Coupled Thermo-hydro-mechanical Simulation of Hydraulic Fracturing in Deep Reservoirs Using Finite-Discrete Element Method

This verification problem deals with the transient heat exchange between the fluid-filled fracture and surrounding matrix. It considers a single horizontal fracture of aperture α = 1 mm with infinite lateral extent, separating two homogeneous, isotropic, impermeable rock blocks. The model depicted in Fig. 7a is created to numerically reproduce this 2D boundary value problem. Initially, the system has a uniform temperature, T₀ = 70 °C. A fluid of temperature T_i = 20 °C is then injected into the fracture at a constant velocity v_f = 0.01 m/s. It is also assumed that (i) the fluid flows only horizontally at the same velocity throughout the fracture at all times, (ii) the physical properties of fluid and rock do not change with temperature, and (iii) the heat conduction through the fluid is negligible. Steady-state hydraulic conditions are obtained first by applying P_A = 24 kPa (inlet) and P_B = 0 (outlet) to the left and right ends of the fracture, respectively, resulting in a flow velocity 0.01 m/s. Then, a constant fluid temperature equal to T_A = 20 °C is applied to the fluid at point A, and the advection in the fracture, conduction in the matrix, and fluid–solid heat exchange across the fracture walls were simulated. Figure 7b illustrates the model meshes. The rock and fluid properties are listed in Table 2.

The analytical solution to this problem is given by Lauwerier (1955), in which the fluid temperature distribution as a function of time and distance from the injection point is given by:while the solid temperature distribution as a function of time and position is given by:where \({T}_{s}^{i}\) is the initial temperature of the system, \({\lambda }_{s}\) is the thermal conductivity of the matrix rocks, \({\rho }_{s, } {\rho }_{f}, {C}_{s} and {C}_{f}\) are the density and heat capacities of water and matrix rock respectively, \({v}_{f}\) is fracture aperture, and erfc is the complementary error function.

Figure 7c shows the simulated temperature distribution in the solid after 46 days of fluid injection. As highlighted in the plots of Fig. 8, a good match is obtained between the numerical and analytical results of the fluid temperature along the fracture as well as the matrix temperature along a horizontal monitoring line. It is worth noting that the analytical solution assumes the fluid and matrix temperature to be equal along the fracture walls. In the numerical model, however, this condition is not explicitly enforced and a convective heat transfer model is adopted instead. A sensitivity analysis to the convective heat transfer coefficient shows that the numerical solution converges with the analytical solution with increasing values of h_conv and that for h_conv greater than 100 the difference becomes negligible.

This example verifies the thermal transport through a saturated fracture of uniform aperture due to fluid advection and conduction. The geometry and boundary conditions of the one-way TH coupled simulation problem are depicted in Fig. 9a. The length and width of the model are 1 m and 0.1 m, respectively. The steady-state flow conditions are first obtained by applying a pressure gradient to the fracture. Then, constant fluid temperatures equal to 50 °C and 20 °C are applied to the two fracture ends, respectively, while the rock block is kept in adiabatic conditions. Fluid density, dynamic viscosity, specific heat capacity and thermal conductivity are set to 1000 kg/m³, 0.001 Pa.s, 4186 J/kg/^oC, and 0.5918 W/m/^oC, respectively. The cases of advection and conduction acting in the same (Case A) and opposite (Case B) directions are considered by changing the direction of the hydraulic pressure gradient. The simulated steady-state flow velocity through the 1 m-long, 0.001 mm-wide fracture is equal to 4.2e-7 m/s.

The steady-state temperature distribution for the verification example described is given by (Ogata and Banks 1961):where T_A and T_B are the temperatures at the fracture end points, L is the length of the fracture, and Pe is the Peclet number defined as:where ρ_f, C_f and q_f are the density, specific heat capacity and thermal conductivity of the fluid, respectively; λ_f is the specific discharge of fluid through the fracture. As depicted in Fig. 9a, b, perfect match was obtained between the analytical and numerical fluid temperature profiles along the fracture. Results also show that thermal advection acts by shifting the temperature distribution in the direction of the fluid flow.

