2.1. Fluid flow in the crack

2.1.1. Lubrication theory

The low Reynolds number flow in the elongated fracture allows us to simplify the Navier–Stokes equation and use lubrication theory. The fluid is injected in a pre-formed crack whose aspect ratio is small:

(2.1)\begin{equation} w \ll R. \end{equation}

The Reynolds number of the flow through the crack is defined for the injected fluid as

(2.2)\begin{equation} Re = \epsilon\,\frac{\rho Uw}{\mu} = \frac{\rho_{in} Q_{in}\,w(r=0)}{2 {\rm \pi}\mu_{in} R^2} \leq 1, \end{equation}

where $\epsilon$ is the aspect ratio of the crack. As a result, the flow of both fluids can be modelled with the lubrication theory, similarly to the single-phase flows in a penny-shaped fracture (Savitski & Detournay Reference Savitski and Detournay2002; Detournay Reference Detournay2004; Detournay & Peirce Reference Detournay and Peirce2014; Lai et al. Reference Lai, Zheng, Dressaire, Wexler and Stone2015; O'Keeffe et al. Reference O'Keeffe, Huppert and Linden2018a). For the two-fluid system, the lubrication equations are

(2.3)$$\begin{gather} \frac{\partial w(r, t)}{\partial t}=\frac{1}{12 \mu_{in}}\,\frac{1}{r}\, \frac{\partial}{\partial r}\left(r\,w^{3}(r, t)\,\frac{\partial p}{\partial r}\right) \quad \text{for}\ 0\leq r \leq R_I, \end{gather}$$

(2.4)$$\begin{gather}\frac{\partial w(r, t)}{\partial t}=\frac{1}{12 \mu_{0}}\,\frac{1}{r}\, \frac{\partial}{\partial r}\left(r\,w^{3}(r, t)\,\frac{\partial p}{\partial r}\right) \quad \text{for}\ R_I\leq r \leq R. \end{gather}$$

2.1.2. Liquid interface

The interface between the two fluids moves outwards during the injection and is described by the dynamics boundary condition

(2.5) \begin{align} & \boldsymbol{n} \boldsymbol{\cdot} \left({-}p^{-} \boldsymbol{I} +\mu_{in}\left(\boldsymbol{\nabla} \boldsymbol{u}^{-}+ (\boldsymbol{\nabla} \boldsymbol{u}^{-})^{\mathrm{T}}\right)\right) \boldsymbol{\cdot} \boldsymbol{n}+\gamma \kappa_c \nonumber\\ &\quad =\boldsymbol{n} \boldsymbol{\cdot} \left({-}p^{+}\boldsymbol{I}+\mu_{0} \left(\boldsymbol{\nabla} \boldsymbol{u}^{+}+(\boldsymbol{\nabla} \boldsymbol{u}^{+})^{\mathrm{T}}\right)\right) \boldsymbol{\cdot} \boldsymbol{n}, \end{align}

where $\boldsymbol {n} = \boldsymbol {e}_r$ is the vector normal to the interface, and $\kappa_c$ is the sum of the principal curvatures of the interface. The $+$ and $-$ exponents indicate that the value of the variable is determined at $r = R_I+ \delta$ and $r = R_I - \delta$ respectively, with $\delta \ll R_I$. We note that $\boldsymbol {{\tau }} = \mu (\boldsymbol {\nabla } \boldsymbol {u}+(\boldsymbol {\nabla } \boldsymbol {u})^{\mathrm {T}})$. We assume that the fluids are perfectly wetting the gel, and neglect the thin film deposited by the outer fluid. The normal stress balance is

(2.6)\begin{align} p^{-} - p^{+} = \sigma_I &= \gamma \left(\frac{1}{R_I} + \frac{2}{w_I}\right) - {\boldsymbol{n}} \boldsymbol{\cdot} \left(\boldsymbol{\tau}^+-\boldsymbol{\tau}^-\right)\boldsymbol{\cdot} {\boldsymbol{n}} \nonumber\\ &= \gamma \left(\frac{1}{R_I} + \frac{2}{w_I}\right) - 2 \mu_{0} \left.\frac{\partial u}{\partial r}\right|_{r= R_I^{+}} - 2 \mu_{in} \left.\frac{\partial u}{\partial r}\right|_{r= R_I^{-}}, \end{align}

where $w_I$ is the width of the fracture at the interface. The expression can be simplified further as $R_I>w_{I}$, and the viscous normal stresses are negligible. Indeed, the capillary number of the invading fluid is

(2.7)\begin{equation} Ca_{in} = \frac{\mu_{in} U}{\gamma} = \frac{\mu_{in} Q_{in}}{2 \gamma {\rm \pi}R\,w(r=0)} \ll 1. \end{equation}

