Hydrogel as a Medium for Fluid-Driven Fracture Study

We will consider a tensile crack (mode I) where fracturing occurs under the linear elastic fracture mechanics (LEFM) assumption of a propagation criterion [13]: where K _I is the stress intensity factor and K _{I C} is the fracture toughness related to the material properties.

Viscous dissipation within the fracture can play a significant role in the shape and speed of the fracture tip. The dominance of viscous dissipation relative to energy dissipation from bond-breaking due to the toughness of the material can be related by a characteristic time. This gives the time in which a fracture transitions from a viscously dominated propagation regime to a material toughness dominated regime [14]: where μ is the dynamic viscosity, Q is the volumetric injection rate, and E ^′ = E/(1 − ν ²) is the plane strain modulus relating Young’s modulus E and Poisson ratio ν.

When the fracture propagates in the toughness dominated regime the crack tip thickness w can be approximated using the LEFM asymptote [15]: where x is the distance from the crack tip and R is the fracture radius.

The hydrogels were prepared by free radical polymerization of acrylamide (Sigma Aldrich) as the monomer and N,N ^′-methylenebis (acrylamide) (Sigma Aldrich) as the cross-linker in aqueous solution. This means any oxygen present in the solution will inhibit the polymerization. The initiator ammonium persulphate (APS) and accelerator N,N,N,N ^′-tetramethylethylenediamine (TEMED) (Sigma Aldrich) then catalyze the reaction. The gels contain differing amounts of acrylamide produced from a 40% w/v (weight/volume) stock solution and bis-acrylamide in solid form, that are added to a certain amount of demineralized water to give the desired % w/v of each chemical. This solution is then degassed for 10-15 minutes to counteract oxygen inhibiton of the reaction, which is crucial for reproducibility of the gel. Degassing is done by placing the solution inside a vacuum chamber which is connected to a pump. A low pressure environment is then created causing air to be removed from the liquid. Next, 0.075% w/v of APS was added and finally 0.05% v/v of TEMED (all volume percentages are of the final volume).

We constructed gels of dimensions 100 × 100 × 77 mm which is equivalent to 770 mL of volume. The total percentage of acylamide monomer and bis-acrylamide cross-linker varied from 8 − 15 % w/v and 2.5 − 6 % wt. (cross-linker weight/monomer weight). Each gel was prepared at room temperature (20 °C) and the solution was left for over 2 hours allowing gelation to occur fully. During this gelation period the chamber containing the solution is covered with parafilm tape, as an additional precaution against oxygen inhibiting the polymerization.

Polyacrylamide gels are recognised as linearly elastic materials in the literature [16, 17]. These gels consist of cross-linked polymer chains. Their elastic properties are determined by the concentration of monomers, acrylamide, and cross-linking molecules, bis-acrylamide. An increase in the monomer % w/v will result in an increase in the Young’s modulus and fracture energy, if the cross-linking % wt. is held constant. Furthermore, increasing the cross-linking % wt. will make the gel more brittle and decrease the fracture surface energy. This complicated relationship means that only a few measurements have been found in the literature for fracture energies at particular gel percentages [18, 19]. Normally these fracture energy values are obtained from peel tests. Here, we approached the measurement of these values using a technique outlined in “Determination of Stress Intensity and Fracture Surface Energy”. It has been shown that an increase in monomer will always lead to an increase in Young’s modulus [6]. However, an increase in cross-linker will also increase the gel modulus up to around 7% wt. cross-linking. If the cross-linking is increased beyond this, the gels become heterogeneous and the value of the Young’s modulus will plateau.

Heavily cross-linked polyacrylamide gels are transparent, elastic and brittle materials. Elasticity and brittleness are two different properties. However, they are not independent for elastic solids. Ideally, fracture toughness K _{I C} follows the relation: where γ _s is the fracture surface energy, which can be defined as the energy required to create one unit of surface area. Equation (4) incorporates both elasticity and brittleness properties in the Young’s modulus and fracture surface energy, respectively.

Rate-dependent fracture surface energies γ _s have been observed in several gel systems. Chemically cross-linked polyacrylamide hydrogels exhibit a rate dependence that decreases with increasing density of chemical cross-links [19]. Since we are using heavily cross-linked hydrogels we will assume the fracture energy to be constant which can be determined from the measurements obtained in “Determination of Stress Intensity and Fracture Surface Energy”. This is an advantage over gelatin systems, where the fracture energy varies linearly with crack velocity and the rate sensitivity increases with the amount of physical cross-linking [20].

The permeability of these gels can be extremely low and comparable to that observed in unconventional hydrocarbon reservoirs. In shale gas formations the expected permeability is in the range 10⁻⁶ − 10⁻⁸ Darcy, while in polyacrylamide hydrogels permeability values can range from 10⁻⁷ − 10⁻⁸ Darcy [21].

Dye attenuation was used to measure fracture width. This is a process in which the absorbance of light is used to correspond to thickness measurements. To do this a red LED light sheet with a diffuser was used as background lighting to provide a uniform monochromatic light source. The injected fluid is dyed with methylene blue, which strongly absorbs at the wavelength of the red light source. The absorption of this background light is then directly related to the amount of fluid through which it passes.

We have conducted a series of compression tests on sample rectangular gels to obtain material properties, such as the Poisson’s ratio, and observe the variability when concentrations are altered. This was done on an Instron 3345, using parallel plates to measure compressive stress and strain. A sample stress-strain graph can be seen in Fig. 4(c) for a 13 - 4% gel. These tests produced an approximate Poisson’s ratio ν ≈ 0.48, which agrees with the literature that the hydrogels can be treated as linearly elastic [23].

