Abstract
Our research is positioned within the framework of subsurface resource utilization for sustainable economies. We concentrate on modeling the underground single-phase fluid flow affected by geological faults using numerical simulations. The study of such flows is characterized by strong uncertainites in the data defing the problem due to the difficulty of taking precise measurements in the subsoil. We aim to demonstrate the feasibility of a reduced order model that is both reliable and computationally efficient, thereby facilitating the incorporation of uncertainties. We account for the uncertainities of the properties of the rock and the geometry of the fault. The latter is achieved by using a radial basis function mesh deformation method. This approach benefits from a mixed-dimensional framework to model the rock matrix and faults as n and \({n-1}\) dimensional domains, allowing for non-conforming meshes. Our primary focus is on a reduced-order model capable of reproducing flow variables across the entire domain. We utilize the Deep Learning Reduced Order Model (DL-ROM), a nonintrusive neural network-based technique, and we compare it against the traditional Proper Orthogonal Decomposition (POD) method across various scenarios. The most relevant contributions of this work are: the proof of concept of the use of neural network for reduced order models for subsoil flow, dealing with non-affine problems and mixed dimensional domain. Additionally, we generalize an existing mesh deformation method for discontinuous deformation maps. Our analysis highlights the capability of reduced order model, highlighting DL-ROM’s capacity to expedite complex analyses with promising accuracy and efficiency, making multi-query analyses with various quantities of interest affordable.
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References
Nordbotten, J.M., Celia, M.A.: Geological storage of \(CO_2\) modeling approaches for large-scale simulation. Wiley (2012)
Lu, C., Sun, Y., Buscheck, T.A., Hao, Y., White, J.A., Chiaramonte, L.: Uncertainty quantification of \(CO_2\) leakage through a fault with multiphase and nonisothermal effects. Greenhouse Gases: Science and Technology 2(6), 445–459 (2012). https://doi.org/10.1002/ghg.1309
Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations. Springer International Publishing (2016). https://doi.org/10.1007/978-3-319-15431-2
Hesthaven, J.S., Rozza, G., Stamm, B.: Certified reduced basis methods for parametrized partial differential equations. Springer International Publishing (2016). https://doi.org/10.1007/978-3-319-22470-1
Brunton, S.L., Kutz, J.N.: Data-driven science and engineering: machine learning, dynamical systems, and control. Cambridge University Press (2019). https://doi.org/10.1017/9781108380690
Benner, P., Mehrmann, V.L., Sorensen, D.C.: Dimension reduction of large-scale systems. Springer (2005)
Maday, Y., Nguyen, N.C., Patera, A.T., Pau, S.H.: A general multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8(1), 383–404 (2009). https://doi.org/10.3934/cpaa.2009.8.383
Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010). https://doi.org/10.1017/s0022112010001217
Kalur, A., Mortimer, P., Sirohi, J., Geelen, R., Willcox, K.E.: Data-driven closures for the dynamic mode decomposition using quadratic manifolds. In: AIAA AVIATION 2023 forum. American Institute of Aeronautics and Astronautics (2023). https://doi.org/10.2514/6.2023-4352
Peherstorfer, B., Willcox, K., Gunzburger, M.: Survey of multifidelity methods in uncertainty propagation, inference, and optimization. SIAM Rev. 60(3), 550–591 (2018). https://doi.org/10.1137/16m1082469
Peherstorfer, B., Willcox, K.: Dynamic data-driven reduced-order models. Comput. Methods Appl. Mech. Eng. 291, 21–41 (2015). https://doi.org/10.1016/j.cma.2015.03.018