The pressure change across the interface is

(2.8)\begin{equation} p^{-} - p^{+} \approx \frac{2\gamma}{w_I}. \end{equation}

As the fluid–fluid interface moves, we can write the following kinematic condition using Reynolds equations:

(2.9)\begin{equation} q^-{=} q^+{=} -\frac{w_I^3}{12\mu_{in}} \left.\frac{\partial p}{\partial r}\right|_{r= R_I^{-}} ={-}\frac{w_I^3}{12\mu_{0}} \left.\frac{\partial p}{\partial r}\right|_{r= R_I^{+}}. \end{equation}

2.1.3. Volume conservation

Finally, through volume conservation, the volume of the crack is equal to the volume injected. We note that $V_0$, the volume of the pre-fracture, is equal to the volume of the outer fluid. The injection begins at $t=0$:

(2.10)\begin{equation} V_0 + Q_{in}t= 2 {\rm \pi}\int_0^{R}r\,w(r,t)\,{\rm d} r \end{equation}

and

(2.11)\begin{equation} Q_{in}t= 2 {\rm \pi}\int_0^{R_I}r\,w(r,t)\,{\rm d} r. \end{equation}

2.2. Fracture equations

2.2.1. Linear elasticity

Similarly to the single-fluid fracture (Savitski & Detournay Reference Savitski and Detournay2002), the linear elasticity equation is

(2.12)\begin{equation} w(r, t)=\frac{8 (1-\nu^2) R}{{\rm \pi} E} \int_{r / R}^{1} \frac{\xi}{\sqrt{\xi^{2}-(r / R)^{2}}} \int_{0}^{1} \frac{x\,p(x \xi R, t)}{\sqrt{1-x^{2}}}\,\mathrm{d} x\,\mathrm{d} \xi. \end{equation}

2.2.2. Fracture propagation

In a small region at the tip of the crack, the material undergoes plastic deformation. The tensile fracture propagates when the mode I stress intensity factor $K_I$ reaches a critical value called the toughness of the material, $K_{IC} = \sqrt {2 E^{'} \gamma _S}$, where $\gamma _S$ is the fracture surface energy of the solid material (Kanninen & Popelar Reference Kanninen and Popelar1985), and $E' = E/(1-\nu ^2)$. For a penny-shaped crack, the stress intensity factor near the tip is defined as (Rice Reference Rice1968)

(2.13)\begin{equation} K_{{I}}=\frac{2}{\sqrt{{\rm \pi} R}} \int_{0}^{R(t)} \frac{p(r, t)}{\sqrt{R^{2}-r^{2}}}\,r\,\mathrm{d} r. \end{equation}

2.3. Boundary conditions at the fracture tip and injection point

At the tip of the crack, the width $w(R)$ goes to zero,

(2.14a,b)\begin{equation} w=0, \quad r=R(t), \end{equation}

and the flow rate also goes to zero,

(2.15)\begin{equation} w^{3}(r, t)\,\frac{\partial p(r, t)}{\partial r}=0, \quad r=R(t). \end{equation}

At the point source, the local flow rate is equal to the injected flow rate:

(2.16)\begin{equation} 2 {\rm \pi}\lim _{r \rightarrow 0} r\,q(r, t)=Q_{in}. \end{equation}

3.1. Toughness regime

For the toughness scaling, we set the non-dimensional groups in (3.3)–(3.6) equal to 1. Indeed, the viscous stresses are negligible, and the crack opening is limited by the toughness of the material. The scaling relations are

(3.7)$$\begin{gather} \frac{W_0E'}{{P_0}R} = 1, \end{gather}$$

(3.8)$$\begin{gather}\frac{K'}{{P_0} \sqrt{R}} = 1, \end{gather}$$

(3.9)$$\begin{gather}\frac{Q_{in}t}{R_I^2 W_0} = 1, \end{gather}$$

(3.10)$$\begin{gather}\frac{V_0}{ W_0} = R^{2} - R_I^{2}. \end{gather}$$

We define the characteristic time scale $T = V_0/Q_{in}$ and the corresponding dimensionless time $\tilde {t} = t/T$. By combining those groups, we obtain the toughness-dominated scaling of the fracture properties:

(3.11)$$\begin{gather} W_0 = \left(\frac{K'}{E'}\right)^{4/5} V_0^{1/5} \left(1+\tilde{t}\right)^{1/5}, \end{gather}$$

(3.12)$$\begin{gather}R = \left(\frac{E' V_0}{K'}\right)^{2/5} \left(1+\tilde{t}\right)^{2/5}, \end{gather}$$

