# Partial Learning Using Partially Explicit Discretization for Multicontinuum/Multiscale Problems with Limited Observation: Dual Continuum Heterogeneous Poroelastic Media Simulation

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## Abstract

**:**

## 1. Introduction

- 1
- Construct the POD projection basis matrix and define the interpolation points using DEIM.
- 2
- Generate the training dataset.
- •
- Generate the input data.
- •
- Solve the poroelasticity problem with the partially explicit discretization at each time step.

- 3
- Train the Deep Neural Network to obtain values of the implicit flow part at the interpolation points at some time steps.

- 1
- The Deep Neural Network obtains the implicit part of the pressure at the interpolation points at some time moments.
- 2
- The POD projection basis matrix restores the complete implicit parts of the flow at some time moments.
- 3
- Linear interpolation over time.
- 4
- For each time step,
- •
- Compute the explicit flow part using the learned and interpolated implicit one.
- •
- Solve the displacement using the implicit and explicit parts of the flow.

## 2. Problem Formulation

## 3. Approximation

- Pressures $({p}_{1}^{n+1},{p}_{2}^{n+1})\in ({W}_{1}\times {W}_{2})$ such that$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {d}_{1}\left(\frac{{u}^{n}-{u}^{n-1}}{\tau},{w}_{1}\right)+{c}_{1}\left(\frac{{p}_{1}^{n+1}-{p}_{1}^{n}}{\tau},{w}_{1}\right)+{b}_{1}({p}_{1}^{n+1},{w}_{1})+{q}_{12}({p}_{1}^{n+1}-{p}_{2}^{n+1},{w}_{1})=0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {d}_{2}\left(\frac{{u}^{n}-{u}^{n-1}}{\tau},{w}_{2}\right)+{c}_{2}\left(\frac{{p}_{2}^{n+1}-{p}_{2}^{n}}{\tau},{w}_{2}\right)+{b}_{2}({p}_{2}^{n},{w}_{2})-{q}_{21}({p}_{1}^{n+1}-{p}_{2}^{n+1},{w}_{2})=l\left({w}_{2}\right),\hfill \end{array}$$
- Displacements ${u}^{n+1}$ such that$$a({u}^{n+1},v)+{g}_{1}({p}_{1}^{n+1},v)+{g}_{2}({p}_{2}^{n+1},v)=0,$$

- Pressures $({p}_{1}^{n+1},{p}_{2}^{n+1})$ such that$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {B}_{1}{p}_{1}^{n+1}+{C}_{1}\frac{{p}_{1}^{n+1}-{p}_{1}^{n}}{\tau}+{D}_{1}\frac{{u}^{n}-{u}^{n-1}}{\tau}+{Q}_{12}({p}_{1}^{n+1}-{p}_{2}^{n+1})=0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {B}_{2}{p}_{2}^{n}+{C}_{2}\frac{{p}_{2}^{n+1}-{p}_{2}^{n}}{\tau}+{D}_{2}\frac{{u}^{n}-{u}^{n-1}}{\tau}-{Q}_{21}({p}_{1}^{n+1}-{p}_{2}^{n+1})=L.\hfill \end{array}$$
- Displacements ${u}^{n+1}$ such that$$A{u}^{n+1}+{G}_{1}{p}_{1}^{n+1}+{G}_{2}{p}_{2}^{n+1}=0.$$

## 4. Discrete Empirical Interpolation Method with Proper Orthogonal Decomposition

- Computing the Proper Orthogonal Decomposition (POD) basis functions;
- Determining the interpolation nodes using the DEIM algorithm.

## 5. Machine Learning Approach

- The first output layer: ${\mathcal{N}}^{1}\left({x}^{0}\right)={Y}^{1}$, i.e., ${Y}^{1}={\sigma}^{1}({W}^{1}x+{b}^{1})$;
- For i’s layer: ${Y}^{i}={\sigma}^{i}({W}^{i}{x}^{i-1}+{b}^{i})$, where i = 1,2,…, L.

- Activation function: ReLU (Rectified Linear Unit) activation function for all layers (first input and hidden layers), no activation function at the last output layer;
- DNN structure: 2 hidden layers, each layer comprises 12 neurons;
- Kernel initializer: normal for input and output layers and he_normal for all hidden layers;
- Training optimizer: Adam.

