A novel flow-resistor network model for characterizing enhanced geothermal system heat reservoir

Abstract

The fracture characteristics of a heat reservoir are of critical importance to enhanced geothermal systems, which can be investigated by theoretical modeling. This paper presents the development of a novel flow-resistor network model to describe the hydraulic processes in heat reservoirs. The fractures in the reservoir are simplified by using flow resistors and the typically complicated fracture network of the heat reservoir is converted into a flow-resistor network with a reasonably simple pattern. For heat reservoirs with various fracture configurations, the corresponding flow-resistor networks are identical in terms of framework though the networks may have different section numbers and the flow resistors may have different values. In this paper, numerous cases of different section numbers and resistor values are calculated and the results indicate that the total number of flow resistances between the injection and production wells is primarily determined by the number of fractures in the reservoir. It is also observed that a linear dependence of the total flow resistance on the number of fractures and the relation is obtained by the best fit of the calculation results. Besides, it performs a case study dealing with the Soultz enhanced geothermal system (EGS). In addition, the fracture numbers underneath specific well systems are derived. The results provide insight on the tortuosity of the flow path between different wells.

Keywords

enhanced geothermal systems / flow-resistor network model / fracture characteristics / heat reservoir

Introduction

The engineered or enhanced geothermal systems (EGSs) aim to explore the enormous thermal energy stored in hot dry rocks (HDR) at depths of 3 to 10 km underground; they have an enormous potential of producing renewable and clean energy and have received increasing attention by the academic and industrial community working in this field across the world [¹]. An EGS is created by first drilling a well into the rock basement, forming in this way an artificial heat reservoir by hydraulic and/or chemical stimulation to produce interconnected fracture networks with high permeability, and then a production well is drilled into the reservoir [¹]. During the operation, cold water, which is injected in the first well (injection well), flows through the reservoir and absorbs heat from the basement rock; the large temperature contrast between the injected water and the rocks, which yields large induced thermal stresses in rocks, and contributes to the stimulation of the reservoir [²,³]. The heated-up water is then brought out from the production well, which may be used for different purposes, but most commonly for power-generation. A steady long-term energy production can be achieved if the hydraulic behavior of the reservoir is well stimulated and to this purpose its full understanding is required.

The fracture characteristics of the reservoir are of great importance to the economic effectiveness of the EGS project. These characteristics can be determined with reasonable approximation by analyzing the experimental data of borehole explorations like chip samples, rock fragments, spectral gamma ray, ultrasonic borehole imagery [⁴–⁸], seismic monitoring [⁹–¹²], and tracer testing [¹³–¹⁵]. The analyses of borehole data, in general, are reliable for evaluating fracture configurations in the near-regions of injection/production wells. For fracture structures in the interior of the reservoir, structures can be obtained from seismic data analyses; however, due to technical limitations, detailed information of the fracture characteristics of a reservoir tend to be somewhat limited with existing measurement techniques.

In addition to direct measurement or analyses of experimental data, numerical simulation can be utilized as an effective tool or method for fracture characterization. Several types of numerical models are classified based on the assumptions that are made to describe the heat reservoir: the regular fracture network model [¹⁶–¹⁸], the stochastic fracture network model [¹⁹–²⁴], and the equivalent porous media model [²⁵–³¹]. The regular fracture network model assumes a single planar fracture or a combination of parallel fracture arrays. It is the simplest type of reservoir model which is suitable for preliminary theoretical analyses with respect to heat exchange in the reservoir; often it lends itself to formulations with an analytical solution. However, actual EGS reservoirs frequently encounter very complex and composite structures with variable fracture widths, lengths, dips and multiple flow paths, which cannot be taken into account by regular fracture network models. Therefore, stochastic fracture network models are of greater interest. In this type of models, the Monte Carlo method is employed to determine the average characteristic parameters and then generate various fracture networks with probably different but statistically equivalent structures. Though approximations to the actual heat reservoirs already have been made in stochastic fracture network models, calculations, in general, represent a major computational effort due to the large number of variables to be solved. One possible avenue to reduce the computational effort is making viable, appropriate simplifying assumptions [²¹–²⁴].

To some extent, the heat reservoir can be treated as a porous media with a definite permeability, and hence the equivalent porous media model is frequently employed in numerical studies of EGS. Generally, the models are sub-categorized as local thermal equilibrium, and local thermal non-equilibrium depending on the method used to deal with the heat transfer process in the porous reservoir between the fluid and the rock matrix. The former assumes the rock matrix and fluid at the same temperature (This assumption greatly simplifies the modeling effort but often overestimates the heat production). To account for the actual heat exchange between the rock and fluid, the local thermal non-equilibrium model with dual [²⁶,²⁷] or multiple porosities [²⁸] considers the fact that the rock and fluid are at different temperatures and renewed interests are being paid to its application. A case in point is the recent development of a porous media model with single porosity that incorporates local thermal non-equilibrium between the rock and fluid [²⁹–³¹]. This model employs two energy equations to describe the heat transfer in the rock matrix and fluid respectively. Generally, the equivalent porous media model, either with single porosity or multi-porosities, is somewhat based on a macroscopic averaging concept with respect to the whole reservoir or a piece of reservoir volume; however, this type of averaging may be valid for only a few heat reservoirs encountered in practice.

