SUPPLEMENTARY MATERIAL FOR
REACTIVE TRANSPORT AND THE THERMO-HYDRO-MECHANICAL COUPLING
IN DEEP SEDIMENTARY BASINS AFECTED BY GLACIATION CYCLES: MODEL
DEVELOPMENT, VERIFICATION AND ILLUSTRATIVE EXAMPLE
Sergio A. Bea(1), K. Ulrich Mayer(2) and Kerry T.B. MacQuarrie(3)
(1)
CONICET-IHLLA, República de Italia 780, Azul, BA C.C. 47 (B7300), Argentina,
sabea@faa.unicen.edu.ar
(2)
Department of Earth, Ocean and Atmospheric Sciences, University of British Columbia, 2207
Main Mall, Vancouver, BC, V5T 1Z4, Canada.
(3)
Department of Civil Engineering, University of New Brunswick, P.O. Box 4400, Fredericton,
NB, E3B 5A3, Canada.
1. FORMULATION OF ONE-DIMENSIONAL VERTICAL STRESS IN THE FLUID
MASS CONSERVATION EQUATION
The fluid mass conservation equation, as implemented in MIN3P-THCm, is:

K
    P  g   Q
t

Equation S1
where  is the fluid density [M L-3], g is the gravity constant [L T-2],  is the porosity [-], P
is the fluid pressure [M L-1 T-2],  is the dynamic fluid viscosity [M L-1 T-1], K is the
permeability tensor [L2], and Q is the volumetric fluid source/sink term [L-3 T-1]. The storage
term in Equation S1 (left term) can be expanded as:




 P 
   

t
t
t
t
P t
t C,T
Equation S2
where  / P [L2 T-2], and  / t C, T [M L-3 T-1] account for density changes induced by liquid
pressure, and the temporal density changes due to concentrations and temperature, respectively.
MIN3P-THCm does not account for the geometric deformation of the grid as a mechanical load
is applied, and it solves the mechanical processes in a simplified way as have several recent
studies of ice-sheet induced sediment-hydrogeologic interactions (e.g. Bense & Person 2008,
Lemieux et al. 2008, Person et al. 2007, McIntosh et al. 2011). In this approach, we assume that
during the ice sheet loading cycle, the load changes are relatively homogeneous per unit of
surface area (especially on geologic time scales). In this situation, the changes in vertical stress
must be considered; however, lateral stress can be neglected. Following Neuzil (2003), and
assuming zero lateral strain, the full fluid mass conservation equation implemented in MIN3PTHCm is:
  
K
 P
Sp    zz   
    P  g   Q
t  t C, T

 t
Equation S3
where  is the one-dimensional loading efficiency coefficient [-], and zz is the vertical stress
[M L-1 S-2]. The mathematical derivation of Equation S3 is described below following Neuzil [0]
(and references therein) and Bea et al. (2011).
Rice and Cleary (1976), among others, have presented the description of porosity changes
in terms of compression moduli as:
 1 1   v  1 1

  P
  
 



t
 K K s  t  K K s K s  t
Equation S4
where K and K s are the drained and solids bulk modulus, respectively [M L-1 T-2], and  v is
the mean normal or mean total stress [M L-1 T-2]. Likewise, fluid density changes induced by
pressure (  / P ) in Equation S2 can be defined as:



P K f
Equation S5
where K f is the bulk modulus of the pore fluid [M L-1 T-2]. Substituting Equations S4 and S5
in Equation S2, and rearranging yields:
 1
   1
1 
1   P
  


  


t
 K f K s   t
  K Ks 
1
1   v

  


t C, T
 K K s  t
Equation S6
Here, it is helpful to introduce the three-dimensional pressure-based specific storage
coefficient ( Sp3 , [M-1 L T2]), and the three-dimensional loading efficiency coefficient (i.e.
Skempton’s coefficient  ,[-]) as reported by Neuzil (2003) and Van der Kamp & Gale (1983):
1
1   1
1 
Sp3   

  

 K Ks   K f Ks 

1
1 
 

K Ks 
Equation S7
Equation S8
S p3
Substituting Equations S7 and S8 into S6, and rearranging yields:


