PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering

Stanford University, Stanford, California, February 24-26, 2014

SGP-TR-202

1

A Heat Extraction Prediction for Multiple Fractures in a Closed-Loop Circulation Enhanced

Geothermal System

Bisheng Wu1, Andrew P. Bunger

2, Xi Zhang

1, Robert G. Jeffrey

1 and Cameron Huddlestone-Holmes

3

1CSIRO Earth Science and Resource Engineering, 71 Normanby Road, Clayton, VIC 3168, Australia

2Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA, USA,

3CSIRO Earth Sciences and Resource Engineering, 1 Technology Court, Pullenvale, QLD 4069, Australia.

Bisheng.wu@csiro.au

Keywords: Optimization, Enhanced geothermal reservoir, Multiple parallel fractures, Dipole wells, Semi-analytical solutions

ABSTRACT

In this paper, we present a semi-analytical model to calculate fluid temperature when it is circulated in an injection-production loop

through multiple fractures. Maximizing the output heat and operational time of an enhanced geothermal system (EGS) is always of

great importance. We propose the use of multiple fractures to provide connectivity among wells and thus target a sufficiently large

volume of rock. Although the temperature of each fracture varies with depth, the duration for higher temperature output can be

prolonged by using a number of fractures rather than a single-fracture reservoir. In addition, considering temperature changes

within the closed-loop circulation system consisting of injection wells, fracture reservoir, and production wells can increase the

applicability of the model. In particular, the model deals with heat advection and the heat exchange between the host rock and the

fluid in both wells and in the fractures. The temperature continuity between the well and the fracture is satisfied. By using potential

flow theory, the computation of temperature in the fracture plane is simplified significantly to be dependent on one variable. Also

heat conduction in the host rock is treated as one dimensional. These simplifications lead to a set of ordinary differential equations

(ODEs) in Laplace space, which can be solved analytically, and Newton’s iteration scheme is implemented to efficiently obtain the

solutions in physical space of temperature changes for the EGS system. Numerical results confirm that thermal energy extraction

rate and the lifetime of fracture reservoirs can be optimized. The results also demonstrate that the effect of the fluid viscosity on the

output temperature of the EGS can be ignored. Parametric studies are performed to identify the controlling parameters to provide

guidance for the operation of an EGS circulation test.

1. INTRODUCTION

Mathematical modeling on thermal energy recovery from hydraulic or natural fractures has been studied extensively. The models

can be classified into three types based on the dimensions of the treatment of heat transfer in the fracture. The simplest model is

based on one-dimensional (1-D) heat diffusion in the fracture and it mainly contains two cases. The first one is that the fluid flow is

uni-directional in the fracture plane. In this case, the fluid velocity is a constant, and this approach was used by Lauwerier (1955),

Bödvarsson (1969) and Gringarten et al. (1975). The second one is that the fluid flow is radial, for example the fluid is injected into

or pumped out from a single well. Due to the symmetry, the fracture can be regarded as a straight line and the variable left is the

radial distance. The fluid velocity can be obtained by considering or not considering the effect of the wellbore size. When the

boundary conditions (such as the temperature and injection rate) are specified at the wall not at the centre of the wellbore, the flow-

rate singularity is avoided (Yang et al., 2008). On the other hand, the flow velocity will have a singularity of 1/r theoretically at

both the injection and pumping locations if they are regarded as source or sink points.

When it comes to the so called 2.5 dimensional model (Kolditz, 1995), which involves 2-D heat transfer within fractures and 1-D

heat transfer within the adjacent rocks, it becomes more difficult to find the analytical solutions due to the complicated coupling of

the fluid flow and heat transfer. There are mainly two ways to obtain the fluid velocity analytically for the cases with multiple

wells, i.e. one method involves calculating the pressure by using the Green’s functions (Gringarten and Ramey, 1973) and the other

involves calculating the velocity potential and stream functions by using the source solution of 2-D Laplace equation (Dacostat and

Bennett, 1960; Grove et al., 1970) if the fluid flow reaches a steady state. At large time, these two methods provide the same

answers. Although the first method can obtain good results for the fluid pressure and thus the fluid velocity, the governing equation

is still two dimensional. To solve it, one has to resort to numerical methods.

