The effect of spatial aperture variations on the thermal performance of discretely fractured geothermal reservoirs
- Don Bruce Fox^{1, 2},
- Donald Lyle Koch^{1, 2} and
- Jefferson William Tester^{1, 2}Email author
Received: 5 August 2015
Accepted: 30 September 2015
Published: 28 October 2015
Abstract
The effect of spatial aperture variations on the thermal performance of discretely fractured geothermal reservoirs was investigated using finite element method solutions to the convective heat transfer in the fracture coupled with a boundary integral description of conductive heat transfer in the rock. The dipolar flow was generated by a source of fluid volume at an injection well and a sink at a production well within a circular fracture. The statistics of the thermal performance of an ensemble of fracture realizations was evaluated. Fractures with self-affine aperture fields where long range correlations dominant over short range correlations were generated. The results showed that spatial aperture variations most frequently lead to diminished thermal performance by creating flow channeling that reduces the heat transfer area. Enhanced thermal performance occurred in the less common cases when the aperture was small in the region between the wellbores, causing fluid flow to sweep out greater areas of the fracture and extract heat from a larger area. The standard deviation of the apertures had the largest influence on the thermal performance, while the spatial correlation of the aperture played a secondary role. Larger values of the standard deviation led to more adverse thermal performance. For the range of standard deviations investigated, the fraction of fractures exhibiting enhanced thermal performance compared to the base case of no aperture variations ranged from 34 to 49 %. The degradation of thermal performance due to aperture variations was largest when the well bore spacing was a larger fraction of the fracture diameter. Reservoirs consisting of two non-intersecting fractures connected to the same injection and production wells were also modeled. The uneven split of the flow between the reservoirs in this case caused a further deterioration of thermal performance compared with the single fracture reservoirs. However, flow control was able to overcome nearly all of the additional loss of thermal performance for the multifracture reservoir.
Keywords
Enhanced geothermal systems Aperture variations Roughness Discrete fractures Heat transport in fractures Coupled convective conductive heat transport Thermal hydraulic modeling Geothermal reservoir performance modelingBackground
Variations in fluid velocity on various flow paths in discretely fractured geothermal reservoirs like those found in Enhanced/Engineered Geothermal Systems (EGS) have a significant effect on the thermal performance of a reservoir. One source of variations in fluid flow is from spatial aperture variations in the fractures. EGS reservoirs are developed in hot rock with low permeability and porosity to extract a portion of the stored thermal energy contained in the rock mass. A reservoir is stimulated by pressurization to open existing sealed fractures (hydro-shearing) or induce new ones by hydraulic fracturing where water may be pumped into and out of the fractured reservoir using a connected set of injection and production wells to “farm” or extract the stored thermal energy. While convective heat transport is the dominant mode of transport within the fractures, it is inherently coupled to conductive heat transport from the surrounding bulk rock. Because of the low thermal diffusivity of the host rock (of order \(10^{-6}\) m^{2}/s), only a small region surrounding the fracture network is effective for heat extraction. Given the finite areal extent of the fracture network, extended use will locally cool down the reservoir. The mass loading per unit area usually expressed as the total flow rate per unit fracture surface area is a critical parameter that often scales directly with the rate of thermal drawdown. Drawdown is a major concern for geothermal developers who must forecast reservoir performance and consider its effect in their techno-economic analysis. This study seeks to determine the effect of fracture aperture variations on the thermal performance of discretely fractured geothermal reservoirs
Because fractures do not have smooth surfaces, spatial aperture variations can lead to maldistributed flow within the fracture network adding a degree of thermal performance uncertainty. Fracture core samples have revealed fracture apertures to have a self-affine spatial correlation governed by a roughness or Hurst coefficient (Brown et al. 1986; Glover et al. 1998; Schmittbuhl et al. 1993, 1995; Plouraboué et al. 1995; Boffa et al. 1999; Ponson et al. 2007). A self-affine spatial correlation is one characterized by long range correlations. The aperture undulations in the core samples are found on all length scales but the aperture variance is dominated by the largest scale. Often self-affine fracture surface variations are referred to as fracture roughness despite the fact that roughness implies variations on small rather than large length scales. In this paper, we have avoided using “rough and roughness” as descriptors as much as possible and instead use terms like aperture or surface variations.
To determine the effect fracture surface variations have on fluid flow and on a reservoir’s thermal performance, a dipole flow in circular fractures between a single injector and producer was selected. Dipole flow is not uniform and is characterized by streamlines where the fluid velocity is faster near the injection and production wells. For simplicity, effects like temperature dependent properties and buoyancy were not considered. We are interested in modeling reservoirs with different degrees of fracture surface variations, roughness/Hurst coefficients, wellbore spacings, and multiple fractures. Our focus is to determine how fracture aperture variations affect reservoir parameters like the production temperature and the likelihood a reservoir will exhibit thermal enhancement or decline through statistical parameters involving an ensemble set of fracture realizations. Through these statistical values, we hope to 1. infer how the extent of aperture variations aid or hinder thermal performance, 2. characterize the degree of reservoir performance uncertainty due to aperture variations, and 3. determine under what scenarios aperture variations need to be modeled. Furthermore, modeling spatial aperture variations provides a method to systematically generate variations in flow fluid rates and their pathway through the fracture system.
