Solute Transport With Chemical Reaction in Single- and Multi-Phase Flow in Porous Media 7
different fluids; it also depends on the internal geometrical structure of the porous medium. A
second consequence of the continuum hypothesis is an uncertainty in the boundary conditions
to be used in conjunction with the resulting macroscopic equations for motion and heat
and mass transfer (Salama & van Geel, 2008b). A third consequence is the fact that the
derived macroscopic point equations contain terms at the lower scale. These terms makes
the macroscopic equations unclosed. Therefore, they need to be represented in terms of
macroscopic field variables though parameters that me be identified and measured.
5. Single-phase flow modeling
5.1 Conservation laws
Following the constraints introduced earlier to properly upscale equations of motion of fluid
continuum to be adapted to the upscaled continuum of porous medium, researchers and
scientists were able to suggest the governing laws at the new continuum. They may be written
for incompressible fluids as:
Continuity
∇·

v
β

= 0 (5)
Momentum
ρ
β
∂

v
β

∂t
+ ρ

−
μ
β
K

v
β

−
ρ
β
F
β
√
K




v
β





v
β

(6)

∂

c
β

β
∂t
+

v
β

·∇

c
β

β
= ∇·

D ·∇

c
β

β

±S (8)
where


, k =(k
M
/(ρC
p
)
f
,
is the thermal diffusivity. From now on we will drop the averaging operator,

, to simplify
notations. The energy equation is written assuming thermal equilibrium between the solid
matrix and the moving fluid. The generic terms, Q and S, in the energy and solute equations
represent energy added or taken from the system per unit volume of the fluid per unit time
and the mass of solute added or depleted per unit volume of the fluid per unit time due to
some source (e.g., chemical reaction which depends on the chemistry, the surface properties
of the fluid/solid interfaces, etc.). Dissolution of the solid phase, for example, adds solute to
the fluid and hence S
> 0, while precipitation depletes it, i.e., S < 0. Organic decomposition or
oxidation or reduction reactions may provide both sources and sinks. Chemical reactions in
porous media are usually complex that even in apparently simple processes (e.g., dissolution),
sequence of steps are usually involved. This implies that the time scale of the slowest step
essentially determines the time required to progress through the sequence of steps. Among
29
Solute Transport With Chemical Reaction in Singleand Multi-Phase Flow in Porous Media
8 Mass Transfer
the different internal steps, it seems that the rate-limiting step is determined by reaction
kinetics. Therefore, the chemical reaction source term in the solute transport equation may
be represented in terms of rate constant, k, which lumps several factors multiplied by the
concentration, i.e.,
S

flow, temperature and concentration fields in porous media are observed to be governed by
complex interactions among the diffusion and convection mechanisms as will be discussed
later. It is assumed that the medium is isotropic with neither radiative heat transfer nor
viscous dissipation effects. Moreover, thermal local equilibrium is also assumed. Physical
model and coordinate system is shown in Fig.4.
The x-axis is taken along the plate and the y-axis is normal to it. The wall is maintained at
constant temperature and concentration, T
w
and C
w
, respectively. The governing equations
for the steady state scenario [as given by (Mulolani & Rahman, 2000; El-Amin, 2004) may be
presented as:
Continuity:
∂u
∂x
+
∂v
∂y
= 0 (10)
30
Mass Transfer in Multiphase Systems and its Applications
Solute Transport With Chemical Reaction in Single- and Multi-Phase Flow in Porous Media 9
Fig. 4. Physical model and coordinate system.
Momentum:
u
c
√
K
ν

Energy:
u
∂T
∂x
+ v
∂T
∂y
=
∂
∂x

α
x
∂T
∂x

+
∂
∂y

α
y
∂T
∂y

(13)
Solute transport:
u
∂C
∂x

ρ
= ρ
∞
[
1 −β
∗
(T − T
∞
) − β
∗∗
(C − C
∞
)
]
(15)
Along with the boundary conditions:
y
= 0:v = 0, T
w
= const., C
w
= const.;
y
→ ∞ : u = 0, T → T
∞
,C → C
∞
(16)
where β
∗


