Hydraulic fracturing pressure of concentric double-layered cylinder in cohesive soil

Abstract

This study aims to investigate hydrofracturing in double-layered soil through theoretical and experimental analysis, as multilayered soils where the difference in mechanical properties exists are generally encountered in practical engineering. First, an analytical solution for fracturing pressure in two different concentric regions of soil was presented based on the cavity expansion theory. Then, several triaxial hydraulic fracturing tests were carried out to validate the analytical solution. The comparison between the experimental and analytical results indicates the remarkable accuracy of the derived formula, and the following conclusions were also obtained. First, there is a linear relationship between the fracturing pressure and confining pressure in concentric double-layered cohesive soil. Second, when the internal-layer soil is softer than the external-layer soil, the presence of internal soil on the fracturing pressure approximately brings the weakening effect, and the greater strength distinction between the two layers, the greater the weakening effect. Third, when the internal-layer soil is harder than the external-layer soil, the existence of the internal-layer soil has a strengthening effect on the fracturing pressure regardless of the proportion of internal-layer soil. Moreover, the influence of strength distinction between the two layers on the fracturing pressure is significant when the proportion of internal-layer soil is less than half, while it’s limited when the proportion is more than half. The proposed solution is potentially useful for geotechnical problems involving aspects of cohesive soil layering in a composite formation.

Graphical abstract

Keywords

hydraulic fracturing pressure / layered / cavity expansion theory / triaxial fracturing test / cohesive soil

1 Introduction

The fracturing phenomenon of geomaterials often occurs in many practical engineering, which is a relatively controversial problem because of its potential impact on the engineering geological environment. On the positive side, hydraulic fracturing is widely used as a technique for oil and gas recovery in rock [ 1– 3] as well as a primary concern on the effectiveness of ground improvement by grouting in ground engineering [ 4]. On the negative side, hydraulic fracture failure of earth dam [ 5] and slurry fracturing during shield tunneling [ 6– 8] may be encountered, which may cause catastrophic results for the safety of engineering.

Hydraulic fracturing is the formation of cracks in soil or rock due to excessive water pressure [ 9– 11], which is a complex problem that involves many aspects like pressure, crack propagation, blow-out, etc. Among them, the fracture initiation pressure, defined as the water pressure when hydraulic fracturing happens, is the one that researchers and engineers are most concerned about [ 5]. Herein, the fracture initiation pressure is referred to as fracturing pressure. Previous studies, including experimental work [ 12– 15], theoretical analysis [ 16– 18], numerical simulation [ 19– 24], in situ measurement [ 25, 26], have been carried out to investigate fracturing pressure and its influencing factors, which is valuable to the research of hydraulic fracturing in soil. In terms of experimental studies, the laboratory test is an essential and effective way to investigate the hydraulic fracturing of soil [ 14]. Decker and Clemence [ 9] developed a laboratory test to study hydraulic fracture initiation in soil samples which were laboratory compacted low plasticity clay. Alfaro and Wong [ 13] presented the results of a laboratory experimental investigation to characterize the mechanisms related to the initiation pressure in low-permeability soils. Ghanbari et al. [ 27] proposed a new empirical criterion to predict the hydraulic fracturing pressure in the core of earth dams based on laboratory studies. Theoretically, the elastic [ 12] or elastic-plastic theory [ 5] and the cavity expansion theory [ 28] have been applied to derive the formula of fracturing pressure based on different assumptions of the failure mechanism, mainly including shear [ 12, 29] and tensile failure [ 30, 31]. Additionally, with the development of computational analysis, the advanced numerical approach is becoming a useful and effective method for modeling the soil’s fracturing phenomenon [ 32]. Tang et al. [ 33] carried out a numerical study of fracture initiation behavior for hydraulic fracturing. Chang and Huang [ 15] discussed the observations of hydraulic fracturing based on the discrete element method and found that hydraulic fracturing is easier to be initiated in anisotropic stress conditions, where the minor principal stress is the key factor. XFEM (Extended Finite Element Method) combined with Biot’s consolidation theory was used by Wang et al. [ 34] to understand the cause and process of hydraulic fracture of soils.

