Impact of Geological Stress on Flow and Transport in ... · Impact of stress on natural fracture...

Peter K. Kang

University of MinnesotaOct 23th, 2018, IMA!

Impact of Geological Stress on

Flow and Transport in Fractured Media

Peter K. Kang

University of MinnesotaOct 17th, 2018, ESCI 8001!

Impact of Geological Stress on

Flow and Transport in Fractured Media

Collaborators

Ruben Juanes

Marco Dentz

Qinghua Lei

Stephen Brown, Michael Fehler, Daniel R. Burns

Yingcai Zheng

Weon Shik Han

Jeffrey Hyman

Intro

Yoon and Kang, in preparation

Why we care about flow and transport through fractures?: Fractures often act as either seal or dominate flow paths

There are many studies on flow and transport through rough fractures

I. Neretnieks et al., WRR, 1982; Y. W. Tsang and C. F. Tsang, WRR, 1987; L. Moreno et al., WRR, 1988; S. Brown et al., JGR, 1998; Y. Méheust and J. Schmittbuhl, GRL, 2000;

Hele-Shaw Rough fracture

Intro

R. L. Detwiler et al., WRR, 2000; H. Auradou et al., PRE, 2001; F. Bauget and M. Fourar., JCH, 2008;Talon et al., EPL 2012; L. Wand and M. B. Cardenas, WRR, 2014

Image: MICROSEISMIC INC.

In reality, fractures are always under confining stress: Hydraulic fracturing

Image: Baker Hughes

Intro

Image: MICROSEISMIC INC.

Impact of confining stress on fluid flow properties has been widely studied

Intro

K. G. Raven and J. E. Gale, 1985; D. D. Nolte et al. 1989; R. W. Zimmerman et al. 1992; D. R. Briggins, 1992;W. A. Olsson and S. R. Brown, 1993; A. J. A. Unger and C. W. Mase, 1993; W. B. Durham and B. P. Bonner, 1994;

C. E. Renshaw, 1995; V. V. Mourzenko et al., 1995;N. Watanabe et al., 2008; L. Zou et al., 2013.T. Koyama et al., 2008.W. Jeong and J. Song., 2005, 2008.H. Auradou, 2009.

Rock physics model – permeability vs stiffness

Unger and Mase, WRR, 1993; Mourzenko et al., PRE, 1997;

Pyrak-Nolte and Morris, 2000

Laboratory experiment

Petrovitch et al, 2013

Numerical experiment

Intro

Seismic equation:

Flow equation (incompressible, plane Poiseuille flow):

, : flux per unit depth

: fourth order stiffness tensor (function of rock properties),

: Cauchy stress tensor

: infinitesimal strain tensor

: displacement vector

,

,

: permeability (fn of fracture rock properties)

Rock physics model links flow and seismic dataIntro

compliance

Rock physics model

perm

eabi

lity

Joint flow-seismic inversion with rock physics model

Kang et al., 2016, Water Resources ResearchHu et al., 2018, Geophysics

Intro

Objectives of this study

1. Study the impact of the geological stress on flow andtracer transport through a single rough fracture scaleto fracture network scale.

2. Develop a parsimonious and effective transport model.

distance [m]0 2 4 6 8 10 12 14 16

dist

ance

[m]

-8

-7

-6

-5

-4

-3

-2

-1

Numerically generate realistic fracture surface (Fourier synthesis method).

1. Construct a rough fracture surface: self-affine fractal

x

z

Top surface

Bottom surface

x

z

a(x)

Top surface

Bottom surface

Fracture aperture

Bottom surfaceTop surface

Assumptions

1. The medium is linearly elastic.

2. Only consider normal stress.

3. Consider non-local interaction between contact areas.

2. Solve the elastic contact problem on rough-walled fractures

: normal displacement

: standard deviation of aperture values at first contact

� = 0

�0

a(x)a(x)δ = 3 σ0

a(x)

δ = 3 σ0

�

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ = 0 δ = 1 .5σ0

δ = 2 .5σ0 δ = 3 .5σ0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ = 0 δ = 1 .5σ0

δ = 2 .5σ0 δ = 3 .5σ0

2. Solve the elastic contact problem

aperture0 0.2 0.4 0.6 0.8 1

prob

abilit

y de

nsity

0

2

4

6

8

10

δ = 3 .5σ0δ = 2 .5σ0δ = 1 .5σ0δ = 0

10-3

10-2

10-1

100

10-2

10-1

10 0

101

aperture

prob

abilit

y de

nsity

Fluid flow in the fracture is calculated by assuming 1) Laminar flow2) The cubic law is locally valid Which gives a series of parallel plates.