(17)
∂p
∂y
= 0 (18)
u
∂T
∂x
+ v
∂T
∂y
=
∂
∂y

α
y
∂T
∂y

(19)
u
∂C
∂x
+ v
∂C
∂y
=
∂
∂y

|
where, α and D are the molecular thermal and solutal
diffusivities, respectively, whereas γd
|
v
|
and ζd
|
v
|
represent dispersion thermal and solutal
diffusivities, respectively. This model for thermal dispersion has been used extensively (e.g.,
(Cheng, 1981; Plumb, 1983; Hong & Tien, 1987; Lai & Kulacki, 1989; Murthy & Singh, 1997)
in studies of non-Darcy convective heat transfer in porous media. Invoking the Boussinesq
approximations, and defining the velocity components u and v in terms of stream function ψ
as: u
= ∂ψ/∂y and v = −∂ψ/ ∂x, the pressure term may be eliminated between Eqs. (17) and
(18) and one obtains:
∂
2
ψ
∂y
2
+
c
√
K
ν
∂
∂y

∂y
=
∂
∂y

α
+ γd
∂ψ
∂y

∂T
∂y

(22)
∂ψ
∂y
∂C
∂x
−
∂ψ
∂x
∂C
∂y
=
∂
∂y

D
+ ζd
∂ψ

w
− T
∞
,φ(η)=
C −C
∞
C
w
−C
∞
(24)
The problem statement is reduced to:
f

+ 2F
0
Ra
d
f

f

= θ

+ Nφ

(25)
θ

+


+ f

φ


−Scλ
Gc
Re
2
x
φ
n=0
(27)
As mentioned in (El-Amin, 2004), the parameter F
0
= c
√
Kα/νd collects a set of parameters
that depend on the structure of the porous medium and the thermo physical properties of
the fluid saturating it, Ra
d
= Kgβ
∗
(T
w
− T
∞
)d/αν is the modified, pore-diameter-dependent
32

=
u
r
x/ν, Sc = ν/D and λ = K
0
αd( C
w
− C
∞
)
n−3
/Kgβ
∗∗
, where the diffusivity ratio Le (Lewis
number) is the ratio of Schmidt number and Prandtl number, and u
r
=

gβ
∗
d(T
w
− T
∞
) is
the reference velocity as defined by (Elbashbeshy, 1997).
Eq. (27) can be rewritten in the following form:
φ

+

0
= 0 corresponds to the Darcian free convection regime, γ = 0
represents the case where the thermal dispersion effect is neglected and ζ
= 0 represents
the case where the solutal dispersion effect is neglected. In Eq. (16), N
> 0 indicates the
aiding buoyancy and N
< 0 indicates the opposing buoyancy. On the other hand, from
the definition of the stream function, the velocity components become u
=(αRa
x
/x) f

and
v
= −(αRa
1/2
x
/2x)[ f − η f

]. The local heat transfer rate which is one of the primary interest
of the study is given by q
w
= −k
e
(∂T/∂y)|
y=0
, where, k
e
= k + k


(0)]θ

(0). Also, the local mass flux at the vertical
wall that is given by j
w
= −D
y
(∂C/∂y)|
y=0
defines another dimensionless variable that is the
local Sherwood number is given by, Sh
x
= j
w
x/(C
w
− C
∞
)D. This, analogously, may also
define another dimensionless variable as Sh
x
/
√
Ra
x
= −[1 + ζRa
d
F


Solute Transport With Chemical Reaction in Singleand Multi-Phase Flow in Porous Media
12 Mass Transfer
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
K
I
F 
F 
F 
Fig. 5. Variation of dimensionless concentration with similarity space variable η for different
χ (Le
= 0.5, F
0
= 0.3, Ra
d
= 0.7, γ = ζ = 0.0, N = −0.1).
(defined in terms of Sherwood number), Fig. 8 illustrates that the parameter ζ enhances the
mass transfer rate with small values of Le

<1.55 and the opposite is true for high values of
Le

>1.55. This may be explained as follows: for small values of Le number, which indicates
0.2
0.4

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Le
Nu
x
/(Ra
x
^0.5)
] 
] 
] 
] 
Fig. 7. Variation of Nusselt number with Lewis number for various ζ (χ = 0.02, F
0
= 0.3,
Ra
d
= 0.7, γ = 0.0, N = −0.1).
that mass dispersion outweighs heat dispersion, the increase in the parameter ζ causes mass
dispersion mechanism to be higher and since the concentration at the wall is kept constant
this increases concentration gradient near the wall and hence increases Sherwood number. As
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Le
Sh
x
/(Ra