In this paper, a flow-resistor network model is developed to simulate the hydraulic process in the HDR reservoir. Similarly to the heat-resistor approach used in heat transfer calculations, the flow-resistor approach for hydraulic analyses also originates from the electric circuit analysis. In the hydraulic process, the relation between mass flux and pressure drop can be described as Q = Dp/R, which is similar to the relation between electric current and voltage described, namely, I = U/R. Based on this generality, the electrical analogy method can be employed to investigate the hydraulic characteristics of HDR reservoirs.

Flow-resistor network model

Physical model

The structure of a single fracture of an EGS reservoir can be very complicated with numerous apertures and lengths, as depicted in Fig. 1(a). However, it can be simplified as a parallel channel consisting of several segments with differing cross-sections, as shown in Fig. 1(b).

Further, the fracture network in a heat reservoir can be described as a parallel channel network consisting of several inter-connected parallel channel segments with differing widths and lengths, as depicted in Fig. 2. To illustrate the basic concept, a two-dimensional analysis is taken without considering the gravity effect and the fracture dips; only two main flow paths are considered to facilitate analysis and discussion. Moreover, the fluid properties are assumed to be constant and the flow in the reservoir fractures is laminar.

The mass flow rate (Q) in parallel channels is described as [³²]

(1)Q={\displaystyle \frac{\mathrm{2}\rho a^{\mathrm{3}}\Delta p}{\mathrm{3}\mu L}}.

where r, m, L, a, and Dp are the density and dynamic viscosity of the fluid, the length and width of the channel, and the flow pressure drop in the channel, respectively. Based on the analogy between fluid flow and electric current, the flow resistance, which is the resistor in the flow-resistor network, is determined as

(2)R={\displaystyle \frac{\mathrm{3}\mu L}{\mathrm{2}\rho a^{\mathrm{3}}}}.

Converting each segmental channel shown in Fig. 2 to a flow resistor and including the injection and production wells, the fracture network of this heat reservoir is then transformed into the flow-resistor network exhibited schematically in Fig. 3. The nodes in Fig. 3 correspond to the intersections of the fractures with corresponding numbers assigned to them, as shown in Fig. 2.

Merging the resistors in serial connection, the flow-resistor network can be further transformed to the network schematically displayed in Fig. 4, which has a construction of M sections (M = 7 for the case presented in Fig. 3). Each section has three branches of resistors except at the end, and each resistor branch consists of one single resistor or multi-resistors in serial connection. Based on the transformation above, all the heat reservoirs regardless of the complicatedness of the fracture structure can be simplified by using the resistor network schematically displayed in Fig. 4 with identical framework but different M values and branch resistor values.

Parameter setting and calculation method

The fracture apertures and lengths of Soultz EGS obey the exponential distribution function, with fracture aperture from 5 to 50 mm and length from 10 to 100 m [⁵,³³]. The probability density functions for fracture apertures (a) and lengths (L) are obtained by data fitting, which are given by

(3)f(a)=\mathrm{0.86}{\mathrm{e}}^{-\mathrm{0.86}a},0\le a\le \infty

(4)f(L)=\mathrm{0.048}{\mathrm{e}}^{-\mathrm{0.048}L},0\le L\le \infty

where a and L are the fracture apertures and lengths in mm and m respectively. For simplicity, the values of a are 0.5 mm, 1 mm, 2 mm, 5 mm, 8 mm and those of L are 10 m, 20 m, 30 m, 50 m, 80 m, which are chosen to cover all the fracture apertures and lengths for model calculations. The fractions of aperture intervals (0, 0.5 mm), (0.5 mm, 1 mm), (1 mm, 2 mm), (2 mm, 5 mm) and (5 mm, ∞) are calculated and assumed to be the fractions of aperture 0.5 mm, 1 mm, 2 mm, 5 mm and 8 mm, respectively, as reported in Table 1.

Similarly, fractions of lengths of 10 m, 20 m, 30 m, 50 m and 80 m are calculated from intervals (0, 10 m), (10 m, 20 m), (20 m, 30 m), (30 m, 50 m) and (50 m, ∞), respectively, as presented in Table 2. Thousands of fractures intersect the well as it can be observed from the borehole data for each well of the Soultz EGS; however, the fractures that constitute the flow path of the reservoir are yet not known.