## 6. Numerical Results

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Meirmanov, A. Mathematical Models for Poroelastic Flows. 2014. Available online: https://link.springer.com/book/10.2991/978-94-6239-015-7 (accessed on 16 December 2013).
- Castelletto, N.; Klevtsov, S.; Hajibeygi, H.; Tchelepi, H.A. Multiscale two-stage solver for Biot’s poroelasticity equations in subsurface media. Comput. Geosci.
**2019**, 23, 207–224. [Google Scholar] [CrossRef] - Iliev, O.; Kolesov, A.; Vabishchevich, P. Numerical solution of plate poroelasticity problems. Transp. Porous Media
**2016**, 115, 563–580. [Google Scholar] [CrossRef] - Vabishchevich, P.N.; Vasil’eva, M.V. Explicit-implicit schemes for convection-diffusion-reaction problems. Numer. Anal. Appl.
**2012**, 5, 297–306. [Google Scholar] [CrossRef] - Quevedo, R.; Roehl, D. A novel and efficient sequential-explicit technique for poroelasticity problems. Comput. Geotech.
**2021**, 138, 104334. [Google Scholar] [CrossRef] - Almani, T.; Kumar, K.; Singh, G.; Wheeler, M.F. Stability of multirate explicit coupling of geomechanics with flow in a poroelastic medium. Comput. Math. Appl.
**2019**, 78, 2682–2699. [Google Scholar] [CrossRef] - Kim, J. Sequential Methods for Coupled Geomechanics and Multiphase Flow. Doctoral Dissertation, Stanford University, Stanford, CA, USA, 2010. [Google Scholar]
- Kim, J.; Tchelepi, H.A.; Juanes, R. Stability and convergence of sequential methods for coupled flow and geomechanics: Drained and undrained splits. Comput. Methods Appl. Mech. Eng.
**2011**, 200, 2094–2116. [Google Scholar] [CrossRef] - Kim, J.; Tchelepi, H.A.; Juanes, R. Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits. Comput. Methods Appl. Mech. Eng.
**2011**, 200, 1591–1606. [Google Scholar] [CrossRef] - Vabishchevich, P.N.; Vasil’eva, M.V.; Kolesov, A.E. Splitting scheme for poroelasticity and thermoelasticity problems. Comput. Math. Math. Phys.
**2014**, 54, 1305–1315. [Google Scholar] [CrossRef] - Akkutlu, I.Y.; Efendiev, Y.; Vasilyeva, M.; Wang, Y. Multiscale model reduction for shale gas transport in poroelastic fractured media. J. Comput. Phys.
**2018**, 353, 356–376. [Google Scholar] [CrossRef] - Vasilyeva, M.; Chung, E.T.; Efendiev, Y.; Kim, J. Constrained energy minimization based upscaling for coupled flow and mechanics. J. Comput. Phys.
**2019**, 376, 660–674. [Google Scholar] [CrossRef][Green Version] - Ammosov, D.; Vasilyeva, M.; Chung, E.T. Generalized Multiscale Finite Element Method for thermoporoelasticity problems in heterogeneous and fractured media. J. Comput. Appl. Math.
**2022**, 407, 113995. [Google Scholar] [CrossRef] - Vasilyeva, M.; Tyrylgin, A. Machine learning for accelerating macroscopic parameters prediction for poroelasticity problem in stochastic media. Comput. Math. Appl.
**2021**, 84, 185–202. [Google Scholar] [CrossRef] - Chaturantabut, S.; Sorensen, D.C. Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput.
**2010**, 32, 2737–2764. [Google Scholar] [CrossRef] - Wang, Z.; Akhtar, I.; Borggaard, J.; Iliescu, T. Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison. Comput. Methods Appl. Mech. Eng.
**2012**, 237, 10–26. [Google Scholar] [CrossRef][Green Version] - Wang, Z.; Akhtar, I.; Borggaard, J.; Iliescu, T. Two-level discretizations of nonlinear closure models for proper orthogonal decomposition. J. Comput. Phys.
**2011**, 230, 126–146. [Google Scholar] [CrossRef] - Efendiev, Y.; Leung, W.T.; Lin, G.; Zhang, Z. HEI: Hybrid explicit-implicit learning for multiscale problems. arXiv
**2021**, arXiv:2109.02147. [Google Scholar] - Boutin, C.; Royer, P. On models of double porosity poroelastic media. Geophys. Suppl. Mon. Not. R. Astron. Soc.
**2015**, 203, 1694–1725. [Google Scholar] [CrossRef][Green Version] - Zhang, J.; Bai, M.; Roegiers, J.C. Dual-porosity poroelastic analyses of wellbore stability. Int. J. Rock Mech. Min. Sci.
**2003**, 40, 473–483. [Google Scholar] [CrossRef] - Barenblatt, G.I.; Zheltov, I.P.; Kochina, I. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. J. Appl. Math. Mech.
**1960**, 24, 1286–1303. [Google Scholar] [CrossRef] - Jänicke, R.; Larsson, F.; Runesson, K. A poro-viscoelastic substitute model of fine-scale poroelasticity obtained from homogenization and numerical model reduction. Comput. Mech.
**2020**, 65, 1063–1083. [Google Scholar] [CrossRef][Green Version] - Tyrylgin, A.; Vasilyeva, M.; Spiridonov, D.; Chung, E.T. Generalized Multiscale Finite Element Method for the poroelasticity problem in multicontinuum media. J. Comput. Appl. Math.
**2020**, 374, 112783. [Google Scholar] [CrossRef][Green Version] - Tyrylgin, A.; Vasilyeva, M.; Ammosov, D.; Chung, E.T.; Efendiev, Y. Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media. Fluids
**2021**, 6, 298. [Google Scholar] [CrossRef] - D’angelo, C.; Quarteroni, A. On the coupling of 1d and 3d diffusion-reaction equations: Application to tissue perfusion problems. Math. Model. Methods Appl. Sci.
**2008**, 18, 1481–1504. [Google Scholar] [CrossRef][Green Version] - Abadi, M.; Agarwal, A.; Barham, P.; Brevdo, E.; Chen, Z.; Citro, C.; Corrado, G.S.; Davis, A.; Dean, J.; Devin, M.; et al. TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems. 2015. Available online: tensorflow.org (accessed on 2 November 2016).
- Geuzaine, C.; Remacle, J.F. A three-dimensional finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods Eng.
**2020**, 11, 79. [Google Scholar] - Logg, A.; Mardal, K.A.; Wells, G. Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012; Volume 84. [Google Scholar]
- Chollet, F.; Watson, M.; Bursztein, E.; Zhu, Q.S.; Jin, H. keras. 2015. Available online: https://github.com/fchollet/keras (accessed on 27 March 2015).
- Ahrens, J.; Geveci, B.; Law, C. Paraview: An end-user tool for large data visualization. Vis. Handb.
**2005**, 717–735. [Google Scholar]