3. Run flow and transport simulation on rough-walled fractures

X

j

Qij = 0

Qij = �b3ij�a

12µ

Pj � Pi�a

Boundary conditions:1. No flow on top and bottom boundaries2. Constant pressure on left and right boundaries

Pressure field

Normalized flow field

Series of terraces

Increase in stress

� = 3.5�0� = 2.5�0� = 1.5�0� = 0

Flow field becomes more tortuous and preferential paths arise as stress increases.

0 10

0 1


Flow fieldAperture field


High stress case:Low stress case: � = 3.5�0� = 0

Will the increase in confining stress

lead to anomalous transport?

The field truth: Anomalous transport is widely observed in subsurface

Extraction boreholeInjection borehole

Photo of Ploemeur field site

Schematic of tracer tests in fractured aquiferField tracer breakthrough curve

Kang et al., Water Resour. Res. (2015)

Fickian transport vs Anomalous transport

Signature of Anomalous Transport

10−3 10−2 10−1 100

10−7

10−6

10−5

10−4

10−3

10−2

time

mea

n sq

uare

dis

plac

emen

t

2

1

1

1

(a) (c)

1 − α

1 + α 11

prob

abilit

y de

nsity

FickianAnomalous

(b)

time

x

y

mean flow direction

Fickiansuperdiffusive

subdiffusive

We can clearly observe transition from Fickian to non-Fickian (anomalous) transport.

4. Analyze flow and transport characteristics with respect to stress

Time evolution of spreading(a) (b)

normalized time101 102 103 104 105

prob

abilit

y de

nsity

10-9

10-7

10-5

10-3

10-1

displacement

peak

arri

val t

ime

100

101

102

0 1 2 3 4

1

3

[σ0 ]

normalized time100 102 104

MSD

10-1

101

103

105

1

1

1.3

1

2

1 displacement0 1 2 3 4

slop

e1

1.1

1.2

1.3

1.4

(a) (b)


prob

abilit

y de

nsity

10-9

10-7

10-5

10-3

10-1

displacement

peak

arri

val t

ime

100

101

102

0 1 2 3 4

1

3

[σ0 ]


MSD

10-1

101

103

105

1

1

1.3

1

2

1

(a) (b)


prob

abilit

y de

nsity

10-9

10-7

10-5

10-3

10-1

displacement

peak

arri

val t

ime

100

101

102

0 1 2 3 4

1

3

[σ0 ]


MSD

10-1

101

103

105

1

1

1.3

1

2

1

δ = 3 .5σ0δ = 2 .5σ0δ = 1 .5σ0δ = 0

δ = 3 .5σ0δ = 2 .5σ0δ = 1 .5σ0δ = 0

5. Develop a predictive transport model

Kang et al., GRL., 2014

3D pore scale

Le Borgne et al., PRL., 2008

Darcy scale

Model development: Spatial Markov Model

Kang et al., WRR. (2015)

Field scale

: follows the transition time distribution ( ) that follows truncated power-law 0(⌧)

vn vn+1 vn+2

⌧n =dx

vn

0(⌧)

transition time100 102 104 106

prob

abilit

y den

sity

10-12

10-10

10-8

10-6

10-4

10-2

1 + �

1

Directly map multiscale heterogeneity onto particle velocity probability distribution: velocity heterogeneity

Berkowitz and Scher, PRL., 1997Dentz et al., AWR, 2004


Velocity correlation structure is Markovian in space

vn vn+1 vn+2

0 100

100

−9

−7

−5

−3

−1

0vn

vn+1

1(v|v0)

Le Borgne et al., Phys. Rev. Lett. (2008)Kang et al., Phys. Rev. Lett. (2011)

current velocity class

next

vel

ocity

cla

ss

0

0.01

0.02

0.5

1

current velocity class

next

vel

ocity

cla

ss

0

no correlation

perfect correlation

log scale


We formulate correlated CTRW that honors velocity heterogeneity and correlation.