In this paper, cases are studied for different numbers (N) of fracture flow paths, specifically, N = 500, 1000, 1500, 2000, 2500 and 3000 are studied, and the corresponding distributions of fractures in terms of the fracture aperture and length are listed in Tables 1 and 2, respectively. By randomly coupling a and L, the random resistor values can be obtained from Eq. (2). The injection temperature of 20°C and the production temperature of 150°C are considered. Therefore, the properties r = 968 kg/m³ and m = 3.35 × 10⁻⁴ Pa·s at the average temperature of 85°C are specified in Eq. (2).

Calculations for cases with different values of M (M = 5, 10, 15, 20, 25 and 30) are performed to investigate the effects of section numbers of the model. For each case, all of the resistor values are randomly distributed to the resistor branch in the model. Each resistor branch contains at least one or more resistors, and its value is given by summing up the values of the resistors that it contains. Finally, the total resistance, with the unit of Pa·s·kg^–1, between the injection and production wells is obtained by the following method.

As demonstrated in Fig. 5, for each node (i, j, k, l,…) in the flow resistor network model, the following equation can be obtained based on mass conservation:

(5){{\displaystyle Q}}_{li}+{{\displaystyle Q}}_{ki}={{\displaystyle Q}}_{ij},

in which Q_ij means the mass flow rate from node i to j, and so on. Replace the mass flow rate Q by the relation Q = Dp/R, Eq. (6) can be obtained for node i.

(6){\displaystyle \frac{{{\displaystyle p}}_l-{{\displaystyle p}}_i}{{{\displaystyle R}}_{li}}}+{\displaystyle \frac{{{\displaystyle p}}_k-{{\displaystyle p}}_i}{{{\displaystyle R}}_{ki}}}={\displaystyle \frac{{{\displaystyle p}}_i-{{\displaystyle p}}_j}{{{\displaystyle R}}_{ij}}},

where p_i means the pressure at node i and R_ij, the flow resistance between node i and j, and so on. Given the pressures at the injection and production wells, all the equations are solved in the MATLAB software, and the pressure at each node and the mass flow rates between pairs of neighboring nodes are obtained, then the total resistance can be obtained.

Results and analyses

General results for the model

Figure 6 depicts the variation of the total resistance (R) of the reservoir with the number of sections (M) of the model for different numbers of the fractures (N). It can be noted that for each value of N the relation R vs. M is practically horizontal with only minor, almost negligible, oscillations, which indicates that the total resistance R is a weak function of the number of sections (M); however, in contrast, the number of fractures (N) has a strong effect on the total resistance (R).

For the sake of clarification, Fig. 7 depicts the variation of the total resistance (R) with the number of fractures (N) for different values of M. The linear dependence between R and N for different values of M can be clearly noticed; as the number of fractures increases, the value of the total resistance increases. This is due mainly to the more tortuous flow path caused by the increased number of fractures in serial connection present in-between the injection and production wells. In Fig. 7, it is clearly noticeable that the curves have a great proximity to each other, in particular for small values of N; therefore, from Figs. 6 and 7, it can be concluded that the total resistance of the heat reservoir is primarily determined by the number of fractures present in the reservoir.

For the fractures in the reservoir, it is intuitively understood that only those fractures that have through-path connections to the main flow path have a contribution to the fluid flow, i.e., those occluded and dead-end fractures essentially have no contribution to the fluid flow. As shown in Figs. 3 and 4, most of the flow resistors must be in serial connection in the network; increasing the resistor number (N) means that more resistors in serial connection are included in the network. Although the section number (M) is also increased, the total flow resistance is primarily determined by the fracture number N rather than the section number M.

By assuming a linear relation between R and N, and by fitting the data obtained from the calculation results, Eq. (7) can be obtained.

(7)R=\mathrm{0.903}\times {\mathrm{10}}^{\mathrm{2}}N.

The fitted curve and the deviation curves for ±5% deviations are presented in Fig. 8, showing good agreement between the calculation results and the fitted curve.

Case study of Soultz EGS

The European enhanced geothermal system research site of Soultz-Sous-Forêts is located in Rhine graben in Alsace, France. This site was selected because high heat flow anomaly was observed during former oil exploration in this region. Since the beginning of this project in 1987, four deep boreholes GPK-1 (3590 m), GPK-2 (5084 m), GPK-3 (5093 m) and GPK-4 (5105 m) have been drilled. Based on the experimental data of borehole explorations, seismic monitoring and hydraulic tests, the fracture structures of the subsurface reservoir were well characterized, and its production pattern in relation to the injection well (GPK-3) and the production wells (GPK-2 and GPK-4) was gradually understood. However, the detailed construction of flow paths between wells and the numbers of fractures connected in flow paths are yet unknown.