 P
 S p3    v
t
t
 t
 
  
t C,T

Equation S9
The storage term in Equation S9 is simplified in the present work by assuming that the
main strain field in a glaciation scenario is purely vertical (as was also assumed by Bense &
Person (2008) and Lemieux et al. (2008). Poroelasticity defines strains in the porous matrix in
terms of displacements, and the three main components of strain are defined by Neuzil (2003):
 xx 
1 

 
 xx 
 kk  
P  TT

2G 
1 
 3K
 yy 
1 

 
 yy 
 kk  
P  TT

2G 
1 
 3K
 zz 
1 

 
 zz 
 kk  
P  TT

2G 
1 
 3K
Equation S10
where  xx ,  yy and  zz are the strains in the x, y and z coordinate directions [-],
  1  (K / K s ) , G is the shear modulus [M L-1 T-2],  is the drained Poisson’s ratio [-],  kk is
the sum of normal stresses, and  T is the linear thermal expansivity of the porous medium [oC1
]. The assumption of purely vertical stress implies that lateral gradients in fluid pressures tend to
be small and that the lateral deformations can be neglected (i.e., εxx = εyy = 0), and Equation S10
can be rearranged to yield (Neuzil 2003):
 xx   yy 

2G
 kk 
P  2G T T
1 
3K
Equation S11
The sum of normal stresses (  kk , Neuzil 2003) can be computed using Equation S11,
and defining and replacing the quantity   2(1  2) /3(1  ) (Neuzil 2003) and the drained
modulus K  2G (1  ) /3(1  2) Rice & Cleary (1976):
 kk   xx   yy   zz  3P 
1    
(1  )
zz 
4(1  )
G T T
(1  )
Equation S12
Multiplying the term with  zz in Equation S12 by 1    / 1    , the following equation is
obtained:
 kk  3P 
1    1      4(1  ) G T
zz
T
(1  ) 1   
(1  )
Equation S13
Note that Equation S13 can be rearranged to yield an explicit expression to mean normal
total stress (  v   kk / 3 ):
 v  P 
1   
31     21  2 
1    zz  4(1  ) G T T
3(1  )
Equation S14
Neglecting the thermal expansion of porous medium in Equation S14 and substituting it
into Equation S9 yields:
 P
 zz
1   

 Sp3 1     
t
31     21  2  t
 t


t C, T



Equation S15
The new quantity Sp , the one-dimensional specific storage coefficient, can be defined as
[0, 0]:
Sp  Sp3 1  
Equation S16
In addition, by defining the one-dimensional loading efficiency as (e.g. Neuzil 2003, Normani
2009):
 
1   
31     21  2 
Equation S17
the full fluid mass conservation equation implemented in MIN3P-THCm is obtained (Equation
S3) by substituting Equations S16 and S17 into S15.
2. FLUID DENSITY AND VISCOSITY
In MIN3P-THCm, density (  ) and viscosity (  ) are considered dependent on both temperature
and solute concentrations. Different equations of state were implemented for both  and  . To
describe the dependence of fluid density on solute concentrations, a commonly employed
approach relates fluid density linearly to the concentration of Total Dissolved Solids (TDS, [M L3
]). This option is also implemented in MIN3P-THCm and the density changes ( C ) can be
computed as:
C 

