Chekalin, A.

Khramchenkov, M.

Konyukhov, V.

Earth-prints Repository

M. Khramchenkov, V. Konyukhov, A. Chekalin 
KARST FAULTS: MECHANISM OF EVOLUTION  
AND ITS MODELLING 
Dr. M. Khramchenkov 
Institute of Mathematics & Mechanics 
Kazan State University, Kazan, Russia 
E-mail: Maxim.Khramchenkov@ksu.ru 
Phone: 7-843-2927524 
Fax:     7-843-2382209 
Address: Prof. Nujin str., 1/37, 420008, Kazan 

MAIN DIFFICULTY OF KARST PROCESSES 
MATHEMATICAL SIMULATION
• KARST PROCESSES SIMULATION = SOLUTION OF 
COMBINATION OF A NUMBER OF NONLINEAR 
PROBLEMS
MAIN NONLINEAR PROBLEMS:
• Chemical dissolution of rock fractures walls
• Hydrodynamics of flow in multi-scale fractured media
• Underground erosion of over-fractured rocks (suffosion –
from lat. suffosio)
KARST TYPS:
• Carbonate karst (dissolution of limestone)
• Sulfate karst (dissolution of gypsum)
• Salt karst (dissolution of halite)
We’ll be concentrated only with carbonate karst (like most 
widely occurring).
Mathematical model of rock dissolution process
Governing equations:
( ) ( )( ) ,( 1,..., ) (1)v si i i u inC J n S i Nt
σ σ∂ +∇ = + =∂
r
( ); div 0 (2)kV p g Vρη= − ∇ + =
r rr
( ) ( )
1 1
(3)
l m
a b
i i i i
i i
A Bν ν
= =
↔∑ ∑
[ ] [ ]( ) ( )( ) ( )( )
1 1
, (4)
a b
i i
k kl m
k
d i r i i i
i i
k A k B
t t
ν νξ ξσ ν
= =
∂ ∂= − =∂ ∂∏ ∏
Dissolution rate
Calcite dissolution rate (White, 1977)
Rate
Region 1
10 000
1 000
100
Region 2
10
Region 3-A
1.0
Region 3-B
10                               20                              30                         
ΔpH
2
lg 0.0COP =
2
lg 1.5COP = −
2
lg 2.0COP = −
2
lg 2.5COP = −
2
lg 2.6COP = −
-2 -1(mg×cm year )
Local equilibrium approach
• Reaction is very fast:
• Chemistry:
• Definitions: 
[ ] [ ] rm
i
i
l
i
i KBA
b
i
a
i =∏∏
== 11
)()( νν
−+ +↔ 332 HCOHCOH
−+− +↔ 233 HCOHHCO
−+ +↔ 232S3 COCa)(CaCO
2 2
3 3 2 3[Ca ] , [H ] , [HCO ] , [CO ] , [H CO ]C x y r m
+ + − −= = = = =
Diffusion limited model
• Equilibrium constants: 
• Governing equations: 
• Balance conditions: 
• Solution: 
),/(10),/(10),/(10 31,102
35,6
1
35,6
1 rxyyxyx aaa γγγγγγγ −−− ===
)/(10 31,102 rxya γγγ−= )/(10 5,83 rCa γγ−=
*
Cz
Cv
t
Cn σ=∂
∂+∂
∂
xxz
xv
t
xn σσ +=∂
∂+∂
∂ *
mmz
mv
t
mn σσ +=∂
∂+∂
∂ *
yyz
yv
t
yn σσ +=∂
∂+∂
∂ *
rrz
rv
t
rn σσ +=∂
∂+∂
∂ *
)(* iiii CC −= βσ /i i u iD S hβ =
****
myrC σσσσ ++= 0=++ myr σσσ
0, ( )n v C r y m
t z
∂Φ ∂Φ+ = Φ = − + +∂ ∂ 0 0C C r y m S⇒ − = + + −
Boundary conditions:
• Dimensionless concentrations and pH 
versus time
• Denotings:
2CO
47.1
00000
10,0,0 PmmwCC
zz
−
== =====
1 2 ;1 , 2 ; 2 ; 3 ; 3 ; 4 ; 4p p p p p p p pC y C y m m pH pH′ ′ ′− − − − − − −
∞
∞
∞
∞
∞
∞
∞
∞
−
−=−
−=−
−=−
−=
mm
mmm
yy
rrr
yy
yyy
yy
CCC pppp
0000
,,,
∞
∞
∞
∞
∞
∞
∞
∞
−
−=−
−=−
−=−
−=
yy
mmm
yy
rrr
yy
yyy
yy
CCC pppp
0000
,,,
Electro-diffusion model (Khramchenkov, 1998)
• Impact of electrical forces in ion’s diffusion
• Chemical reactions:
• Near the chemical equilibrium:
( )( ) ( ) ( ) ,vi i i i i
nC VC C j n
t
α σ∂ +∇ =∇ ∇ + +∂
r
( ), div 0,kV p g Vρη= ∇ + =
r rr
( ) ( )( ) ,v si i i u i
nC j n S
t
σ σ∂ + = +∂ ( ),
i i
i i i i
z C Fj C C
RT
β δϕ= − + 0,i i
i
zC =∑ 0.i i