To reduce the problem dimension, Muskat (1937) proposed an orthogonal set of curvilinear coordinates when there is a permanent

set of equipotential and streamline surfaces such as exist in potential flow. By using this transformation, the number of variables

can be reduced by one. Gringarten & Sauty (1975) were first to apply this concept to solve the dipole well problem of fluid flow

and heat diffusion in an infinite fracture by using a velocity potential and stream functions based on the results of Dacostat and

Bennett (1960). In the Gringarten & Sauty model (1975), the flow path can be treated as independent channels leaving a particular

injection well, in which temperature can be solved analytically. Later Rodemann (1979), Schulz (1987), Heuer et al. (1991) and

Ogino et al. (1999) extended Gringarten & Sauty’s (1975) approach to the problems related to heat extraction from a finite fracture

or multiple fractures with a recharge-discharge well pair and obtained large-time analytical solutions.

Even though significant progress has been made, most studies do not consider the effect of the geothermal gradient on the

temperature evolution and also consider only one single fracture. The multiple fracture geometry is simplified by the spatial

periodicity so that only one fracture is considered (Gringarten et al., 1975; Bödvarsson et al., 1982; Heuer et al., 1991). The

existence of geothermal gradients becomes more important in the presence of multiple fractures as each fracture may experience

different heat sources.

Wu et al.

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Apart from calculating the geothermal gradient, we also employ Muskat’s method to calculate the temperature changes in the

fracture plane. Instead of one single fracture, we will use it for multiple fractures. In addition, a scaling is used to help determine

the controlling parameters.

2. PROBLEM FORMULATION

The geometry for the present model is shown in Fig. 1. Initially, there exist N (N≥1) evenly-spaced parallel penny-shaped fractures,

which divide the half space into N+1 layers, and the whole system (fluid and rock formation) is in an equilibrium state with the

temperature distribution T=A0z+B0, where A0 is the geothermal gradient, B0 is the surface soil temperature and z is the depth. The

surface temperature is fixed with a value of B0 for all time. When time t>0, a cold fluid with a constant injection rate Q and constant

temperature Tin is injected from the injection points and this same rate is pumped out from the production wells. The axis of the

injection well coincides with z axis of the coordinate system and the production well is located at the position (xo, yo). The initial

velocity potential for flow in a fractured reservoir is denoted as 0

where ℓ denotes the ℓth fracture counted from the bottom and.

ρw is the fluid density.

The geometry and size of the system are defined as follows: the well and reservoir radii are rw and re, respectively; the depth of the

bottom fracture is H and the spatial distance between the fracture reservoirs is d; and the width (aperture) of the ℓth fracture is

denoted by ωℓ.

Some assumptions are made for the present model: (1) the fluid is incompressible with Newtonian rheology and the rock is

impermeable, homogenous and isotropic; (2)the stress-induced change of the fracture aperture is ignored; (3) the mechanical

properties (density, specific heat capacity and thermal conductivity) of the rock and the fluid are constant and independent of the

temperature; and (4) after a long time, the injection temperature for each fracture is the same and the injection rate for each fracture

is equal to the total injection rate Q divided by the number of fractures N.

2.1 Pressure diffusion in planar Fractures

As the fluid diffusion through the fractures is much quicker than the heat diffusion in the reservoir, fluid flow can be assumed to be

at steady state. This means that the fluid pressure or velocity potential is only a function of the coordinates, i.e. x and y. Generally,

the fracture aperture is very mall (less than 1 mm), the governing equations for the fluid flow obeys Darcy’s law

, = 1f f

N q , (1)

where f

q denotes the discharge flowing through the ℓth fracture per unit length, f

is the velocity potential function for the fluid

in the ℓth fracture , and ∇ is the gradient operator.

In the present model, only one injection well and one production well are considered. The continuity equation for the flow of an

incompressible Newtonian fluid is expressed as

, ,f i i i o o o

Q x x y y Q x x y y q, (2)

where ∇∙ is the divergence operator and δ denotes the Dirac delta function, i

Q is the injection rate at point ( , )i i

x y in the th

fracture and o

Q is the extraction rate at point ( , )o o

x y in the th fracture. For the purpose of simplicity, the locations of the

injection ( , )i i

x y and extraction ( , )o o

x y points are assumed to be at (-re, 0) and (re, 0), respectively. The origin of the coordinate

system is set to be in the middle of the line segment connecting the centers of the two wells on the ground surface.

The outlet temperature for each fracture can be different than that from its neighbours. In the present model, the outlet temperature

TAOT for the whole reservoir system is then given by the average value of the outlet temperatures of all the fractures, i.e.

A O T o u t

1

1,

N

T TN

(3)

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where o u t

T is the output temperature of the ℓth fracture.