Ahmed, H.F., Farooq, H., Akhtar, I., Bangash, Z.: Machine learning-based reduced-order modeling of hydrodynamic forces using pressure mode decomposition. Proc. Inst. Mech. Eng., Part G: J. Aerosp. Eng. 235(16), 2517–2528 (2021). https://doi.org/10.1177/0954410021999864
Im, S., Lee, J., Cho, M.: Surrogate modeling of elasto-plastic problems via long short-term memory neural networks and proper orthogonal decomposition. Comput. Methods Appl. Mech. Eng. 385, 114030 (2021). https://doi.org/10.1016/j.cma.2021.114030
Fu, J., Xiao, D., Fu, R., Li, C., Zhu, C., Arcucci, R., Navon, I.M.: Physics-data combined machine learning for parametric reduced-order modelling of nonlinear dynamical systems in small-data regimes. Comput. Methods Appl. Mech. Eng. 404, 115771 (2023). https://doi.org/10.1016/j.cma.2022.115771
Gonzalez, F.J., Balajewicz, M.: Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems. arXiv:1808.01346 (2018). https://doi.org/10.48550/ARXIV.1808.01346
Murata, T., Fukami, K., Fukagata, K.: Nonlinear mode decomposition with convolutional neural networks for fluid dynamics. J. Fluid Mech. 882 (2019). https://doi.org/10.1017/jfm.2019.822
Hasegawa, K., Fukami, K., Murata, T., Fukagata, K.: Machine-learning-based reduced-order modeling for unsteady flows around bluff bodies of various shapes. Theoret. Comput. Fluid Dyn. 34(4), 367–383 (2020). https://doi.org/10.1007/s00162-020-00528-w
Fresca, S., Dede’, L., Manzoni, A.: A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs. J. Sci. Comput. 87(2) (2021). https://doi.org/10.1007/s10915-021-01462-7
Fresca, S., Manzoni, A.: Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models. Fluids 6(7), 259 (2021). https://doi.org/10.3390/fluids6070259
Fresca, S., Manzoni, A.: POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition. Comput. Methods Appl. Mech. Eng. 388, 114181 (2022). https://doi.org/10.1016/j.cma.2021.114181
Fresca, S., Fatone, F., Manzoni, A.: Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models. Math. Eng. 5(6), 1–36 (2023). https://doi.org/10.3934/mine.2023096
Franco, N., Manzoni, A., Zunino, P.: A deep learning approach to reduced order modelling of parameter dependent partial differential equations. Math. Comput. 92(340), 483–524 (2022). https://doi.org/10.1090/mcom/3781
Franco, N.R., Fresca, S., Manzoni, A., Zunino, P.: Approximation bounds for convolutional neural networks in operator learning. Neural Netw. 161, 129–141 (2023). https://doi.org/10.1016/j.neunet.2023.01.029
Fu, R., Xiao, D., Navon, I., Fang, F., Yang, L., Wang, C., Cheng, S.: A non-linear non-intrusive reduced order model of fluid flow by auto-encoder and self-attention deep learning methods. Int. J. Numer. Meth. Eng. 124(13), 3087–3111 (2023). https://doi.org/10.1002/nme.7240
Regazzoni, F., Pagani, S., Salvador, M., Dede’, L., Quarteroni, A.: Latent dynamics networks (LDNets): learning the intrinsic dynamics of spatio-temporal processes (2023). https://doi.org/10.48550/ARXIV.2305.00094
Wangen, M.: Physical principles of sedimentary basin analysis. Cambridge University Press (2009)
Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications. Springer Series in Computational Mathematics. Springer, Berlin Heidelberg (2013)
Martin, V., Jaffré, J., Roberts, J.E.: Modeling fractures and barriers as interfaces for flow in porous media. SIAM J. Sci. Comput. 26(5), 1667–1691 (2005). https://doi.org/10.1137/s1064827503429363
D’Angelo, C., Scotti, A.: A mixed finite element method for Darcy flow in fractured porous media with non-matching grids. Mathematical Modelling and Numerical Analysis 46(02), 465–489 (2012). https://doi.org/10.1051/m2an/2011148
Flemisch, B., Berre, I., Boon, W., Fumagalli, A., Schwenck, N., Scotti, A., Stefansson, I., Tatomir, A.: Benchmarks for single-phase flow in fractured porous media. Adv. Water Resour. 111, 239–258 (2018). https://doi.org/10.1016/j.advwatres.2017.10.036
