Research Paper
A composite element solution of convection-conduction heat transfer in fractured rock mass

https://doi.org/10.1016/j.compgeo.2021.104217Get rights and content

Abstract

A reliable and efficient numerical model is required for predictions of convection-conduction behavior in fractured rock mass to obtain a better understanding of heat transfer processes within it. This study presents a composite element formulation that addresses the problem of heat transfer via fluid flow in fractured rock mass. The three-dimensional convection-conduction equation is converted to an equivalent variational principle, and the governing equations of the rock sub-elements and fracture segments are subsequently deduced based on the composite element method (CEM). The developed CEM algorithm takes into account the heat transfer inside the fracture fluid and inside the rock matrix, respectively, as well as the heat exchange between the fracture fluid and the adjacent rock blocks. The CEM model allows for the simulation of the discontinuous characteristics of temperature across the fractures via a simplified computational mesh that is generated without restrictions, and the fractures are explicitly and automatically embedded into the mapped composite element via the CEM pre-process program. The composite element contains fracture segments exhibiting arbitrary shapes, the temperatures of fractures are interpolated from their corresponding mapped nodal temperatures of rock sub-elements delimited by the fractures, and, therefore, the temperature of the fracture is not necessarily continuous. The performance of the developed CEM algorithm is verified by comparisons with analytical solutions, a case study of lab experiments and numerical results, which demonstrate the validity and advantages of the composite element solution.

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):Tt+Tv=aT+f

where tis the time, vis the fluid velocity, f is a source term, and a is the thermal diffusivity coefficient. For a rock matrix of density ρr and heat capacity Cr, a=λr/(ρrCr), where λr 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):TT0=12erfcx-vt4at+expxva·erfcx+vt4at

where T0 is a constant initial temperature at x=0.

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)

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    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.

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