In general, it can be found that most of the previous studies were focused on single-layered soils, while the fracturing pressure in the non-single layer is still premature. However, multi-layered soils were usually encountered in practical engineering [ 35]. A difference in mechanical properties between two layers could influence the stress state, which means the conventional single-layered solutions could not predict the fracture pressure in layered media [ 17, 36]. Besides, mechanical differences also exist in some structures like dams, roadbeds due to reinforcement or layered filling, indicating that their response to external pressure is not the same as in a homogeneous state [ 37, 38]. Therefore, it is necessary to carry out investigations of fracturing in double-layered or multilayered soil. Sayed and Hamed [ 39] were the first to apply the cavity expansion method of concentrically layered media to the field of geomechanics. They were also the first to present theoretical formulations to the problem of expansion of spherical and cylindrical cavities in a layered elastic system. However, the plastic behavior of soils was not taken into consideration. Mo et al. [ 17] treated the soil as an isotropic dilatant elastic-perfectly plastic material with the Mohr−Coulomb yield criterion and derived the elastoplastic solution to expand two different concentric regions of soil. However, the external-layer soil in this study extended to infinity which may be not suitable for finite layered soil. Zhuang et al. [ 35] constituted a phase-field framework to examine hydraulic fracture propagation in naturally-layered porous media. However, the hydraulic fracture initiation pressure was not focused on.

This paper focuses on the fracturing pressure of concentric double-layered cohesive soil. An analytical solution based on cavity expansion theory was derived. Then laboratory tests of hydraulic fracturing pressure in the concentric double-layered cylinder were carried out to validate the analytical solution. Through the comparative analysis, it can be found that the analytical solution keeps in line with the experimental value indicating the accuracy of the derived formula. Meanwhile, based on the analytical solution, parametric analysis was carried out, and this solution’s application was discussed.

2 Analytical solution

Cavity expansion theory has been extensively developed and widely used for many engineering problems, and many analytical solutions have been proposed based on it [ 18, 29, 39]. The equations of stresses in a cylinder with a cavity have been deduced by Yu [ 40] based on the elastic theory. Herein, it is applied to consider a cavity embedded within a profile of two different concentric regions of soil.

2.1 Basic assumption

Considering the sample section in the plane strain condition as an infinitely long and concentric double-layered cylinder under the confining pressure of P₀ as shown in Fig. 1.

Within the sample, there is a cylinder cavity with borehole radius R _i, and the water pressure P _i is applied to its surface. The radii, elastic modulus and Poisson’s ratio of the external-layer and internal-layer are R ₀, E ₀, μ ₀, and R _i, E _i, μ _i, respectively. The pressure at the interface is given by P _m. Define the ratio of layer diameter as the radius of the internal-layer to the radius of the external-layer, the ratio of borehole diameter as the radius of the borehole to the radius of the sample.

The soil was assumed to be homogeneous, isotropic, elastic, and perfectly plastic, obeying Hooke’s law before yielding. With the injection borehole pressure increasing, the soil around the hole begins to yield, and there is a plastic zone around the borehole. Thus, the borehole pressure is considered fracturing pressure once hydraulic fracturing occurs. Assuming that the internal-layer soil is in the elastoplastic stage, while the external-layer soil was still in the elastic stage when the fracturing occurred. The displacement of soil is neglected since the initiation of fracturing is relatively instantaneous, and generally, the yield of the soil would occur under a very small deformation [ 28]. In this paper, the criterion which was applied in Mori’s study is suitable for cohesive soil and is used in this study [ 41]:

(1) {\sigma }_{\mathrm{r}} - {\sigma }_{\text{θ}} = q_{\mathrm{u}} ,

where σ _r, σ _θ are stresses acting in the radial and tangential direction, respectively.

2.2 Solution for the concentric double-layered cylinder

1) Stress distribution in external-layer soil ( R_{\mathrm{m}} ⩽ r ⩽ R_0)

As illustrated in Fig. 1, for an arbitrary radial distance r under conditions of radial symmetry, the stresses within the soils around the cavity must satisfy the following equation of equilibrium [ 5, 17, 29]:

(2) \frac{\mathrm{\partial }{\sigma }_{\mathrm{r}}}{\mathrm{\partial }r} + \frac{{\sigma }_{\mathrm{r}}-{\sigma }_{\text{θ}}}{r} = 0.