The rate of change in the fluid injection can significantly affect the HF path. This section assesses this parameter's influence on the HF's development pattern. Two in situ stress states are considered while changing the flow rate. First, an isotropic stress state of σ_H = σ_h = 6 MPa is established before the fluid injection. In the second scenario, an anisotropic stress state of σ_H = 15 MPa and σ_h = 5 MPa is considered. The flow rates are increased from 1 to 2 and 3 kg/s with a constant fluid kinematic viscosity of 1e-6 m²/s. DFNs with low fracture density have no overlapping or intersection between the DFNs, while high fracture density DFNs refere to DFNs with frequent intersection.

Figure 11a illustrates the developed HFs in an isotropic confining stress state of 6 MPa under various flow rates for the models with low frequency DFNs. Under the low flow rate of 1 kg/s, long HFs propagate from the injection point with similar development lengths. However, increasing the flow rate to higher values leads to the propagation of numerous short and long HFs from the injection point. In addition to the increased number of HFs under the higher flow rates, the HFs in the models with high flow rates exhibit more crossing types, while the activation and dilation + activation types dominate the interaction types under low flow rates. Also, the higher flow rates cause higher fluid pressure around the injection point, which in turn causes many short HFs and excessive rock mass breakage.

Figure 11b shows the developed patterns of the HFs under anisotropic confining stresses for the models with low frequency DFNs. Under the low flow rate of 1 kg/s, HFs tend to develop in the direction of the maximum principal stress, while increasing the flow rate to higher values leads to the propagation of the HFs in all directions (but still preferably in the direction of the maximum principal stress). This trend indicates that the injection of fluids with higher flow rates minimises the influence of the confining in situ stresses. The dominant interaction type under low flow rates is observed to be crossing which is more under the influence of the confining stresses than the flow rate. However, the higher flow rates impose higher fluid pressure on the joints surfaces, which not only dilates them but also facilitates the HF propagation speed under higher stress concentration in the joints tips and thus causes more dilation + activation as the interaction type.

In order to better investigate the combined influence of the flow rate and DFN on the HF development pattern, models with high fracture density DFNs and the same dimensions as those in Fig. 11 are built. There are two sets of DFNs with the following mean and standard deviation values for dip (angle), length (m) and fracture density (m⁻¹), respectively; set one 30,5; 5,1; 1,0.2 and set two 120,5; 5,1; 1,0.2. All other properties including materials and fluid properties remain unchanged.

Figure 12 shows the results of the HF development pattern for these models under the flow rates of 1, 2 and 3 kg/s. For the isotopic confining stress conditions under the 1 kg/s flow rate as shown in Fig. 12a, the injected fluid creates some new HFs around the injection point, then fluid flows from the HFs into the DFNs, and finally, a uniform fluid pressure distribution occurs. An increase in the fluid flow rate to 2 kg/s shows more HFs’ generation around the injection point and exhibits nonuniform fluid pressure distribution. This behaviour is also observed in the model with a 3 kg/s flow rate but with excessive nonuniformity in the fluid pressure distribution and the generation of more HFs around the injection point. It is observed that under the higher flow rate, the excessive fragmentation around the injection point suspends the fluid flow and leads to the high fluid pressure accumulation and shorter HFs, and thus decreased efficiency. In the anisotropic confining in situ stress regime as shown in Fig. 12b, under the low 1 kg/s flow rate, new HFs develop in the direction of the maximum principal stress, fluid then flows in a small perimeter around the injection point and a few DFNs are activated. The generation of new HFs, fluid flow distance and DFN activation for fluid flow are observed to increase once the flow rate rises to 2 kg/s. However, a high flow rate of 3 kg/s shows excessive fracturing around the injection point, leading to more DFN activation and pushing the fluid to flow further into the surrounding DFNs. Moreover, a relatively large area in the direction of the maximum principal stress is observed to have fluid pressure. Therefore, the high fracture density DFNs cause more HF arrestation, but further DFN activation and thus large fluid extension.