2.2 Heat transport in closed fractures

According to Cheng et al. (2001), the heat transport in the fractures contains four parts, i.e. heat storage, advection, dispersion and

conduction from the fracture walls, and can be expressed as

1

2

00 a t ( 1)

f r r

w w w w f f w w L f r

T T Tc c T c D T z z H d

t z z

q , (4)

where ∇2 is the Laplace operator, f

T is the fluid temperature along the ℓth fracture, ρw and cw

are the mass density and specific

heat, respectively, of the fluid, DL is the fluid dispersion coefficient and λr is the thermal conductivity of the rock.

The heat diffusion equation for rock is

2 r

r r

TT

t

, (5)

where κr=λr/ρrcr, ρr and cr are the mass density and specific heat capacity, respectively, of the rock formation.

2.3 Initial and boundary condition

The initial and boundary conditions for the whole system are listed as follows

0 0

0

1

0

0

1

0

, o n = 0 ,

, o n = 0 ,

o n ,

, a t ( , ) ,

= , a t 0 .

f r

f

r r f

f in e

N

r

T T A z B t

t

T T T z z

T T r z

T B z

(6)

3. FINAL FORM OF GOVERNING EQUATIONS

To simplify the above equations and initial and boundary conditions, it is more convenient to have a scaling with fixed

characteristic lengths. Letting

0

0

2

00

0 0

0 2

, ,

22 ,

, , ,

,

=

, , , .2

w r e

e

fr

r f e e

r

f f

e w w

f f

T BT Bx r X y r Y

z Hz HZ Z

d d

r N rtQ d

Q N

A H A H

r r H cd

N

Q d

q Q

(7)

The fluid flow problem is uncoupled from the thermal problem. As the injection rates for each fracture are assumed to be equal, the

fluid flow displays the same behavour and thus the superscript for the variables, such as discharge vector, velocity potential and

streamline function, is omitted

,

2 ( 1,Q ) )2 .( 1,X Y f

X Y X Y (8)

If desired, the potential field can be recovered up to an additive constant using

,

Q .f X Y

(9)

As for the thermal problem we have

,

2

1

0

2, o n 1 ( = 1 ,2 , + 1 ) ,

· o n ( = 1 ,2 , ) ,Q 0 ,

e

X Y

r r

r r

f f

tN

rZ

Z Z NZ Z

(10)

with the corresponding boundary and initial conditions

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2 2 2

0

0

1

0

0

, o n ( 1 )

o n ( 1)

1) , ( ,2 ,

, 1 ) ,

1

10 ,

| ( ,2 ,

,

a t .

o n 0 ,r

N

in

f i

r

r

n

f

X Y

Z

Z

Z

T BN

A H

N

Z

(11)

4. SOLUTION METHOD

The solution starts with the points on the injection well and tracks along points on the streamline function Ψ. By decreasing the

fluid velocity potential Φ, as the streamline function is constant, the coordinates for the new point are obtained. Similarly, we

obtain the information of the next point along the same streamline until it reaches the production well. In the same way, we obtain

the temperature of all the points on the streamlines emanating from the injection well. As each streamline which flows into the

production well has a different temperature at the production well, the average temperature of all the points on the production well

is regarded as the output temperature of a specific well. The Stehfest method (1970) is used to convert the results from Laplace

domain back into the time domain.

5. METHOLOGY, VERIFICATION AND RESULTS

For verification, a comparison is made between the current method and the analytical solution for the case N=1. Figures 2 (a) and (b) display the fluid temperature along the streamline connecting the two wells and that to the left of the injection well, respectively. It can be seen that the results from the current approach, the analytical solution and the numerical Stehfest method (1970) are in excellent agreement.

Table 1 Parameters for the calculations

Parameter Value Parameter Value

Wellbore radius rw (m) 0.1 Fracture aperture w (m) 0.001

Well separation re (m) 500 Depth of bottom fracture H (m) 4500

Granite thermal conductivity λs (W m-1K-1) 2.4 Injection temperature ( oC) 90

Granite mass density ( Kg/m3) 2700 Geothermal gradient A0 (oC/m) 0.047

Granite specific heat cs (J kg- K-1) 790 Surface temperature B0 ( oC) 27

Fluid thermal conductivity λs (W m-1K-1) 0.6 Total flow rate Q (m3/s) 0.1

Fluid mass density ( Kg/m3) 1000 Fracture number N varies

Fluid specific heat cs (J kg- K-1) 4200 Fracture spacing d (m) varies

Figure 3 shows the temperature evolution along the shortest streamline in the first fracture when d=5m and N=4. It should be noted that the temperature at X=1 shown in the figure does not equal the output temperature of the fracture, since fluid temperature along one streamline may be different than others. As there are four fractures, the injection rate for each fracture is Qin/N=0.025 m3/s. From Figure 3, the output temperature after one year of circulation is equal to the initial formation temperature. As it takes a longer time for the fluid along other streamlines to reach the production well, it is expected that the output temperature is the same for all streamlines. With time increasing, the cooling front propagates towards the production well and eventually this leads to a decrease in output temperature.