(3.13)$$\begin{gather}R_I = \left(\frac{E' V_0}{K'}\right)^{2/5} \tilde{t}^{1/2} \left(1+\tilde{t}\right)^{{-}1/10} \end{gather}$$

and

(3.14)\begin{equation} {P_0} = K' \left(\frac{K'}{E' V_0}\right)^{1/5} \left(1+\tilde{t}\right)^{{-}1/5}. \end{equation}

3.2. Viscous regime

For the viscous scaling, we set the non-dimensional groups in (3.2)–(3.3) and (3.5)–(3.6) equal to 1. Here, the propagation of the fluids follows the lubrication equation, and the viscous dissipation limits the propagation of the fracture:

(3.15)$$\begin{gather} {\frac{W_0E'}{P_{0}R}= 1}, \end{gather}$$

(3.16)$$\begin{gather}{\frac{\mu'_{e} R^2}{W_0^2 P_{0}\left(t+V_0/Q_{0}\right) } = 1}, \end{gather}$$

(3.17)$$\begin{gather}{\frac{Q_{in}t}{ R_I^2 W_0} = 1}, \end{gather}$$

(3.18)$$\begin{gather}{\frac{V_0}{ W_0} = R^{2} - R_I^{2}}. \end{gather}$$

We define the effective viscosity of the volume of fluid in the fracture as a weighted average. The effective viscosity depends on the viscosities of the two fluids in the fracture and their relative volumes at time $\tilde {t}$:

(3.19)\begin{equation} \mu'_e = \frac{\mu'_{0}+\mu'_{in}\tilde{t}}{1 + \tilde{t}}. \end{equation}

We obtain the viscous-dominated scaling of the variables:

(3.20)$$\begin{gather} {W_0 = \left(\frac{\mu'_{e}Q_{in}}{E'}\right)^{2/9} V_0^{1/9} \left(1+\tilde{t}\right)^{1/3}\left(\alpha+\tilde{t}\right)^{{-}2/9},} \end{gather}$$

(3.21)$$\begin{gather}{R = \left(\frac{E'}{\mu'_{e}Q_{in}}\right)^{1/9} V_0^{4/9} \left(1+\tilde{t}\right)^{1/3} \left(\alpha+\tilde{t}\right)^{1/9},} \end{gather}$$

(3.22)$$\begin{gather}{R_I = \left(\frac{E'}{\mu'_{e}Q_{in}}\right)^{1/9} V_0^{4/9} \tilde{t}^{1/2} \left(1+\tilde{t}\right)^{{-}1/6} \left(\alpha+\tilde{t}\right)^{1/9},} \end{gather}$$

(3.23)$$\begin{gather}{P_{0} = \left(\frac{\mu'_{e}Q_{in}E^{\prime 2}}{V_0}\right)^{1/3}\left(\alpha + \tilde{t}\right)^{{-}1/3}}, \end{gather}$$

with $\alpha = Q_{in}/{Q_{0}}$. The results are summarized in table 2.

Table 2.

Scaling relations for a penny-shaped fracture driven by a displacement flow. The time evolution of the geometrical properties of the fracture depends on the dominant resisting stress, which can be viscous or toughness-related.

4.1. Material preparation and characterization

The properties of the gelatin and the injected fluids control the fracture dynamics and are therefore characterized systematically. The gelatin is prepared by first heating ultra-pure water to $60\,^{\circ }$C. While stirring the heated water, we slowly add gelatin powder (Gelatin type A; Sigma-Aldrich, USA) at a mass fraction of 10 % to 30 % to vary Young's modulus of the resulting gel. The gelatin then cools down to room temperature and sets over 24 hours prior to testing or fracturing. The Young's modulus of gelatin is measured with a custom-built displacement-controlled load frame. The sample is compressed by a ballscrew stage whose speed is set by a stepper motor Parker Compumotor OS22B controlled by a controller Parker Compumotor ZL6104. A load cell, Eaton 3108-10 (10 lb capacity) measures the force generated by the compressed sample. We record force–displacement values for cylindrical samples of gelatin of diameter and height equal to 1 in, and compute the stress–strain curves of the material. At small strain, all gelatin samples exhibit a linear elastic response to the compression. As listed in table 3, Young's modulus ranges between $15$ and 116 kPa with a measurement error of ${\pm }10\,\%$. The fracture energy and Poisson's ratio of the gelatin are assumed constant in our experiments, with $\gamma _S \approx 1$ J m$^{-2}$ and $\nu \approx 0.5$ (Menand & Tait Reference Menand and Tait2002).

Table 3.

List of experiments: symbols and parameters.