Boon, W.M., Nordbotten, J.M., Yotov, I.: Robust discretization of flow in fractured porous media. SIAM J. Numer. Anal. 56(4), 2203–2233 (2018). https://doi.org/10.1137/17m1139102
Berre, I., Boon, W.M., Flemisch, B., Fumagalli, A., Gläser, D., Keilegavlen, E., Scotti, A., Stefansson, I., Tatomir, A., Brenner, K., Burbulla, S., Devloo, P., Duran, O., Favino, M., Hennicker, J., Lee, I.H., Lipnikov, K., Masson, R., Mosthaf, K., Nestola, M.G.C., Ni, C.F., Nikitin, K., Schädle, P., Svyatskiy, D., Yanbarisov, R., Zulian, P.: Verification benchmarks for single-phase flow in three-dimensional fractured porous media. Adv. Water Resour. 147 (2020). https://doi.org/10.1016/j.advwatres.2020.103759
Formaggia, L., Fumagalli, A., Scotti, A., Ruffo, P.: A reduced model for darcy’s problem in networks of fractures. ESAIM: Mathematical Modelling and Numerical Analysis 48(4), 1089–1116 (2014). https://doi.org/10.1051/m2an/2013132
Nordbotten, J.M., Boon, W.M., Fumagalli, A., Keilegavlen, E.: Unified approach to discretization of flow in fractured porous media. Comput. Geosci. 23(2), 225–237 (2018). https://doi.org/10.1007/s10596-018-9778-9
Blazek, J.: Computational Fluid Dynamics: Principles and Applications, 3rd edn. Butterworth-Heinemann, Oxford (2015). https://doi.org/10.1016/B978-0-08-099995-1.09986-3
Hirish, C.: Numerical Computation of Internal and External Flows. Elsevier (2007). https://doi.org/10.1016/b978-0-7506-6594-0.x5037-1
Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6(3), 405–432 (2002). https://doi.org/10.1023/a:1021291114475
Nordbotten, J.M., Keilegavlen, E.: In: Polyhedral Methods in Geosciences, pp. 119–158. Springer International Publishing (2021). https://doi.org/10.1007/978-3-030-69363-3_4
Starnoni, M., Berre, I., Keilegavlen, E., Nordbotten, J.M.: Consistent MPFA discretization for flow in the presence of gravity. Water Resour. Res. 55(12), 10105–10118 (2019). https://doi.org/10.1029/2019wr025384
Stefansson, I., Berre, I., Keilegavlen, E.: Finite-volume discretisations for flow in fractured porous media. Transp. Porous Media 124(2), 439–462 (2018). https://doi.org/10.1007/s11242-018-1077-3
Keilegavlen, E., Berge, R., Fumagalli, A., Starnoni, M., Stefansson, I., Varela, J., Berre, I.: Porepy: an open-source software for simulation of multiphysics processes in fractured porous media. arXiv:1908.09869 (2019). https://doi.org/10.48550/ARXIV.1908.09869
Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015). https://doi.org/10.1137/130932715
Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1(3), 211–218 (1936). https://doi.org/10.1007/bf02288367
Schmidt, E.: Zur theorie der linearen und nichtlinearen integralgleichungen. Math. Ann. 63(4), 433–476 (1907). https://doi.org/10.1007/bf01449770
DeVore, R.A., Howard, R., Micchelli, C.: Optimal nonlinear approximation. Manuscripta Math. 63(4), 469–478 (1989). https://doi.org/10.1007/bf01171759
de Boer, A., van der Schoot, M., Bijl, H.: Mesh deformation based on radial basis function interpolation. Computers & Structures 85(11), 784–795 (2007). https://doi.org/10.1016/j.compstruc.2007.01.013
Forti, D., Rozza, G.: Efficient geometrical parametrisation techniques of interfaces for reduced-order modelling: application to fluid-structure interaction coupling problems. Int. J. Comput. Fluid Dyn. 28(3–4), 158–169 (2014). https://doi.org/10.1080/10618562.2014.932352
Aubert, S., Mastrippolito, F., Rendu, Q., Buisson, M., Ducros, F.: Planar slip condition for mesh morphing using radial basis functions. Procedia Eng. 203, 349–361 (2017). https://doi.org/10.1016/j.proeng.2017.09.819
Kingma, D.P., Ba, J.,: Adam: a method for stochastic optimization. arXiv:1412.6980 (2014). https://doi.org/10.48550/ARXIV.1412.6980