While little work has been done on the coupled convective-conductive heat transport of the fracture and surrounding rock in a reservoir with spatial aperture variations, researchers have modeled fluid flow and other transport properties using a self-affine aperture field. They have observed that self-affine aperture fields lead to flow channeling where most of the fluid will travel through several preferential channels (Brown 1987; Tsang and Tsang 1989; Glover et al. 1998; Drazer and Koplik 2000; Méheust and Schmittbuhl 2001, 2003; Auradou et al. 2006). Méheust and Schmittbuhl (2001) determined that three parameters are required to model the flow, the mean aperture, the root mean square of the aperture, and the Hurst coefficient. They further showed how the mean flow rate was either enhanced or inhibited, depending on the orientation of the applied pressure drop. Transport properties such as dispersion have also been studied in self-affine fractures (Plouraboué et al. 1998; Drazer and Koplik 2002; McDermott et al. 2009; Auradou et al. 2010). Numerical studies exhibit non-Gaussian transit time distribution with long tails (Drazer and Koplik 2002) and experiments with Newtonian and shear thinning fluids showed that geometric/hydraulic dispersion is dominant at low Péclet numbers with the dispersion coefficient on the order of the Péclet number (Auradou et al. 2010).
Other researchers have used flow channeling to reduce numerical computation by reducing flow in 2D fractures to an equivalent 1D flow (Cacas et al. 1990; Nordqvist et al. 1992; Bruel et al. 1994) or to reduce 3D flow into an equivalent 2D system (Bruderer-Weng et al. 2004). The above mentioned studies have focused on the effects fracture surface variations have on hydraulic performance and particle/solute transport and do not cover the consequence on convective heat transport. However, these studies mention preferential flow paths which suggest that surface variations may have a negative impact on thermal exchange.
Kolditz (1995) suggested that local aperture variations may play a strong role in a reservoir’s thermal performance. Jupe et al. (1995) attempted to model spatial aperture variations by modeling separate flow paths. They used the multifracture parallel rectangular reservoir model described by Nicol and Robinson (1990) and treated the different flow paths as separate fractures. Their work focused on determining the wellbore spacing needed to obtain an economic EGS reservoir. By incorporating multiple flow paths, the wellbore spacing needed to obtain their desired performance criteria went from 700 m (no channeling) to 1080 m.
The uncertainty of reservoir parameters like heat capacity and permeability have been modeled for a heterogeneous porous media geothermal system (Watanabe et al. 2010). The spatial correlation of the reservoir parameters were modeled using a spherical variogram. Using a hundred realizations of the reservoir, Watanabe et al. (2010) determined that permeability was the most sensitivity parameter, with the production temperature having an uncertainty range of 40 K for the 15 years. Since the aperture can be viewed as an analog to permeability, we would expect spatial aperture variations to substantially control a reservoir thermal performance.
The thermal effect of surface variations has recently been investigated in a rectangular fracture where the base flow is uniform (Neuville et al. 2010a, b). However, Neuville et al. (2010a, b) used a simplified heat transfer model that only considers the heat transfer in the fracture and not in the surrounding rock; the surrounding rock was assumed to be permanently at the original reservoir temperature. As a consequence, their results can be interpreted as the early thermal behavior of the reservoir and not the behavior after extended reservoir use. The assumption of constant rock temperature is valid for the time scale of \(w^2/\alpha\), where w is the fracture aperture and \(\alpha\) is the thermal diffusivity of the rock. For an aperture value from 1 to 5 mm and a thermal diffusivity of \(1\times 10^{-6}\) m^{2}/s, the time scale will be on the order of 1–10 s. Neuville et al. (2010b) work focused on developing relationships between a reservoir’s hydraulic performance and its thermal performance. They concluded heat exchange in self-affine fractures is less efficient compared to a constant aperture case. As a case study, Neuville et al. (2010a) modeled the Soultz-sous-Forêts EGS reservoir as a single fracture and concluded that merely knowing the mean aperture is not sufficient to determine the thermal heat exchange and that fracture surface variations always led to inhibited thermal exchange. Although Neuville et al. (2010a, b) have modeled variations in convective heat transport in the presence of aperture undulations, there is still a need to model the coupled convective heat transport in the fracture with that of the conductive transport in the surrounding rock.
Our approach fully couples the heat transfer in the fracture to the surrounding bulk rock by using a finite element method in the fracture and boundary integral equations to describe the temperature in the bulk rock. Additionally, our study aims to determine how much deviation in production temperature in the absence of aperture undulations is anticipated after prolonged reservoir utilization. The use of statistical values of an ensemble of realizations including the mean and standard deviation of production temperatures will offer a better and more practical understanding of the effects fracture surface variations have on discretely fractured geothermal reservoirs.
Methods
The following section outlines our numerical approach in modeling self-affine aperture fields and the thermal performance of the resulting reservoirs. We first introduce the mathematical theory of self-affine aperture fields and describe the algorithm used to generate the fracture surfaces. Then, we detail the method used to solve for the fluid flow in the fracture and finally, we describe our numerical approach in solving for the heat transport in the reservoir and the transient temperature field.