To study displacement flows in fluid-filled fractures, we use immiscible Newtonian liquids. Silicone oils of different viscosity are used to form the pre-fracture. An aqueous solution composed of water, glycerol or corn syrup is then injected. The silicone oil is displaced outwards, further expanding the fracture in the gelatin. We measure the viscosity of the fluids and the surface tension at the oil/water and oil/syrup interfaces. Viscosity measurements are conducted using an MCR 92 Anton Parr rheometer with a parallel plate measuring system at $20\,^{\circ }$C. The values obtained have a measurement error of ${\pm }1\,\%$ and are listed in table 3. Surface tension measurements are conducted using the pendant drop method with an Attension Theta Flex tensiometer: the surface tension between silicone oil and water is $\gamma _{o/w} \approx 35 \pm 2$ mN m$^{-1}$, and between silicone oil and syrup is $\gamma _{o/s} \approx 50 \pm 2$ mN m$^{-1}$. The fluid properties are selected to study the propagation of the pre-fracture and fracture in a single regime during the experiment. For example, the values of $\kappa$ at the end of the injection forming the pre-fracture, for experiments 1–10, vary as $35 \leq \kappa \leq 55$. Those experiments are expected to be in the toughness-dominated regime. Similarly, for experiments 11–16, $1.7 \leq \kappa \leq 1.9$ at the end of the formation of the pre-fracture: these experiments target the viscous-dominated propagation. These values are consistent with previous work on the formation of fluid-driven fractures in a block of gelatin. During the formation of the fracture through the displacement flow, we estimate the pressure ratio in the viscous regime $({\Delta P_m}/{\Delta P_v})_v$ at the end of the injection. The ratio varies between 24 and 58 for experiments 1–10, which are therefore expected to be in the toughness regime. The ratio varies between 1.9 and 2.5 for experiments 11–16, which are therefore expected to be in the viscous regime.

4.2. Set-up

The injection experiments are conducted in a large block of gelatin to avoid boundary effects on the propagation of the fracture. The gelatin is set in a cubic clear container ($15\,{\rm cm} \times 15\,{\rm cm}\times 15\,{\rm cm}$) around a blunt needle as represented in figure 2(a). We use a thin needle (inner diameter $ID = 1$ mm) for low-viscosity injections, and a wider needle for high-viscosity injections ($ID = 2.15$ mm). In our system, the fracture propagates horizontally, in the direction that opposes the least resistance. To avoid small tilts of the fracture that would compromise the quality of the recording, a plastic washer (outer diameter $OD=6.5$ mm) is placed at the tip of the needle to initiate the fracture in the horizontal plane. Two fluids are successively pumped into the gelatin. For each fluid, we use a kdScientific Legato 200 syringe pump to set the injection flow rate at a value between $0.1$ and $20$ ml min$^{-1}$ with accuracy ${\pm }0.35\,\%$. Both fluid-filled syringes are connected to the same injection needle using a switch valve. First, a silicone oil of viscosity $\mu _0$ is injected at a constant volumetric flow rate $Q_0$ in the gelatin matrix to form the pre-fracture. The injection stops when a volume $V_0$ of silicone oil has been dispensed. The valve is then switched to inject the aqueous phase of viscosity $\mu _{in}$ into the pre-fracture at flow rate $Q_{in}$. The injection stops when the fracture tip is within 2 cm of the container walls to avoid confinement effects (Bunger, Gordeliy & Detournay Reference Bunger, Gordeliy and Detournay2013). The two fluids are dyed to allow visualizing the propagation of the fracture and interface between the two liquids in the clear gelatin: we use a blue water-based food dye for the aqueous phase, and a red oil-based food dye for the oil phase, as shown in figure 2. The propagation is recorded using a Nikon D5300 camera with a Phlox LED panel ensuring uniform backlighting. The images are processed using a custom-made MATLAB code to determine the radius of the fracture $R$ and the position of the interface between the two liquids $R_I$.

Figure 2.

(a) Schematic of the experimental set-up. (b–e) Time evolution of the fracture formed in experiment 1. The water dyed with blue food colour is injected at $Q_{in} = 0.5$ ml min$^{-1}$ in a pre-fracture formed with silicone oil dyed with red food colour. The recording starts when the water injection begins, and the experimental images are taken at (b) $t =50$ s, (c) $t = 300$ s, (d) $t = 1300$ s, and (e) $t =2800$ s. Supplementary movies 1 and 2 (available at https://doi.org/10.1017/jfm.2022.954) show the complete time evolution of the pre-fracture and fracture, respectively.