J 
Fig. 9. Variation of Nusselt number with Lewis number for various γ ( χ = 0.02, F
0
= 0.3,
Ra
d
= 0.7, ζ = 0.0, N = −0.1).
Le increases (Le
> 1), heat dispersion outweighs mass dispersion and with the increase in ζ
concentration gradient near the wall becomes smaller and this results in decreasing Sherwood
number. Fig. 9 indicates that the increase in thermal dispersion parameter enhances the heat
transfer rates.
6. Multi-phase flow modeling
Multi-phase systems in porous media are ubiquitous either naturally in connection with,
for example, vadose zone hydrology, which involves the complex interaction between three
phases (air, groundwater and soil) and also in many industrial applications such as enhanced
oil recovery (e.g., chemical flooding and CO
2
injection), Nuclear waste disposal, transport of
groundwater contaminated with hydrocarbon (NAPL, DNAPL), etc. Modeling of Multi-phase
flows in porous media is, obviously, more difficult than in single-phase systems. Here we
have to account for the complex interfacial interactions between phases as well as the time
dependent deformation they undergo. Modeling of compositional flows in porous media is,
therefore, necessary to understand a number of problems related to the environment (e.g.,
CO
2
sequestration) and industry (e.g., enhanced oil recovery). For example, CO
2
injection
in hydrocarbon reservoirs has a double benefit, on the one side it is a profitable method

(
ρ
α
u
α
)
+
q
α
(30)
Momentum conservation in phase α:
u
α
= −
Kk
rα
μ
α
(
∇
p
α
+ ρ
α
g∇z
)
(31)
Energy conservation in phase α:
∂
∂t

Mass conservation of component i in phase α:
∂
(φcz
i
)
∂t
+ ∇·
∑
α
c
α
x
αi
u
α
= ∇·

φD
i
α
∇(cz
i
)

+ F
i
, i = 1, ···, N (33)
where the index α denotes to the phase. S, p, q, u,k
r
,ρ and μ are the phase saturation, pressure,

represent the effective thermal conductivity of the phase α and
the interphase heat transfer rate associated with phase α, respectively. Hence,
∑
α
¯
q
α
= q, q is
an external volumetric heat source/sink (Starikovicius, 2003). The phase enthalpy k
α
is related
to the temperature T by, h
α
=

T
0
c
pα
dT + h
0
α
. The saturation S
α
of the phases are constrained
by, c
pα
and h
0
α

q
j
δ(x − x
j
) (36)
The index j represents the points of sources or sinks. Eq. (35) represents sources and q
j
represents volume of the fluid (with density ρ
j
) injected per unit time at the points locations
x
j
, while, Eq. (36) represents sinks and q
j
represents volume of the fluid produced per unit
time at x
j
.
On the other hand, the molar density of wetting and nonwetting phases is given by,
c
α
=
N
∑
i=1
c
αi
(37)
where c
αi

φ
∑
α
c
α
x
αi
S
α

+ ∇·
∑
α
c
α
x
αi
u
α
= ∇·

φc
α
S
α
D
i
α
∇x
αi

S
α
=
∑
α
c
αi
S
α
, i = 1, ···, N (43)
If one uses the total mass variable X of the system (Nolen 1973; Young and Stephenson 1983),
X
=
∑
α
c
α
S
α
(44)
Therefore,
38
Mass Transfer in Multiphase Systems and its Applications
Solute Transport With Chemical Reaction in Single- and Multi-Phase Flow in Porous Media 17
1 =
∑
α
c
α
S

(47)
The capillary pressure is the the difference between the pressures for two adjacent phases α
1
and α
2
, given as,
p
cα
1
α
2
= p
α
1
− p
α
2
(48)
The capillary pressure function is dependent on the pore geometry, fluid physical properties
and phase saturations. The two phase capillary pressure can be expressed by Leverett
dimensionless function J
(S), which is a function of the normalized saturation S,
p
c
= γ