Self-affine aperture field
With a self-affine behavior, larger scale variations in h (small frequencies, long wavelengths) are more important than small scale (large frequencies, short wavelengths) variations. The large scale variations become more strongly dominant when \(\zeta\) is increased. However, the large scale variations are the primary determinant of the aperture variations for any positive \(\zeta\). A self-affine aperture field is analogous to fractional Brownian motion, if one views the spatial displacement \(|\mathbf {r'}|\) as a time-like variable and the change of aperture with displacement as equivalent to the spatial step of the Brownian motion. With this analogy, \(\zeta <1/2\) corresponds to a subdiffusive or antipersistent process, \(\zeta = 1/2\) corresponds to a normal diffusive process, and \(1/2<\zeta <1\) corresponds to a super diffusive process with \(\zeta = 1\) representing ballistic motion (Mandelbrot and Ness 1968). Interested readers are advised to consult the following references for more details on the theory of self-affine behavior and other applications, (Feder 1988; Peitgen and Saupe 1988; Falconer 2003).
The algorithm used to generate spatially varying aperture fields, which was adapted from Méheust and Schmittbuhl (2001), uses random seeding and Fast Fourier Transforms to obtain isotropic self-affine fracture aperture fields on rectangular fractures. In order to generate circular fractures with a self-affine aperture field, a square fracture of length 2R was first generated and only the area of an inscribed circle was used for the analysis. Any shaped fracture could have been generated by first generating a large enough rectangular fracture to inscribe the desired shape of the fracture. Only the inscribed area formed by the arbitrary shaped fracture is used for the analysis. To obtain the desired \(\sigma _h\) of the fracture, the aperture value h is rescaled.
Real reservoirs are not static; there will be dynamic changes to the aperture and permeability fields due to thermal, hydraulic, mechanical, and chemical (THMC) effects. In earlier works, these effects have been incorporated to model a reservoir’s thermal performance for both porous media (McDermott et al. 2006; Taron and Elsworth 2009) and discrete fractures (Ghassemi and Zhou 2011; Rawal and Ghassemi 2014). To achieve viable treatments, researchers have to specify quantitative deterministic models of THMC effects which themselves require further additional assumptions. The objective of our study was different and solely directed towards understanding how stochastic spatial aperture variations affect reservoir performance—not to incorporate coupled THMC effects.
Fluid flow
Often, Eq. (9) is referred to as the Reynolds equation and is derived using an approximate lubrication analysis. As such, Eq. (9) is valid when \(\mathrm {Re}(w/L_{x,y}) \ll 1\) and \(w/L_{x,y} \ll 1\) (Deen 1998). \(L_{x,y}\) refers to the characteristic length scale of aperture variations in the x and y direction and may be considered a “wavelength” of aperture variations. Here, the Reynolds number is defined as \(\text{Re}\equiv (\rho _w U_c w)/\mu\), where \(U_c\) is the characteristic velocity. For a mass flow of 40 kg/s, a wellbore spacing L of 400 m, a width of 1 mm, and a characteristic velocity of \(U_c \sim \dot{m}/(\rho _w w L)\), then \(\text{Re}\sim 100\), well below the critical Reynolds number for turbulence transition. The lubrication criteria \(\text{Re} (w/L_{x,y}) \ll\) and \(w/L_{x,y} \ll 1\) will also be satisfied when \(\text{Re}\sim 100\) because \(L_{x,y} > 1\) m because large scale aperture variations dominate when the aperture fluctuations are self-affine. Eq. (9) has been extensively used to model fluid flow in fractures (Brown 1987; Zimmerman and Bodvarsson 1996; Méheust and Schmittbuhl 2001; Watanabe et al. 2008). Because the aperture is raised to the third power, flow modeled using Eq. (9) is sometimes said to follow the “cubic law”.
The pressure field is first solved using Eq. (9) with the finite element method (FEM). After solving for the pressure, the velocity is obtained from Eq. (8). The injection and production wells are modeled as circles of radius 0.15 m to avoid singularities in pressure. The mass flux was evenly distributed along the circumference of the injection and production well. We specify a no mass flux condition on the boundary of the fracture and in the direction normal to the surface in order to model the fracture having a limited areal extent with no leak off, that is the surrounding rock matrix is assumed to be impermeable.
Heat transport
The numerical model we have developed is a hybrid finite element and boundary element method for discretely fractured geothermal reservoirs. The three-dimensional heat conduction in the rock matrix is captured through a boundary element treatment while the fluid flow and thermal convection are solved via the FEM. As stated earlier, the rock matrix surrounding the discrete fractures is assumed to be impermeable with respect to fluid flow. The numerical model relies on 2D discretization of only the fracture surface and not the entire domain. The 2D discretization offers faster numerical run times over a 3D discretization of the entire reservoir and allows one to invest more computational effort on fracture scale details. The approach to use hybrid FEM and boundary integral representations is not new and our developed model was influenced by such works as Zhou et al. (2009), which studied the effect of thermoelastic stresses in a geothermal reservoir. Other hybrid numerical and analytical methods have been derived before, Cheng et al. (2001) used their developed methodology to model multi-dimensional heat effects in geothermal reservoirs and McDermott et al. (2009) used their hybrid model to solve for the flow, mass, and heat transport in a fracture, taking advantage of not having to discretize a three dimensional domain.
