Darcy’s Law, valid or not in acontaminant-impacted zone?

(Part 1: membrane porous media)

Dr Xiaohui ChenLecturer in Geotechnics

Academic Leader of BioSAS consortium (UoL, UoS, UoY)Joint Leader of Nuclear Group at Civil (UoL)

1

1. Introduction

2. Methodology

3. New MatrixContentsContents

4. Modelling

5. End

2

Note: The pictures taken from internet are only used to demonstrate the research idea. These pictures will not be used for publication or commercial purpose

Revisit Darcy’s Law (1856)

beds of sand

Water

kgrad p

v= u

160 years, forming the basis of hydrogeology, cross-disciplinary usedin geotechnics, geochemistry etc.

3

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World has changed, and is changing

Nuclear: Fukushima (2011) Landfill example Waste Geological Disposal

4

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Revisit Darcy’s Law (1856)…needs to be changed

beds of sand

kgrad p

v= u

Darcy, H. (1856). Les fontaines publiques de la ville de Dijon. Paris: Dalmont.

Assumption of Darcy’s Law

water

No chemicals (only water) High permeability, non-reactive

X XMay Not applicable to pollutionand contaminated lands

5

Environmentalgeochemistry

Environmentalmineralogy

Geomechanics

Geomicrobiology

Soils/Rocks system

Fe2+, Mg, Na, Mn, SO4,K, NH4, Ni, Cd, Cr and Zn

Question: how to incorporate such coupled influence into theoretical prediction?

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Molecular-scale coupled influence？

6

Conceptual model？

Multiple minerals transfer

Multiple chemicals reaction

Gas/CO2/Oil interactions with water

Soil/rock particles deformation

Multiple microbes process

Chemical OsmosisThermal Osmosis

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Main Theory: balance laws

Why low permeability matters?

http://www.keepoklahomabeautiful.com/21st-century-landfills

http://nucl-mater.eng.hokudai.ac.jp/overview-en/theme-en/

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8

Main Theory: balance laws

Osmosis

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9

1. Introduction

2. Methodology

3. New MatrixContentsContents

4. Modelling

5. End

10

Mixture Theory

Blood in tissues

≈groundwater insoils/rocks

Mixture Coupling Theory“a systematic approach…,which canhelp build coupled formulationsbased on a single unified theory,between geophysics andgeochemistry”

More realisticMore systematic

CHEN, X., PAO, W., THORNTON, S. & SMALL, J. 2016. Unsaturated hydro-mechanical–chemical constitutive coupled model based on mixturecoupling theory: Hydration swelling and chemical osmosis. International Journal of Engineering Science (IF:3.165), 104, 97-109.

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11

Entropy

Second law of thermodynamicsThe entropy of an isolated system not inequilibrium will tend to increase over time,approaching a maximum value at equilibrium

Professor Stephen Hawking: Black holeInterpreted as the degree of disorder or randomness in the system

http://www.michelecoscia.com/?p=1041

Freedom change

https://en.wikipedia.org/wiki/Chemical_reaction

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the Fourth Law of Thermodynamics

• phenomenological equations:A set of functions are used to express the linear dependence of

these two flows on the corresponding forces.

How to give the relationships between flows

Lars OnsagerNobel Prize in Chemistry in 1968

J1 = L11F1 + L12F2 +……..+L1kFk

J2 = L21F1 + L22F2 +……..+L2kFk

Jk = Lk1F1 + Lk2F2 +……..+LkkFk

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1. Introduction

2. Methodology

3. New MatrixContentsContents

4. Modelling

5. End

14

Solid and fluid

Dissipation generated by frictionalresistance at solid and fluid interface

Entropy production

0 w w c cTγ µ µ≤ = − ⋅∇ − ⋅∇I I

0 ( )w w c cT pγ µ µ≤ = − ⋅∇ − ⋅∇ + ⋅∇u J J

( )fβ β βρ= −J v v%Note: Since the diffusion fluxes of the water and chemical relative to the barycentric motion can be written as

