Homogenization theory and nonlinearities in Darcy's law

Guerra, Karl Igor Martins; Romanel, Celso

doi:10.28927/SR.2024.012622

Karl Igor Martins Guerra ^##Corresponding author. E-mail address: karligor@tecgraf.puc-rio.br

conceptualization

data curation

visualization

writing – original draft

Pontífica Universidade Católica do Rio de Janeiro, Departamento de Engenharia Civil e Ambiental, Rio de Janeiro, RJ, Brasil.

https://orcid.org/0000-0002-0253-6648

Celso Romanel

supervision

validation

writing – original draft

Pontífica Universidade Católica do Rio de Janeiro, Departamento de Engenharia Civil e Ambiental, Rio de Janeiro, RJ, Brasil.

https://orcid.org/0000-0002-8812-3456

#Corresponding author. E-mail address: karligor@tecgraf.puc-rio.br

Declaration of interest

The authors have no conflicts of interest to declare. All co-authors have observed and affirmed the contents of the paper and there is no financial interest to report.

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Figure 1
Periodically perforated domain.

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Figure 2
Unit cell.

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Figure 3
Homogenized domain.

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Figure 4
Particle size distribution of soils

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Figure 5
Limits for a(ε) and σ(ε).

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Figure 6
Representation of the unit cell with more than one grain.

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Figure 7

P_{C e l} - N u m

domain.

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Figure 8
Discretization of the finite element space.

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Figure 9
T2-type element.

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Figure 10
DNS convergence for Darcy.

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Figure 11
DNS convergence for Navier-Stokes.

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Figure 12
DNS convergence for Darcy with calculated limit.

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Figure 13
DNS convergence for Navier-Stokes with calculated limit.

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Figure 14
Relation between A and C_u/C_c.

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Figure 15
Comparison of laboratory results and model prediction for K.

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Figure 16
Loss of linearity in function of the hydraulic gradient for different augmented viscosities.

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Figure 17
Relation between the ratio C_u/C_c and the critical hydraulic gradient.

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Figure 18
Relation between the estimated hydraulic conductivity and the critical hydraulic gradient.

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Table 1
Laboratory tests used for calibration of the model.

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Table 2
Relation between augmented viscosity and the PDE limit.

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Table 3
Critical Reynolds number in function of the ratio C_u/C_c.

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(48)

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Reference	e	ф	D₅₀ (mm)	C_c	C_u	A	K-Lab (cm/s)
Yang et al. (2016)Yang, B., Liu, Y., Wan, F., Yang, T., Feng, J., Zhao, X., & Zheng, D. (2016). Experimental study on influence of particle-size distribution on permeability coefficient of sand. Journal of Southwest Jiaotong University, 51(5), 855-861.	0.525	0.34426	0.5	0.75	3	0.0055	1.84e-3
	0.525	0.34426	1	1.03	5.6	0.002	3.83e-3
	0.525	0.34426	2	1.45	11	0.0015	3.68e-3
	0.559	0.35856	2	0.75	5.6	0.0017	1.23e-3
	0.559	0.35856	0.5	1.03	11	0.001	3.35e-3
	0.559	0.35856	1	1.45	3	0.0011	8.99e-3
	0.572	0.36387	1	0.75	11	0.002	1.36e-3
	0.572	0.36387	2	1.03	3	0.0006	3.18e-2
	0.572	0.36387	0.5	1.45	5.6	0.0002	9.44e-4
Akbulut et al. (2014) Akbulut, N., Wisniewski, M., & Calabar, A.F. (2014). Influences of grain shape and size distribution on permeability. In Proceedings of the TC105 ISSMGE International Symposium on Geomechanics from Micro to Macro (pp. 1451-1454). Cambridge.	0.664	0.39904	1.27	2.14	7.41	0.001	1.89e-1
	0.796	0.44321	1.63	0.89	1.35	0.075	1.26
	0.892	0.47146	0.84	0.93	1.41	0.05	4.18e-1
	0.641	0.39062	1.27	2.14	7.41	0.007	1.73e-1
	0.788	0.44072	1.63	0.89	1.35	0.073	1.23
	0.843	0.45741	0.84	0.93	1.41	0.05	3.75e-1
	0.495	0.3311	1.27	2.14	7.41	0.001	1.39e-1
	0.645	0.3921	1.63	0.89	1.35	0.07	1.07
	0.707	0.41418	0.84	0.93	1.41	0.05	3.96e-1
	0.966	0.49135	1.27	2.14	7.41	0.001	2.69e-1
	1,059	0.51433	1.63	0.89	1.35	0.09	1.46
	1,153	0.53553	0.84	0.93	1.41	0.07	5.97e-1
Calabar & Akbulut (2016)Calabar, A. F., Akbulut, N. (2016). Evaluation of actual and estimated hydraulic conductivity of sands with different gradation and shape. SpringerPlus, 5, 820. https://doi.org/10.1186/s40064-016-2472-2.	0.86	0.46237	3.1	0.93	1.5	0.03	1.61
	0.82	0.45055	1.63	0.89	1.35	0.036	0.35
	0.77	0.43503	0.84	0.93	1.41	0.034	0.13
	0.72	0.4186	0.5	1	1.19	0.02	0.05
	0.6	0.375	0.35	0.98	1.16	0.018	0.03
	0.87	0.46524	1	0.73	4.39	0.011	0.03
	0.82	0.45055	0.69	1.01	3.63	0.006	0.02
	0.79	0.44134	0.5	1.44	3.47	0.011	0.02
	0.7	0.41176	0.39	1.74	3.27	0.0004	0.02
	0.63	0.3865	0.32	1.29	3.09	0.0015	0.01
	0.56	0.35897	0.16	0.92	2.07	0.012	0.01
	0.74	0.42529	2.2	0.98	2.6	0.015	0.26
	0.67	0.4012	1.18	0.88	1.93	0.022	0.12
	0.52	0.34211	2	1.22	3.28	0.0083	0.14
	0.7	0.41176	0.99	0.8	2.41	0.025	0.06
	0.55	0.35484	0.7	0.93	1.71	0.016	0.04