By working in the Laplace space, time discretization is explicitly avoided along with singularities at \(t=0\) and accuracy issues associated with early time discretization. Equations (13) and (14) are solved simultaneously to obtain the surface heat flux and the temperature profile of the fracture at different time steps of interest. The inverse Laplace transform is computed using the Gaver–Stehfest algorithm (Stehfest 1970a, b) which determines the value of f(t) by computing several terms in a series involving \(\bar{f}(s)\) at different values of s.
The matrix \(\bar{G}_{ij}(x-x',y-y',z)\) is computationally intensive to construct because every entry must be evaluated. The computational cost is \(O(N^2)\) where N is the number of nodes. Computations can be greatly reduced by noticing that many of the entries in \(\bar{G}_{ij}\) are virtually zero and can be skipped in populating \(\bar{G}_{ij}\). For determining \(\bar{G}_{ij}\) for a given s, entries that correspond to evaluating points that are farther than \(20\sqrt{\alpha /s}\) are skipped where \(\alpha\) refers to the thermal diffusivity of the rock. The computational intensity can then be reduced to \(O(N \log N)\). The value 20 is arbitrary and can be changed to modify accuracy. This method is akin to a single term Fast Multipole Method as described by Greengard and Rokhlin (1997). For a 2500 node mesh, the computational time was reduced by a factor of about 20 with only a difference in the fifth significant figure for the outlet temperature.
Alternatively, since the conduction is dominated by the direction perpendicular to the fracture, the 1D Green’s function can be used instead of the full 3D form. Doing so results in a less than 1 % error and over a 20-fold decrease in simulation time compared to using the truncation approximation of \(\bar{G}_{ij}\). Additionally, the 1D simplification corresponds to a 400-fold decrease in simulation time compared to the fully populated \(\bar{G}_{ij}\) matrix. Fox et al. (2013) showed that 1D conduction is a reasonable approximation when \(\beta ^2 \ll 1\). Here, \(\beta\) is a dimensionless parameter describing the ratio of the extent of thermal depletion normal to the plane of the fracture to the extent of depletion in the plane of the fracture. For dipole flow, \(\beta\) can be defined as \(\beta = 2k_r L/(\dot{m} c_w)\), where L is the spacing between the injection and production well. For typical flow conditions of 40 kg/s and a wellbore separation of 400 m, \(\beta ^2 = 2 \times 10^{-4}\). Nonetheless, for completeness, we used the full 3D heat conduction in this study.
Results and discussion
Results for one fracture
For an idealized single circular fracture with self-affine aperture variations, our modeling provides the statistical distribution of the dimensionless production temperatures as a function of wellbore spacing, duration of extraction, the variance of fracture aperture variations, and the Hurst coefficient. The extraction time for geothermal production selected for our study corresponded to a particular level of the thermal performance. In our study, this occurred when the dimensionless production temperature reached 0.8 for the case without aperture variations, which we call the base case. The variable \(\Theta _{p,0}\) refers to the base case production temperature. The drawdown criteria will vary depending on the end-use of the thermal energy; direct thermal use operations can afford a greater drawdown compared to electricity production. However, using \(\Theta _{p,0}=0.8\) is a realistic value for the drawdown criteria in many practical situations.
For \(\sigma _h/\left\langle w \right\rangle\), \(\zeta\), and L/(2R), we chose a range of possible values for these parameters to model their impacts on thermal performance. The range of \(\sigma _h/\left\langle w \right\rangle\) was chosen to be from 0.015 to 0.35. Granite core samples have been shown to have a value of \(\sigma _h/\left\langle w \right\rangle\) of 0.42 (Hakami and Larsson 1996) but also low values around \(5\times 10^{-4}\) (Sausse 2002). Because of the uncertainty in the standard deviation of the local aperture values and the difficulty in concluding how representative core sample values will be of practical in situ conditions in geothermal reservoirs, we believe the range chosen is adequate for simulating a large class of fractures. Above \(\sigma _h/\left\langle w \right\rangle = 0.35\), it becomes difficult to generate fractures that do not have zero or negative aperture values. Thus, the aperture realizations generated represent fractures that have been sufficiently propped open due to the combination of “self-propping” due to shear displacement and inflation as a result of higher fluid pressures within the fracture. As previously reported, the Hurst coefficient has been observed to be always above 0.5 and is often reported to be around 0.8. However, the Hurst coefficient is a model specific parameter and is not often reported. It is plausible to think that there are instances where the Hurst coefficient is less than 0.5. Given its theoretical range of 0 to 1, the range of the Hurst coefficient chosen was from 0.1 to 1. The dimensionless wellbore spacing L/(2R) can vary anywhere from 0 to 1, depending on where the injection and production wells have been placed relative to the fracture size. In this study, we investigated the range from 0.2 to 0.8.
We first present the results where the standard deviation of the aperture is varied. For the range of values investigated, five thousand fracture realizations were generated for each value of \(\sigma _h/\left\langle w \right\rangle\) with L/(2R) and \(\zeta\) kept constant at 0.4 and 0.8, respectively. In a second set of simulations, we repeated the process but now varying the Hurst coefficient and keeping \(L/(2R)=0.4\) and \(\sigma _h/\left\langle w \right\rangle =0.25\). Finally, in a third set, the wellbore spacing was varied and \(\sigma _h/\left\langle w \right\rangle\) and \(\zeta\) are set to 0.25, and 0.8, respectively. The statistical values of the thermal performance of most interest are the mean dimensionless production temperature \(\bar{\Theta }_p\) of the ensemble, the standard deviation of production values \(\sigma _{\Theta }\), and the fraction of cases with a higher production temperature than the base case F. Five-thousand was chosen to have an adequate sample size but also to be able to generate results in a timely manner. Using ten-thousand fracture realizations did not result in any significant change in the derived results.