For the external-layer, which is an elastic region, with Hooke’s law as shown in Eq. (3):

(3) \begin{array}{rl}& {\xi }_{\mathrm{r}}=\frac{1-{\mu }^2}{E}\left({\sigma }_{\mathrm{r}}-\frac{\mu }{1-\mu }{\sigma }_{\text{θ}}\right),\\ & {\xi }_{\text{θ}}=\frac{1-{\mu }^2}{E}\left({\sigma }_{\text{θ}}-\frac{\mu }{1-\mu }{\sigma }_{\mathrm{r}}\right).\end{array}

The solutions are subject to two stress boundary conditions:

(4) \begin{array}{rl}& {\sigma }_{\mathrm{r}(r=R_{\mathrm{m}})}=p_{\mathrm{m}},\\ & {\sigma }_{\mathrm{r}(r=R_0)}=p_0.\end{array}

According to Eqs. (2)–(4), the solutions for the stresses distribution in external-layer soil can be obtained as follow:

(5) \begin{array}{rl}& {\sigma }_{\mathrm{r}}=P_0-{\displaystyle \frac{(P_0-P_{\mathrm{m}})\left({\displaystyle \frac{{R_0}^2}{r^2}}-1\right)}{\left({\displaystyle \frac{{R_0}^2}{{R_{\mathrm{m}}}^2}}-1\right)}},\\ & {\sigma }_{\text{θ}}=P_0+{\displaystyle \frac{(P_0-P_{\mathrm{m}})\left({\displaystyle \frac{{R_0}^2}{r^2}}+1\right)}{\left({\displaystyle \frac{{R_0}^2}{{R_{\mathrm{m}}}^2}}-1\right)}}.\end{array}

2) Stress distribution in the elastic region of internal-layer soil ( R_{\mathrm{c}} ⩽ r ⩽ R_{\mathrm{m}})

Similarly, according to Eq. (5), the stress distribution in the elastic region of internal-layer soil can be expressed as follow:

(6) \begin{array}{rl}& {\sigma }_{\mathrm{r}}=P_{\mathrm{m}}-{\displaystyle \frac{(P_{\mathrm{m}}-P_{\mathrm{y}})\left({\displaystyle \frac{{R_{\mathrm{m}}}^2}{r^2}}-1\right)}{\left({\displaystyle \frac{{R_{\mathrm{m}}}^2}{{R_{\mathrm{c}}}^2}}-1\right)}},\\ & {\sigma }_{\text{θ}}=P_{\mathrm{m}}+{\displaystyle \frac{(P_{\mathrm{m}}-P_{\mathrm{y}})\left({\displaystyle \frac{{R_{\mathrm{m}}}^2}{r^2}}+1\right)}{\left({\displaystyle \frac{{R_{\mathrm{m}}}^2}{{R_{\mathrm{c}}}^2}}-1\right)}},\end{array}

where P _y is the elastic limiting pressure of the internal soil.

The yielding first occurred at the borehole wall. Substituting the criterion Eq. (1) into Eq. (6) leads to:

(7) P_{\mathrm{y}} = P_{\mathrm{m}} + \frac{q_{\mathrm{u}}}{2} \left(1-\frac{{R_{\mathrm{i}}}^2}{{R_{\mathrm{m}}}^2}\right) .

Thus, the solutions for the stress distribution in the elastic region of internal-layer soil can be expressed as:

(8) \begin{array}{rl}& {\sigma }_{\mathrm{r}}=P_{\mathrm{m}}+\frac{q_{\mathrm{u}}{R_{\mathrm{c}}}^2}{2{R_{\mathrm{m}}}^2}\left(\frac{{R_{\mathrm{m}}}^2}{r^2}-1\right),\\ & {\sigma }_{\text{θ}}=P_{\mathrm{m}}-\frac{q_{\mathrm{u}}{R_{\mathrm{c}}}^2}{2{R_{\mathrm{m}}}^2}\left(\frac{{R_{\mathrm{m}}}^2}{r^2}+1\right).\end{array}

According to the deformation coordination between the internal and external-layer soil, the stress on the interface between the two layers can be obtained as follow:

(9) P_{\mathrm{m}} = \frac{{\displaystyle \frac{2{R_0}^2{P}_0(1-{\mu }_0^2)}{E_0({R_0}^2-{R_{\mathrm{m}}}^2)}}+{\displaystyle \frac{q_{\mathrm{u}}{R_{\mathrm{c}}}^2(1-{\mu }_{\mathrm{i}}^2)}{E_{\mathrm{i}}{R_{\mathrm{m}}}^2}}}{{\displaystyle \frac{(1+{\mu }_0){R_0}^2+(1+{\mu }_0)(1-2{\mu }_0){R_{\mathrm{m}}}^2}{E_0({R_0}^2-{R_{\mathrm{m}}}^2)}}+{\displaystyle \frac{(1+{\mu }_{\mathrm{i}})(1-2{\mu }_{\mathrm{i}})}{E_{\mathrm{i}}}}} .