Berre, I., Boon, W.M., Flemisch, B., Fumagalli, A., Gläser, D., Keilegavlen, E., Scotti, A., Stefansson, I., Tatomir, A., Brenner, K., Burbulla, S., Devloo, P., Duran, O., Favino, M., Hennicker, J., Lee, I.H., Lipnikov, K., Masson, R., Mosthaf, K., Nestola, M.G.C., Ni, C.F., Nikitin, K., Schädle, P., Svyatskiy, D., Yanbarisov, R., Zulian, P.: Verification benchmarks for single-phase flow in three-dimensional fractured porous media. Adv. Water Resour. 147, 103759 (2021). https://doi.org/10.1016/j.advwatres.2020.103759
Winter, R., Valsamidou, A., Class, H., Flemisch, B.: A study on darcy versus forchheimer models for flow through heterogeneous landfills including macropores. Water 14(4), 546 (2022). https://doi.org/10.3390/w14040546
Feinberg, J., Langtangen, H.P.: Chaospy: an open source tool for designing methods of uncertainty quantification. J. Comput. Sci. 11, 46–57 (2015). https://doi.org/10.1016/j.jocs.2015.08.008
Saltelli, A., Tarantola, S., Campolongo, F., Ratto, M.: Sensitivity Analysis in Practice, p. 232. Wiley (2004)
Scott, D.W.: Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley (1992). https://doi.org/10.1002/9780470316849
Storn, R., Price, K.: Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11(4), 341–359 (1997). https://doi.org/10.1023/a:1008202821328
Virtanen, P., Gommers, R., Oliphant, T., et al.: SciPy 1.0: fundamental algorithms for scientific computing in python. Nature Methods 17(3), 261–272 (2020). https://doi.org/10.1038/s41592-019-0686-2
Funding
Open access funding provided by Politecnico di Milano within the CRUI-CARE Agreement. The research has been supported by the Italian Ministry of Universities and Research (MUR) under the project “PON Ricerca e Innovazione 2014-2020” and was carried out in collaboration with Eni S.p.A. The last three authors also gratefully acknowledge the support of the ”Dipartimento di Eccellenza 2023-2027”. All authors warmly thank Andrea Manzoni, Stefano Micheletti, and Nicola Rares Franco for many insightful discussions.
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A Nomenclature
A Nomenclature
- a:
-
Trainable parameter of PReLU
- A:
-
Matrix describing the discrete equation
- b:
-
Discrete right-hand side
- \(C, C_d, C_{df}, C_s, C_{sf}\):
-
Control points sets
- \(\overline{C}_s\):
-
Set of surfaced where sliding conditions are applied
- d(x, y):
-
Euclidean distance between x and y
- D:
-
Spatial dimension
- e:
-
Number of parameters
- \(e_{min}, e_{max}, e_{ave}\):
-
Minimum, maximum, averaged relative errors between the FOM and ROM solution
- \(f [{1/s}]\):
-
Scalar source or sink term
- \(f_\gamma [{1/s}]\):
-
Scalar source or sink term in the fault
- G:
-
Matrix of displacement constraint
- g:
-
Radial basis function dependent on the distance, d, of two points, \(x_1, x_2\)
- \(g^*\):
-
Radial basis function dependent on \(x_1, x_2\)
- \(g^\dag \):
-
Modified radial basis function
- \(H_k\):
-
Matrix of no tangential contribution constraint applied to surface k
- \(\mathcal {I}\):
-
influence function
- l:
-
Number of control points
- N:
-
Number of degrees of freedom of the full order problem
- n:
-
Number of degrees of freedom of the reduced problem
- \(n_{br}\):
-
number of intersecting branches
- \(n_{\min }\):
-
Minimal latent dimension
- \(p [{\text {Pa}}]\):
-
Pressure
- \(\overline{p} [{\text {Pa}}]\):
-
Pressure on boundaries
- \(p_\gamma [{Pa}]\):
-
Pressure in the fault
- \(p_\iota [{Pa}]\):
-
Pressure at the intersection
- \(\overline{\Delta p} [{Pa}]\):
-
Mean value of \(\Delta p\)
- \(q [{m/s}]\):
-
Darcy velocity
- \(\overline{q} [{m/s}]\):
-
Darcy velocity on boundaries
- \(K [{m^{3}s/kg}]\):
-
Intrinsic permeability scaled by the dynamic viscosity
- \(K_{n} [{m^{3} s / kg}]\):
-
Normal fault permeability
- \(K_{\tau } [{m^{3} s / kg}]\):