Varying strength of aperture fluctuations
For a range of \(\sigma _h/\left\langle w \right\rangle\) from 0.015 to 0.35, five thousand fracture realizations were generated for each value of \(\sigma _h/\left\langle w \right\rangle\). For each five thousand fracture realization data set, the values of \(\bar{\Theta }_p\), \(\sigma _{\Theta }\), and F are reported based on the thermal performance for all realizations at the time when the dimensionless production temperature reached 0.8 in the absence of aperture variations. A depiction of the fracture surface temperature for the base case, thermally under performing, and thermally over performing (compared with the base case) and their corresponding aperture fields is shown in Fig. 1. The dimensionless heat transfer area accessed at a certain time is defined as \(\int (1-\Theta )\ \mathrm {d}A/A=1-\left\langle \Theta \right\rangle\). Thus, the mean fracture surface temperature \(\left\langle \Theta \right\rangle\) is a measure of the thermally accessed area of the fracture at a given extraction time. Comparing \(\left\langle \Theta \right\rangle\) for different fracture aperture distributions, a fracture with a lower value of \(\left\langle \Theta \right\rangle\) was able to access more area of the reservoir for heat transfer. For dipole flow, the majority of the flow occurs in the region surrounding the injection and production wells with less flow in the periphery of the fracture. The base case mean value for \(\left\langle \Theta \right\rangle\) is 0.816 when \(\Theta _{p,0} = 0.8\). For a fracture with smaller apertures in the region between the wells, the fluid is pushed to higher aperture regions in the periphery and the effective heat transport area increases, \(\left\langle \Theta \right\rangle =0.787\) when the dimensionless production temperature reaches 0.8. As expected, this less channeled, beneficial flow field leads to a higher production temperature and slows the rate of thermal drawdown. On the other hand, when there exists a high aperture region in between the wells, the flow is further channeled from the injection to production well and the effective heat transfer area becomes narrower than the base case; the mean dimensionless fracture surface temperature is 0.854 when the dimensionless production temperature is 0.8. Consequently, the production well cools down rapidly. How the low and high aperture regions are arranged in the fracture is clearly important in determining whether the reservoir will exhibit poor or good thermal performance.
The linear behavior of Fig. 4 is expected due to the before mentioned linear behavior of \(\Theta _p\) to small variations of h. In particular, \(\Theta _0\) has zero standard deviation, while \(\Theta _1\) has an O(1) standard deviation so that the standard deviation of \(\Theta _p \approx \Theta _0+ \epsilon \Theta _1\) is proportional to \(\sigma _h\). The quadratic relationship between the mean production temperature and the aperture standard deviation is also expected owing to the fact that \(\bar{\Theta }_1\) must be zero based on linearity. Thus, \(\bar{\Theta }-\Theta _0 = (\sigma _h/\left\langle w \right\rangle )^2 \bar{\Theta }_2\), where \(\bar{\Theta }_2\) is a constant of proportionality.
Geothermal operations are sensitive to temperature drawdown and a premature drop in production temperature would reduce the economic attractiveness of the project. Each fracture in the ensemble has a lifetime corresponding to the time when \(\Theta _p\) reaches 0.8. When \(\sigma _h/\left\langle w \right\rangle =0.1\), which resulted in \(\sigma _{\Theta }=0.031\), the 25 and 75 % quantiles of the distribution of the ratio of reservoir life to the reservoir life of the base case are equal to 0.83 and 1.16, respectively. For \(\sigma _h/\left\langle w \right\rangle =0.35\), corresponding to \(\sigma _{\Theta }=0.092\), the results are more drastic as the 25 and 75 % quantiles become 0.47 and 1.28, respectively. On the other hand, aperture variations are less important for lower values of \(\sigma _h/\left\langle w \right\rangle\); when \(\sigma _h/\left\langle w \right\rangle =0.05\), the 25 and 75 % quantiles of reservoir life are closer together and equal to 0.91 and 1.08, respectively.
The results shown indicate that the magnitude of aperture variations has a large effect on the thermal performance of the reservoir. For a specified standard deviation, the production temperature can be modeled as a Gaussian distribution. The fraction of fracture whose thermal performance benefits from a self-affine aperture field monotonically decreases with respect to the strength of the aperture variations and is never higher than 0.5. It ranged from 0.49 to 0.34 for the range of \(\sigma _h/\left\langle w \right\rangle\) investigated. Thus, there are more ways to arrange isotropic self-affine aperture fluctuations that lead to a flow field which produces adverse rather than beneficial thermal performance.