i
I z j= =∑
−+ +↔+ 322 HCOHCOOH
−++ +↔+ 323 HCOCaHCaCO
−−+ ++↔+ OHHCOCaOHCaCO 3223
Dissolution rate (Dreybrodt, 1988)
• Measured dissolution rate as a function of 
• Measured curves:
A:
B:
C:
Dissolution rate
10
I
8
II
6
4
III
2
А
B  
C  ΔpH
0            0.2          0.4          0.6          0.8 1.0  
-2 -1(mg×cm year )
pHΔ
2
lg 0.0COP =
2
lg 1.5COP = −
2
lg 2 .0C OP = −
Pure kinetics model
• Chemical reactions:
• Dissolution rate equation (modified Plummer – Wigley kinetics):
−+ +↔ 332 HCOHCOH
−+− +↔ 233 HCOHHCO
−+ +↔ 232S3 COCa)(CaCO
−+
−++
+↔+
+↔+
3
2
32S3
3
2
S3
HCO2CaH)(CaCO
HCOCaH)(CaCO
CO
00
2
13
2
2
3
2
31 ,11)(
1 CC
m
Cy
aa
amk
x
Cy
a
akxk
dt
dC
S tu
=⎥⎦
⎤⎢⎣
⎡
⎟⎟⎠
⎞
⎜⎜⎝
⎛−+⎥⎦
⎤⎢⎣
⎡
⎟⎟⎠
⎞
⎜⎜⎝
⎛−+= =
Comparison with experimental data of  Erga, Terjesen, 1956:
Plot of complex                  versus C:
Denoting: 
/uS dC dt
2CO
0, 0952 M PaP− =
2CO
0,0135 MPaP− =
Comparison with experimental data of Lekhov et al., 1974
• Plot of complex              versus IAP/[ ]R H+
1.2
1
0.8
0.6
0.4
0.2
0                       0.05                 0.1      0.15                  0.2                   0.25   0.3   LC
IAP
Conclusions (“chemical” part):
• Principal problems are solved
• Some problems are waiting for solution 
(construction of universal chemical and hydro-
mechanical model)
Hydrodynamics of flow in multi-scale fractured media
• Main problem: transformation of fissured media into conduit media (media 
with domination of relatively small number of transport fractures)
v
Evolution of transport fractures in consequence of fracture wall dissolution
Hydrodynamics of flow in multi-scale fractured media
• Main publications (karst fracture’s evolution models):
Bauer et al. (2003), Baedke & Krothe (2001), Birket et al. (2003), 
Dreybrodt,(1988, 2000), Jeannin (2001), Kaufmann & Braun (1999, 
2000), Kaufmann (2003), Liedl et al. (2003).
Simplest model: dissolution of single fracture
x
y
b
00
u y( ) C x y( , )dx
dy
Hydrodynamics of flow in multi-scale fractured media
• Governing equations:
• Boundary condition:                            Approach: integral correlations
• Symmetrical condition:
• Wall’s dissolution                                   Characteristic point      :
2
2 ( ) (5)
C CD u y
y x
∂ ∂=∂ ∂
2( ) 1.5 1 ( / ) (6)bu y u y b⎡ ⎤= −⎣ ⎦
0),0( CyC =
0)0,( =∂
∂
y
xC
( , )
s
C x b dbD
y dt
ρ∂ = −∂
(5) ( ), (5)( ), : ( ) 0, ( ) 0, 0
b
h
Cy b y b h C b b b
y
∂× − − = = ≥∂∫
40 /(40 )h x gb Dν∗= ⇒ =
x∗
Hydrodynamics of flow in multi-scale fractured media
Equation for fracture’s size distribution:
First case:          small
Normal distribution with variance
Second case:          large
Normal distribution with variance
Turbulent regime of flow – normal distribution transforms in log-
normal distribution
, 0 1 (7)e
db f k sf s
dt b
∂ = − ≤ ≤∂
x∗
40 exp[ ( )], 60 /(11 )
s
DC x x D gb
ks
χσ χ νρ
∗− −= =
x∗
0
3 ( / )s
DC
ks x x
σ ρ ∗=
Hydrodynamics of flow in multi-scale fractured media
Numerical simulation
1. Kinetic regime of dissolution, laminar flow – moderate competition 
between fractures with different size (finish of calculations – an increase 
