Abstract
Efficient heat exploitation strategies from geothermal systems demand for accurate and efficient simulation of coupled flow-heat equations on large-scale heterogeneous fractured formations. While the accuracy depends on honouring high-resolution discrete fractures and rock heterogeneities, specially avoiding excessive upscaled quantities, the efficiency can be maintained if scalable model-reduction computational frameworks are developed. Addressing both aspects, this work presents a multiscale formulation for geothermal reservoirs. To this end, the nonlinear time-dependent (transient) multiscale coarse-scale system is obtained, for both pressure and temperature unknowns, based on elliptic locally solved basis functions. These basis functions account for fine-scale heterogeneity and discrete fractures, leading to accurate and efficient simulation strategies. The flow-heat coupling is treated in a sequential implicit loop, where in each stage, the multiscale stage is complemented by an ILU(0) smoother stage to guarantee convergence to any desired accuracy. Numerical results are presented in 2D to systematically analyze the multiscale approximate solutions compared with the fine scale ones for many challenging cases, including the outcrop-based geological fractured field. These results show that the developed multiscale formulation casts a promising framework for the real-field enhanced geothermal formations.
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Acknowledgments
Hadi Hajibeygi acknowledges the grant from Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg and SimTech Centre of Stuttgart University.
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Appendices
Appendix A: Fluid model
All water properties are calculated using curve fitting approach with the correlations provided by the International Association for the Properties of Water and Steam Industrial Formulation 1997 (IAPWS-If97) [43]. Following the literature [30], water density, internal energy and enthalpy read
and
Here, ρws and uws are, respectively, water density and internal energy at saturation condition (ps and Ts). In this work, after curve fitting, the saturation density function reads
The isothermal compressibility values are calculated as a function of T as
The saturation pressure, ps, is calculated based on IAPWS-If97 [43] as
with the validity range of 273.15 K ≤ T ≤ 647.096 K (critical point). Here,
and
hold, with
Moreover, the empirical parameters, ni, are shown in Table 6.
The combination of uws = 420 kJ/kg, Cpw = 4.2 kJ/kg, and Ts = 373 K was found to provide the best fitting values for internal energy calculation. More precisely, compared with the data, the density relative error norms were below 1% in most regions and 2.2 % near the critical point. Similarly, the internal energy errors were less than 6%. Therefore, the presented fitted function has reasonable accuracy at temperature range of 273.15 K ≤ T ≤ 647.096 K and ps(T) ≤ p ≤ 22.064 MPa, which is the single phase liquid region. Finally, the water viscosity [44] reads
Appendix B: Fracture coordinates for the test cases
In this appendix, the fracture coordinates for test cases 1–2 are presented. For test case 3, readers are referred to [39] due to the high number of fractures in the model.
The fractures are defined by two points: A and B and the x and y coordinates are given in the tables.
2.1 B.1 Fracture coordinates for test case 1
The fracture coordinates for test case 1 are listed in Table 7.
2.2 B.2 Fracture coordinates for test case 2
The fracture coordinates for test case 2 are listed in Table 8.
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Praditia, T., Helmig, R. & Hajibeygi, H. Multiscale formulation for coupled flow-heat equations arising from single-phase flow in fractured geothermal reservoirs. Comput Geosci 22, 1305–1322 (2018). https://doi.org/10.1007/s10596-018-9754-4
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DOI: https://doi.org/10.1007/s10596-018-9754-4