The HF formation behaviour and pattern is largely influenced by the fracturing kinematic viscosity (μ). It is known that high fluid kinematic viscosity leads to the development of larger volume and wider fractures in the reservoirs (Nagaso et al. 2015). In a numerical study by Sarris and Papanastasiou (2015), the influence of fluid viscosity and the injection rate on a fluid-driven fracture in cohesive poroelastic and poroelastoplastic weak formations was investigated. Their result showed that under the injection of fluid with a small value, the action of the viscous fluid flow does not impose any critical effect on the fracture profiles. The fracture is wider when it is driven with highly viscous fluids. The formed aperture with the low-viscosity fluids is narrower. Also, Wrobel et al. (2021) showed that the rheological properties of fluids considerably influence the process of hydraulic fracture not only by the viscosity values but also by the range of fluid shear rates over which variation of viscosity occurs. The influence of the temperature on the fluid viscosity is not considered in this study and a constant kinematic fluid viscosity is considered in all simulations. Thermal-to-hydraulic (T → H) processes, such as the dependency of fluid physical properties (e.g., density, viscosity) on temperature, are not accounted for in the current version of the software. Moreover, the degree of the kinematic viscosity of an injected fluid into the reservoirs alters as the temperature and the pressure change. Therefore, it is of great importance to assess this factor on the behaviour of the HF propagation. In this regard, this section analyses the effect of the fluid kinematic viscosity in the models with the two mentioned DFN sets. The isotropic and anisotropic confining in situ stresses are considered. The kinematic viscosity varies from 1 to 2, 3 and 4 mm²/s. The flow rate is kept at 1 kg/s. Figure 13 illustrates the development of HFs under the kinematic viscosity of 2, 3 and 4 mm²/s for the low fracture density DFNs under both isotropic and anisotropic confining stresses. Under the isotropic confining stress condition and the 2 mm²/s fluid kinematic viscosity as shown in Fig. 13a, individual HFs are observed to propagate in all directions and crack development density around the injection point is low. Increasing the fluid kinematic viscosity to 3 mm²/s leads to severe breakage around the injection point and then the development of a large number of HFs in all directions. The case with μ = 4 mm²/s exhibits severe breakage around the injection point and the development of many Hfs from the breakage zone. Moreover, the length of the HFs under higher fluid kinematic viscosity is shorter compared with the lower fluid kinematic viscosity models. The observed HF propagation behaviour in the isotropic stress conditions is also true for the models with the anisotropic confining stresses in Fig. 13b, but with the eminent directional preference in the direction of the maximum principal stress. Furthermore, under the anisotropic confining stress conditions, the length of the HFs is observed to be longer than those in the isotropic stress state due to the combined action of the fluid flow in the new HFs in the direction of σ_H and the higher stress concentration in this direction. Therefore, under the higher fluid kinematic viscosities, the fluid pressure overshadows the high anisotropic in situ stresses effects, and the HFs propagate in all directions, reducing the HF treatment efficiency.

In the high fracture density DFN models under the isotropic confining stress condition, as shown in Fig. 14a, the low viscous fluid (2 mm²/s) flows quickly and causes rapid fluid pressure rise in the DFNs close to the borehole and also creates some new HFs, which facilitate the fluid flow through the DFNs. Under this viscosity, the fluid affected area is large. Increasing the kinematic viscosity to 3 mm²/s reduces the fluid flow speed which is marked by a smaller area under the fluid pressure raise. Once the kinematic viscosity is increased to 4 mm²/s, the fluid is unable to flow quickly through the DFNs and as seen in Fig. 14b, a tiny perimeter around the injection point is affected by the fluid flow and fluid pressure rise is limited to the vicinity DFNs. The HF patterns in the models with anisotropic confining stresses of σ_H = 15 MPa and σ_h = 5 MPa under various fluid viscosities are shown in Fig. 14b. The fluid injection with low viscosity of 2 mm²/s shows fewer new HF initiations from the injection point and the HFs preferably propagate towards the direction of σ_H and also exhibit more activation and crossing types. However, the high viscous fluids (3 mm²/s) generate more new HFs around the injection point and severe breakage occurs at the immediate vicinity of the injection. However, the high viscous fluid is unable to flow further and fluid pressure rise in the direction of σ_H is observed to be within a short distance. Therefore, low viscous fluid injection with a higher flow rate in deep hot rocks is recommended for an efficient fluid injection project.

Among the physical and mechanical properties of the DFN, the aperture plays a crucial role in the fluid path through the pre-existing fractures. It thus significantly influences the HF development pattern and efficiency. To study the effect of DFN aperture on the HF propagation pattern and extension, four numerical models with both the low and high fracture density DFN sets are built with the joint aperture varying from 0.2 to 0.3, 0.4 and 0.5 mm. The injection flow rate is fixed at 1 kg/s for all the models and the fluid kinematic viscosity is kept constant at 1 mm²/s. Once again, two confining stress scenarios are considered: isotropic and anisotropic confining stresses.