3) Stress distribution in plastic region of internal-layer soil ( R_{\mathrm{i}} ⩽ r ⩽ R_{\mathrm{c}})

Combining the yield criteria in the equation of equilibrium, the stress components in the plastic region can be derived from Eqs. (1) and (2):

(10) \begin{array}{rl}& {\sigma }_{\mathrm{r}}=P_{\mathrm{m}}+\frac{q_{\mathrm{u}}}{2}+q_{\mathrm{u}}\mathrm{I}\mathrm{n}\left(\frac{R_{\mathrm{c}}}{r}\right)-\frac{q_{\mathrm{u}}{R_{\mathrm{c}}}^2}{2{R_{\mathrm{m}}}^2}\\ & {\sigma }_{\text{θ}}=P_{\mathrm{m}}-\frac{q_{\mathrm{u}}}{2}+q_{\mathrm{u}}\mathrm{I}\mathrm{n}\left(\frac{R_{\mathrm{c}}}{r}\right)-\frac{q_{\mathrm{u}}{R_{\mathrm{c}}}^2}{2{R_{\mathrm{m}}}^2}\end{array} .

Hence, the fracturing pressure of concentric double-layered soil is:

(11) P_{\mathrm{f}}^{\mathrm{d}} = {\sigma }_{\mathrm{r}(r=R_{\mathrm{i}})} = P_{\mathrm{m}} + \frac{q_{\mathrm{u}}}{2} + q_{\mathrm{u}} \mathrm{I}\mathrm{n} \left(\frac{R_{\mathrm{c}}}{R_{\mathrm{i}}}\right) - \frac{q_{\mathrm{u}}{R_{\mathrm{c}}}^2}{2{R_{\mathrm{m}}}^2} .

It should be noted that the fracturing pressure of single-layered soil P_{\mathrm{f}}^{\mathrm{s}} can be obtained by replacing P _m with P ₀ and R _m with R ₀.

3 Validation with laboratory test

Even though it is usually expensive and time-consuming, the experiment is still an effective way to investigate the fracturing phenomenon in the soil. Many laboratory studies have been carried out on different geomaterials [ 42– 44] in the shape of cylindrical [ 12, 45], cubic [ 16, 46], and rectangular [ 47], etc. However, the fracturing pressure of double-layered soil in the laboratory condition is still underexplored. To further validate the analytical solution, several laboratory tests were carried out to investigate the fracturing pressure in cohesive soil. Additionally, to demonstrate the difference of double-layered soil, the test of single-layered was also added in this research.

3.1 Soil sample

Among all laboratory tests, most of the cases were conducted using axisymmetric specimens under axisymmetric stress. In this study, test samples were artificially prepared cohesive soil composed of kaolin, gypsum, silty clay, and water. They were mixed, and the proportion of water volume was adjusted to achieve different unconfined compressive strengths. All samples were cut down in the designed size and were consolidated in the mold. The size of the double/single-layered sample is illustrated in Fig. 2.

The external radius and height of the sample are 90 and 190 mm, respectively. The radius of the borehole in the double-layered sample was 7.5 mm, while the radius of the internal-layer soil, which was also the radius of the borehole in the single-layered sample, was set between 7.5–90 mm. The unconfined compressive strength of the internal and external-layer soil was 77 and 183 kPa, respectively, as shown in Table 1.

3.2 Apparatus and procedure

The test of samples under the conventional triaxial loading condition cannot precisely simulate soil fracturing. Therefore, it’s necessary to develop a new triaxial fracturing apparatus that can inject liquid into the sample from the bottom of the device. The schematic diagram of the fracturing apparatus is shown in Fig. 3, which was modified from the conventional triaxial instrument. The plastic gum was packed on the top, and the bottom of the borehole assisted with a non-permeable grease layer to prevent the leakage of injection liquid. The internal pressure, confining pressure, and axial pressure can be controlled independently through the servo mechanism.

