**1. Introduction**

Heat contained in the upper 10 km of the Earth's crust represents a large, accessible, lowemission energy source that can substitute for other energy sources that produce significantly more greenhouse gas. Geothermal energy has become one of the most promising energy alternatives in the future. For example, Australia has a large volume of identified high heat

© 2013 Wu et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

producing granites within 3 to 5km of the surface. In some places at 5kms the temperature is more than 250<sup>0</sup> C. One cubic kilometer of hot granite at 250<sup>0</sup> C has the stored energy equivalent of 40 million barrels of oil (Geodynamics website). Therefore, developing methods to capture this resource of clean energy will help realize the potential of commercial EGS. In addition to the physical experiments and field exploitation, theoretical studies such as mathematical modeling are important in developing ways to maximize heat extraction from geothermal reservoirs treated by hydraulic fracturing.

well is treated separately and the two-dimensional heat diffusion problem in the fracture plane is simplified to be one-dimensional. Later Rodemann [13], Schulz [14], Heuer et al. [15] and Ogino et al. [16] extended Gringarten and Sauty's [12] approach to the problems related to heat extraction from a finite fracture or multiple fractures with a recharge-discharge well pair and

An Efficient and Accurate Approach for Studying the Heat Extraction from Multiple Recharge and Discharge Wells

The third type is the complete 3D problem for the heat transfer in the fracture and the rock. In these more complicated cases, numerical schemes, such as finite element [7], Markerand-Cell method [17] and finite difference approaches [18, 19], have to be used to obtain the thermal behavior of the fluid and reservoir. Normally, finding the solution is computa‐

Besides applying the method to a particular set of boundary conditions for EGS reservoirs, the advance of the present model is in that the semi-analytical solutions for the rock and fluid temperatures are obtained, thus significantly reducing computational cost. It also provides an efficient and accurate way to further study the effect of some factors, such as the number of wells, the well spacing, the injection rates and production rates, to maximize the heat extraction

The geometry considered by the present model is shown in Fig. 1. There exist *M* (*M*≥1) vertical recharge well and *N-M* (*N*≥2) vertical discharge wells, which intercepts the fracture containing the injected fluid. The whole system (liquid and rock formation) is initially in an equilibrium state with the uniform temperature *T* 0. When time *t*>0, a cold fluid with a constant injection

(*i*=1,..*M*) and the heated fluid is pumped out at a constant production rate *Q <sup>i</sup>* (*i*=*M*+1, *M*+2...*N*),

It is assumed that after a hydraulic fracturing treatment, there is a connected flat fracture plane intercepted by all wells. The geometry of the fracture is defined by its radius In particular, the

**1.** The fluid is incompressible with Newtonian rheology and the rock is impermeable,

**3.** The material properties (density, specific heat capacity and thermal conductivity) of the

**4.** The heat condution in the fluid is neglected when the injection rate is sufficiently great.

) (*i*=*M*+1, 2...*N*).

*<sup>i</sup>* is injected from the injection points (*x* <sup>i</sup>

, *y <sup>i</sup>* )

http://dx.doi.org/10.5772/56373

931

obtained good analytical solutions.

tionally demanding.

from the EGS reservoirs.

**2. Problem description**

from the discharge wells (*x* <sup>i</sup>

fracture radius is infinite in this paper.

homogenous and isotropic;

rate *Q <sup>i</sup>* (*i*=1, 2...*M*) and constant temperature *Tin*

Some assumptions are made for the present model:

, *y <sup>i</sup>*

**2.** The stress-induced change of the fracture aperture is ignored;

rock and the fluid are constant and independent of the temperature;

Mathematical modeling of thermal problems associated with oil recovery or geothermal heat extraction from reservoirs containing hydraulic or natural fractures has been studied exten‐ sively. The models can be classified into three types based on the dimension of the heat transfer problem. The first type is based on one-dimensional (1-D) heat diffusion in the fracture and it mainly contains two cases. The first one considers 1D fluid flow in a 2D fracture. In this case, the fluid velocity is uniform and no singular point in pressure exists at the injection or production well [1-3]. The second one considers fluid flow that is radial and axisymmetric, for example the fluid is injected into a single well and produced from a ring of production wells that are all at the same radial distance from the injection well [4]. Due to the symmetry, the problem can be treated as one-dimensional based on the radial distance. The fluid velocity can be obtained either by considering the effect of the wellbore size or neglecting it. When the injection or pumping at the well is regarded as a point source or sink, the velocity will have a singularity of 1/r theoretically at the injection and production points. Fortunately, this singularity is overcome by the geometrical symmetry [5, 6] for the case with a single well.

The second type is so-called 21 /2 dimensional model [7] which results from 2-D heat transfer within fractures and 1-D heat transfer within the adjacent rocks, it becomes more difficult to find analytical solutions for these cases because of the coupling of the steady fluid flow and heat transfer. There are mainly two ways to obtain the fluid velocity analytically for the cases with single, dipole or multiple wells; one method calculates the pressure by using a Green's function [8], and the other method calculates the velocity potential and stream functions by using the source solution for the 2-D Laplace equation [9, 10]. From the perspective of modeling the long lifetime of a geothermal reservoir, the above two approaches are functionally the same. Although the first method can obtain accurate results for the fluid pressure and thus the fluid velocity, there still exists two coordinates in the governing equation and the difficulty of solving this equation is not reduced. In addition, the singularity issues make it invalid to use the boundary condition (i.e. injection temperature) when solving the equations. For this case, solutions still require the use of numerical methods.

In order to overcome the above issues including the computational difficulty and singularity at the source or sink points, Muskat [11] proposed a method in which an orthogonal set of curvilinear coordinates is introduced that correspond to the permanent set of equipotential and streamline surfaces. By using this transformation, the terms involving cross products related to the fluid diffusion and heat transfer are greatly simplified. Gringarten and Sauty [12] were the first to apply this concept to solve the dipole well problem of fluid flow and heat diffusion in an infinite fracture by using velocity potentials and stream functions based on the results of Dacosta and Bennett [9]. In their model, each channel leaving a particular injection well is treated separately and the two-dimensional heat diffusion problem in the fracture plane is simplified to be one-dimensional. Later Rodemann [13], Schulz [14], Heuer et al. [15] and Ogino et al. [16] extended Gringarten and Sauty's [12] approach to the problems related to heat extraction from a finite fracture or multiple fractures with a recharge-discharge well pair and obtained good analytical solutions.

The third type is the complete 3D problem for the heat transfer in the fracture and the rock. In these more complicated cases, numerical schemes, such as finite element [7], Markerand-Cell method [17] and finite difference approaches [18, 19], have to be used to obtain the thermal behavior of the fluid and reservoir. Normally, finding the solution is computa‐ tionally demanding.

Besides applying the method to a particular set of boundary conditions for EGS reservoirs, the advance of the present model is in that the semi-analytical solutions for the rock and fluid temperatures are obtained, thus significantly reducing computational cost. It also provides an efficient and accurate way to further study the effect of some factors, such as the number of wells, the well spacing, the injection rates and production rates, to maximize the heat extraction from the EGS reservoirs.