φ
K

1

Kk
rα
μ
α
(
∇
p
α
+ ρ
α
g∇z
)
α = w, n (51)
Mass conservation of component i in phase α:
∂
(φcz
i
)
∂t
+ ∇·
(
c
w
x
wi
u
w
+ c
n
x

component. The saturation S
α
of the phases are constrained by,
39
Solute Transport With Chemical Reaction in Singleand Multi-Phase Flow in Porous Media
18 Mass Transfer
(a) Corey approximation (b) LET approximation
Fig. 10. Relative permeabilities.
S
w
+ S
n
= 1 (53)
The normalized wetting phase saturation S is given by,
S
=
S
w
−S
0w
1 −S
nr
−S
0w
0 ≤ S ≤ 1 (54)
where S
0w
is the irreducible (minimal) wetting phase saturation and S
nr
is the residual

rn
(S = 0) is the endpoint relative permeability to the non-wetting phase.
For example, for the Corey power-law correlation, a
= b = 2, k
0
rn
= 1, k
0
rw
= 0.6, for water-oil
system see Fig.10a. Another example of relative permeabilities correlations is LET model
which is more accurate than Corey model. The LET-type approximation is described by three
empirical parameters L,E and T. The relative permeability correlation for water-oil system has
the form,
k
rw
=
k
0
rw
S
L
w
S
L
w
+ E
w
(1 −S)
T

the oil-water system. Correlation of the imbibition capillary pressure data depends on the
type of application. For example, for water-oil system, see for example, (Pooladi-Darvish &
Firoozabadi, 2000), the capillary pressure and the normalized wetting phase saturation are
correlated as,
p
c
= −B ln S (59)
where B is the capillary pressure parameter, which is equivalent to γ

φ
K

1
2
, in the general
form of the capillary pressure, Eq. (49), thus, B
≡−γ

φ
K

1
2
and J( S) ≡ lnS. Note that J(S) is
a scalar non-negative function. Capillary pressure as a function of normalized wetting phase
(e.g. water) saturation is shown in Fig. 11. Also, the well known (van Genuchten, 1980; Brooks
& Corey, 1964) capillary pressure formulae which can be written as,
p
c
= p

41
Solute Transport With Chemical Reaction in Singleand Multi-Phase Flow in Porous Media
20 Mass Transfer
u = u
w
+ u
n
(63)
the total mobility is given by,
m
(S)=m
w
(S)+m
n
(S) (64)
the fractional flow functions are,
f
w
(S)=
m
w
(S)
m(S)
, f
n
(S)=
m
n
(S)
m(S)

are the molar densities of component i in the wetting phase and nonwetting
phase phases, respectively. Therefore, the mole fraction of component i in the respective phase
is given as,
x
wi
=
c
wi
c
w
, x
ni
=
c
ni
c
n
, i = 1, ···, N (68)
The mole fraction balance implies that,
N
∑
i=1
x
wi
= 1,
N
∑
i=1
x
ni

(
c
w
x
wi
u
w
+ c
n
x
ni
u
n
)
=
F
i
, i = 1, ···, N (71)
F
i
may be written as,
F
i
= x
wi
q
w
+ x
ni
q

, i = 1, ···, N (73)
If one uses the total mass variable X of the system (Nolen, 1973; Young & Stephenson, 1983),
X
= c
w
S
w
+ c
n
S
n
(74)
42
Mass Transfer in Multiphase Systems and its Applications
Solute Transport With Chemical Reaction in Single- and Multi-Phase Flow in Porous Media 21
Therefore,
1
=
c
w
S
w
X
+
c
w
S
n
X
= C

ni
, i = 1, ···, N (77)
The pressure equation can be obtained, using the concept of volume-balance, as follows,
φC
f
∂p
∂t
+
N
∑
i=1
¯
V
i
∇·
(
c
w
x
wi
u
w
+ c
n
x
ni
u
n
)
=

, x
w2
,···, x
wN
)= f
ni
(p
n
, x
n1
, x
n2
,···, x
nN
), i = 1,···, N (79)
The fugacity of the i
th
component is defined by,
f
αi
= p
α
x
oi
φ
αi
, α = w,n, i = 1, ···, N (80)
φ
αi
,α = w, n is the fugacity coefficient of the i