-
In-plane fault permeability
- \(K_{\iota } [{m^{3} s / kg}]\):
-
Representative permeability at intersection
- r:
-
Map from \(\gamma \) to \(\partial _{in}\Omega \)
- S:
-
Snapshot matrix
- \(s [{m}]\):
-
Displacement
- \(\overline{s} [{m}]\):
-
Known displacement
- \(\tilde{s}\):
-
First order sensitivity index
- t, b:
-
Non-parallel tangent unit vectors of sliding surface
- \(u_N\):
-
Full order model solution
- \(\tilde{u}_N\):
-
Reconstructed solution
- \(u_n\):
-
Reduced order model solution
- U:
-
Left singular vector matrix
- \(U_{tr}\):
-
Left singular vector matrix truncated
- V:
-
Right singular vector matrix
- z:
-
Unknown of mesh deformation linear system
- \(\mathscr {L}\):
-
Loss function
- \(\mathcal {M}\):
-
Map from full order model space to reduced space
- \(\mathcal {S}\):
-
Solution manifold
- \(\mathcal {V}_n\):
-
Reduced problem solution space
- \(\mathcal {V}_N\):
-
Full order model solution space
- \(\alpha , \beta \):
-
User-defined loss function weights
- \(\beta \):
-
Side function
- \(\gamma \):
-
Fault domain
- \(\partial _p \gamma \):
-
Boundary of \(\gamma \) where Dirichlet boundary condition for the pressure is applied
- \(\partial _q \gamma \):
-
Boundary of \(\gamma \) where Neumann boundary condition is applied
- \(\partial _{ex}\gamma \):
-
Boundary of \(\gamma \) in contact with \(\partial \Omega \)
- \(\partial _{in}\gamma \):
-
Boundary of \(\gamma \) not in contact with \(\partial \Omega \)
- \(\gamma ^+ (\gamma ^-)\):
-
Additional interfaces between the matrix domain, \(\Omega \) and fault domain, \(\gamma \)
- \(\delta _n\):
-
Non-linear counterpart of Kolmogorov n-width.
- \(\epsilon [{\text {m}}]\):
-
Fault aperture
- \(\zeta \):
-
Unknown coefficients of the linear combination of radial functions
- \(\eta \):
-
Exponent defining the permeability
- \(\lambda ^+ (\lambda ^-) [{m/s}]\):
-
Volumetric fluid flux exchanged between subdomains
- \(\lambda _\gamma [{m/s}]\):
-
Volumetric fluid flux exchanged between branches of a intersection
- \(\mu \):
-
Parameters
- \(\mu _{geom}\):
-
Geometrical parameters
- \(\mu _{phy}\):
-
Physical parameters
- \(\nu \):
-
Normal of a sliding surface
- \(\rho \):
-
Activation function
- \(\sigma \):
-
Right-hand side of mesh deformation system
- \(\tilde{\sigma }\):
-
Standard deviation
- \(\theta \):
-
Scalar function \(\mu \)-dependent
- \(\Theta \):
-
Parameter space
- \(\Sigma \):
-
Singular values matrix
- \(\upsilon \):
-
Unit vector
- \(\upsilon _\gamma \):
-
Unit vector associated to \(\gamma \)
- \(\hat{\upsilon }\):
-
Unit vector aligned with the fault
- \(\varphi \):
-
Map \(\varphi : \Theta \rightarrow \mathcal {V}_n\). In the DL-ROM approach, it is represented by the reduced map network
- \(\Phi \):
-
Transition matrix
- \(\Psi \):
-
Map \(\Psi : \mathcal {V}_n \rightarrow \mathcal {V}_N\). In the DL-ROM approach, it is represented by a decoder
- \(\Psi '\):
-
Map \(\Psi ' : \mathcal {S} \rightarrow \mathcal {V}_n\). In the DL-ROM approach, it is represented by an encoder
- \(\Omega \):
-
Matrix domain
- \(\partial \Omega \):
-
Boundary of \(\Omega \)
- \(\partial _p \Omega \):
-
Boundary of \(\Omega \) where Dirichlet boundary condition for the pressure is applied
- \(\partial _q \Omega \):
-
Boundary of \(\Omega \) where Neumann boundary condition is applied
- \(\partial _{ex} \Omega \):
-
External boundary of \(\Omega \)
- \(\partial _{in} \Omega \):
-
Internal boundary of \(\Omega \)
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Ballini, E., Formaggia, L., Fumagalli, A. et al. Application of deep learning reduced-order modeling for single-phase flow in faulted porous media. Comput Geosci (2024). https://doi.org/10.1007/s10596-024-10320-y
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DOI: https://doi.org/10.1007/s10596-024-10320-y