4.3. Measurement of the fracture width

To measure the thickness of the fracture during the propagation, we use a light absorption method (Bunger Reference Bunger2006; Bunger & Detournay Reference Bunger and Detournay2008). This method consists in selecting a soluble dye and the corresponding optical filter. The filter should transmit light to the camera at the wavelength at which the dye absorbance $A$ is maximum, $A_{\lambda }$. The absorbance follows Beer's law:

(4.1)\begin{equation} A_{\lambda} ={-} \log_{10}\left(\frac{I_{\lambda}}{I_{\lambda,0}}\right)= \epsilon_{\lambda} c h, \end{equation}

where $I_{\lambda,0}$ is the background intensity, and $I_{\lambda }$ is the intensity when light passes through a liquid layer of thickness $h$, with dye concentration $c$. The parameter $\epsilon _{\lambda }$ characterizes the absorbance of the dye at the selected wavelength and is obtained through calibration. To measure the thickness of both liquids in the fracture, we use two dyes, one water-soluble and one oil-soluble, and record the absorbance using a single optical filter. To get accurate measurements, we select dyes with a large absorbance at the same wavelength. In all experiments, the water-soluble dye is nigrosin (Sigma-Aldrich) at 0.05 g l$^{-1}$. Nigrosin is a black dye that absorbs at all wavelengths. We use different dyes depending on the viscosity of the silicone oils, because of their solubility limit. We dilute Sudan red (Sigma-Aldrich) in the low-viscosity silicone oils, i.e. 10 and 20 mPa s silicone oils, at 0.05 g l$^{-1}$, and Nile red (Sigma-Aldrich) in high-viscosity silicone oils, i.e. 10 300 and 30 000 mPa s silicone oils, at 0.2 g l$^{-1}$. For experiments with Sudan red in the oil phase and nigrosin in the aqueous phase, we use a 520 nm optical filter. When Nile red dyes the oil phase and nigrosin the aqueous phase, we use a 632 nm optical filter.

To measure the thickness of the fracture, we first conduct calibration experiments for each dye solution. For the low-viscosity solutions (water, 10 and 20 mPa s silicone oils), we use a glass wedge with an aperture that increases linearly from 0 to 10 mm, as shown in figure 3. The wedge is placed on the LED panel and the light intensity is obtained by taking a picture of the wedge with the optical filter mounted on the camera. The background intensity corresponds to an empty wedge. The light intensity of a liquid-filled wedge decreases as the thickness of the aperture increases (see figure 3b). The grey values are used to determine the absorbance as a function of the liquid thickness, as plotted in figure 3(c). Using a linear fit, we get $1/{\epsilon _{\lambda } c} = 47.2$ mm for Sudan red in oil, and $1/{\epsilon _{\lambda } c} = 32.6$ mm for nigrosin in water, for $\lambda = 520$ nm. For the high-viscosity samples, we use rectangular cells that are easier to fill. The cell thickness or height ranges from 0.3 to 3 mm. For each cell, we measure the background intensity of the empty cell and the intensity of the cell filled with the viscous fluid, i.e. syrup, or 10 000 or 30 000 mPa s silicone oil. We obtain the absorbance for a set of thickness values as shown in figure 3. Using a linear fit, we get $1/{\epsilon _{\lambda } c} = 56.5$ mm for Nile red in oil, and $1/{\epsilon _{\lambda } c} = 44$ mm for nigrosin in water, for $\lambda = 632$ nm. The fitting parameters obtained are then used to obtain the width of the fracture using Beer's law:

(4.2)\begin{equation} h = \frac{A_{\lambda}}{\epsilon_{\lambda} c} = \frac{-\log_{10}\left(\dfrac{I_{\lambda}}{I_{\lambda,0}}\right)}{\epsilon_{\lambda} c}. \end{equation}

Figure 3.

Calibration. (a) Schematic of the calibration experiment for low-viscosity fluids. (b) The intensity gradient was recorded for a wedge filled with water dyed with nigrosin at 0.05 g l$^{-1}$ through the 520 nm filter. (c) Absorbance measured for nigrosin-dyed water at $\lambda = 520$ nm ($\blacktriangle$), nigrosin-dyed syrup at $\lambda = 632$ nm (), Sudan-red-dyed 10 mPa s siliconeoil at $\lambda = 520$ nm ($\vartriangle$), and Nile-red-dyed 10 Pa s silicone oil at $\lambda = 632$ nm (). The solid lines are the best linear fit for each calibration data set.

The concentrations of dyes chosen for this study, some of which are limited by solubility, are all sufficiently low for the absorbance to vary linearly with the sample thickness or fracture aperture. The accuracy of the measurements is limited by the noise due to low absorbance values.