Figure 15 shows the HF pattern for the low fracture density DFNs under both isotropic and anisotropic in situ stress states. As can be seen in Fig. 15a, under isotropic confining stresses, the HFs propagate in all directions with a similar pattern; however, the length of the HFs is considerably longer in the low aperture DFN model. Under the low DFN aperture, the fluid interaction with the DFNs decreases and fluid interaction increases with the rock matrix, resulting in faster rock breakage and, thus longer HFs. However, when the aperture increases, the influence of the DFNs on the fluid path increases and the fluid pressure accumulates in the DFNs, dilates them and causes more activation, crossing and offset crossing. Thus more fluid flow and energy are required to generate longer HFs. Under anisotropic confining stresses, as shown in Fig. 15b, the variation in the DFN aperture does not impose a significant change in the HF propagation pattern, and the high anisotropic confining stresses influence the fluid diffusion.

Figure 16a illustrates the effect of the DFN aperture on the HF patterns under the isotropic confining stresses in the models with the high fracture density DFNs. As can be seen, the model with 0.2 mm aperture shows a larger perimeter with the high fluid pressure (marked with red and yellow contours) around the injection point. However, the fluid pressure rise is only observed up to a limited distance to the injection point (marked with blue contours). This behaviour is reversed with an increase in the DFN aperture. Once the DFN aperture increases, the area under the higher fluid pressure decreases and the area under the fluid flow extends further to a larger perimeter, as marked by blue counters of fluid pressure which is extended to all the DFN areas in the model with a 0.5 mm aperture. The reason is that when the aperture is small, the fluid can flow over a limited distance, which causes the fluid pressure to rise considerably inside the DFNs. Then, the fluid flow is stopped due to the small aperture and narrow channels. However, once the aperture increases, the DFNs can allow the fluid to flow easily inside the fractures over longer distances, decreasing fluid pressure rise in the vicinity of the injection point.

Figure 16b depicts the HF patterns with varying DFN apertures under the anisotropic confining stresses. In these models, similar to the models shown in Fig. 17 under the isotropic confining stresses, the high fluid pressure builds up in the areas close to the injection point and the extension of the high fluid pressure region is larger for the low aperture DFNs. However, this extension is not uniform (different from the uniform extension of fluid pressure in Fig. 16a) and extends in the direction of σ_H. Once again, the low aperture DFNs do not allow the fluid to flow over a large distance, while the DFNs with a higher aperture do allow the fluid to flow conveniently over a longer length due to the large fracture channels, which in turn avoids the high fluid pressure accumulation at the vicinity of the injection point. In Fig. 16b, it is interesting to note that under a low DFN aperture, the HFs propagate over a long distance in the direction of σ_H while the high aperture DFNs case exhibits short HFs. This is because when the aperture is small, the fluid pressure rises quickly due to the difficulty in flowing in the narrow channels, and the high fluid pressure generates HFs in the direction of σ_H with the high stress concentrating at the newly formed HFs, which further helps the HFs to grow faster. On the other hand, the DFNs with larger apertures provide wider flow channels where the fluid can flow easily and avoid high fluid pressure build-up in the areas close to the injection point.

It is worth studying the temperature field distribution in these models. Figure 17 shows the temperature fields of some of the selected DFN models in Figs. 11, 12, 13, 14, 15, 16 It can be seen that the tmeperature does not experience a significant change in the models with the DFNs. The reason is that since the matrix is impermeable and fluid is only flows through the newly formed HFs and also the DFNs, the cold fluid exchanges heat with the hot dry rock as soon as it flows into a fracture at the immidiate vicinity of the injection point and heat up to 200 °C quickly. Therefore, the temperature of the fluid raises to the rock’s temperature quickly. This trend is also demosntrated and observed in other studies in Lou et al. (2022), Wang et al. (2022), and Liu et al. 2019.