Varying the Hurst coefficient
Thermal performance with varying the Hurst coefficient
\(\zeta\) | \(\bar{\Theta }_p\) | \(\sigma _{\Theta }\) | F |
---|---|---|---|
0.1 | 0.786 ± 0.002 | 0.063 ± 0.001 | 0.41 ± 0.01 |
0.2 | 0.785 ± 0.002 | 0.065 ± 0.001 | 0.40 ± 0.01 |
0.3 | 0.784 ± 0.002 | 0.067 ± 0.001 | 0.40 ± 0.01 |
0.4 | 0.784 ± 0.002 | 0.069 ± 0.001 | 0.40 ± 0.01 |
0.5 | 0.784 ± 0.002 | 0.069 ± 0.001 | 0.39 ± 0.01 |
0.6 | 0.784 ± 0.002 | 0.071 ± 0.001 | 0.40 ± 0.01 |
0.7 | 0.783 ± 0.002 | 0.072 ± 0.001 | 0.39 ± 0.01 |
0.8 | 0.783 ± 0.002 | 0.072 ± 0.001 | 0.39 ± 0.01 |
0.9 | 0.784 ± 0.002 | 0.073 ± 0.001 | 0.40 ± 0.01 |
1 | 0.784 ± 0.002 | 0.073 ± 0.001 | 0.40 ± 0.01 |
Varying wellbore separation
The effect of self-affine aperture variations on different base flows can be understood by changing the value of the wellbore separation value, L/(2R). As L/(2R) gets smaller the streamlines becomes more varied and diverse. On the other hand, the streamlines become more uniform as the injection and production wells are pulled apart. For values of L/(2R) of 0.2, 0.4, 0.5, 0.6, and 0.8, the thermal performance of the five thousand fracture simulations for each value of L/(2R) were all performed for an end time when the respective L/(2R) base case production temperature reaches 0.8. To reach the \(\Theta _p=0.8\), the dimensionless time \(t/\tau\) for L/(2R) of 0.2, 0.4, 0.5, 0.6, and 0.8 is equal to 4.7, 3.5, 2.8, 2.2, and 1.2 respectively. Although increasing the wellbore spacing increases the time to reach the drawdown criteria, the dimensionless time \(t/\tau\) to reach the drawdown criteria drops for increasing values of wellbore spacing because \(\tau\) is proportional to \(L^4\). The values of \(\zeta\) and \(\sigma _h/\left\langle w \right\rangle\) were kept constant at 0.8 and 0.25, respectively.
Thermal performance with varying wellbore spacing L/(2R)
L/(2R) | \(\bar{\Theta }_p\) | \(\sigma _{\Theta }\) | F |
---|---|---|---|
0.2 | 0.793 ± 0.002 | 0.057 ± 0.001 | 0.42 ± 0.01 |
0.4 | 0.783 ± 0.002 | 0.072 ± 0.001 | 0.39 ± 0.01 |
0.5 | 0.779 ± 0.002 | 0.073 ± 0.001 | 0.38 ± 0.01 |
0.6 | 0.774 ± 0.002 | 0.068 ± 0.001 | 0.36 ± 0.01 |
0.8 | 0.763 ± 0.002 | 0.053 ± 0.001 | 0.28 ± 0.01 |
The mean dimensionless production temperature is closest to the base case value when the wellbore spacing is small and there is less opportunity for the conductance to vary in the inter-wellbore region. The thermal performance distribution is most varied when \(L/(2R)=0.4\) because it corresponds to the typical length scale of one “peak” or “valley”, as can be see in Fig. 6. When the dimensionless wellbore spacing is smaller than 0.4, then the region in between the wells will have less aperture variations which results in a lower \(\sigma _{\Theta }\). The trend of lower values of F for larger values of wellbore spacing can be rationalized by considering the heat transfer area. The mean fracture surface temperature \(\left\langle \Theta \right\rangle\) is a measure of the accessed area for heat transfer at a point in time; smaller values of \(\left\langle \Theta \right\rangle\) mean more area has been accessed. The base case values for \(\left\langle \Theta \right\rangle\) are 0.95, 0.82, 0.74, 0.67 and 0.57 for \(L/(2R)=0.2, 0.4, 0.5, 0.6,\) and 0.8, respectively. For larger wellbore spacing values, the base case is already harvesting energy from a large portion of the fracture and any deflection of the base flow caused by aperture variations will tend to decrease the heat transfer area as seen in the results for F, the fraction of fractures with better performance than the base case. In contrast, smaller wellbore spacing values, the base case heat transfer area is small and perturbations of the flow field are more likely to increase the heat transfer area.
Results for multiple fractures
The previous section described the effect of a self-affine aperture field on the thermal performance when there is only one fracture. A relevant situation to further model would be a reservoir consisting of two fractures. We consider a reservoir consisting of two fractures with equal radii that do not intersect with one another, but rather are hydraulically connected via the injection and production well. The pressure drop along the wellbore is assumed negligible because \(w/r \ll 1\), where the wellbore radius r is around 10 cm and the aperture is around 1 mm. The mass flow was doubled to 80 kg/s, but the value of \(\dot{m}/A\) remains the same as that in the one fracture system because the total surface area of the system is twice as large.
With two fractures the flow field exhibits both inter- and intrafracture flow variations. As has been seen in the one fracture reservoir, local aperture variations disturb the baseflow creating intrafracture flow variations. However, aperture variations can also modify the ratio of the mean flow rate between fractures through altering the flow impedance or resistance to mean flow of each fracture and affect the reservoir’s overall pressure drop. The flow impedance of the fractures in the reservoir is what drives the global distribution of flow in the reservoir.