: characterized by the transition time distributionand one-step transition probability density

0(⌧) 1(⌧ |⌧ 0)

xn+1 = xn + dx, tn+1 = tn + ⌧ , ⌧ =dx

v

St+1 = f( ,�, St)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AAACCHicbVDJSgNBFHwTtxi3qEcPNgYhYggzXvQiCF48RmIWyEZPpydp0rPQ/UYIQ45e/BUvHhTx6id482/sLKAmFjQUVa94/cqNpNBo219Waml5ZXUtvZ7Z2Nza3snu7lV1GCvGKyyUoaq7VHMpAl5BgZLXI8Wp70pecwfXY792z5UWYXCHw4i3fNoLhCcYRSN1sofldoKnzohcEi/fjLQokKY08S4tkHIbTzrZnF20JyCLxJmRHMxQ6mQ/m92QxT4PkEmqdcOxI2wlVKFgko8yzVjziLIB7fGGoQH1uW4lk0NG5NgoXeKFyrwAyUT9nUior/XQd82kT7Gv572x+J/XiNG7aCUiiGLkAZsu8mJJMCTjVkhXKM5QDg2hTAnzV8L6VFGGpruMKcGZP3mRVM+KjuG3P21AGg7gCPLgwDlcwQ2UoAIMHuAJXuDVerSerTfrfTqasmaZffgD6+MbSnqXKg==AAACCHicbVC7SgNBFJ2Nrxhfq5YWDgYhYgi7NtoIQRvLSMwDspswOzubDJl9MHNXCEtKG3/FxkIRWz/Bzr9x8ig0emDgcM493LnHSwRXYFlfRm5peWV1Lb9e2Njc2t4xd/eaKk4lZQ0ai1i2PaKY4BFrAAfB2olkJPQEa3nD64nfumdS8Ti6g1HC3JD0Ix5wSkBLPfOw3s3g1B7jSxyUnETxMnaEjvukjOtdOOmZRatiTYH/EntOimiOWs/8dPyYpiGLgAqiVMe2EnAzIoFTwcYFJ1UsIXRI+qyjaURCptxsesgYH2vFx0Es9YsAT9WfiYyESo1CT0+GBAZq0ZuI/3mdFIILN+NRkgKL6GxRkAoMMZ60gn0uGQUx0oRQyfVfMR0QSSjo7gq6BHvx5L+keVaxNb+1itWreR15dICOUAnZ6BxV0Q2qoQai6AE9oRf0ajwaz8ab8T4bzRnzzD76BePjG3HUl6w=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

5. Develop a predictive transport model: Correlated CTRW


MSD

10-1

101

103

105

1

1

1.3

1

2

1


MSD

10-1

101

103

105

1

1

1.3

1

2

1

Kang et al., EPSL. (2016)

Field trip to obtain natural fracture surfaces

Visual laboratory experiment on stressed rough fracture

Experimental set up: Dynamic imaging with transparent micromodels

Laboratory experiment results: rough fracture

High stressLow stress

1. Stress induces anomalous transport at a single fracture scale: transition from Fickian to non-Fickian

2. Spatial Markov model can predict anomalous transportthrough stressed rough fractures.

Part 1 Conclusions

distance [m]0 2 4 6 8 10 12 14 16

dist

ance

[m]

-8

-7

-6

-5

-4

-3

-2

-1

Impact of stress on natural fracture network scale flow and transport

Natural fracture networks of the Bristol fractured reservoir

• A natural fracture network is extracted from the geological map of a rock outcrop.• Aperture values are sampled from a aperture distribution that follows log normal

distribution with mean 1 [mm] and variance 1.• Aperture values are randomly assigned to each link.

