Research PaperA composite element solution of convection-conduction heat transfer in fractured rock mass
Introduction
The convection-conduction heat transfer between the fracture fluid and rock matrix is a critical phenomenon in rock engineering, such as in enhanced geothermal systems (EGS), which are one of the few renewable energy resources that can provide continuous base-load power with minimal visual and other environmental impacts (Tester et al., 2006). The calculation of convection-conduction heat transfer in fractured rock mass plays an important role in the efficiency of heat extraction, the formulation of mining strategies, and the sustainable utilization of geothermal reservoirs.
Many experiments have been conducted to investigate the heat transfer processes between the fluid flow and the rock matrix in fractured rock mass. By comparing two analytical models with different assumptions to simulate the heat transfer of granite fractures via experiments (Zhao, 1994), fluid temperature was found to increase non-linearly along the fracture surface (Zhao, 2014). A series of experimental studies were conducted with a focus on the effect of fracture geometric features and roughness on the fluid flow and heat transfer (Lu, 2012, Luo et al., 2016, Li et al., 2017, Ma et al., 2019), and the results showed that the heat transfer and heat transfer coefficients primarily depend on the fracture surface roughness. Some empirical equations for the heat transfer coefficient were obtained through experiments (Ogino et al., 1999, Wang and Horne, 2000, Zhao, 2014). Some investigations on the performance and characteristics of heat transfer during heat extraction within an EGS reservoir have also been conducted via field experiments (Richards et al., 1994, Stefansson, 1997, Sasaki, 1998, Brown, 2009). Because laboratory studies and field experiments, however necessary, are limited in space and time, numerical solutions provide a powerful tool with which to study the convection-conduction heat transfer processes in fractured rock mass.
Many researchers have explored fluid flow and heat transfer using analytical solutions (Gringarten et al., 1975, Ghassemi et al., 2003, Bai et al., 2017, Abbasi et al., 2019), and several numerical investigations on heat transfer in fractured rock mass have also been conducted. Pruess (Pruess and Narasimhan, 1985) established a practical model for heat transfer using a dual- permeability approach, which was adopted in a two-dimensional mountain-scale model (Haukwa et al., 2003); a three-dimensional mountain-scale model was ultimately developed (Buscheck et al., 2002, Wu et al., 2006). Early numerical simulation for addressing the heat transfer problem used a continuum approach (Zhao et al., 2000, Shaik et al., 2008). Saghir et al. (Saghir et al., 2001) introduced a simulation procedure for heat transfer by using the Navier-Stokes equation in the fracture flow while maintaining the Brinkman equation in the porous medium; thus, the incorrect boundary conditions for the interface between the fracture and porous medium were not required. Costa et al. (Costa et al., 2019) proposed a very high-order accurate finite volume scheme addressing heat transfer problems with arbitrary curved interfaces and imperfect thermal contacts. Vasilyeva et al. (Vasilyeva et al., 2019) developed a generalized multiscale finite element method to describe heat transfer processes by reducing the dimensions of equations. Because the direct use of discrete fractures in the continuum approach is too exhaustive to estimate the fluid flow, a new method, called the “discrete fracture network” (Shaik et al., 2011) was developed, in which only the region surrounding fractures, rather than the entire rock matrix is discretized. Based on the stochastic discrete fracture model and GEOFRAC software, Li et al. (Li et al., 2013) introduced a new approach to address the heat transfer problem, and derived a simple thermal drawdown equation for a geothermal reservoir. Lee et al. (Lee et al., 2018) proposed a numerical model in a naturally fractured geothermal reservoir based on the discrete fracture network; however, they did not consider the fluid flow and heat transfer in the rock matrix.
Numerical methods used in the study of heat transfer between the fracture flow and the rock matrix generally fall into two categories: (i) the fracture continuum model (Zhao et al., 2000, Shaik et al., 2008, Vasilyeva et al., 2019, Reeves, 2006, Yang and Yeh, 2009), and (ii) the discrete fracture model (Vasilyeva et al., 2019, Lee et al., 2018, Kodlitz, 1995, Yan et al., 2019). The continuum model treats the fractured rock mass as a continuum without considering the real fracture geometry; the property of the fracture is homogenized, and the nodal variables of the fracture are the average values of the adjacent rock matrix. This approach is easy to implement and computationally efficient, especially in large-volume geothermal reservoir problems. However, too much important information of the fracture is lost, and, in a typical enhanced geothermal system, there exists a large temperature gradient between the fluid and rock matrix (Vasilyeva et al., 2019). The discrete model exactly simulates the geometrical and mechanical properties of fractures, and it can reveal detailed characteristics of fractures. However, it treats the rock mass as an impermeable block, and it is often computationally limited for large reservoirs. The discrete approach is often used to simulate fractures by reducing the dimension of the fracture (Shaik et al., 2011).
An innovative method termed as CEM (Chen et al., 2004) combines the merits of the aforementioned approaches, and can provide a detailed description of discontinuous fractures using a simple computational mesh. The fractured rock domain is first discretized as a continuum, and, fractures, holes, and other discontinuous segments with detailed geometric information are then automatically and explicitly embedded within the mapped elements by the CEM pre-process program. In doing so, the computational mesh generation with a large number of discontinuities can be greatly simplified, and a sound topological configuration of the mesh can be guaranteed.