The testing procedures mainly include three stages which are described as follow.

1) The sample preparation: test materials were mixed according to the designed proportion and were poured into molds with different diameters. It should be noted that the internal and external soil were separated by a film that was removed when the sample initially condensed. Then the specimen was taken out of the mold after 4 h and placed on the pedestal of the fracturing apparatus. Grease was also used at this point of time to cover the sample’s top and bottom to prevent the leakage of injection liquid. Then, the sample was wrapped by a rubber membrane that was tightened at both ends. Finally, install the pressure chamber and prepare the servo pressure control system.

2) The sample pressurizing: the axial pressure was gradually increased to 40 kPa and then kept constant in the following test. To prevent the borehole from being broken by the confining pressure, the borehole pressure and confining pressure were applied simultaneously stepwise to 30 kPa with the same pressure rate. The specimen was consolidated under pressure until the specimen stress state was stable.

3) The sample fracturing: under the constant axial and confining pressure, the borehole pressure was increased with a pressurizing rate of 2 kPa/s until the fracturing occurred in the sample.

3.3 Results and analysis

According to the previous studies [ 41], the fracturing initiation can be identified by the rapid increase of the inflow volume accompanied by the slight decrease of borehole pressure. A typical result of the variation of borehole pressure with inflow volume is illustrated in Fig. 4.

The peak value indicated by the arrow is the fracturing pressure, and the results are listed in Table 2. The data for 2.5 in double-layered sample was missing due to the record operation error during the test. In terms of 9 in single-layered sample, it is an ideal condition that the water pressure (fracturing pressure) can be equal to the confining pressure because there is actually no soil sample that exists.

It’s deemed that the greater the ratio of borehole diameter, the easier the fracturing can occur. Thus, the fracturing pressure would increase with the decrease of the ratio of borehole diameter. Figure 5 lists the test results of single-layered samples and analytical results with a plastic radius of once and twice the borehole radius. According to the results based on experimental and theoretical analysis obtained by Mori et al. [ 48], the plastic radius when fracturing occurred was approximately equal to 1.2–1.8 times the borehole radius while the value is about 1–2 times in this paper. It’s approximately consistent, and the slight deviation of the plastic radius may be owed to the difference of the injecting material, which was water in this study and was slurry in Mori’s research. Furthermore, it’s evident that the fracturing pressure increases with the decrease of borehole radius R_m, and the plastic radius when fracturing occurred would increase correspondingly.

By fitting the test results, the relationship between the plastic radius R _c and the radius of the sample R ₀ as well as the borehole radius R _m can be expressed as follow:

(12) R_{\mathrm{c}} = {\mathrm{e}}^{\frac{R_0}{20R_{\mathrm{m}}}} R_{\mathrm{m}} .

To verify the fitted curve results from Ref. [ 48] were also used to conduct comparative analysis, as shown in Fig. 6. It can be seen that all test values are near the curve, indicating a high curve fit.

In terms of double-layered samples, the test results and analytical results with a plastic radius of 1.2–1.8 times the borehole radius are illustrated in Fig. 7. It can be found that with the rise of the internal-layer radius, the fracturing pressure measured in tests of which the law of change is basically in line with the analytical solution, which drops rapidly at first and then gradually becomes steady. Furthermore, the experimental value is closer to the theoretical curve with a larger plastic radius when the radius of the internal-layer soil is relatively smaller. In general, the experimental value is relatively consistent with the theoretical value when the plastic radius is 1.5 times the borehole radius.

4 Discussion

4.1 Difference between double and single layer

The dashed curve in Fig. 8 is the analytical results of the single-layered sample by combining Eq. (12) in the calculation formula, while the solid curve is the analytical results of the double-layered sample with a plastic radius of 1.5 times the borehole radius. It can be found that the theoretical calculation results of both single and double-layered samples are in good agreement with the laboratory values. Additionally, the distinction between double and single-layered samples is also apparent: when the ratio of R_m to R₀ is approximately smaller than 0.6, the fracturing pressure of single-layered sample is greater than that of double-layered sample, which means the existence of the internal-layer soil would reduce the fracturing pressure of the specimen. However, when that ratio is greater than 0.6, the single-layered samples tend to become thin-walled, of which the fracturing pressure is no longer more than double layered samples and would eventually reduce to the confining pressure.