u*	M₀	C_c	a(ε) (mm)	ε (mm)	A	σ²	σ² - fit	Limit model
0.001	12848.4	27.3	1	6	14.14	912.12	908.6	NSI
0.01	12848.4	14.2	1	6	4.49	289.82	285.95	NSI
0.1	12848.4	7.3	0.99	6	1.39	90.21	91.86	NSI
1	12848.4	3.8	0.99	6	0.44	28.72	28.87	NSI
10	12848.4	1.95	0.97	6	0.14	9.05	9.3	Brinkman
100	12848.4	1	0.84	6	0.04	3.01	3	Darcy
1000	12848.4	0.5	0.66	6	0.012	1	1.01	Darcy
10000	12848.4	0.25	0.53	6	0.003	0.33	0.3415	Darcy

C_c	C_u	C_u/C_c	i-critical	μ*	K (cm/s)	u-critical (m/s)	R_e	Limit model
27.3	1	0.03663	1.00E-05	0.001	70.71	7.00E-07	7.07E-04	Navier-Stokes
14.2	1	0.07042	1.00E-04	0.01	22.25	2.23E-06	2.23E-03	Navier-Stokes
7.3	1	0.13699	1.00E-03	0.1	7.149	7.15E-06	7.15E-03	Navier-Stokes
3.8	1	0.26316	1	1	2.247	2.25E-03	2.23E+00	Navier-Stokes
1.95	1	0.51282	1.00E+02	10	0.724	7.24E-02	7.24E+01	Brinkman
1	1	1	1.00E+04	100	0.233	2.34E+00	2.34E+03	Darcy
0.5	1	2	1.00E+06	1000	0.078	7.88E+01	7.88E+04	Darcy

\begin{matrix} \begin{array}{l} \int_{Ω_{ε}} D (u) \cdot \nabla v d V = \int_{Ω_{ε}} D (u) \cdot \frac{\nabla v}{2} d V + \\ \int_{Ω_{ε}} D (u) \cdot \frac{\nabla v^{T}}{2} d V = \int_{Ω_{ε}} D (u) \cdot D (v) d V \end{array} \end{matrix}

\begin{matrix} \begin{array}{l} - \int_{Ω_{ε}} p_{ε} \nabla \cdot v d V + 2 μ \int_{Ω_{ε}} D (u_{ε}) \cdot D (v) d V - \\ \int_{Γ_{2,4}} n^{T} (- p_{ε} I + 2 μ D (u_{ε})) n v \cdot n d S \end{array} \\ + \int_{Γ_{2,4}} n^{T} (- p_{ε} I + 2 μ D (u_{ε})) t v \cdot t d S = \int_{Ω_{ε}} f \cdot v d V \end{matrix}

\begin{matrix} \begin{array}{l} - \int_{Ω_{ε}} p_{ε} \nabla \cdot v d V + 2 μ \int_{Ω_{ε}} D (u_{ε}) \cdot D (v) d V + \\ \int_{Γ_{2,4}} n^{T} (- p_{ε} I + 2 μ D (u_{ε})) t v \cdot t d S \end{array} \\ = \int_{Ω_{ε}} f \cdot v d V \end{matrix}

\begin{matrix} \begin{array}{l} \int_{Ω_{ε}} ϕ \nabla u_{ε} \cdot \nabla w_{ε}^{k} d V + \int_{Ω_{ε}} u_{ε} \nabla ϕ \cdot \nabla w_{ε}^{k} d V = \\ - \int_{Ω_{ε}} ϕ u_{ε} \cdot Δ w_{ε}^{k} d V \end{array} \end{matrix}