The effects of fracture aperture variations in two and three fracture systems was studied by determining the thermal performance of an ensemble of fracture realizations. The wellbore spacing and the Hurst coefficient were kept constant and the standard deviation of the aperture was varied over the same range considered before. Finally, we examined the efficacy of flow control methods in minimizing any negative effects aperture variations have on the thermal performance of multifracture systems.
Varying strength of aperture fluctuations
A portion of the decreased thermal performance in the two fracture reservoir results solely from the existence of an unequal flow split between the fractures. The reduction in the production temperature for two smooth fractures with increasing flow split ratio is indicated by the solid line in Fig. 10 and we will refer to this function as \(g(\phi )\). A metric of the additional loss of thermal performance in the two fracture reservoir compared with the case of two smooth fractures with unequal apertures is \(F^*\), which we define as the fraction of two fracture reservoirs with \(\Theta _p > g(\phi )\), i.e., the fraction that beat the smooth fracture reservoir with the same flow split. The results for \(F^{*}\) are higher than F because the smooth case thermal performance drops with respect to increasing \(\phi\). However, \(F^{*}\) is closer to the results of F for the two fracture results than to the single fracture results. These results demonstrate that the primary determinant of the diminished performance of the two fracture system is the aperture variation and flow channeling within each reservoir rather than being simply a result of the difference of hydraulic aperture of the two fractures.
The reduced thermal performance compared to a one fracture system can be explained by the relationship between pressure drop and the thermal performance in an individual fracture. In general, a fracture with a small aperture region in between the injection and production wells will exhibit enhanced thermal performance because the flow will expand and sweep through the reservoir more effectively. The reverse is true; a high aperture region in between the wells will promote flow short circuiting and the flow will primarily harvest thermal energy in a narrow area. This behavior was illustrated in Fig. 1, where the temperature of the fracture surface was plotted along side their respective aperture fields. Furthermore, a low aperture wellbore region requires a higher pressure drop to push the fluid through the constricted region surrounding both the injection and production wells. The covariance of \(w_H\) and \(\Theta _p\) for the single fracture is \(-0.65\) when \(L/(2R)=0.4\), \(\zeta =0.8\), and \(\sigma _h/\left\langle w \right\rangle =0.25\), which supports our qualitative argument.
For single fracture reservoirs, poor performing fractures exhibit a smaller pressure drop. However, in a two fracture reservoir, the poor thermally performing reservoir will receive a larger portion of the flow and the reservoir with good thermal performance receives a smaller portion of the flow. Thus, it is not simply the existence of an unequal flow split but the fact that the higher flow goes preferentially to the fracture with more channeling that accounts for the poorer performance of two fracture reservoirs compared with single fracture reservoirs. However, the negative impact on thermal performance can be mitigated if the flow split between the reservoirs can be manipulated.
Three fracture system
Flow control
To mitigate the adverse impact of fracture surface variations on the interfracture flow variations, flow control methods can modify the flow split and reduce the negative effects associated with flow split disparity. In a real reservoir, flow control is achieved by using interflow control devices (ICD). An ICD works by increasing the resistance of flow from the wellbore to the formation or fracture. The increased resistance is achieved using nozzles, orifices, and helical channels on the base pipe. For more information on ICD, please refer to Al-Khelaiwi et al. (2010) and Birchenko et al. (2010).
The fact that flow control was able to bring F near the single fracture results indicates that interfracture flow variations inhibit thermal performance more than intrafracture flow variations. The results in Fig. 12 showed that these detrimental interfracture flow variations consist of flow preferentially going to poor thermal performing fractures rather than simply involving unequal flow splits. Nonetheless a comparison of our two flow control strategies indicates that an effort to divert flow preferentially away from fractures exhibiting channeling yields only a marginal improvement over a flow control strategy that yields an equal flow split.
The flow control results reveal that for equally sized fractures, a 50/50 flow split is good practice. Knowing the optimal split does not result in a large improvement in thermal performance. Flow control would be limited if the reservoir consists of fracture-to-fracture intersections as ICD can only control flow in the fractures that are directly connected to the wellbore. Fracture-to-fracture intersections, therefore, open up new possible flow paths rather than just partitioning the flow between two independent flow paths where the path with the least impedance takes the majority of the flow. Thus, the presence of fracture-to-fracture intersections may not result in a diminished thermal performance, as was shown in the case of multiple fractures only hydraulically connected via the wellbores.
