Rheology of rocks

•• Paul BonsPaul Bons

•• TTüübingen Universitybingen University

•• Paul.bons@uni-Paul.bons@uni-tuebingentuebingen.de.de

This lecture

• Discuss exercise last week

• Make a deformation mechanism map

• Look at relationship stress and grain size: piezometry

• Introduce last deformation mechanism

• Dissolution-precipitation creep• Diffusion control

• Reaction control

• Dislocation creep:

• GRAIN-SIZE INSENSITIVE (GSI)

• Strain rate is independent of grain

size for dislocation creep

• Horizontal lines

• Diffusional creep:

• GRAIN-SIZE SENSITIVE (GSS)

• Strain rate is dependent of grain

size for diffusional creep

Add strain rate lines

NH creep

Coble creep

Dislocation creep

!

log "( ) = 21.8 + log ˙ # ( ) + 2log g( )

!

log "( ) = 24.0 + log ˙ # ( ) + 3log g( )!

log "( ) = 5.67 +1

3log ˙ # ( )

!

log ˙ " ( ) = #21.8 + log $( ) # 2log g( )

!

log ˙ " ( ) = #24.0 + log $( ) # 3log g( )!

log ˙ " ( ) = 3log #( ) $ 3 % 5.67 • Shear zones (mylonites)

• Higher strain rate

• Smaller grain size

• Is this because of differentdeformation mechanisms?

• A: Dislocation creep in wall rock:

• with grain size 10 mm

• Strain rate is 10-15 s-1

• B: Cobble creep in shear zone:

• with grain size 0.1 mm

• Strain rate is 10-12 s-1

Using the map: shear zones

AB

Can we freely change grain size?

• Dynamic recrystal-lisation tends todecrease grain size

• Surface-energy drivenrecrystallisationincreases grain size(grain growth)

• Somewhere there is abalance:

Stable grain size

Grain size piezometer

• The stable grain size depends on:

• Grain size reduction

• Dynamic recrystallisation

• Depends on dislocation density

• Depends on stress

• Grain size increase (grain growth)

• Static recrystallisation

• Depends on amount of surfaces

• Depends on grain size

• Empirical relationship:

• If we know the grain size, we know the stress!

• Hence the name palaeo-piezometer

!

g " c #$%plog(!)

log(g

)

-p

• According to the piezometer

• Stress has a fixed relationship tograin size

• Example: "If the stress is !5 MPa

Grain size is !2 mm"

• If wall rock and shear zoneexperience the same shear stress

• Their grain size should be the same

• Their strain rate should be the same

• There can be no shear zone!

• But notice that piezometer is veryclose to mechanism boundary

Adding the palaeopiezometer

AB!

g " c #$%p

• Coarse-grained material

• Dislocation creep induces dynamicrecrystallisation:

• Grain size decreases

• Material moves to left towards Coblecreep field

Dislocation creep: grain-size reduction

AB!

g " c #$%p

• Fine-grained material

• Coble creep induces NO dynamicrecrystallisation:

• Grain size increases (grain growth)

• Material moves to right towardsdislocation creep field

Coble creep: grain-size increase

AB!

g " c #$%p

Piezometer ! mechanism boundary

AB!

g " c #$%p

• Mechanism boundary is line where

• dynamic rexx grain-size reduction

• static rexx grain-size increase

• Balance

• How to get away from piezometer line?

• Sudden change in stress

• Time/strain delay to reach equilibrium

• Inhibition of grain growth

• E.g. by pinning of grain boundaries

Dissolution-precipitation creep

• DPC involves mass transfer in fluid

• Short distance: from one side of grain to another

• Long distance: from stylolite to vein

Equilibrium concentration

• All minerals can dissolve in fluids (water)

• There is an equilibrium concentration Ceq

• If actual concentration is lower than Ceq

• Mineral dissolves

• concentration increases towards Ceq

• If actual concentration is higher than Ceq

• Mineral precipitates

• concentration decreases towards Ceq

• Finally, equilibrium concentration is reachedconcentration

Chemicalpotential

Ceq

Chemical potential and pressure

• The chemical potential (µ) is a function of pressure:

• Concentration is proportional to µ, giving:!

"µ

"P=#P

• So the equilibrium concentration is:

!

"Ceq =C0

+#P!

C"µ #$C

$P"$µ

$P=%P

Flow by DP-creep

• Equilibrium concentration is a function of pressure

• If there are pressure gradients

• You get concentration gradients

• You get diffusional transport of matter

• You get strain

• To know the flow law, we need to know how

• Pressure gradients

• Relate to differential stress

!

Ceq =C0

+"P

Effective pressure on the surface of grains

• Under a differential stress grain boundaries havedifferent effective pressures

• Perpendicular to !1 : Peff+ = P + "!n/2

• Perpendicular to !3 : Peff- = P - "!n/2

• This drives transport from compressional grainboundaries to extensional grain boundaries

-"!n/2

+"!n/2

g x

Peff

compression extension

x Pressure = mean stress

Equilibrium concentration on the surface ofgrains

• Under a differential stress grain boundaries havedifferent equilibrium concentrations

• Perpendicular to !1 :

• Perpendicular to !3 :

• This drives transport from compressional grainboundaries to extensional grain boundaries

-"!n/2

+"!n/2

g x

Ceq

compression extension

x Average Ceq

!