0

1

2 [mm]

distance [m]0 2 4 6 8 10 12 14 16

dist

ance

[m]

-8

-7

-6

-5

-4

-3

-2

-1

3 MPa 3 MPa

3 MPa

3 MPa

Impact of stress magnitude on fracture network-scale transport

σ'x = 3 MPa, σ'y = 3 MPa

distance [m]0 2 4 6 8 10 12 14 16

dist

ance

[m]

-8

-7

-6

-5

-4

-3

-2

-1

15 MPa 15 MPa

15 MPa

15 MPaσ'x = 15 MPa, σ'y = 15 MPa

Impact of stress magnitude on fracture network-scale transport

Impact of stress orientation on fracture network-scale transport

distance [m]0 2 4 6 8 10 12 14 16

dist

ance

[m]

-8

-7

-6

-5

-4

-3

-2

-1

15 MPa 15 MPa

5 MPa


distance [m]0 2 4 6 8 10 12 14 16

dist

ance

[m]

-8

-7

-6

-5

-4

-3

-2

-1

5 MPa 5 MPa

15 MPa



• The Cauchy linear momentum equation is solved using finite-discrete element method (FEMDEM) (Munjiza, 2004).

• Joint constitutive model (JCM) captures joint closure, shearing, shear dilatancy by taking fracture roughness into account.

Geomechanical modelling of fracture networks

Lei et al., Int. J. Rock Mech. Min. Sci. (2014)

σ'x = 15 MPa, σ'y = 5 MPaσ'y

σ'x

Example of an aperture field under stress

• For each scenario, an ensemble of 20 realizations are generated and studied.

Aperture field: no stress

0

1

2 [mm]

0

1

2 [mm]Aperture field:

Fluid flow in the fracture is calculated by assuming 1) Laminar flow2) The cubic law is locally valid Which gives a series of parallel plates.

Solve for fluid flow through stressed discrete fracture networks

X

j

Qij = 0

1. No flow on top and bottom boundaries2. Constant pressure on left and right boundaries

Boundary conditions:No flow

No flow

Pin Pout

Qij = �a3ij12µ

Pj � Pilij

l ij

aijP i P j

0

1

2

3

4

5

6

7Flow field: 15MPa-5MPa

0

0.5

1Pressure field: 15MPa-5MPa

Solve pressure equation to obtain flow field

Pressure field: no stress

0

0.5

1

Flow field: no stress

0

1

2

3

4

5

6

7

Run tracer transport simulation on obtained flow fields

Park et al., Water Resour. Res., 2001Kang et al., Adv. Water Resour., 2017

• Flux-weighted Injection at the inlet:

• Complete mixing at intersections:

• Particle tracking simulation: σ'x = 15 MPa, σ'y = 15 MPa


σ'x

σ'yσ'x




σ'x

σ'yσ'x

Stress induces anomalous transport in fracture network-scale transport

We can observe features of non-Fickian (anomalous) transport:

- Emergence of late-time tailing- Anomalously early arrival of tracers for the 15MPa-5MPa case.

10-1 100 101 102 103

normalized time

10-8

10-6

10-4

10-2

100

conc

entra

tion

nostress3MPa3MPa15MPa15MPa

10-1 100 101 102 103

normalized time

10-8

10-6

10-4

10-2

100

conc

entra

tion


1

4

1

3

1

4

1

3

(a) (b)

What are the origins of anomalous transport?

No stress

15MPa-15MPa

5MPa-15MPa3MPa-3MPa

15MPa-5MPa

0

0.05

0.1 [mm]

- Stress significantly increases small aperture values

Origin of the emergence of late-time tailing

Origin of the emergence of late-time tailing

Aperture distributions

- Geologic stress broadens aperture distribution- Increase in the variance of the aperture distribution makes

heterogeneous velocity field

10-2

10-1

100

101

aperture [mm]

10-3

10-2

10-1

100

prob

abilit

y de

nsity

10-2 10-1 100 101

aperture [mm]

10-3

10-2

10-1

100

prob

abilit

y de

nsity

(a) (b)



Normal stress on fractures

0

1

2107[Pa]



Shear dilation

0

0.5

1 [mm]



Origin of the anomalously early arrival

• Stress anisotropy opens / closes different fracture sets• Interplay between fracture geometry (connectivity, direction) and

stress condition controls shear dilation

σ'yσ'x

Impact of geologic stress on flow field

- Flow values are normalized with the mean flow rate of the no stress case- Shear dilation causes strong preferential paths

No stress

15MPa-15MPa

5MPa-15MPa3MPa-3MPa

15MPa-5MPa

01234567

σ'yσ'x

Can we develop a parsimonious upscaled transport model?