Conclusions
Isotropic self-affine fracture aperture variations limit a reservoir’s thermal performance. There are more ways to arrange aperture variations that causes flow paths that lead to premature drawdown than flow pathways that enhance heat transfer. We have shown that the dimensionless standard deviation of the aperture \(\sigma _h/\left\langle w \right\rangle\) is more important in controlling thermal performance than the Hurst coefficient \(\zeta\). Specifically, both the uncertainty in thermal performance, \(\sigma _{\Theta }\), and the number of fracture realizations that benefit from aperture variations have linear relationships to the root mean square of the aperture variations. However, the effects of aperture variations become nearly insignificant when \(\sigma _h/\left\langle w \right\rangle < 0.1\). For the investigated range of \(\sigma _h/\left\langle w \right\rangle\) from 0.015 to 0.35, 34 to 49 % of the fracture realizations benefited from aperture variations. Reservoirs with a larger wellbore separation have more uniform base flow, flow in the absence of aperture variations, that is more negatively affected by aperture variations. A reservoir with a large wellbore separation already accesses a larger area of the fracture for heat exchange and any flow channeling induced by aperture variations will more likely decrease the effective heat exchange area. Our finding that aperture variations will lead to adverse thermal performance, more often than not, is in agreement with the previous work of Neuville et al. (2010a, b), despite the fact that their study did not couple heat transport in the fracture with that in the surrounding rock and used a uniform base flow in a rectangular fracture. Additionally, our use of statistical values for an ensemble of realizations, including the mean and standard deviation of production temperatures, offers a better and more practical understanding of the effects fracture aperture variations have on discretely fractured geothermal reservoirs.
We also demonstrated that fracture aperture variations alter the interfracture flow split in a system with multiple parallel fractures. The number of reservoirs exhibiting diminished thermal performance increased compared a reservoir with one fracture and the same mass flow per fracture area. Simply having a higher flow rate in one fracture will more than likely lead to a worse thermal performance because the fracture receiving most of the flow will experience thermal breakthrough more quickly. Further deterioration in thermal performance arises because fractures with a larger flow impedance usually have better thermal performance but receive a smaller portion of the flow. However, applying flow control can alleviate the problems with undesirable flow splits, suggesting a need for inflow control devices specifically suited for geothermal applications rather than “off the shelf” devices made for oil and natural gas operations.
Our analysis suggests that a major cause of poor thermal performing reservoirs is due to interfracture flow variations. With more fractures, the reservoir is more susceptible to adverse thermal performance. In general, more fractures in the reservoir lead to a drop in the variability of thermal performance but also leads to a higher likelihood of having a worse production temperature compared to the base case scenario of no aperture variations. The multiple fracture systems simulated in this study were only hydraulically connected via the injection and production well. Real EGS reservoirs, however, like the one at Soultz-sous-Forêts will have fractures that intersect with each other (Sausse et al. 2010). With fracture-fracture intersections, flow control devices become less effective because they can only control well-to-fracture flow.
The present study dealt only with a static reservoir where the spatially varying aperture field stayed constant with time. Real reservoirs, however, will experience thermal, hydraulic, mechanical, and chemical (THMC) changes that alter the aperture field throughout the energy extraction process. In addition to THMC effects, fracture shearing/slipping during heat extraction will also change the aperture field. In general, aperture change due to THMC effects will increase permeability/aperture in the region enclosed by the injection and production well (Taron and Elsworth 2009; Ghassemi and Zhou 2011). Fracture slippage will lead to an increase in the standard deviation of the aperture fluctuations and result in preferential flow channeling in the direction perpendicular to the shear displacement (Yeo et al. 1998; Plouraboué et al. 2000; Watanabe et al. 2008). Although our work did not incorporate fracture slippage and THMC effects it, nonetheless, provides a framework through which to understand how spatial aperture fluctuations affect thermal performance. It shows that fracture surface variations are a reservoir characteristic that researchers should consider in modeling discretely fractured geothermal reservoirs and performing techo-economic analysis (Beckers et al. 2014; Held et al. 2014). Our systematic approach for generating spatially varying apertures can be used in conjunction with modeling the effects that THMC and fracture slippage have on reservoir thermal performance, by providing a base aperture field that already has a spatially varying aperture field.
The developed numerical framework and the lessons learned from this study are currently being applied to a field laboratory at the Altona Flat Rocks in Plattsburgh, New York. The Altona reservoir represents a meso-scale field site, larger than a bench scale, artificial reservoir but smaller than a full sized geothermal reservoir. The Altona reservoir consists of Cambrian Potsdam Sandstone formation, where a five spot well system has been drilled in a 10 m by 10 m area, intersecting an isolated horizontal fracture 7.6 m deep (Becker and Tsoflias 2010). Ground penetrating radar has been used to determine the spatial aperture of the fracture (Tsoflias and Becker 2008), creating a reservoir with a characterized subsurface. The site has been used to conduct tracer tests, demonstrating the presence of flow channelizations caused by the spatially varying aperture field (Castagna et al. 2011; Becker et al. 2015; Hawkins et al. 2015). The site resembles the reservoir model used in this study, with an induced dipole flow, Gaussian distribution of aperture, and a self-affine spatial correlation. Future experimentation at the site will include deployment of thermally degrading and sorbing tracers and reservoir heating through injection of hot water as an analog of the geothermal energy extraction process.
Declarations
Authors’ contributions
This study was performed by DBF with advice on the planning, design, and geothermal application of the model provided by DLK and JWT. DBF wrote the manuscript and it was revised and the final version approved by all three authors. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the National Science Foundation IGERT program (contract 0966045), the U.S. Department of Energy (contract DE-EE0000842), and the Cornell Energy Institute for partial financial support for this work. They would also like to thank their colleagues at Cornell University, specifically Koenraad Beckers, Adam Hawkins, Maciej Łukawski, and Calvin Whealton for their input and suggestions about this work. The authors appreciate the feedback and advice received from the two anonymous reviewers. Their input helped strengthen this work.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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