i.e., l_c = b_h; u_c = Q/b_hW, where Q is the macroscopic flow rate, and W is the fracture width). In our tests ( s/m²; W = 38.1 mm), this macroscopic Re ranges between 0.3 and 5.9 (for flow rates Q = 0.01–0.20 mL/s). Note, however, that the Re as defined in Equation A1 is a sole function of the fluid properties (ρ/μ), the macroscopic flow rate (Q), and the fracture width (W); and that it does not evolve as the fracture deforms and the flow channelizes. Moreover, it does not contain any information regarding the local aperture structure and the flow field, which do evolve with increasing normal stress as shown in Figures 6 and 13.

While the use of this expression for the macroscopic Re (Equation A1) is convenient in experimental studies due to its simplicity, when the aperture (Figure 6) and flow fields are available (Figure 13), other formulations that capture the evolving flow structure may be preferred. Here, we propose a flux-averaged version of the macroscopic Reynolds number, , given by,

(A2)

(A3)

where Re_ij, q_ij, b_ij, and u_ij are the local Re, fluxes, apertures, and fluid velocities rates in the 2D ij grid, and n_x and n_y are the total number of grid blocks along the x- and y-directions (see Section 4.3.1 and Text 10 in Supporting Information S1). This flux-averaged Reynolds number, , does consider the internal geometric structure of the fracture and the flow field; and it evolves as a function of the applied normal stress (Figure A1)—in fact, it does so in a fashion similar to the contact area (Figure 7b). For the experiments reported in this study, the flux-averaged Reynolds number, , ranged between 0.5 (Specimen A, Q = 0.01 mL/s, and lowest σ_eff) and 30 (Specimen B, Q = 0.20 mL/s, and highest σ_eff), that is, up to five-fold those determined using the expression in Equation A1. These Reynolds numbers can be considered in the low range for groundwater flows (Re_groundwater ∼ 0–400, see Zou et al. (2017)), and as suggested by the linearity of pressure versus flow rate curves (Figure 9a), they are representative of linear flows in our tests. Finally, note that the ability of to account for the flow structure may help to reduce the uncertainties in the definition of critical Re, especially when the flow regime evolves as a consequence of fracture deformation.

Figure A1

Evolution of the flux-averaged Reynolds number, , as defined in Equation A2 for Specimens A and B under different normal stress levels. As the fracture deforms, the flow field channelizes into preferential pathways that are dominated by larger local velocities resulting in larger macroscopic Reynolds numbers. Note that the is normalized by the inlet flow rate, Q_in (Q_in = 0.01–0.20 mL/s).

Open Research

Data Availability Statement

The experimental data supporting this work are publicly available online via Mendeley Data, and listed as Villamor-Lora et al. (2022).

The authors would also like to share the in-house numerical codes developed for (a) the digital fabrication of fractures (Villamor-Lora, 2022a) and (b) the simulations of fracture flow (Villamor-Lora, 2022b). These are provided under the terms of the Creative Commons Attribution License, which permits use, distribution, and reproduction in any medium, provided the original work is properly cited.