A coupled thermalhydraulicmechanical (THM) model is developed to simulate the combined effect of fracture fluid flow, heat transfer from the matrix to injected fluid, and shearing dilation behaviors in a coupled fracturematrix hot volcanic reservoir system. Fluid flows in the fracture are calculated based on the cubic law. Heat transfer within the fracture involved is thermal conduction, thermal advection, and thermal dispersion; within the reservoir matrix, thermal conduction is the only mode of heat transfer. In view of the expansion of the fracture network, deformation and thermalinduced stress model are added to the matrix node’s in situ stress environment in each time step to analyze the stability of the matrix. A series of results from the coupled THM model, induced stress, and matrix stability indicate that thermalinduced aperture plays a dominant role near the injection well to enhance the conductivity of the fracture. Away from the injection well, the conductivity of the fracture is contributed by shear dilation. The induced stress has the maximum value at the injection point; the deformationinduced stress has large value with smaller affected range; on the contrary, thermalinduced stress has small value with larger affected range. Matrix stability simulation results indicate that the stability of the matrix nodes may be destroyed; this mechanism is helpful to create complex fracture networks.
With the increase of energy consumption, the exploration and development of oil and gas has gradually shifted from the conventional to unconventional reservoir, such as volcanic reservoirs [
Injectivity index during stimulations in hydrocarbon and geothermal granitic formations.
Different 
Project  Depth 
Lithology  Max. Inj. 
Inj. 
Total 
Injectivity 
Reference 

Oil and Gas  Liaohe: #1  3022  Granite 
6.4  0.0218  572  1.62  / 
Liaohe: #2  4772  5.5  0.0145  498  1.33  /  
Liaohe: #3  4188  4.4  0.0205  480  0.85  /  


Geothermal  GPK2  3600  Granite  3.6  0.0036  30000  4.62  (Baria, 2009) 
Habanero 1  4421  Granite  1.8  0.0147  20000  1.00  [  
Habanero 4  4205  Granite  3.2  0.0114  27000  16.00  [  
RRG9 ST1  1524  Granite  1.0  0.0036  /  3.03  [ 
The purpose of fracturing is to improve the conductivity of natural fractures and possibly extend or develop new fractures [
The injected water flow pattern of the volcanic reservoir is mainly controlled by the network of preexisting and induced natural fractures and is affected by changes in conductivity resulting from heat transfer, shear dilation, and chemical processes. Many detailed studies of the precipitation/dissolution processes and their impact on fracture aperture variation in fractured volcanic reservoirs can be found in the literature [
Because of the complexities of the numerical modeling, deformation and thermalinduced stress are often neglected and the matrix nodes assume stability and no secondary fractures form [
The hot and fractured volcanic oil & gas reservoir is always considered nonpermeability, and natural fractures within the matrix are the channels for fluid flow and heat exchange. Based on the simple principle of reducing the extremely complex system to an idealized one, an idealized system for integrating hot matrix and natural fracture is established to study the fracture deformation and their induced stress. The geometry of the conceptually idealized model corresponding to a fracturematrix coupled system is shown in Figure
Idealized hot matrixnatural fracture system.
It is assumed that the fracture aperture varies in space and time, and the interface of the fracture does not leakoff so fluid flows along the natural fracture. The fracture is assumed to correspond to a parallelplate system, and the laminar flow is valid. Without considering the slip effect, the velocity is integrated in the fracture cross section [
where
where
Temperature differences drive the heat transfer between the hot reservoir matrix and injected cold water. The main mechanisms involved are thermal conduction, thermal advection, and thermal dispersion within the fracture, conductionlimited thermal transport from the reservoir matrix to the fracture. Within the reservoir matrix, thermal conduction is the only mode of heat transfer [
The following general assumptions are used in this simulation for calculating the fracture’s temperature profile in each time step:
where
The rough surface of a natural fracture is always undulating. During the hydroshearing stimulation, the effective normal stress of the fracture surface will continue to decrease, and the permanent shear displacement will be generated. The interface of the natural fracture will no longer completely overlap under normal stress and the dilation aperture
Graphs of the fracture aperture caused by shear displacement and dilation.
In 1985, Barton [
where
where
where
The elastic displacement discontinuity method (DDM) is an indirect boundary element method to cope those problems involving pure elastic nonporous media containing discontinuous surfaces or thin fractures [
The local coordinate distribution after the discretion of natural fracture boundary.
The deformationinduced stress in the field by the shear and dilation displacement of
where
where the field point
The deformationinduced stress is approximated as the sum of the all the displacement fracture unit, as shown in the following:
During the hydroshearing stimulation process, a large heat transfers from the reservoir matrix into the injected cold fluid. The temperature change causes thermalinduced stress and displacement within the rock and fracture. The thermalinduced stress model under the condition of the continuous heat source is deduced by using Green’s function [
The displacement of the granite matrix can be given by the following Navier equation [
where
where
The Bodvarsson analytical solution [
Assuming that the initial temperature of the granite rock is
where
The induced stress will be generated through the deformation of single fracture, which will lead to redistribution of the stress state around the rock matrix. As a result, the rock matrix nodes and unconnected natural fractures may fail under the new stress environment, and the underground fracture network of hot rock is formed by the communication between the natural fractures.
Criterion of rock matrix failure is
Criterion of unconnected fracture failure is
where
The high temperature and fractured granite reservoir system is considered two sets of coupled partial differential equations for the natural fracture and the matrix. All the various input parameters used in the simulations are reported in Table
Parameters used in numerical calculations.
Parameters  Value 

