A Design Nomogram for a Horizontally-Fractured Geothermal Reservoir to determine the Production Temperature

T.C. Ekneligoda Abstract: The paper presents a graphical technique for the rapid analysis of the complex, nonlinear equation that describes the variation of the production temperature of a horizontally-fractured geothermal well. The nomogram presented for the evaluation of the production temperature incorporates the mass flow rate, fracture width, fracture length, number of conductive fractures, host rock temperature, and the production time. The range of the fracture length considered in the nomogram varies from 100 m to 1000 m whereas host rock temperature can be selected from a range of 100 oC to 260 oC. The production temperature predicted by the nomogram agrees well with the analytical solution. The most attractive feature of the proposed technique is its extreme simplicity and speed of operation.


Introduction
Among the renewable energy options, geothermal energy has the potential to contribute significantly towards alleviating the world's energy-climate predicament by providing baseload power (Tester et al., 2007). In order for geothermal power generation to become a major energy source, the development in non-volcanic are as becomes essential (IEA, 2011). The Enhanced Geothermal Systems(EGS) is a universal technology( Figure 1) that can be applied in non-volcanic areas, and EGS circulates the water between the injection and the production wells, extracting the stored geothermal energy in the reservoir at depths of 3-5 km. Forced fluid flow occurs both in natural and artificial fracture systems, and heat is extracted by convective heat transfer. One of the main challenges is ensuring that the reservoir has sufficient permeability for the process to function appropriately, so hydraulic stimulation is used initially to enhance the permeability of the host rock.
Both analytical and numerical studies have been conducted to determine the coupled thermal, mechanical, and hydraulic behavior of fractured systems (Tsang,1991). Although numerical models offer the advantage of enabling the consideration of complex geometry and boundary conditions, analytical studies are important because they provide a basis for the use of numerical models. Analytical solutions in geothermal engineering incorporate the mechanism of forced, convective heat flow in the fractures. The analytical solutions presented by Gringarten et al.(1975) were analyzed in parallel with equidistant vertical fractures. The solution can be easily converted in to horizontal fractures in a thermal isotropic media using simple arithmetic.
In fact, their equivalenttemperature approach can be incorporated to highlight the importance of the multi-fracture concept in the extraction of geothermal energy. The analytical model developed by Yang and Yeh (2011) can be used to obtain the geothermal production temperature. Their approach involved numerical inversion of the Laplace transform to determine the fluid temperature. The importance of the spacing of wells, the radii of the wells, the thickness of the reservoir, and the flow rates in a multi-well system was illustrated. A more general form of a solution for forced convection and conduction was presented by Carslaw and Jaeger(1959) and Bodvarsson(1969 In this article, we present a special graphic and nomographic solution for the problem.
Nomography (Otto, 1963) is one approach that can be used to construct graphical representations of mathematical relationships. Nomography was invented in the late nineteenth century (Evesham, 1982   PyNomo also supports more complicated equations that are in general in the determinant form, so it can produce an output for any equation that can be plotted as a nomogram.

2.
A Nomogram to determine the production temperature The nomogram for the determination of production temperatures was constructed by assuming that the thermodynamic properties of rock and water were constant. These standard thermodynamic properties are given in Table 1. The original form of Bodvarsson equation which has been derived for a host rock temperature of 0 o C is given in Eq.1.
∆ where x is the distance from the injection well, t is the time that has passed after the injection of water, T o is the initial temperature difference between the injected fluid and the rock,q is the mass flow rate, k is the thermal conductivity of the rock, is the thermal diffusivity of the rock, erfc and inverfs are the error function and the inverse of the error function. In order to simplify Eq.1, the mass flow rate term is substituted by the product of aperture − ∆ where x is the distance from the injection well,t is the time that has passed after the injection of water, −∆ is the initial temperature difference between the injected fluid and the rock, h is the fracture aperture, vi s the flow velocity, k is the thermal conductivity of the rock, is the thermal diffusivity of the rock,  f is density of the water, c w is the specific thermal capacity of water, N and W are the number of conductive fractures and fracture

Conductive Fractures
width respectively, and m is the mass flow rate. Figure 2 illustrates the different geometrical variables together with the flow rate.

Figure 2 -Illustration of different geometrical variables and the flow rate in Bodvarsson's equation
It is understood that the key variables that affect the final production temperature are the initial rock temperature, mass flow rate, fracture width, fracture length, number of conductive fractures, and the operation time, assuming that the thermal properties of the rock and the injected water remain constant. After rearranging some of the terms in Eq.(2), the final form of the equation used to construct the nomogram was formed by taking the log of Eq. (3), which produced a sum of the separate variables on the right-hand side of the equation, which is a form that is suitable for constructing a nomogram.