To investigate the influences of the blocks in a matrix in conglomerate reservoirs on the HF propagation, three sets of models are considered. Each model contains a single type of block. Here, three sets of blocks with various strength characteristics are simulated to investigate the effects of block strengths on the HF development pattern. The shape of the blocks is randomly built to reflect the real dimensions and shapes of a conglomerate rock mass. Table 5 presents the input data for the matrix and the blocks. It should be mentioned that the matrix is considered permeable while the blocks are considered impermeable. The permeability of the matrix is 3e-15 m². Moreover, fluid can flow through the new HF. Mechanical interfaces are defined as the areas of the model where two different material properties are (or may become) in contact during the simulation and the mechanical properties of these interfaces can be explicitly defined. By default, the weaker property values of the two materials are used as the interface properties, although this choice can be overwritten by explicit assignment.

To better assess the HF development process in such rock masses, three models, each containing a single block type (block 1, block 2 or block 3, weak to strong) are built under an isotropic confining stress state of 6 MPa with a flow rate of 1 kg/s. Figure 18 illustrates the development stages of the HFs in the three models. In the model containing blocks 1 in Fig. 18a, at the early stages of the fluid injection, some blocks at the immediate vicinity of the injection point break and the new HFs cross them. Moreover, due to the low strength of the blocks, the working fluid not only breaks the blocks but also branches into the broken blocks to form more HFs. The HFs then grow further and break most of the blocks in their propagation path and, on some occasions, break the interface between the blocks and the matrix. In the models with medium strength blocks (block 2) shown in Fig. 18b, the working fluid breaks the interface between the blocks and matrix at the vicinity of the injection point and propagates further. In the propagation path, most of the fracturing is of intergranular type with some minor transgranular breakage within a few blocks. In models with hard blocks (block 3 in Fig. 18c), excessive hydraulic fracturing takes place around the injection point because the high strength blocks do not allow the working fluid to break them. As a result, the fluid pressure rises significantly in this area and breaks the matrix. In this case, no transgranular HFs are observed; all the HFs are of the intergranular type, and the interface breakage dominates the failure.

Figure 19 shows the fluid pressure contours in the three models. As shown in Fig. 19a for the models with the low and medium strength blocks (block 1 and block 2), the fluid pressure contours exhibit a relatively uniform distribution around the injection point and the fluid pressure rise is limited to a small circumference. The medium strength blocks (block 2) arrest the fluid from flowing further and, therefore, fluid pressure rises around the injection point more than that of the block 1 case. When the blocks are hard as shown in Fig. 19a, c, very high fluid pressure accumulates at the vicinity of the injection point and the hard blocks do not allow the fluid to cross the matrix or break them. The accumulated high fluid pressure causes excessive breakage of the matrix and the formation of intergranular HFs. As the result, the energy of the fluid is consumed in this area to break the matrix excessively and thus high fluid pressure spreads over a short distance.

Figure 20 shows the contours of temperature around the injection point. For the low and medium strength blocks in Fig. 20a and b, the low-temperature fluid flows over longer distances and thermal exchange due to conduction heats up the fluid. In these models, the thermal conduction develops over a larger area because the low and medium strength blocks allow for easy fluid flow inside the blocks and matrix, and heat exchange occurs frequently. The blocks with different thermal properties to the matrix cause some thermal cracking due to shrinkage deformation of the block, but their length is short compared to the main HFs (shown in Fig. 20d). In Fig. 20c for the hard blocks, since the hard blocks confine the fluid at the area close to the injection point, the thermal conduction occurs in a limited area, and the fluid heats up in this narrow area. In this model, the accumulation of cold water around the injection point leads to severe thermal shrinkage cracking, as shown in Fig. 20d. By comparing the fluid pressure and temperature fields in Figs. 19 and 20, the heat is only exchanged in the HFs and the low fluid pressure distribution in the matrix without HF generation does not influence the heat exchange. Therefore, the strength of the blocks plays a significant role in the heat exchange and conduction, which affects the efficiency of the HF operation in hot dry rocks.