Injected water density (kg/m^{3})  1000 
Injected water heat capacity (J/Kg/K)  4180 
Hot dry rock fracture roughness  15 
Hot dry rock Young’s modulus (Pa)  4E10 
Hot dry rock Poisson’s ratio  0.28 
Hot dry rock density (kg/m^{3})  2650 
Hot dry rock initial temperature (°C)  200 
Hot dry rock specific heat capacity (J/Kg/K)  1070 
Hot dry rock thermal conductivity(J/s/m/K)  2.60 
Fracture begin shear expansion coefficient  0.3 
The problem boundary condition of the fracture is prescribed along with the fracture direction and the rock matrix boundary is specified at the interface of the fracturematrix. The initial and boundary conditions are (
In this study, the system is solved numerically using a secondorder centraldifference finitedifference scheme. The solution is iterated in each time step to satisfy the continuity at the matrixfracture interface. The grid size in the fracture is maintained uniform whereas a nonuniform size is adopted in the rock matrix; it is to be noted that all the fracture and matrix grid size does not change with time, shown as in Figure
Discretization of the solution domain satisfying both the cellbased finite volume and the nodalbased finitedifference schemes [
The heat transfer model is used to calculate the temperature profile through the coupling of the fracture and the matrix. The pressure distribution within the fracture is updated from the thermalinduced aperture change. The fracture deformation due to the effective pressure change will be analyzed through the fracture constitutive model. The coupling between thermal, deformation, and hydroflow is iterated at each time step. The convergence criterion for heat transfer is that the temperature difference is less than 0.01K. The convergence criterion for THM coupling is that the aperture difference is less than 0.00001mm. After the THM coupling, the new stress environment of each matrix nodes is updated from the calculation of induced stress. The stability analysis of matrix nodes is also completed. The coupling between THM, induced stress, and matrix stability is illustrated in Figure
Coupled thermalhydraulicmechanical model and induced stress.
As previously described in this paper, the shear displacement and dilation behavior of a single fracture are compared with a numerical solution, as shown in Figure
Comparison between the lab test and the numerical model for shear stress and displacement.
During the verification of the heat transfer model, the temperature in both the rock matrix and natural fracture is a relative temperature which is defined as the ratio of current temperature to the initial reservoir temperature. The initial temperature of the matrix is 200°C and the injection water is 20°C. The water velocity between the inlet and outlet is maintained at a constant value of 50 m per day, and the other thermal parameters of the water/matrix are given in Table
Comparison the normalized temperature distribution between numerical model and Cheng’s solution in the fracture.
Comparison the thermalinduced aperture along the fracture between numerical model and Ghassemi’s solution.
RRG9 ST1 well is located in Cassia County, southwestern Idaho. The open hole section of the well, from 5551 to 5900 ft MD, consists of granite and minor diabase. The tested initial minimum horizontal stress is 22MPa and the maximum is 32MPa [
Hydroshearing stimulation schedule of the RRG9 ST1 well.
Hydroshearing stimulation  Injected water temperature, °C  Injection rate, m^{3}/min  Pumping pressure, Psi  Time, day 