Impact of Stress on Lagrangian Velocity Distributions

Geologic stress has major impacts on Lagrangian velocity distribution

10-6 10-4 10-2 100

normalized velocity

10-4

10-2

100

prob

abilit

y de

nsity


10-6 10-4 10-2 100

normalized velocity

10-4

10-2

100

prob

abilit

y de

nsity


(a) (b)

Impact of Stress on Lagrangian Velocity Correlation

1 2 3 4 5 6 7 8 9 10velocity class

1

2

3

4

5

6

7

8

cond

ition

al v

eloc

ity c

orre

latio

n [m

]


1 2 3 4 5 6 7 8 9 10velocity class

1

2

3

4

5

6

7

8

cond

ition

al v

eloc

ity c

orre

latio

n [m

]


(a) (b)

Conditional correlation length:{Ref: Le Borgne et al., 2007, WRR

Spatial Markov Model with average correlation length

10-1 100 101 102 103

normalized time

10-8

10-6

10-4

10-2

100

conc

entra

tion


10-1 100 101 102 103

normalized time

10-8

10-6

10-4

10-2

100

conc

entra

tion


(a) (b)

SMM prediction SMM prediction

The model with a single correlation length shows good predictability except for the 15MPa-5MPa scenario.

Emergence of strong correlation length due to shear dilation

Two correlation length model is proposed:

1 2 3 4 5 6 7 8 9 10velocity class

1

2

3

4

5

6

7

8

cond

ition

al v

eloc

ity c

orre

latio

n [m

]


1 2 3 4 5 6 7 8 9 10velocity class

1

2

3

4

5

6

7

8

cond

ition

al v

eloc

ity c

orre

latio

n [m

]


(a) (b)

1 2 3 4 5 6 7 8 9 10velocity class

1

2

3

4

5

6

7

8

cond

ition

al v

eloc

ity c

orre

latio

n [m

]


1 2 3 4 5 6 7 8 9 10velocity class

1

2

3

4

5

6

7

8

cond

ition

al v

eloc

ity c

orre

latio

n [m

]


(a) (b)

Tij = ai�ij + (1� ai)1� �ijN � 1

where is a function of velocity class.ai

Spatial Markov Model with dual correlation lengths

• We honor the emergence of strong velocity correlation with the two correlation length model.

• The model nicely predicts 15MPa-5MPa scenario.

10-1 100 101 102 103

normalized time

10-8

10-6

10-4

10-2

100

conc

entra

tion

10-1 100 101 102 103

normalized time

10-8

10-6

10-4

10-2

100

conc

entra

tion

(a) (b)nostress3MPa3MPa15MPa15MPa1 CorrL model2 CorrL model

nostress5MPa15MPa15MPa5MPaSMM prediction2 CorrL model

Kang et al., submitted. (2018)

1. Interplay between stress and fracture geometry can induce anomalous transport in fractured media.

2. Stress fundamentally changes velocity correlation and velocity heterogeneity.

3. Spatial Markov model with dual correlation length can predict anomalous transport through fractured media.

Conclusions

σ'x = 15 MPa, σ'y = 5 MPaσ'x = 5 MPa, σ'y = 15 MPaσ'y

σ'x

σ'yσ'x

Impact of Variable-Density Flow on the Value-of-Information of Pressure and Concentration Data for Saline Aquifer Characterization

Seonkyoo Yoon, Peter Kang

Heterogeneous aquifer

?

Managed aquifer recharge

Thank youResearch website: www.pkkang.com

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