In 2005, a five-month circulation test coupled with a tracer test was performed between the injection well GPK-2 and the production wells GPK-3 and GPK-4 [¹³]. The cold water with a total volume of 209000 m³ was injected from GPK-3, and 165500 m³ and 40500 m³ were discharged from GPK-2 and GPK-4, respectively. The injection rate was estimated to be 15 L/s, with 11.9 L/s coming from GPK-2 and 3.1 L/s from GPK-4. Tracer injection began 8 days after the circulation test started. 150 kg of 82.5% pure fluorescein was injected into GPK-3 at a rate of about 50 L/h. Fluorescein was first detected in the fluid produced from GPK-2 about 4 days after injection, peaked at the maximum concentration between 9 and 16 days and then decreased gradually during the next 5 months. However, fluorescein was first detected in the fluid produced from GPK-4 about 28 days after injection, and the concentration was very low compared to that in GPK-2. Moreover, after about 5 months of test, no maximum concentration was reached in GPK-4.

The different hydraulic behaviors between the wells GPK-2 and GPK-4 were noticed and investigated for understanding the flow paths between the injection and production wells [¹³–¹⁵]. To date, it is generally accepted that there exists a short, fast and direct hydraulic connection between the wells GPK-3 and GPK-2; in contrast, GPK-3 and GPK-4 are not well connected and the flow path is relatively long as compared to that between GPK-3 and GPK-2. Although the flow paths between wells have been generally patterned as the above, more details about the fractures in paths are still appreciated.

In this section, the fracture network configuration is investigated by the proposed flow resistor network model. First, the total resistances between GPK-3, GPK-2 and GPK-4 are calculated based on the experimental data; and the fracture number is calculated by Eq. (7) based on the flow resistor network model; finally, the fracture construction is analyzed from the calculation results.

Resistance calculation

For the purpose of determining the permeability between the wells, the productivity index (PI) is defined as [¹]

(8)\mathit{PI}={\displaystyle \frac{Q}{\Delta p}},

where Q is the mass flux from the production well and Dp is the pressure drop between the injection and production wells. It can be noted that the productivity index and the total resistance are reciprocal, that is R = 1/PI. It was determined [¹] that the productivity index between GPK-3 and GPK-2 is PI_GPK3,2 = 3.5 kg/(s·MPa), and hence the total resistance is R_GPK3,2 = 2.85 × 10⁵Pa·kg⁻¹·s. During the circulation test in 2005, the pressure drop of GPK-3 and GPK-4 is very close to that of GPK-3 and GPK-2 [¹⁵], and the production of the former is about a quarter of the latter [¹³]. Therefore, the productivity index between GPK-3 and GPK-4 is PI_GPK3,4 = 0.875 kg/(s·MPa), and hence the total resistance R_GPK3,4 = 11.4 × 10⁵ Pa·kg⁻¹·s.

Fracture number

The numbers of fractures for the path from GPK-3 to GPK-2 and GPK-4 are derived from the total resistances by using the proposed relation, Eq. (7), and the values are N_GPK3,2≈ 3150 and N_GPK3,4≈ 12620.

Analyses of fracture constructions

The number of fractures in the path from GPK-3 to GPK-2 is almost 4 times of that in path from GPK-3 to GPK-4, which indicates that the flow path from GPK-3 to GPK-4 is more tortuous and longer than that from GPK-3 to GPK-2. The calculation results are supported by the hydraulic behaviors of the circulation and tracer tests. In the earlier years, the three-loop conceptual model was proposed to analyze the different hydraulic behaviors between the wells in the tracer tests [¹³–¹⁵]. In this generally adopted model, there is a short loop with rather a good hydraulic connection between GPK-3 and GPK-2; but the connection between GPK-3 and GPK-4 is very poor because of the presence of a longer and a slower circulation loop. In other words, the fractures in worse connection between GPK-3 and GPK-4 are more than those between GPK-3 and GPK-2, and the flow path from GPK-3 to GPK-4 is more tortuous and longer than that from GPK-3 to GPK-2.

Summary and conclusions

A novel flow-resistor network model for modeling the hydraulic processes in heat reservoir is developed and proposed in the present work. It is advanced that heat reservoirs with different characteristics can be simplified through the flow-resistor network model using the same framework with different section numbers and resistor values. To demonstrate the capability of the model, a comprehensive calculation was performed and the results show that the total resistance between the injection and production wells is primarily determined by the number of fractures in flow path(s) between the wells, and based on the data generated through the numerical results, is further proposed an equation, i.e. Eq. (7), relating the total resistance to the number of fractures. This particular equation using the productivity index values can predict the numbers of fractures in the paths from GPK-3 to GPK-2 and GPK-4 for the Soultz EGS.

For a practical EGS field, applying the flow-resistor network model can give more characteristic information about the interior structure of the reservoir, which enables detailed modeling and analyses on the EGS performance and even can serve for further reservoir engineering work.

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