Using the principle of determination of Young’s modulus by Hertzian contact, or indentation with a solid sphere [24, 25], we calculated a range of elastic moduli for different monomer to cross-linker ratios. This method was carried out using a TA.XT Texture Analyser (Stable Micro Systems). A sphere of radius a = 0.63 cm is indented a certain distance onto the surface of the hydrogel as seen in Fig. 4(a) and the force measured. For a perfectly elastic solid, if the surface is displaced a distance δ under a force F, the plane strain modulus E ^′ is given by the relation: However to exclude surface tension effects in the initial stages of indentation, the slope of the linear section of a δ ^3/2 versus F plot, an example of which can be seen in Fig. 4(b), is used to calculate the Young’s modulus E:

The results of these tests can be seen in Fig. 4 and Table 1. We conducted two indentation experiments at different places on the gel surface for each sample. The maximum deviation from the average value was always less than 2.5%. The error bars are estimated from measurement uncertainty on the slope calculation. From the figures we can clearly see that increases in the amount of monomer and cross-linker, in the ranges shown, both contribute to an increase in Young’s modulus. From Fig. 4 it is evident that monomer percentage has a larger effect on the elastic modulus than cross-linking. The linear fits seen in Fig. 4(d), (e) and (f) have a slope m and intercept c. It has been shown that higher percentages of cross-linking can lead to a plateau in the elastic modulus values and the relationship is more complex [6].

Experimentally we can obtain a wide range of values for the Young’s modulus, with values varying from at least 50 − 700 kPa. This was one of our main motivations for conducting experiments in this medium, as it allowed us considerable variation in the transition time (equation (2)) derived for hydraulic fractures.

We can use our measurements of fracture thickness, crack-tip behaviour and radial extent to consider the stress intensity factor [26]. This can be done in the toughness limiting regime where the crack tip LEFM asymptote (equation (3)) relates the fracture thickness, distance from the tip and stress intensity. We consider only mode I opening where we can use the stress intensity factor K _I to estimate the fracture toughness K _{I C} of the hydrogel matrix, under the LEFM assumption that propagation occurs when (equation (1)) is satisfied.

Plotting the opening over 60% of the fracture radius normalised according to equation (3) produces Fig. 5, where the stress intensity factor K _I is the value averaged over the crack tip region. The thickness is plotted for 10 time steps with the black line representing the average of these experimental curves. This particular experiment corresponds to a gel concentration of 10% w/v monomer and 2.5% wt. cross-linker. From the propagation criterion we assume K _I = K _{I C} and we can deduce that K _{I C} = 631 ± 36 Pa m ^1/2. Thus, we can use equation (4) to estimate the fracture surface energy of this particular gel configuration to be γ _s ≈ 1.4 J/m ². The noise near x/R = 0 is due to the minuscule thickness near the crack tip. The dye attenuation value is extremely small in this area and is comparable to the noise in the experiment due to the limitations of our calibration. However, the data clearly follow a specific trend before the noise becomes significant. The observed constant stress intensity value observed all along the fracture tip, retrieved from using LEFM, suggests that the assumption of a homogeneous brittle material is valid.

This technique was then applied to a number of gel concentrations, the results of which can be seen in Table 1 and Fig. 6. In Fig. 6(a) we can see that the stress intensity value increases with monomer percentage as expected. K _I is similar for both 2.5 and 6% wt. cross-linking, with 2.5% wt. having marginally higher values.

The similar behaviour in stress intensity values means that the fracture energies for the higher cross-linked gels, calculated using equation (4), will be lower due to larger moduli measurements. The values of fracture energy γ _s can be seen in Table 1, which are similar to previous values reported from peel testing [18]. In Fig. 6(b), the fracture energy γ _s for 2.5% wt. cross-linking varies with larger amounts of monomer. However, at 6% wt. cross-linking the fracture energy settles to a cross-linking value of γ _s ∼ 0.9 J m ⁻². This is an extremely useful property of these gels for our hydraulic fracturing experiments, allowing us to change the Young’s modulus independent of fracture energy. It allows us to tune the characteristic time for transition (equation (2)) appropriately, so that the viscosity dominated regime may be explored. Otherwise, this would be very difficult experimentally, as we would need to increase both the injection rate Q and fluid viscosity μ significantly in order to analyse the same parameter space.

Moreover, this technique of determining fracture energy can become extremely useful when trying to characterise heterogeneous or layered gel systems.

The step-line patterns on the fracture surfaces also form fractal networks. These beautiful fractal patterns originate from the distribution of these crack surface markings. The fractal dimension of these patterns was computed using the classical box-counting technique [35]. The results of this analysis can be seen in Fig. 11. It is clear from this graph that gels with a cross-linking of 1.9 − 2.5% and those with cross-linking ∼ 6% follow two different curves, in relation to the fractal dimension at specific box sizes. The fractal dimensions are d f ∼ 1.624 ± 0.143 and 1.89 ± 0.08 for 1.9 − 2.5% and ∼ 6% cross-linking, respectively. The fractal dimension for 6% cross-linking is similar to that of the theoretical logarithmic spirals in Fig. 9, which have a fractal dimension of d f ∼ 1.85 ± 0.15.

Interestingly, the range of these fractal values is consistent with those observed in the literature for plastic shear bands [29]. It has also been observed that the fractal dimension depends on the dimensionless parameter B in equation (7). The fractal dimension df is consistently larger in systems with lower B values. This is observed in our experiments, if we assume the confining pressure and ratio of sound to boundary velocity to be constant for each gel. Hydrogels with higher percentages of monomer and cross-linking have a larger elastic modulus, and therefore a smaller B and larger fractal dimension.