To further verify the accuracy of theoretical analysis, the comparative analysis of laboratory and analytical results were carried out, as shown in Table 3. It should be noted that the test value in the table was the average value in the experiment. As it can be seen, the maximum error between the analytical solution and the experimental value is 13.5% and 13.9% in double-layered and single-layered samples, respectively, an indication that the analytical method can accurately predict fracturing pressure.

4.2 Parametric analysis

For double-layered specimens, the relative size of the diameter and strength of the internal and external-layers has a significant influence on the fracturing pressure. Thus, the variation of fracturing pressure with confining pressure, R _m/ R _0, and E _m/ E ₀ is discussed in the following.

1) Confining pressure

Take the condition of the laboratory sample as an example; the borehole diameter is 7.5 mm, the unconfined compressive strengths of the internal and external-layer soil are 77, 183 kPa, respectively. The Poisson’s ratio of internal and external-layer soil is identical. The plastic radius is 1.5 times the borehole pressure in the double-layered sample, while in single-layered, it fits the Eq. (12). The variation of fracturing pressure with confining pressure is depicted in Fig. 9. It shows an apparent linear relationship between the fracturing pressure and confining pressure which harmonizes with what was suggested in Refs. [ 12, 41]. Moreover, the dashed and solid lines are always parallel, which means the relative magnitude of the fracturing pressure in the single and the double-layered samples is not affected by the confining pressure.

2) R_{\mathrm{m}} / R_0

For the single-layered sample, R _m/ R ₀ represents the ratio of borehole diameter, while in the double-layered sample, it means the ratio of layer diameter. Similarly, take the condition of the laboratory sample as an example, the borehole diameter is 7.5 mm, and the plastic radius is 1.5 times the borehole, which means R _m should start from 1.125 mm ( R _m/ R ₀ starts from 0.125). The confining pressure and the unconfined compressive strengths of the internal-layer soil are 30 and 77 kPa, respectively. The Poisson’s ratio of internal and external-layer soil is identical, and the plastic radius in the single-layered fits the Eq. (12).

As presented in Fig. 10, it is apparent that when the strength of external-layer soil is identical with that of double-layered soil, which represented by the black line, with the increase of R_m/ R₀, the fracturing pressure remains the same in double-layered while in single-layered it gradually reduces to the confining pressure 30 kPa. Then, when the external-layer soil is weaker than the internal-layer soil, the fracturing pressure of single-layered samples is always lower than that of double-layered samples, which means the existence of the internal-layer soil has a strengthening effect that would strengthen the fracturing pressure regardless of the change in the ratio of layer diameter.

However, when the internal-layer soil is weaker than the external-layer soil, the law of change has become more complicated, so that it needs to be discussed separately. When R _m/ R ₀ is less than a specific value, the fracturing pressure of the sample with internal-layer soil is lower than that without internal-layer soil, indicating that the existence of the internal-layer soil has a weakening effect that would reduce the fracturing pressure of the sample. When R _m/ R ₀ is greater than that specific value, the single-layered samples tend to become thin-walled, resulting in a smaller fracturing pressure than double-layered samples. Furthermore, the specific value where the fracturing pressure of the double-layered sample begins to be greater than that of the single-layered sample would rise with the increase of the ratio of elastic modulus. In other words, the more significant the difference in strength between them, the greater the weakening effect of the existence of internal soil on the fracturing pressure of the sample.

3) The ratio of elastic modulus ( E_0 / E_{\mathrm{i}})

Take the same condition of section (2), the effect of a distinct change in the ratio of elastic modulus on the fracturing pressure is significant when the ratio of layer diameter is no more than 0.5, as shown in Fig. 11. However, when the ratio of layer diameter is greater than 0.5, which means the proportion of internal-layer soil in the sample gradually increases, the elastic modulus ratio has little effect on the fracturing pressure in the double-layered sample. Furthermore, it can be seen that with the increase of the ratio of elastic modulus, the fracturing pressure of the double-layered sample increases nonlinearly, which is sharply in the initial stage and then gradually slow down.

