Rheology of rocks This lecture - uni-tuebingen.de · Rheology of rocks ¥Paul Bons ¥T bingen...
Transcript of Rheology of rocks This lecture - uni-tuebingen.de · Rheology of rocks ¥Paul Bons ¥T bingen...
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Rheology of rocks
•• Paul BonsPaul Bons
•• TTüübingen Universitybingen University
•• [email protected]@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
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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 #$%p
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• Fine-grained material
• Coble creep induces NO dynamicrecrystallisation:
• Grain size increases (grain growth)
• Material moves to right towardsdislocation creep field
Coble creep: grain-size increase
AB!
g " c #$%p
Piezometer ! mechanism boundary
AB!
g " c #$%p
• Mechanism boundary is line where
• dynamic rexx grain-size reduction
• static rexx grain-size increase
• Balance
• How to get away from piezometer line?
• Sudden change in stress
• Time/strain delay to reach equilibrium
• Inhibition of grain growth
• E.g. by pinning of grain boundaries
Dissolution-precipitation creep
• DPC involves mass transfer in fluid
• Short distance: from one side of grain to another
• Long distance: from stylolite to vein
Equilibrium concentration
• All minerals can dissolve in fluids (water)
• There is an equilibrium concentration Ceq
• If actual concentration is lower than Ceq
• Mineral dissolves
• concentration increases towards Ceq
• If actual concentration is higher than Ceq
• Mineral precipitates
• concentration decreases towards Ceq
• Finally, equilibrium concentration is reachedconcentration
Chemicalpotential
Ceq
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Chemical potential and pressure
• The chemical potential (µ) is a function of pressure:
• Concentration is proportional to µ, giving:!
"µ
"P=#P
• So the equilibrium concentration is:
!
"Ceq =C0
+#P!
C"µ #$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$%
!
Ceq =C0
+"P
Remember:
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Equilibrium concentration on the surface ofgrains
• Under a differential stress grain boundaries havedifferent equilibrium concentrations
• Diffusion tries to equalise concentration " transport
• Compressional faces: under-saturated " dissolution
• Extensional faces: over-saturated " precipitation
-"!n/2
+"!n/2
g x
Cactual
compression extension
x
Diffusional transport
Cactual < Ceq
Under-saturated
Cactual > Ceq: Over-saturated
DPC: three steps
• DPC involves three sequential steps:
1. Dissolution reaction
2. Transport by diffusion through grain-boundary fluid
3. Precipitation reaction
• Because this is a sequential process (chain process)
• The slowest step determines the rate of the process
• And it determines the flow law
Case 1: diffusion is rate controlling
• Reaction is very fast, relative to diffusion
• Whole "Ceq is used to drive diffusional transport
-"!n/2
+"!n/2
g x
Cactual
compression extension
x
Cactual ! Ceq
Cactual ! Ceq
!
"C = Ceq
+#Ceq
#
!
"C =Ceq
+#Ceq
#=C
0+$
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=#$u
g2J
!
"C = Ceq
+#Ceq
#
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Case 1: diffusion is rate controlling
• Equation for strain rate:
• Fick's law for diffusion:
-"!n/2
+"!n/2
g x
Cactual
compression extension
x
Cactual ! Ceq
Cactual ! Ceq
!
˙ " =#$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"$• For "C we found before:
!
˙ " =R
g
#
2$%
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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
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Cementation and overgrowths
Repeatedprecipitation inthin fracture
Cementation and overgrowths
Quartz precipitationIn thin cracks
Dissolution
Partly dissolved micro-fossil
Dissolution
Partly dissolved micro-fossil
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Dissolution
Dissolution seam(Stylolite)
Dissolution
Cemented pore
Dust rim
Dissolution seam(Stylolite)
Grain indentation
Dissolution
Dissolution seam(Stylolite)
Dissolution
Dissolution seam(Cleavage)