(a) (b)

Figure 2: Comparison of fluid temperature along the (a) streamline connecting the two wells and (b) the streamline that

emanates from the leftmost point of the injection well when N=1. Blue circle denotes injection well and red circle

denotes production well.

X

-1.0 -0.5 0.0 0.5 1.0

0.26

0.28

0.30

0.32

0.34

0.36

0.38

0.40

Present

Stehfest

Analytical

X

-3.0 -2.5 -2.0 -1.5 -1.0

0.4

0.6

0.8

1.0

Present

Stehfest

Analytical

Wu et al.

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Figure 3: Fluid temperature along the streamline connecting the two wells in a straight line when d=5 m and N=4.

Figure 4 plots the output temperature of each fracture at large and small fracture spacings when N=4. When the fracture spacing is large enough, for example d=80 m as show Figure 4(a), the four fractures are independent of one another. In this case, the reservoir can be regarded as one which is made of four independent fractures with the flow rate Qin/4=0.025 m3/s for each fracture. However, when the fracture spacing is d=5m as shown Figure 4(b), the output temperature shows a faster decrease in time. Although the initial temperatures of all the four fracture in the latter case have a higher initial temperature, due to the effect of mutual interaction between fracture, the thinner layer cannot maintain the higher heat flux to the outlet. Therefore, appropriate fracture spacing is needed to maximize the extraction duration with higher output temperature.

(a)

2D Graph 1

X-1.0 -0.5 0.0 0.5 1.0

f

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.00 years

4.00 years

9.00 years

14.0 years

19.0 years

24.0 years

30.0 years

Time (years)5 10 15 20 25 30

Te

mp

era

ture

0.75

0.80

0.85

0.90

0.95

1.00

1st fracture

2nd fracture

3rd fracture

4th fracture

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(b)

Figure 4: Output temperature of each fracture when (a) N=4, d=80 m and (b) N=4, d=5 m.

The contours for the fluid temperature on each fracture plane when N=4 and d=5m are shown in Figure 5. The blue circle denotes

the injection well and the red circle denotes the production well. It must be noted that the contour plots are in the XY plane, but

they are arranged vertically for the purpose of presentation. Due to the symmetry of contours with respect to the line connecting

injection and production wells, only half of the fracture plane is considered.. It can be seen clearly that the temperature around the

production well decreases with time.

Now we try to answer the question if there is an optimal fracture number and spacing to maximize the lifetime of EGS. One aim of

the EGS design is to make the system work as long as possible with higher output temprature. Figure 6 shows the dependence of

the higher output temperature ratio over the initial formation temperature on the number of fracture and the fracture spacing. When

around 10 years of lifetime is required, there are many choices for d and N to maintain the output temperature ratio over 0.95, as

shown in Figure 6(a). In this case, 4 fractures with fracture spacing 80m would be the best choice as the cost increases with

generating more fracture reservoirs. But more fractures may be needed to obtain adequate flow between wells for economic power

generation. When the expected operation lifetime is 20 years, both N=6, d=50 and N=10, d=30m work. If we intend the EGS to

work for more than 30 years, we have to reduce our expectation on the temperatiure ratio to choose a reasonable fracture number

and spacing, as shown in Figure 6(c). The optimal value for the number of fracture and fracture spacing should be N=7 and d=100

m.

Time (years)5 10 15 20 25 30

Te

mp

era

ture

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1st fracture

2nd fracture

3rd fracture

4th fracture

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(a)

(b)

(c)

Figure 5: Contours for the fluid temperature when N=4 and d=5 m. Blue circle denotes the injection well and red circle the

production well.

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(a)

(b)

(c)

Figure 6: Relationship between the fracture number and fracture spacing.

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6. CONCLUSION

This paper considers the heat extraction efficiency from multiple parallel fracture reservoirs based on a semi-analytical method. By

using the velocity potential and streamline functions, the heat balance equation for fluid flow in a fracture is converted into a

system of ODEs, which are solved by using fourth-order Runge-Kutta method. The following brief conclusions are obtained:

(1) More accurate results are obtained by the proposed model, which provide a benchmark for numerical methods; (2) The reservoir’s lifetime can be significantly increased when multiple fractures are created, compared to a single

fracture reservoir; (3) There exist optimal fracture number and fracture spacing to meet the designed reservoir lifetime.

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