4.3 Comments on the analytical solution

In the proposed solution, there is an assumption that the internal-layer soil is in the elastoplastic stage while the external stage was in the elastic stage when fracturing occurred. Thus, an important premise can be inferred that the pressure P_m at the interface of the two layers should always be smaller than the elastic limiting pressure of external soil. Take the laboratory condition as an example; as illustrated in Fig. 12, the dashed black curve is the elastic limiting pressure of the external-layer soil varies with the ratio of R_m to R₀ while the solid curve is P_m under different plastic radii.

As it can be seen, when the plastic radius is twice the borehole radius which equals 1.5 cm, if the radius of internal-layer soil R _m is smaller than 1.5 cm ( R ₀/ R _m is larger than 6), the pressure P _m at the interface of the two layers becomes greater than the elastic limiting pressure P _y of the external-layer soil. This means the external-layer soil has entered the plastic stage, and the assumption of the formula is no longer valid. Similarly, when the plastic radius is 1.5 times the borehole radius which equals 1.125 cm, if the radius of internal-layer soil R _m is smaller than 1.125 cm ( R ₀/ R _m is larger than 8), the elastic limit pressure P _y is no longer greater than the pressure P _m at the interface. Therefore, it can be concluded that the derived formula is applicable for the condition that the radius of the internal-layer soil is not less than the plastic radius that when fracturing occurred. Nevertheless, if the radius of internal soil is too small and even smaller than the plastic radius at which fracturing occurs, it can be considered as a single-layered sample. Thus, the proposed method is suitable for most double-layered situations.

Keep the radius of internal-layer soil always smaller than the plastic radius that when fracturing occurred, it’s evident that no matter how the radius of internal-layer soil changes, the pressure P_m is always smaller than the elastic limiting pressure of the external-layer soil, as demonstrated in Fig. 13.

The relative magnitude of P_m and P_y with the variation of the ratio of elastic modulus is shown in Fig. 14. When the radius of the internal-layer soil is not small enough to approach the plastic radius when fracturing occurs, the pressure P_m at the interface of the two layers is always smaller than the elastic limiting pressure of external soil regardless of the change of elastic modulus. However, when the radius of the internal-layer soil is exactly equal to the plastic radius when fracturing occurs, the two pressures are very close if the distinction in strength between the two layers is not great. Nevertheless, in this case, where one of them has a small proportion, and the strength of both layers is similar, it is easier to simplify it into a single-layered sample. Therefore, the pressure P_m at the interface is basically smaller than the elastic limiting pressure of the external-layer soil with elastic modulus variation.

In conclusion, the results indicated that the cavity expansion theory could be effectively used to predict fracturing pressure in two concentric regions of different soils. In some scenarios, the concentric assumption may prove suitable for applying this method in layered soils [ 36]. Therefore, the proposed solution is potentially useful for geotechnical problems involving aspects of soil layering, such as the fracturing phenomenon during shield tunneling and the fracturing grouting in a composite clay formation.

5 Conclusions

An analytical solution for fracturing pressure in two different concentric regions of soil was presented and validated against a laboratory experiment where triaxial hydraulic fracturing tests of concentric double and single layer cylinder samples with different radius were carried out. Then, based on the derived formulas, the effect of variation of radius and strength of both soils on fracturing pressure was illustrated, and the use of the proposed solution was discussed. The obtained results highlighted the capability of the analytical solution, and the following conclusions can be drawn.

1) The good agreement between theoretical and experimental results indicates the remarkable accuracy of the analytical solution based on the cavity expansion method. Consistent with previous research, there is a linear relationship between the fracturing pressure and confining pressure.

2) When the internal-layer soil is softer than the external-layer soil, the presence of internal soil on the fracturing pressure approximately brings the weakening effect, and the greater strength distinction between the two layers, the greater the weakening effect. When the internal-layer soil is harder than the external-layer soil, the existence of the internal-layer soil has a strengthening effect on the fracturing pressure regardless of the proportion of internal-layer soil.

3) The influence of strength distinction between the two layers on the fracturing pressure is significant when the proportion of internal-layer soil is less than half, while it’s limited when the proportion is more than half. The fracturing pressure of the double-layered sample increases nonlinearly with the increase of the ratio of elastic modulus.

4) The proposed solution is applicable for the condition that the radius of the internal-layer soil is not less than the plastic radius that when fracturing occurred and is potentially useful for various geotechnical problems involving aspects of soil layering.

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