Unstable pumping stage  40  0.160.530.99  270540810  35 
12  0.96  740520  6  


Stable pumping stage  29  0.5  270  90 


Efficient pumping stage  45  0.81.9  260  110 
At the beginning of the fracturing, high temperature (40°C) water was injected into the fracture. The permeability of the fracture was very low and nearly equal to the initial conductivity while the surface pumping pressure increases with the injection rate. After unstable pumping stage, the permeability of the natural fracture is greatly improved; the ground injection pressure is maintained at a constant value. The pumping pressure fitting shows that the simulation results are in good agreement with the well site fracturing date, as shown in Figure
Pressure fitting on RRG9 ST1 well hydroshearing simulation.
Having verified the basic coupled fracturematrix, heat transfer model, and fracture deformation equations, the total aperture change along the fracture is computed. Both models that affect the aperture change have been used in this work and the results are shown in Figure
Change in fracture total aperture along the fracture due to coupling of thermal and mechanical.
The combined influence of thermal stress and decrease in net pressure induced deformation in a fracturematrix coupled system on fracture aperture is illustrated in Figure
Change in thermalinduced aperture and dilation aperture along the fracture.
The temperature profiles along the fracture axis after three different pumping stages are illustrated in Figure
Change in temperature profile along the fracture in different pumping stage.
After the calculation of fracture total aperture and heat transfer in each time step, the deformationinduced stress and thermalinduced stress are calculated. Figure
The variation of deformationinduced stress in minimum and maximum horizontal stress direction in different pumping stages.
Minimum horizontal stress direction
Maximum horizontal stress direction
The thermalinduced stress in both minimum and maximum horizontal stress directions are negative, which means that tensile stress is formed near the injection area, as shown in Figure
The variation of thermalinduced stress in minimum and maximum horizontal stress direction in different pumping stage.
Minimum horizontal stress direction
Maximum horizontal stress direction
After the given of induced stress, the new stress state of the matrix’s nodes can be updated. The new stress state consists of three parts: initial stress, deformationinduced stress, and thermalinduced stress. The new stress state will affect the stability of the rock matrix. In other words, when the new stress state of the matrix nodes is located above the strength envelope, the stability of the matrix node will be destroyed. Figure
The relationship between the new stress state and the strength envelope of the matrix (left), the failure modes, and their position in different pumping stages (right).
A coupled THM model is developed to simulate the combined effect of fracture fluid flow, heat transfer from the matrix to injected fluid, and shearing dilation behaviors in a coupled fracturematrix hot volcanic reservoir system. In view of the expansion of the fracture network, deformation and thermalinduced stress models are added to the matrix node’s in situ stress environment in each time step to analyze the stability of the matrix.
The coupled THM modeling results indicate that, in these case conditions, the change in fracture aperture due to thermoelastic stress occurs near the injection well, thus increasing the fracture opening and playing a dominant role. Away from the injection well, the conductivity of the fracture is contributed by shear dilation. Therefore, the best way to enhance the conductivity of the underground hundreds of meters’ fracture area is the using of shear dilation.
The induced stress simulation results indicate that deformationinduced stress is a compressive stress with a positive and thermalinduced stress is tensile stress with a negative value. The induced stress has the maximum value at the injection point; the deformationinduced stress has a large value with smaller affected range; on the contrary, thermalinduced stress has a small value with larger affected range.
The new stress state of the matrix nodes is updated by initial stress, deformationinduced stress, and thermalinduced stress. The matrix stability simulation results indicate, in this case, some matrix nodes are located above the strength envelope after hydroshearing stimulation, which means the stability of the matrix node will be destroyed. This mechanism is helpful to create complex fracture networks.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The authors would like to acknowledge the support of the National Science Fund for Distinguished Young Scholars (ID: 51525404) and Foundation of China Scholarship Council (File no. 201608510077).