α
)
, α = w,n (81)
a
α
=
N
∑
i=1
N
∑
j=1
x
iα
x
jα
(1 −κ
ij
)

a
i
a
j
, b
α
=
N
∑
j=1

i
= Π
ib
RT
ic
p
ic
, i = 1, ···, N (83)
T
ic
and p
ic
are the critical temperature and pressure,
43
Solute Transport With Chemical Reaction in Singleand Multi-Phase Flow in Porous Media
22 Mass Transfer
Π
ia
= 0.45724, Π
ib
= 0.077796, α
i
=

1
−λ
i

1
−

2
α
)Z
α
−(A
α
B
α
− B
2
α
− B
2
α
)=0, α = w, n (85)
where Z
α
is the compressibility factor given by,
Z
α
=
b
α
V
α
RT
, α
= w, n (86)
(Chen, 2007) explained how to solve the cubic algebraic equation, Eq. (64). The fugacity
coefficient φ

∑
N
j
=1
x
jα
(1 −κ
ij
)
√
a
i
a
j
−
b
i
b
α

·ln

Z
α
+(1+
√
2)B
α
Z
α

w
−Δρg∇z

+ u

+ q
w
(88)
Alternatively, in terms of pressure the flow equations may be rewritten in the form,
∂
(φρ
w
S
w
)
dp
c

∂p
n
∂t
−
∂p
w
∂t

= ∇·ρ
w

Kk

∂t

= ∇·ρ
n

Kk
rn
μ
n
(
∇
p
n
−ρ
n
g∇z
)

+ q
n
(90)
Both models, Eq. (88) and Eqs. (89)- (90) are used intensively especially in the field of oil
reservoir simulations.
6.2 Three-phase compositional flow
In three-phase compositional flow the governing equations will not has a big difference from
the two-phase case. In this section we introduce the main points which distinguish the
three-phase flow. On the other hand, we consider the black oil model as an example of the
three-phase compositional flow instead of considering the general case to investigate such
kind of complex flow. The black oil model is water-oil-gas system such that water represents
the aqueous phase and oil represents oleic phase. The hydrocarbon in a reservoir is almost

u
o
+ c
g
x
gi
u
g

= F
i
, i = 1, ···, N (91)
or
∂
∂t

φ

c
w
x
wi
S
w
+ c
o
x
oi
S
o

wi
q
w
+ x
oi
q
o
+ x
gi
q
g
, i = 1, ···, N
(92)
Following (Stone, 1970; 1973) we assume that the water-oil and oil-gas relative permeabilities
are given as the two-phase case,
k
rw
(S
w
)=k
0
rw

S
w
−S
wc
1 −S
wc
−S

org
is the residual oil saturation to gas, S
orw
is the
residual oil saturation to water, S
gr
is the residual gas saturation to water. S
w
= 1 −S
orw
. The
intermediate-wetting phase (oil phase) relative permeabilities are given by,
k
row
(S
w
)=k
0
row

1
−S
w
−S
orw
1 −S
wc
−S
orw


ro
(S
w
,S
g
)=
k
row
k
ro g
k
norm
(97)
k
norm
may be setting as one or given by another formula as in the literature which will not
mention here for breif.
6.3 Numerical methods for multi-phase flow
Much progress in the last three decades in numerical simulation of multi-phase flow with
compositional and chemical effect. Both first-order finite difference and finite volume
methods are used. First-order finite difference schemes has numerical dispersion issue, while
the first-order finite volume has powerful features when used for two-phase flow simulation
(Leveque, 2002). However, the later one has some limitations when applied to fractured
media (Monteagudo & Firoozabadi, 2007). Also, higher-order methods have less numerical
dispersion and more accurate flow field calculations than the first-order methods. The
combined mixed-hybrid finite element (MHFE) and discontinuous Galerkin (DG) methods
have been used to simulate two-phase flow by (Hoteit & Firoozabadi, 2005; 2006; Mikyska
& Firoozabadi, 2010). In the combined MHFE-DG methods, MHFE is used to solve the
pressure equation with total velocity, and DG method is used to solve explicitly the species
transport equations. Therefore, the parts are coupled using scheme such as the iterative