TDS
TDS
Equation S18
where TDS are changes in TDS [M L-3], and  / TDS [-] is assumed to be a constant
independent of fluid composition. Values ranging from 0.688 to 0.714 are used to simulate
seawater-freshwater interactions.
Although a linear relationship between density and TDS is commonly assumed when
NaCl dominates the salinity (e.g., seawater), the presence of CaCl2-enriched brines in many
sedimentary basins requires a more sophisticated model for density calculations, because density
will depend on the elemental composition of the fluids. MIN3P-THCm also allows for the
computation of C using a non-linear relationship as a function of composition. For instance,
C can be computed as the difference between a reference density and a density calculated
based on Pitzer's equations according to Monnin (1994) and Bea et al. (2010). Using this
approach, density calculations are based on the total volume of the solution that contains 1 kg of
water ( V ) (Monnin 1994):
V  Vid  Vex  1000 w   mi Vio  Vex
i
Equation S19
where Vid is the ideal volume based on the molar volume of solutes and Vex represents the
total excess volume of a multicomponent electrolyte solution.  w is the specific volume of
pure water (L3 M-1), m i is the molality of the ith aqueous species [mol M-1], and V io is the
standard partial volume of the ith solute (L3 mol-1). The term Vex in Equation S5 can be
expressed as a virial expansion of the solute molalities:
Vex
v (I)  2
v
v
 f DH
  mc ma (Bca  ( mc z c )Cca
RT
c a
c
Equation S20
In this expression, m c is the molality of cation c (of charge z c ), and ma that of anion a.
v is the Debye-Hückel term that is a function of the ionic strength, and B v and C v are the
f DH
ca
ca
second virial coefficients for the volume that accounts for the interactions among ions. The
density of the solution (  ) is computed according to:

1000   mi Wi
i
Equation S21
V
where m i and Wi are the molality and molecular weight [g mol-1] of the ith solute,
respectively.
The dependence of density on temperature ( T ), can be computed using a linear
relationship:
T 

T
T
Equation S22
where  / T [M L-3 K-1] is a constant, and T is the temperature change with respect to the
reference temperature at a reference density.
With regard to the fluid viscosity, the temperature and concentration-dependent terms are
empirically defined and can be computed as (e.g., see Bea et al. 2011, Diersch & Kolditz 1998):
f ,C 
1  1.85  4.12  44.53
1  1.85f  4.1f2  44.53f
Equation S23
1  0.7063f  0.04832 3
f
Equation S24
f ,T 
1  0.7063  0.04832 3
where  and f are the solute mass fractions in the fluid for the actual and reference
viscosities, respectively, and   (T  150) / 100 ,with T provided in units of oC. Alternatively,
the viscosity-temperature dependence ( f,T ) can be computed based on the expression
presented by Voss & Provost (2008):
f , T 
 248 .37 


Tf 133 .15 

10
 248 .37 


10  T 133 .15 
Equation S25
where Tf is the temperature of the reference water, again with T provided in units of oC.
3. MIN3P-THCm ILLUSTRATIVE EXAMPLE: IMPACT OF A CONTINENTAL
GLACIATION ON REGIONAL GROUNDWATER FLOW IN SEDIMENTARY
BASINS - MODEL PARAMETERIZATION
In the MIN3P-THCm application example, initial porosity was considered to vary with depth
based on the expression proposed by Bahr et al. [0]:
e cg(s  )z
(z) 
e
 cg(s  )z
K1 
 K1
1  z 0 
z  0
Equation S26
Equation S27
where z 0 is the porosity at the surface, c is the compaction rate [M-1 L T2], s and  are the
densities for solid and fluid, respectively [M L-3].
The initial porosity distribution was computed according to the parameters shown in
Table S3 using Equation S26 and S27.
The hydraulic conductivity was also considered to vary with depth as a function of
porosity based on the Karman-Kozeny model:
K (z) 
3

2 
1   ref 


1 2
 3ref
 
K ref
Equation S28
where K ref is the reference hydraulic conductivity tensor [L T-1], and  ref [-] is the reference
porosity corresponding to K ref (Table S3). Vertical hydraulic conductivity was considered to be
one order of magnitude less than the horizontal component.
The one-dimensional specific storage coefficient ( Ss ), was computed based on
mechanical parameters for the different geological units that are shown in Table S4 according to
(e.g., see Neuzil 2003):
 1 1 
 1
1 
Ss   f g   1        
 K K s 
 K f K s  
Equation S29
where K and K s are the drained and solids bulk moduli, respectively [M L-1 T-2], and K f is
the bulk modulus of the pore fluid [M L-1 T-2]. The remaining terms in Equation S29 are defined
as:

2(1  2 )
3(1  )
  1
K
K
Ks
E
3(1  2)
Equation S30
Equation S31
Equation S32
where  is the Poisson’s ratio [-] for drained conditions, and E is Young's modulus representing
the stiffness of a rock material [M L-1 T-2] (Table S4). The latter is defined, for small strains, as
the ratio of the change of stress with strain.
The one-dimensional loading efficiency coefficient (  ), is computed as:

(1  )
3(1  )  2 (1 2 )
Equation S33
where  is the Skempton’s coefficient [-], computed according to:
1 1 
  
 K Ks 

1 1   1
1 
 
    
 K Ks   Kf Ks 
Equation S34
Following an assumption made by Normani [0], the rock formations were considered
incompressible in this context ( Ks   ). The bulk modulus of the pore fluid ( K f ) in Equation
9
S29 and S34 was set to that of seawater ( K f  3x10 Pa ), independent of fluid composition and
pore water pressure.
For the parameterization of the thermal properties, MIN3P-THCm employs a harmonic
average for bulk thermal conductivity ( λ ) computed as:
λ  λ l  (1  )λ s
Equation S35
where λ l and λ s are the fluid and solid thermal conductivities [E L-1 T-1 oC-1], respectively. The
initial solid thermal conductivity distribution was also computed to vary with depth as a function
of porosity using parameters shown in Table S5 and Equations S26 and S27.
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Table S1. Chemical composition of initial and boundary solutions for the verification example:
Replacement of gypsum by polyhalite (Bea et al. 2010).
Parameter
Boundary water
pH
7.3
Ca(*)
3.8x10-3
Mg(*)
2.2
Cl(*)
6.48
K(*)
7.43
Na(*)
2.72
Br(*)
10-6
TIC (*, **)
5.1x10-5
SO4(*)
5.6x10-1
Ionic strength(*)
13.83
-1
(*) Concentrations in [mol l ].
(**) Total Inorganic Carbon.
Initial water
8.1
1.1x10-2
6.3x10-1
5.94
1.2x10-1
5.178
1.0x10-9
1.7x10-5
2.3x10-1
7.35
Table S2. Model parameters for the verification example involving positive and negative thermohaline convection (Geiger et al. 2006).
Parameter
Value
Unit
Positive initial buoyancy
Initial parcel temperature
Initial parcel brine mass fraction
Initial parcel density
300
0.14
831
o
C
[-]
[kg m-3]
Negative initial buoyancy
o
Initial parcel temperature
Initial parcel brine mass fraction
Initial parcel density
250
0.47
919
C
[-]
[kg m-3]
Reservoir temperature
Reservoir initial mass fraction
Reservoir liquid density
200
0
875
C
[-]
[kg m-3]
Porosity
Permeability
Molecular diffusion
Formation conductivity
Specific heat capacity of rock
Specific heat capacity of water
Density of rock
0.1
5x10-14
10-8
1.8
1000
4184
2650
[-]
[m2]
[m2 s-1]
[J s-1 m-1oC-1]
[J kg-1oC-1]
[J kg-1oC-1]
[kgm-3]
o
Table S3. Model parameters for the illustrative sedimentary basin example: Porosity, horizontal
and vertical hydraulic conductivities for the different geological materials.
Flow parameters
Unit
Dol3
Sh3
Sand4
Dol2
Sand3
Sh2
Ev
Dol1
Sh1
Sand2
Lim
Sand1
Lithology