Based on the HF paths observed in Fig. 18, the following interaction types between the fluid-block-matrix are identified and presented in Fig. 21. The crossing is the dominant interaction type observed in the models with low strength blocks (Fig. 21a). When the blocks are of high strength, interface breakage governs the interaction type (Fig. 21b). Once the fluid has flown over a longer distance, a block may stop the further growth of a HF (Fig. 21c). In the models containing medium strength blocks (block 2), a propagating HF may cross the interface and break the block (Fig. 21d). In the blocks close to the injection point where the fluid pressure is high, the flowing fluid not only can cross a block but can also branch into more HFs inside the block (Fig. 21e). On some occasions, the fluid can branch within the matrix (Fig. 21f). The matrix branching is usually formed under two conditions: high flow pressure and the presence of high strength blocks, which do not permit the fluid to generate intergranular HFs. So fluid pressure rise in the matrix causes excessive branching. Lastly, the interaction between the cold fluid and hot rock causes rock shrinkage due to thermal shock, which results in the initiation of some short cracks from the matrix-block interfaces.

The effect of flow rate on the HF propagation pattern is investigated in this section. Each model (models with block 1, block 2 and block 3) is simulated under two values of injection rate: 2 and 3 kg/s. Figure 22 shows the results of the fluid injection for each model. Figure 22a represents the HF pattern and fluid pressure in the low strength blocks (block1). To better visualize the HF patterns, an enlarged view around the injection point is cropped as shown in the yellow frame in Fig. 21a. Under the 2 kg/s flow rate, the newly formed HFs propagate uniformly in all directions and mainly break the blocks, while interface breakage occurs in a few blocks. Both the intergranular and transgranular HFs contribute almost equally. The fluid pressure distributes uniformly but with a smaller perimeter and a higher flow rate because a high flow rate consumes more energy to break more blocks and thus cannot flow longer distances. Under the high flow rate of 3 kg/s, the high pressure fluid breaks the matrix around the injection point and excessive matrix branching is observed. Moreover, the transgranular HFs are seen to be the dominant cracking type with the blocks close to the injection point undergoing severe fracturing, reducing the HF treatment's efficiency. In the medium strength blocks shown in Fig. 22b, under both flow rates, severe matrix breakage occurs around the injection point and between some blocks, and matrix branching occurs quite frequently. However, under the injection rate of 2 kg/s, intergranular HFs dominate, while under the flow rate of 3 kg/s transgranular HFs govern the cracking type. In these models, the contours of fluid pressure show high pressure rise around the injection point with less development than those in the block 1 models. In the models with hard blocks in Fig. 22c, the fluid flow at the rate of 2 kg/s excessively branches the matrix around the injection point and intergranular HFs develop around the nearby blocks. The hard blocks do not allow the fluid to propagate over longer distances or break them. By increasing the flow rate to 3 kg/s, in addition to the severe matrix branching around the injection point and between the blocks, many intergranular HFs grow over longer distances, and a few transgranular HFs are observed due to the high energy fluid flow. In these grounds, the fluid pressure rises considerably high around the injection point, resulting in excessive matrix breakage and thus less fluid flow defuse.

The three models are also examined under two different values of fluid kinematic viscosity, 2 and 3 mm²/s. Figure 23 presents the simulation results. In the low strength block model (block 1) shown in Fig. 23a, with the fluid of 2 mm²/s kinematic viscosity, the newly formed HFs propagate in all directions and break the majority of the blocks in their way with the intergranular and transgranular HFs contributing equally to create the HFs. Once the kinematic viscosity increases to 3 mm²/s, severe matrix breakage and branching occur around the injection point and the blocks at the vicinity of the injection point experience crossing, branching and interface breakage. In this case, the transgranular HFs occur more frequently than the intergranular one since the high viscous fluid is arrested around the injection point and is unable to flow further with sufficient energy to break the blocks; instead break the weakest part, which is the interface. In Fig. 23b for the model with medium strength blocks, the 2 mm²/s viscous fluid heavily breaks the matrix and some blocks around the injection point, while the 3 mm²/s viscous fluid excessively breaks and branches both matrix and blocks, resulting in a severe fragmentation zone. In the case of strong blocks in Fig. 23c under the 2 mm²/s kinematic viscosity, the fluid excessively breaks the matrix around the fluid injection point and the interface breakage occurs frequently, and the HFs propagate over a short distance. In this case, a few intergranular HFs are generated while the transgranular HFs dominate with most of the nearby blocks remaining unbroken. However, the 3 mm²/s viscous fluid breaks the matrix and the interfaces severely, and interestingly, many nearby blocks are also broken. In this regard, the developed distances of the HFs are short, and the high viscous fluid is unable to flow further. Furthermore, injecting the high viscous fluid causes more cold fluid accumulation due to the slow diffusion of the fluid, resulting in more heat exchange between the fluid, matrix and blocks and thus more thermal shrinkage cracking.