**Figure 2.**Flow charts of the offline and online stages of partial learning using partially explicit discretization with limited observation.

**Figure 4.**Elasticity parameter E (

**left**) and heterogeneous permeabilities ${k}_{1}$ (

**center**) and ${k}_{2}$ (

**right**).

**Figure 5.**Distribution of pressure for the first continuum at different times, ${t}_{m}$ with $m=0.4,0.8,1$ (from

**left**to

**right**). (

**The first row**): the coupled explicit–implicit solution. (

**The second row**): the split explicit–implicit solution. (

**The third row**): the proposed approach’s solution.

**Figure 6.**Distribution of pressure for the second continuum at different times, ${t}_{m}$ with $m=0.4,0.8,1$ (from

**left**to

**right**). (

**The first row**): the coupled explicit–implicit solution. (

**The second row**): the split explicit–implicit solution. (

**The third row**): the proposed approach’s solution.

**Figure 7.**Distribution of displacement in ${x}_{1}$ direction at different times, ${t}_{m}$ with $m=0.4,0.8,1$ (from

**left**to

**right**). (

**The first row**): the coupled explicit–implicit solution. (

**The second row**): the split explicit–implicit solution. (

**The third row**): the proposed approach’s solution.

**Figure 8.**Distribution of displacement in ${x}_{2}$ direction at different times, ${t}_{m}$ with $m=0.4,0.8,1$ (from

**left**to

**right**). (

**The first row**): the coupled explicit–implicit solution. (

**The second row**): the split explicit–implicit solution. (

**The third row**): the proposed approach’s solution.

**Figure 9.**Relative ${L}_{2}$ errors in % for the reference mesh (6561 vertices) with other meshes (121, 441, and 1681 vertices) for the first and second continuum and displacement (from

**left**to

**right**).

**Figure 10.**Distributions of the stress at the final time. The coupled explicit–implicit solution, the split explicit–implicit solution, and the proposed approach’s solution (from

**left**to

**right**).

**Figure 11.**Distributions of the strain at the final time. The coupled explicit–implicit solution, the split explicit–implicit solution, and the proposed approach’s solution(from

**left**to

**right**).

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**MDPI and ACS Style**

Tyrylgin, A.; Stepanov, S.; Ammosov, D.; Grigorev, A.; Vasilyeva, M.
Partial Learning Using Partially Explicit Discretization for Multicontinuum/Multiscale Problems with Limited Observation: Dual Continuum Heterogeneous Poroelastic Media Simulation. *Mathematics* **2022**, *10*, 2629.
https://doi.org/10.3390/math10152629

**AMA Style**

Tyrylgin A, Stepanov S, Ammosov D, Grigorev A, Vasilyeva M.
Partial Learning Using Partially Explicit Discretization for Multicontinuum/Multiscale Problems with Limited Observation: Dual Continuum Heterogeneous Poroelastic Media Simulation. *Mathematics*. 2022; 10(15):2629.
https://doi.org/10.3390/math10152629

**Chicago/Turabian Style**

Tyrylgin, Aleksei, Sergei Stepanov, Dmitry Ammosov, Aleksandr Grigorev, and Maria Vasilyeva.
2022. "Partial Learning Using Partially Explicit Discretization for Multicontinuum/Multiscale Problems with Limited Observation: Dual Continuum Heterogeneous Poroelastic Media Simulation" *Mathematics* 10, no. 15: 2629.
https://doi.org/10.3390/math10152629