In this paper, the general convection-conduction equation is converted to a variational principle for systems that do not involve mixed partials via the equivalent stationary principle. A composite element solution of convection-conduction heat transfer in fractured rock mass is then developed based on the CEM (Chen et al., 2004, Chen et al., 2010, Xue et al., 2014). The objectives of the developed composite element algorithm are to investigate heat transfer processes associated with (i) fluid flow in the rock matrix and fractures, (ii) convection-conduction heat transfer in the rock matrix, (iii) convection-conduction heat transfer in the fracture fluid, and (iv) heat exchange between the fracture fluid and rock matrix. Comparative analyses with analytical solutions and two case studies of complicated fractures are performed, from which the advantages and reliability of the proposed CEM algorithm are illustrated.
Section snippets
Concept of CEM
In the CEM, the fractured rock domain is discretized into the conventional finite element (FE) mesh without restrictions of fractures. Subsequently, based on the geometry of fractures (i.e., fracture strike, dip, aperture, and spacing), the fractures are automatically embedded into the corresponding finite elements via the CEM pre-process program to generate composite element computation, where the fractures are assumed as penetrating fractures. The embedded elements are termed as composite
Mathematical Model Based on CEM
The general convection-conduction equation of transient temperature is (Guymon, 1970, Ruiz Martínez et al., 2014):
where is the time, is the fluid velocity, is a source term, and is the thermal diffusivity coefficient. For a rock matrix of density and heat capacity , , where is the thermal conductivity coefficient of the rock.
For the resulting matrices to have desirable properties for the numerical solution, Eq. (11) is converted to a variational principle
Rock Block Case: Validation through an Analytical Solution by Guymon
The most obvious one-dimensional for rock block case is to consider a rectangular region and set one convection parameter, the schematic diagram of the boundary conditions is shown in Fig. 4. The exact solution of Eq. (11) is (Guymon, 1970, Guymon et al., 1970):
where is a constant initial temperature at .
Because the numerical solution more closely approximated the exact solution as the ratio vh/a became small (Guymon, 1970), where h was the distance
Conclusions
In this paper, an innovative numerical procedure for the investigation of the convection-conduction heat transfer processes between fracture fluid and the rock matrix was presented. The three-dimensional convection-conduction equation was first converted to a variational principle for a system that avoids mixed partials by using an equivalent stationary principle. Based on the CEM, the equivalent variational principle was solved by dividing the domain of interest into an arbitrary number of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (51209097, 51508203, 51974135), the State Key Lab of Subtropical Building Science of South China University of Technology (2020ZB23), the Foundation of Guangdong Key Laboratory of Oceanic Civil Engineering (LMCE202005), and the State Key Research Development Program of China (2016YFC0600802).
References (40)
- et al.
Analytical model for heat transfer between vertical fractures in fractured geothermal reservoirs during water injection
Renew. Energ.
(2019) - et al.
Composite element method for the seepage analysis of rock masses containing fractures and drainage holes
Int. J. Rock Mech. Min.
(2010) - et al.
Very high-order accurate polygonal mesh finite volume scheme for conjugate heat transfer problems with curved interfaces and imperfect contacts
Comput. Method. Appl. M.
(2019) - et al.
Modeling thermal–hydrological response of the unsaturated zone at Yucca Mountain, Nevada, to thermal load at a potential repository
J. Contam. Hydrol.
(2003) - et al.
Heat transfer between fluid flow and fractured rocks
GRC Transactions.
(2013) - et al.
Experimental research on the convection heat transfer characteristics of distilled water in manmade smooth and rough rock fractures
Energy.
(2017) - et al.
The role of fracture surface roughness in macroscopic fluid flow and heat transfer in fractured rocks
Int. J. Rock Mech. Min.
(2016) - et al.
Experimental study of the heat transfer by water in rough fractures and the effect of fracture surface roughness on the heat transfer characteristics
Geothermics.
(2019) - et al.
Heat transfer from hot dry rock to water flowing through a circular fracture
Geothermics.
(1999) - et al.
The performance and characteristics of the experimental hot dry rock geothermal reservoir at Rosemanowes, Comwall (1985–1988)
Geothermics.
(1994)
Characterristics of microseismic events induced during hydraulic fracturing experiments at Hijiori hot dry rock geothermal energy site, Yamagata
Japan. Tectonophysics.
Numerical simulation of fluid-rock coupling heat transfer in naturally fractured geothermal system
Appl. Therm. Eng.
Geothermal reinjection experience
Geothermics.
Multiscale modeling of heat and mass transfer in fractured media for enhanced geothermal systems applications
Appl. Math. Model.
Boiling flow in a horizontal fracture
Geothermics.
A mountain-scale thermal–hydrologic model for simulating fluid flow and heat transfer in unsaturated fractured rock
J. Contam. Hydrol.
Modeling heat extraction from hot dry rock in a multi-well system
Appl. Therm. Eng.
Finite element analysis of heat transfer and mineralization in layered hydrothermal systems with upward throughflow
Comput. Method. Appl. M.
Geothermal testing and measurements of rock and rock fractures
Geothermics.
On the heat transfer coefficient between rock fracture walls and flowing fluid
Comput. Geotech.
Cited by (1)
Multiscale roughness influence on hydrodynamic heat transfer in a single fracture
2021, Computers and GeotechnicsCitation Excerpt :In addition, Chen et al. (2004) proposed an innovative treatment method for fractured rock mass, which allows a detailed description of discontinuous fractures with a simple computational mesh. On this basis, Xue et al. (2021) developed a composite element solution of convection-conduction heat transfer in fractured rock mass. These works provide an important guidance for the accurate prediction and evaluation of the hydrodynamic heat transfer process within the fractured geothermal reservoirs.