of total discharge by two order).
0 5 10 15 20 25 30
i <x*3+1, ì >
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
R, ì
Hydrodynamics of flow in multi-scale fractured media
Numerical simulation
2. Diffusion regime of dissolution, laminar flow – more high competition 
between fractures with different size.
0 5 10 15 20 25 30
i <x*3+1, ì >
0.0002
0.0004
0.0006
0.0008
0.001
R, ì
Hydrodynamics of flow in multi-scale fractured media
Numerical simulation
3. Kinetic regime of dissolution, turbulent flow – moderate competition between 
fractures with different size.
0 5 10 15 20 25 30
i
0.01
0.02
0.03
0.04
0.05
R, ì
Hydrodynamics of flow in multi-scale fractured media
Numerical simulation
4. Diffusion regime of dissolution, turbulent flow – more high competition 
between fractures with different size.
0 5 10 15 20 25 30
i
0.01
0.02
0.03
0.04
0.05
R, ì
Conclusions (“hydrodynamics” part):
Character of karst fractures growth depends on:
• Regime of flow (laminar or turbulent);
• Regime of dissolution (kinetic or diffusion);
• Correlation between discharge, aperture and length of fracture.
KARST FAULTS
• Karst fractures drain the water from over-fractured rocks
• Simulation of suffosion: suufosion starts when the head of gradient stays 
more then critical meaning
Distribution of the head of gradient  and flow velocity near the point 
of contact fractured and over fractured rocks (point of initial erosion)
KARST FAULTS
Equation of suffosion (Khramchenkov et al., 2006):
We find        , if we solve the integral
Realistic view of          is 
Variation of N is 
N - volume fraction of substance of the ground, capable to suffosion
0/ ,| | ( ) (8)N t N H I N∂ ∂ =−Ψ( ∇ − )
Ψ 0
( )
(9)
N
N t
dN t=Ψ∫
Ψ 0 0(| | ),  | | , 0 (10)N H I H I NγΨ = ∇ − ∇ > >
0
2
0
0
0
1,           
( ) (1 ) ,      , 0 1/
0,          , 1/
| H | I
N t Bt | H | I t B
N
| H | I t B
∇ ≤
= − ∇ > < <
∇ > >
0 0( ) /(2 )B | H | I Nγ= ∇ −
KARST FAULTS
Analytical solution
Bounds of suffosion zone for different
2 2 2 2( ) Re arsch( ) , ( ) Im arsch( ) .T kTI T kTIi ix y
Q Q
e eθ θθ θπ π
⎛ ⎞ ⎛ ⎞= ⋅ = ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
02 kI T
KARST FAULTS
Numerical simulation: dynamics of porosity changing near the point of initial 
erosion for different time: a – d - =  1, 20, 140, 950 c. u.
a b
KARST FAULTS
c d
KARST FAULTS
Numerical simulation for the number of points of initial erosion:
a) isotropic case
KARST FAULTS
Distribution of mobile particles saturation s and porosity m for density difference s-w 0,05ρ =
KARST FAULTS
b) anisotropic case
KARST FAULTS
Arrows – field of fluid velocity V
Conclusions:
In conclusion I should like to say that 
mathematical simulation is useful addition 
to classical methods of karst investigation, 
but…
really I just like to do it!
Thank You for Your attention!
Work was supported by ISTC, project 3225.


Karst faults: mechanism of evolution and its modelling

http://www.earth-prints.org/bitstream/2122/2686/1/Kramchenkov.pdf

Karst faults: mechanism of evolution and its modelling

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