\begin{matrix} \begin{array}{l} \int_{Ω_{ε}} ϕ \nabla u_{ε} \cdot \nabla w_{ε}^{k} d V = - \int_{Ω_{ε}} u_{ε} \nabla ϕ \cdot \nabla w_{ε}^{k} d V - \\ \int_{Ω_{ε}} ϕ u_{ε} \cdot Δ w_{ε}^{k} d V \end{array} \end{matrix}

\begin{matrix} \begin{array}{l} \int_{Ω} (\nabla q_{ε}^{k} - Δ w_{ε}^{k}) \cdot ϕ \tilde{u_{ε}} d V = \\ (\nabla q_{ε}^{k} - Δ w_{ε}^{k}, ϕ {\tilde{u_{ε})}}_{H^{- 1} (Ω), H^{1} (Ω)} \end{array} \end{matrix}

\begin{matrix} \begin{array}{l} - \int_{Ω_{ε}} p_{ε} \nabla \cdot (ϕ w_{ε}^{k}) d V + 2 μ \int_{Ω_{ε}} D (u_{ε}) \cdot D (ϕ w_{ε}^{k}) d V = \\ \int_{Ω_{ε}} f \cdot ϕ w_{ε}^{k} d V \end{array} \end{matrix}

\begin{matrix} \begin{array}{l} \int_{Ω_{ε}} D (u_{ε}) \cdot \nabla v d V = - \frac{1}{2} \int_{Ω_{ε}} \nabla \cdot (\nabla u_{ε} + \nabla u_{ε}^{T}) \cdot \\ v d V + \frac{1}{2} \int_{\partial Ω_{ε} = Ã_{2,4}} (\nabla u_{ε} + \nabla u_{ε}^{T}) \cdot v \cdot n d S \end{array} \end{matrix}

\begin{matrix} \begin{array}{l} \int_{Ω_{ε}} D (u_{ε}) \cdot \nabla v d V = - \int_{Ω_{ε}} Δ u_{ε} \cdot v d V = \\ - \int_{Ω_{ε}} Δ u_{ε} \cdot ϕ w_{ε}^{k} d V \end{array} \end{matrix}

\begin{matrix} \begin{array}{l} - \int_{Ω_{ε}} p_{ε} \nabla \cdot (ϕ w_{ε}^{k}) d V - μ \int_{Ω_{ε}} Δ u_{ε} \cdot ϕ w_{ε}^{k} d V = \\ \int_{Ω_{ε}} f \cdot ϕ w_{ε}^{k} d V \end{array} \end{matrix}

\begin{matrix} e^{\frac{r}{2}} \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1} r^{n}}{n! 2^{n - 1}} \sum_{k = 0}^{\frac{n - 1}{2}} \frac{1}{2 k + 1} \to α r^{p} e^{\frac{r}{2}} \end{matrix}

\begin{matrix} w_{ε}^{k} (r, θ) = \sum_{k = 1}^{\infty} \begin{matrix} \begin{array}{l} - \frac{ε i}{k} (γ + \ln (\frac{i k r}{ε}) + α i k r^{p} e^{\frac{i k r}{2 ε}}) \\ + \frac{ε^{2}}{k^{2}} (γ + \ln (\frac{i k r}{ε}) + α i k r^{p} e^{\frac{i k r}{2 ε}}) \end{array} \\ + C_{1} \log (r) + C_{2} \end{matrix} \end{matrix}

\begin{matrix} \begin{array}{l} w_{ε}^{k} (r, θ) = \sum_{k = 1}^{\infty} (- \frac{ε i}{k} + \frac{ε^{2}}{k^{2}}) \\ (γ + \ln (\frac{i k r}{ε}) + α i k r^{p} e^{\frac{i k r}{2 ε}}) + C_{1} \log (r) + C_{2} \end{array} \end{matrix}

\begin{matrix} w_{ε}^{k} (r, θ) = \sum_{k = 1}^{\infty} \begin{matrix} (- \frac{ε i}{k} + \frac{ε^{2}}{k^{2}}) (\begin{array}{l} γ + \ln (\frac{i k β a_{ε}}{ε^{2}}) + \\ α i k {(\frac{β a_{ε}}{ε})}^{p} e^{\frac{i k β a_{ε}}{2 ε^{2}}} \end{array}) \\ + C_{1} \log (\frac{β a_{ε}}{ε}) + C_{2} \end{matrix} \end{matrix}

\begin{matrix} M_{0} = \int_{Ω_{u n i t}} {(\partial_{x} u)}^{2} + {(\partial_{y} u)}^{2} + {(\partial_{x} v)}^{2} + {(\partial_{y} v)}^{2} d Ω_{u n i t} \end{matrix}

[1] #Corresponding author. E-mail address: karligor@tecgraf.puc-rio.br

Brasil

Brasil

Homogenization theory and nonlinearities in Darcy's law

Abstract