Ceq

+=C

0+"

2#$

!

Ceq

"=C

0"#

2$%

!

Ceq =C0

+"P

Remember:

Equilibrium concentration on the surface ofgrains

• Under a differential stress grain boundaries havedifferent equilibrium concentrations

• Diffusion tries to equalise concentration " transport

• Compressional faces: under-saturated " dissolution

• Extensional faces: over-saturated " precipitation

-"!n/2

+"!n/2

g x

Cactual

compression extension

x

Diffusional transport

Cactual < Ceq

Under-saturated

Cactual > Ceq: Over-saturated

DPC: three steps

• DPC involves three sequential steps:

1. Dissolution reaction

2. Transport by diffusion through grain-boundary fluid

3. Precipitation reaction

• Because this is a sequential process (chain process)

• The slowest step determines the rate of the process

• And it determines the flow law

Case 1: diffusion is rate controlling

• Reaction is very fast, relative to diffusion

• Whole "Ceq is used to drive diffusional transport

-"!n/2

+"!n/2

g x

Cactual

compression extension

x

Cactual ! Ceq

Cactual ! Ceq

!

"C = Ceq

+#Ceq

#

!

"C =Ceq

+#Ceq

#=C

0+$

2"% #C

0+$

2"%

!

"#C =$#%

• Flux proportional to concentration gradient (Fick's law)

!

"C

"x#$C

$x=%&

$'

g

# = shape factor

g = grain size

Case 1: diffusion is rate controlling

• All atoms have to move through area ug

• (u = grain boundary width)

-"!n/2

+"!n/2

g x

Cactual

compression extension

x

Cactual ! Ceq

Cactual ! Ceq

!

" = ugJ• Number $ of atoms going through area ug is:

!

V ="# ="ugJ• Volume V of atoms going through area ug is:

• Whole volume V arrives at

extensional side, adding a layer of width w:

!

w =V

g2

="ug

g2J =

"u

gJ

• Producing a strain rate of:

!

˙ " =#w

g=#$u

g2J

!

"C = Ceq

+#Ceq

#

Case 1: diffusion is rate controlling

• Equation for strain rate:

• Fick's law for diffusion:

-"!n/2

+"!n/2

g x

Cactual

compression extension

x

Cactual ! Ceq

Cactual ! Ceq

!

˙ " =#$u

g2J

!

J = "D#C

#x

!

˙ " =#u

g2D$C

$x

• Concentration gradient was derived as:

!

"C

"x=#$

%&

g

!

"C = Ceq

+#Ceq

#

• Finally giving:

!

˙ " =#$2uD

%&

g3

Diffusion-controlled DPC-creep

• Flow law : or simply:

• Linear (Newtonian) viscous creep:

• Strongly grain-size sensitive: (like Coble creep)

• Thermally activated (diffusion):

• Diffusion-controlled DPC-creep is important in

• Very fine-grained rocks

• Wet rocks

• Soluble minerals (calcite, quartz)

!

˙ " =#$2uD

%&

g3

!

˙ " = AdcD#$

g3

!

˙ " #$%

!

˙ " # g$3

!

˙ " #D#exp

$Q

RT

%

& '

(

) *

Case 2: reaction is rate controlling

• Reaction is very slow, relative to diffusion

• Whole "Ceq is used to drive the reaction

-"!n/2

+"!n/2

g x

Cactual

compression extension

x Cactual ! Caverage

!

"C = Ceq

+#Caverage

!

"C =Ceq

+#Caverage =C

0+$

"%

2#C

0

!

"#C =$

2#%

Cactual ! Caverage

Case 2: reaction is rate controlling

• Dissolution rate is proportional to under-saturation "C

-"!n/2

+"!n/2

g

x

!

w = R"C• Per second a layer w dissolves with:

• Producing a strain rate of:

!

˙ " =w

g=R

g#C

x

Cactual

compression extension

Cactual ! Caverage

!

"C = Ceq

+#Caverage

Cactual ! Caverage

!

"C =#

2"$• For "C we found before:

!

˙ " =R

g

#

2$%

Reaction-controlled DPC-creep

• Flow law : or simply:

• Linear (Newtonian) viscous creep:

• Weakly grain-size sensitive:

• Thermally activated (reaction):

• Reaction-controlled DPC-creep is important in

• Fine-grained rocks

• Wet rocks

• Soluble minerals (calcite, quartz)

!

˙ " = ArcR#$

g

!

˙ " #$%

!

˙ " # g$1

!

˙ " # R#exp

$Q

RT

%

& '

(

) *

!

˙ " =#

2R$%

g

DPC microstructures

Cementation and overgrowths

Cemented pore

Dust rim

Cementation and overgrowths

Mica beard

Cementation and overgrowths

Repeatedprecipitation inthin fracture

Cementation and overgrowths

Quartz precipitationIn thin cracks

Dissolution

Partly dissolved micro-fossil

Dissolution

Partly dissolved micro-fossil

Dissolution

Dissolution seam(Stylolite)

Dissolution

Cemented pore

Dust rim

Dissolution seam(Stylolite)

Grain indentation

Dissolution

Dissolution seam(Stylolite)

Dissolution

Dissolution seam(Cleavage)