It has been extensively studied and demonstrated that the in situ stresses (both direction and magnitude) have a crucial role in a hydraulic fracturing operation (Shi et al. 2022; Zhang et al. 2018). Based on the previous studies, it is evident that the HFs tend to propagate in the direction parallel to the maximum principal stress. The previous sections showed the HFs propagate in all directions with no directional preference when the confining in situ stresses are isotropic. To evaluate the effect of in situ confining stresses on the HF development behaviour in the blocky reservoirs, the three models (each containing a single type of block) are modelled under anisotropic in situ stresses of σ_H = 15 MPa and σ_h = 5 MPa. Figure 24 shows the HF pattern developed in the models with different blocks. In the models with low strength blocks (block 1) in Fig. 24a, relatively straight HFs propagate in the direction of σ_H, and intergranular HFs dominate. In the model with block 2 shown in Fig. 24b, the direction of the HFs is observed to be almost parallel to the σ_H, however, matrix branching occurs frequently in this direction due to the combined action of the fluid flow through the HFs and the high anisotropic in situ stresses. The model with block 3 in Fig. 24c shows that the high strength properties of the blocks suppress the high anisotropic in situ stresses effect, and the nearby blocks deviate the fluid to propagate in all directions with a minor tendency in the direction of σ_h, with interface breakage being the dominant HF type. Moreover, once the strength of the blocks increases, the fluid accumulation occurs more in the matrix causing high fluid pressure to rise in the area. Furthermore, this fluid accumulation facilitates better heat exchange, thus, more thermal shrinkage cracking. The contours of the fluid pressure demonstrate that block strength increase causes excessive fluid pressure around the injection point (23.8 MPa for block 3 compared to 16.8 MPa for block 1) and less fluid pressure distribution over longer distances, indicating inefficiency of the hydraulic fracturing treatment.

Figure 25 illustrates the temperature field for the blocky reservoirs. It should be mentioned that the models in Fig. 25 are the temperature fields of the models shown in Figs. 22, 23, 24 with the identical model parameters and injection time. For the models under the 2 and 3 kg/s fluid flows as shown in Fig. 25a, b, the HFs propagate over long distances in the matrix containing the low to medium strength blocks (block1 and block2), and some blocks are fragmented as well. Thus the heat exchange between the fluid inside the HFs and the matrix occurs frequently. The cold fluid far from the injection point is also heated to the reservoir temperature. When the block strength is high (block 3) in Fig. 25a–-b, the cold fluid is suppressed in the area close to the injection point due to the increased resistance of blocks against the fluid flow. Therefore, the low-temperature field is only limited to the area close to the injection point and the temperature in the long HFs is 200 °C due to the rapid thermal conduction and convective heat transfer. A similar temperature field is observed for the models with high kinematic viscous fluid, as shown in Fig. 25c, d. However, as the kinematic viscosity is a measure of the fluid’s resistance to deformation and movement, this fluid cannot flow for long distances, meaning the fluids accumulates in the limited zone around the injection point. Therefore, the fluid pressure rises significantly, and the breakage is excessive, resulting in frequent heat exchange in this area. Thus, the fluid in longer distances is a heated fluid; therefore, the low temperature is only observed in the small perimeter around the injection point. Under the high anisotropic confining stresses shown in Fig. 25e, the strength of the blocks significantly impacts the temperature field. In particular, the low to medium strength blocks allow the cold fluid to flow over a longer distance and transfer more fluid for heat exchange. However, the high strength blocks suppress the fluid flow from creating long HFs, and thus the convective heat transfer is seen to happen in a narrow area. Overall, the temperature rise in the fluid due to the conduction between the hot blocks and the fluid is evident in the high strength blocks model, and shrinkage cracking often occurs close to the injection point. In the model with block 3, the thermal exchange between the high strength blocks and strong conduction causes frequent shrinkage cracking and less high-temperature fluid distribution over longer distances. Since the strength of block 1 is low and the fluid can break them and flow over longer distances, the fluid temperature rise is mainly through the main HFs and shrinkage cracking occurs less often.