[-]
(1)
0.02
(1)
0.11
(2)
0.04
(1)
0.02
(2)
0.04
(3)
0.11
(1)
0.08
(1)
0.02
(1)
0.11
(2)
0.04
(1)
0.02
(2)
0.04
Dolostones
Shales
Sandstones
Dolostones
Sandstones
Shales
Evaporites
Dolostones
Shales
Sandstones
Limestones
Sandstones
Weathered
Crw
*0.01
crystalline rocks
Cr
Crystalline rocks (1,4)0.01
*Estimated.
(1)
Normani (2009).
(2)
Medina et al. (2011).
(3)
Curtis (2002).
(4)
Sykes et al. (2008).
(5)
Avis (2009).
(6)
Harrison et al. (2009).
(7)
McIntosh et al. (2011).
(8)
Westjohn & Weaver (1998).
(9)
Raven et al. (1998).
(10)
Harrison et al. (2009).
(11)
Birkholzer & Zhou (2009).
zref
[m]
724
559
0
724
0
559
394
724
559
4000
724
4000
KH
[m s-1]
(6) -7
10
(7)
3.2x10-11
(8)
5.3x10-6
(10) -7
10
(8)
5.3x10-6
(7)
3.2x10-11
(5) -13
10
(5) -12
10
(5) -13
10
*10-5
(9) -12
10
(8) -5
10
Kv
[m s-1]
*10-8
*3.2x10-12
*5.3x10-6
*10-8
*5.3x10-6
*3.2x10-12
*10-14
*10-13
*10-14
(11) -6
10
*10-13
(11) -6
10
zref
[m]
1064
0
0
1064
0
0
394
1064
559
0
724
0
861
(8)
*10-11
0
861
(1,4)
(1,4)
861
10-10
8x10-12[0]
8x10-13
Table S4.Model parameters for the illustrative sedimentary basin example: mechanical
parameters, estimated based on lithology.
Mechanical parameters
Unit
Lithology
Dol3
Sh3
Sand4
Dol2
Sand3
Sh2
Ev
Dol1
Sh1
Sand2
Lim
Sand1
Dolostones
Shales
Sandstones
Dolostones
Sandstones
Shales
Evaporites
Dolostones
Shales
Sandstones
Limestones
Sandstones
Weathered
crystalline rocks
Crystalline rocks
Crw
Cr
E
[GPa]
50
15
25
50
25
15
16.5
50
15
25
45
25
Poisson's ratio
[-]
0.15
0.1
0.14
0.15
0.14
0.1
0.35
0.15
0.1
0.14
0.3
0.14
15
0.26
55
0.24
Table S5.Model parameters for the illustrative sedimentary basin example: thermal and chemical
parameters.
Unit
Dol3
Sh3
Sand4
Dol2
Sand3
Sh2
Ev
Dol1
Sh1
Sand2
Lim
Sand1
Lithology
Thermal parameters
Volumetric heat
Thermal
capacity
conductivity
[J m-3 oC-1]
[W m-1 oC-1]
[0]
[0]
6
1.87x10
3.8
1.87x106
2
6
1.87x10
3
1.87x106
3.8
1.87x106
3
6
1.87x10
2
1.87x106
3.8
1.87x106
3.8
1.87x106
2
6
1.87x10
3
1.87x106
2.5
1.87x106
3
Dolostones
Shales
Sandstones
Dolostones
Sandstones
Shales
Evaporites
Dolostones
Shales
Sandstones
Limestones
Sandstones
Weathered
Crw
crystalline
1.87x106
rocks
Crystalline
Cr
1.87x106
rocks
(*)Estimated based on lithology.
Chemical Parameters
Bulk
CEC
density
[meq 100gr-1]
[kg m-3]
(*)
[0]
2900
2.5
2650
5
2800
1
2900
2.5
2800
1
2650
5
2900
2.5
2900
2.5
2650
5
2800
1
2700
2.5
2800
1
2.6
2750
1
2.6
2750
1
Table S6.Geochemical reactions considered in the illustrative sedimentary basin example.
Homogeneous reactions
Ref.

2HSO 
4  H + SO 4
(*)
MgOH   H   H 2 O  Mg 2 
CaOH   H   H 2 O
(*)
(*)
OH   H   H 2 O
(*)
HCO 3  CO 32   H 
(*)
CO 2 (aq)  H 2 O  CO32   2H 
(*)
CaCO3(aq)  CO32   Ca 2 
(*)
MgCO 3(aq)  CO32   Mg 2 
(*)
CaSO 4 (aq)  SO 24   Ca 2 
(*)
CaCl   Ca 2   Cl 
(*)
MgHCO 3  Mg 2   H   CO32 
(*)
Cation-Exchange reactions