− ∆
The form of the nomogram was the combination of type 5 (contour plot) and type 3 (N-parallel lines) forms of nomograms, as defined by Doerfler (2009). Figure 2 shows the nomogram that was developed for determining the production temperature. As shown in Figure 3, the mass flow rate can be varied from 10 kg/s to 60 kg/s, and the fracture width from 10 m to 1000 m. The range of the fracture length goes from 100 m to 1000 m, and the range of the number of conductive fractures from 10 to 60. The production period can be set from 1 to 30 years. Five different contour lines shown in the production temperature contour curves illustrate five different rock temperatures(260, 240, 220, 200, 180 o C). Numbers that appear in the production temperature curves 200, 180,160,140,120 denote the initial temperature differences between the inlet fluid(60 o C) and the rock temperature.
The isopleth curve illustrates the use of the nomogram for one possible case to determine the production temperature. Reading the nomogram starts with the given mass flow rate (e.g., 30 kg/s) and the fracture width (e.g., 60 m).

Verification of the present work
To verify the Bodvarsson's solution, it was compared with the values obtained by the finiteelement method for a convective and conductive, numerical heat transfer model. In this simulation, we used a mass flow rate of 40 kg/s, a fracture aperture of 0.01 m, a fracture length of 600 m, and a fracture width of 100 m. From these values, we determined that the basic flow velocity was 0.04 m/s. Table 2 provides the values that we used for the remaining thermodynamic properties. The production temperatures derived by the Bodvarsson solution and by the finite-element method at different times are compared in Figure 4 and they reach a good agreement. The verification of a nomogram is a key component in its development. To verify our nomographic results, we compared them with analytical solutions provided by a MATLAB program. Several MATLAB calculations with different combinations of variables were conducted to determine the production temperatures at different times. The results of our verification calculations are summarized in Table 2. The error percentages in the production temperatures of nomographic values were computed by comparing with the production temperatures derived by the MATLAB. Some combinations of variables were outside the values predicted by the nomogram. The error percentages associated with these combinations of variables are denoted as 'out of the range' in the Table. The agreement between the MATLAB results(Eq. (2)) and the results derived from the nomogram was excellent; the maximum error was less than 1%. Similar types of error calculations were made using the same combinations of variables as in Table 1, but with different fracture lengths, i.e., 500 m, 600 m, and 1000 m. We performed the same error calculation for 1500 cases, and the maximum error was always less than 3%.
In principle, an analytical equation can be used to compute the production temperature at a given time, but solving Bodvarsson's equation is not easy without the aid of computers. So, it would also be useful if the production temperatures could be calculated using a simpler approach, such as a nomogram, without elaborate analytical or numerical calculations.
In designing geothermal wells, it would be of interest to determine the limits of different variables, such as the fracture length, fracture width, and mass flow rate, for a required production temperature. Our nomographic solution can also be used easily to determine the limits of different variables without having to solve complex equations several times. This point can be used as the limiting value for the selection of other variables. Another possible advantage of a nomogram is that it provides a simple, graphical tool that can be used to study the effects of different variables on the production temperature. Since our nomogram covers most possible ranges, the selection of different variables can be made easily by drawing a set of different lines parallel to the isopleths. This helps designers select the most appropriate and economical parameters for a given geothermal reservoir.
As shown in the validation tables, some combinations of variables lie outside the range of the nomogram, especially after 20 years of production with a flow rate of 50 kg/s. Although this may seem to limit the usefulness of the nomogram approach, this is not true in practice because such cases represent greater temperature drops after the production period, and they do not warrant further interest or investigation because their costs would be prohibitive. Therefore, as illustrated, the range of production temperatures predicted by the nomogram probably covers most cases of practical and economic interest.

Conclusions
The Bodvarsson solution, which provides the basis for the nomogram, was first verified by a forced convection/conduction numerical model. We designed and constructed a nomogram for determining the production temperature for a forced-convection/conduction, coupled system, incorporating the mass flow rate, fracture width, fracture length, number of conductive fractures, and the production time. The proposed nomogram was verified against Bodvarsson's solution using several possible combinations of variables. The nomogram developed in this study can handle a large number of possible combinations of different variables that affect the production temperature.