nuclear waste packages buried in partially saturated geological media, Int. J. Heat
Mass Transfer 31: 79–90.
El-Amin, M. F. (2004). Double dispersion effects on natural convection heat and mass transfer
in non-darcy porous medium, Appl. Math. Comp. 156: 1–17.
El-Amin, M. F., Aissa, W. A. & Salama, A. (2008). Effects of chemical reaction and double
dispersion on non-darcy free convection heat and mass transfer, Transport in Porous
Media 75: 93–109.
Elbashbeshy, E. M. A. (1997). Heat and mass transfer along a vertical plate with variable
surface tension and concentration in the presence of magnetic field, Int. J. Eng. Sci.
Math. Sci. 4: 515–522.
Hassanizadeh, S. M. & Gray, W. (1979a). General conservation equations for multi-phase
systems 2. mass, momenta, energy and entropy equations, Advances in Water
Resources 2: 191–208.
Hassanizadeh, S. M. & Gray, W. (1980). General conservation equations for multi-phase
systems 3. constitutive theory for porous media, Advances in Water Resources 3: 25–40.
Hassanizadeh, S. M. & Gray, W. G. (1979b). General conservation equations for multi-phase
systems 1. averaging procedure, Advances in Water Resources 2: 131–144.
Herwig, H. & Koch, M. (1991). Natural convection momentum and heat transfer in saturated
highly porous mediaan asymptotic approach, Heat Mass Transfer 26: 169–174.
Hong, J. T. & Tien, C. L. (1987). Analysis of thermal dispersion effect on vertical plate natural
convection in porous media, Int. J. Heat Mass Transfer 30: 143–150.
46
Mass Transfer in Multiphase Systems and its Applications
Solute Transport With Chemical Reaction in Single- and Multi-Phase Flow in Porous Media 25
Hoteit, H. & Firoozabadi, A. (2005). Multicomponent fluid flow by discontinuous galerkin
mand mixed methods in unfractured and fractured media, Water Resour. Res.
41: W11412.
Hoteit, H. & Firoozabadi, A. (2006). Compositional modeling by the combined discontinuous
galerkin mand mixed methods, SPE Journal .
Lai, F. C. & Kulacki, F. A. (1989). Thermal dispersion effect on non-darcy convection from

Stone, H. (1970). Probability model for estimating three-phase relative permeability, JPT 214.
Stone, H. (1973). Estimation of three-phase relative permeability and residual oil data, J. Cdn.
Pet. Tech. 13: 53.
Sun, S., Riviere, B. & Wheeler, M. F. (2002). A combined mixed finite element and
discontinuous galerkin method for miscible displacement problems in porous media,
Proceedings of International Symposium on Computational and Applied PDEs, Zhangjiajie
National Park of China, 321-341.
Sun, S. & Wheeler, M. F. (2005a). Discontinuous galerkin methods for coupled flow and
reactive transport problems, Appl. Num. Math. 52: 273–298.
Sun, S. & Wheeler, M. F. (2005b). Symmetric and nonsymetric discontinuous galerkin methods
for reactive transport in porous media, SIAM J. Numer. Anal. 43: 195–219.
47
Solute Transport With Chemical Reaction in Singleand Multi-Phase Flow in Porous Media
26 Mass Transfer
Sun, S. & Wheeler, M. F. (2006). Analysis of discontinuous galerkin methods for
multi-components reactive transport problem, Comput. Math. Appl. 52: 637–650.
Telles, R. S. & V.Trevisan, O. (1993). Dispersion in heat and mass transfer natural convection
along vertical boundaries in porous media, Int. J. Heat Mass Transfer 36: 1357–1365.
Udell, K. S. (1985). Heat transfer in porous media considering phase change and capillarity –
the heat pipe effect, Int. J. Heat Mass Transfer 28: 485–495.
van Genuchten, M. (1980). A closed-form equation for predicting the hydraulic conductivity
of unsaturated soils, Soil Sci. Soc. Am. J. 44: 892–898.
Wheeler, M. F. (1987). An elliptic collocation finite element method with interior penalties,
SIAM J. Numer. Anal. 15: 152–161.
Whitaker, S. (1967). Diffusion and dispersion in porous media, AIChE 13: 420–427.
Whitaker, S. (1977). Simultaneous heta, mass, and momentum transfer in porous media: A
theory of drying, Advances in Heat Transfer 13: 119–203.
Young, L. C. & Stephenson, R. E. (1983). A generalized compositional approach for reservoir
simulation, SPE Journal 23: 727–742.
48