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Gibbs-duhem equation

Extending ofFick’s diffusion law and Darcy’s laws

( )f c

crw tw c

kk p r C

C C

ρ µ

θ

∂= − ∇ − ∇

∂u

c f f ct t

pL D C

pρ ρ

∇ = − ∇ J

( )

11

2rw

ft

k Lk

θ ρ=

21

11

Lr

L= − ( )

21

2ft

L pL

ρ=

( )

22

2

c

cw ft

LD

CC

µ

ρ

∂=

∂

where

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Darcy’s Law and Fick’s Law

f f s

d s

ssf

f

krk

grad pv vc c

grad cLD

p

ρ µ

ρρ

∂

∂ = −

u

J

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CHEN, X., PAO, W., THORNTON, S. & SMALL, J. 2016. Unsaturated hydro-mechanical–chemical constitutive coupled model based on mixturecoupling theory: Hydration swelling and chemical osmosis. International Journal of Engineering Science (IF:3.165), 104, 97-109.

0

0s

sf

kgrad p

vgrad c

Dρ

= −

u

J

Darcy’s Law and Fourier's law of heatconduction

q f

q f

krk

grad pv vT

L grad T

p

ρ

ρλ

= −

u

q

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CHEN, X., PAO, W. & LI, X. 2013. Coupled thermo-hydro-mechanical model with consideration of thermal-osmosis based on modified mixturetheory. International Journal of Engineering Science, 64, 1-13.

0

0

kgrad p

vgrad T

λ

= −

u

q

1. Introduction

2. Methodology

3. New MatrixContentsContents

4. Modelling

5. End

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(Experiment set up + chemical transport)

Invalidate Darcy’s Law (1: Chemical osmosis)

St i f f c o n s t r a i n t

1 5 m m

x

F r i c t i o n l e s s i mp e r m e a b l e

b o u n d a r y

30m

m

U n sa t u r a t e d r o c kC h e m i c a l co n c e n t r a t i o n

F re e p e r m e a b l e

b o u n d a ry A

Fi x e d i m p e r me a b l e

b o u n d a r y B

F r i c t i o n l e s s im p e r m e a b le

b o u n d a r y

20

Fig. 3 Evolution of pore water pressuredistribution with time during first day

Fig. 4 Evolution of degree of saturation

distribution with time during first day 21

Invalidate Darcy’s Law (2: Thermo osmosis)

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1. Introduction

2. Methodology

3. New MatrixContentsContents

4. Modelling

5. End

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Intermediate/Low Level nuclear waste: Radionuclides transport through sandstone in groundwater

(2) Conceptual model

Corkhill, C; Bridge, J; Chen, X; Hillel, P; Thornton, S; Romero-Gonzalez, M; Banwart, S; Hyatt, N. (2013) Real-time gamma imaging of technetium transport through naturaland engineered porous materials for radioactive waste disposal, Environmental Science & Technology, 47(23): 13857-1864.Chen, X., Thornton, S. F. & Small, J. Influence of Hyper-Alkaline pH Leachate on Mineral and Porosity Evolution in the Chemically Disturbed Zone Developed in the Near-Field Host Rock for a Nuclear Waste Repository. Transport in Porous Media 107, 489-505, doi:10.1007/s11242-014-0450-0 (2015).

Nuclear Waste Concept

Conclusion: multi-minerals dissolution strongly affects the pathway and chemistry of groundwater flow .

End, but not the End ….

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Darcy’s Law, valid or not in a contaminant-impacted zone?(Part 2: heavily reactive porous media)

Acknowledgement

New theory could boost engineering solutions/innovations…… gap between theory and engineering application….

Geochemistry

Geomechanics

Groundwater

Nuclear Waste

Agriculture Soil

x.chen@leeds.ac.ukContact information:

Current funded project:BioSAS consortium (around £580k)

Available for consultancy;

Welcome industry partner to join our teamto develop links with China industry andacademy for applications to Innovate UKand Newton fund in the area ofcontaminated lands.