MgX 2  2Na  2 NaX  Mg 2 
CaX 2  2Na   2NaX  Ca 2 
KX  Na   NaX  K 
Dissolution/precipitation reactions
(**)
(**)
(**)
Enthalpy
[kcal mol-1]
Halite  Na  + Cl -
-0.918
(*)
Calcite  CO32-  Ca 2 
2.585
(*)
8.29
(*)
Dolomite  2CO32-  Ca 2   Mg 2 
3.769
Anhydrite  SO 24-  Ca 2 
(*) Equilibrium constant taken from Wolery & Daveler (1992).
(**) Selectivity coefficients taken Appelo & Postma (1993).
(*)
Table S7.Initial mineral contents considered in the illustrative sedimentary basin example. For the purpose of the illustrative example
considered in the present simulation, a highly simplified mineralogy was considered. Clay minerals, which are known to be abundant
present in shale were not considered beyond the enhanced effect of ion exchange (see Table S5).
Mineral
Cr Crw
Sand1 Lim1 Sand2 Sh1(*) Dol1 Ev Sh2(1) Sand3 Dol2 Sand4 Sh3(1)
Calcite
0
0
1
90
1
9
10
4
9
1
10
1
9
Dolomite
0
0
9
10
9
1
90
36
1
9
90
9
1
Halite
0
0
0
0
0
0
0
50
0
0
0
0
0
Anhydrite
0
0
0
0
0
0
0
10
0
0
0
0
0
Nonreactive
100 100
90
0
90
90
0
0
90
90
0
90
90
minerals
(*) Based on rock composition for Queenston, Georgian Bay and Blue Mountain Formations reported by Mazurek [0].
Dol3
10
90
0
0
0
Table S8.Chemical composition of brines and brackish waters considered in the illustrative
sedimentary basin example (chemical compositions taken from Hobbs et al. (2011).
Formational and boundary water compositions
(2)
(3)
(4)
6
6
6
pH
(1)
6
Ca(6)
1.5
0.77
0.2
0.19
6.2x10-4
6.5x10-3
6
5.4x10-4
3.4x10-2
1.2x10-4
2.85
1.8 10-2
4x10-2
4.15
4x10-3
5.7x10-2
3.2x10-4
2.49
9.7x10-3
8.6x10-3
6.24
1.1x10-2
0.07
5.2x10-4
5.77
7.4x10-3
10-2
3.61
5.2x10-5
0.12
1.8x10-3
3
5.6x10-3
8.2x10-4
2.9x10-3
2.6x10-4
6.4x10-5
1.7x10-5
5.2x10-4
4x10-5
Ca-Na-Cl
Na-Cl
Na-Cl
Na-Cl
-
7.6
4.97
6.48
3.82
5.2x10-3
0.34
0.24
0.37
0.21
2x10-4
1217
1150
1204
1128
997
-0.78
0
0
-2.62
-7.47
-4.44
-7.64
-2.76
Mg(6)
Cl(6)
SO4(6)
K(6)
TIC(6,7)
Na(6)
Br(6)
Chemical
signature
Ionic
strength(6)
TDS(8)
Density(9)
SIHal
-0.22
-0.71
SICal
0
0
SIDol
0
0
SIAnhy
-0.82
-0.25
(1) Units Cr, Crw, Sand1 and Sand2.
(2) Units Lim1,Sh1 and Dol1.
(3) Unit Ev.
(4) Units Sh2, Sand3, Sand4, Sh3, Dol2 and Dol3.
(5) Meteoric water.
(6) Units in [mol l-1].
(7) Total Inorganic Carbon.
(8) Units in [kg l-1].
(9) Units in [kg m-3].
0
0
0
0
(5)
6
Figure S1.A), B) and C) Distribution of CaX2 in equivalent fractions at 10,000, 20,000 and 30,000
years, respectively. D) Temporal evolution of equivalent fractions of NaX, CaX2, MgX2 and KX at a
selected location in the shallow part of the Sand4 unit.
Figure S2. A) Distribution of the porosity changes [%] after 32,500 years. B) Temporal evolution
porosity changes [%] at two selected locations.