Abstract
In this work, we consider a single-phase flow and heat transfer problem in fractured geothermal reservoirs. Mixed dimensional problems are considered, where the temperature and pressure equations are solved for porous matrix and fracture networks with transfer term between them. For the fine-grid approximation, a finite volume method with embedded fracture model is employed. To reduce size of the fine-grid system, an upscaled coarse-grid model is constructed using the nonlocal multicontinuum (NLMC) method. We present numerical results for two-dimensional problems with complex fracture distributions and investigate an accuracy of the proposed method. The simulations using upscaled model provide very accurate solutions with significant reduction in the dimension of problem.
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Acknowledgements
The authors thank the anonymous reviewers for their constructive comments.
Funding
MV and VA’s works are supported by the grant of Russian Scientific Fund N17-71-20055 and the mega-grant of Russian Federation Government (N 14.Y26.31.0013). MB’s contribution was made through CUHK-UoM Research Fund. EC and MB’s work are also partially supported by Hong Kong RGC General Research Fund (Project 14304217), CUHK Direct Grant for Research 2017-18, and CUHK-UoM Research Fund.
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Appendix
Appendix
In this Appendix, we present the detailed definitions of the fine-scale matrices introduced in Section 3. Our solution strategy includes:
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Solve the pressure equation and find p = (pm,pf)T
$$ M^{\text{flow}} \frac{p - \check{p} }{\tau} + A^{\text{flow}} p = G^{\text{flow}} + S^{\text{flow}} \frac{T - \check{T} }{\tau}, $$where
$$ M^{\text{flow}} = \left( \begin{array}{cccc} M^{\text{flow}}_{m} & 0 \\ 0 & M^{\text{flow}}_{f} \end{array}\right), \quad S^{\text{flow}} =\! \left( \begin{array}{cccc} S^{\text{flow}}_{m} & 0 \\ 0 & S^{\text{flow}}_{f} \end{array}\right), \quad G^{\text{flow}} = \left( \begin{array}{cccc} G^{\text{flow}}_{m} \\ G^{\text{flow}}_{f} \end{array}\right), $$$$ A^{\text{flow}} = \left( \begin{array}{cccc} A^{\text{flow}}_{m} + Q & -Q \\ -Q & A^{\text{flow}}_{f}+Q \end{array}\right), $$ -
Solve the temperature equation and find T = (Tm,Tf)T
$$ M^{\text{heat}} \frac{T - \check{T} }{\tau} + A^{\text{heat}}(p) T = G^{\text{heat}} + S^{\text{heat}} \frac{p - \check{p} }{\tau}, $$where
$$ M^{\text{heat}} = \left( \begin{array}{cccc} M^{\text{heat}}_{m} & 0 \\ 0 & M^{\text{heat}}_{f} \end{array}\right), \quad S^{\text{heat}} = \left( \begin{array}{cccc} S^{\text{heat}}_{m} & 0 \\ 0 & S^{\text{heat}}_{f} \end{array}\right), \quad G^{\text{heat}} = \left( \begin{array}{cccc} G^{\text{heat}}_{m} \\ G^{\text{heat}}_{f} \end{array}\right), $$$$ A^{\text{heat}} = \left( \begin{array}{cccc} A^{\text{heat}}_{m} + C^{UP}_{m}(p_{m}) + L^{UP}(p_{m}, p_{f}) + {\Theta} & -{\Theta} + L^{UP}(p_{m}, p_{f})\\ -{\Theta} - L^{UP}(p_{m}, p_{f}) & A^{\text{heat}}_{f} + C^{UP}_{f}(p_{f}) - L^{UP}(p_{m}, p_{f}) +{\Theta} \end{array}\right), $$.where
$$ M^{\text{flow}}_{m} = \{m^{\text{flow},m}_{ij}\}, \quad m^{\text{flow},m}_{ij} = \left\{\begin{array}{cccc} a_{m,i} |\xi_{i}| & i = j, \\ 0 & i \neq j \end{array}\right. , $$$$ M^{\text{flow}}_{f} = \{m^{\text{flow},f}_{ln}\}, \quad m^{\text{flow},f}_{ln} = \left\{\begin{array}{cccc} a_{f,l} |\iota_{l}| & l = n, \\ 0 & l \neq n \end{array}\right. , $$$$ M^{\text{heat}}_{m} = \{m^{\text{heat},m}_{ij}\}, \quad m^{\text{heat},m}_{ij} = \left\{\begin{array}{cccc} (c \rho)_{m,i} |\xi_{i}| & i = j, \\ 0 & i \neq j \end{array}\right. , $$$$ M^{\text{heat}}_{f} = \{m^{\text{heat},f}_{ln}\}, \quad m^{\text{heat},f}_{ln} = \left\{\begin{array}{cccc} (c \rho)_{f,l} |\iota_{l}| & l = n, \\ 0 & l \neq n \end{array}\right. , $$$$ S^{\text{flow}}_{m} = \{s^{\text{flow},m}_{ij}\}, \quad s^{\text{flow},m}_{ij} = \left\{\begin{array}{cccc} s_{m,i} |\xi_{i}| & i = j, \\ 0 & i \neq j \end{array}\right. , $$$$ S^{\text{flow}}_{f} = \{s^{\text{flow},f}_{ln}\}, \quad s^{\text{flow},f}_{ln} = \left\{\begin{array}{cccc} s_{f,l} |\iota_{l}| & l = n, \\ 0 & l \neq n \end{array}\right. , $$$$ S^{\text{heat}}_{m} = \{s^{\text{heat},m}_{ij}\}, \quad s^{\text{heat},m}_{ij} = \left\{\begin{array}{cccc} c_{T,i} \check{T}_{m,i} |\xi_{i}| & i = j, \\ 0 & i \neq j \end{array}\right. , $$$$ S^{\text{heat}}_{f} = \{s^{\text{heat},f}_{ln}\}, \quad s^{\text{heat},f}_{ln} = \left\{\begin{array}{cccc} c_{T,l} \check{T}_{f,l} |\iota_{l}| & l = n, \\ 0 & l \neq n \end{array}\right. , $$$$ Q = \{q_{il}\}, \quad q_{il} = \left\{\begin{array}{cccc} \sigma & i = l, \\ 0 & i \neq l \end{array}\right. , \quad {\Theta} = \{\theta_{il}\}, \quad \theta_{il} = \left\{\begin{array}{cccc} \beta & i = l, \\ 0 & i \neq l \end{array}\right. , $$and \(A^{\text {flow}}_{m} = \{W^{m}_{ij}\}\), \(A^{\text {flow}}_{f} = \{W^{f}_{ln}\}\), \(A^{\text {heat}}_{m} = \{D^{m}_{ij}\}\), \(A^{\text {heat}}_{f} = \{D^{f}_{ln}\}\), \(G^{\alpha }_{m} = \{ g^{\alpha }_{m,i} |\xi _{i}|\}\), \(G^{\alpha }_{f} = \{g^{\alpha }_{f,l} |\iota _{l}| \}\), where α = heat or flow. For convection terms \(C^{UP}_{m}\) and \(C^{UP}_{f}\), we use upwind scheme and LUP is the convective mass transfer between matrix and fracture with upwind approximation.
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Vasilyeva, M., Babaei, M., Chung, E.T. et al. Upscaling of the single-phase flow and heat transport in fractured geothermal reservoirs using nonlocal multicontinuum method. Comput Geosci 23, 745–759 (2019). https://doi.org/10.1007/s10596-019-9817-1
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DOI: https://doi.org/10.1007/s10596-019-9817-1