– external mechanical loads.
The significant physical (and chemical) processes are:
a) thermal deformation,
b) autogenous shrinkage,
c) carbonation,
d) elastic and creep deformation,
e) additional thermal deformation,
f) drying shrinkage and swelling.
3
2 Mass Transfer
In the first period of intense hydration a), accompanied by b), is dominant. In the later
period the role of a) decreases, but the effect of c) has to be taken into account. The external
mechanical loads cause d) (creep especially in the earliest age), the external temperature
changes simultaneously force e), modified by f).
The traditional approach to the modelling of such complex physical and technical problems
is the phenomenological one, as discussed in (Ba
ˇ
zant, 2001): the effect of changes of density,
porosity, permeability, compressive strength, etc. on material behaviour is lumped together to
some model parameters, which must be identified by long-lasting tests in the whole range of
model applicability. On the contrary, the so-called CCBM (“Computational Cement-Based
Material”) approach, suggested in (Maruyama et al., 2001), develops the original idea of
(Tomosawa, 1997): the slight generalization of its (seemingly simple) form
˙

= Φ(,
∗
),
˙


Γ :
=
μ
h
μ
h
∞
where μ
h
denotes the (usually increasing) mass of skeleton (and corresponding sink of liquid
water mass) ans μ
h
∞
the final mass of hydrated (chemically combined) water in a volume unit;
alternatively (cf. (Gawin et al., 2006a), p. 309)
Γ :
=
Q
h
Q
h
∞
in terms of the heat Q
h
released during hydration and of its final value Q
h
∞
. However, it
is difficult to guarantee above sketched model assumption in building practice, applying
also (not single-sized) additional aggregate; thus Γ is usually quantified from macroscopic

−4
m),
III) mortar scale (about 10
−2
m),
IV) macroscale (about 10
−1
m).
The analysis of capillary depression at scale I) (considering membrane forces on solid/liquid,
solid/gas and liquid/gas interfaces), of ettringite formation at scale II), of autogenous
deformation at scales II) and III), referring to the Hill homogenization lemma (see (Dormieux
et al., 2006), p. 105), must be completed by the interpretation of such multiscale results at
scale IV). However, different physical and chemical processes studied at particular scales do
not admit proper and physically transparent mathematical analysis of two- and more-scale
convergence, as discussed in (Cioranescu & Donato, 1999), (Vala, 2006) or (Efendiev et
al., 2009), including its non-periodic (formally complicated) generalization, introduced in
(Nguentseng, 2003-4).
The approach (Gawin et al., 2006a) applies certain mechanistic-type method to obtain the
governing equations only, using the averaging hybrid mixture theory: the developments starts
at the micro-scale and balance equations for particular phases and interfaces are introduced
at this level and then averaged for obtaining macroscopic balance equations. Four phases are
distinguished: solid skeleton, liquid water, vapour and dry air, whose densities are considered
(under the passive air assumption) as constants; the whole hygro-thermo-chemo-mechanical
process is then studied as the time evolution of capillary pressure, gas pressure, temperature
and displacement of points related to the reference (initial) configuration, driven by balance
equations of classical thermodynamics and conditioned by corresponding constitutive laws.
The detailed geometrical analysis (Sanavia et al., 2002) (without phase changes) offers the
possibility to extend such considerations beyond the assumption of small deformations and
involve some elements of fracture mechanics.
The development, laboratory testing and computational simulations of new materials, namely

– dry air, identified by an index a.
In addition to partial derivatives of scalar quantities ψ with respect to time, i. e.
˙
ψ :
= ∂ψ/∂t ,
we shall introduce also the partial derivatives of such quantities with respect to x
i
, i ∈{1,2,3},
x
=(x
1
, x
2
, x
3
) being a Cartesian coordinate system in the three-dimensional Euclidean space
R
3
,
ψ
,i
:= ∂ψ/∂x
j
.
In the case of real vector variables with values in
R
3
we shall write ψ briefly instead of
(ψ
1

s
,ρ
w
,ρ
v
,ρ
a
),
– 12 components of phase velocities V
=(v
s
i
,v
w
i
,v
v
i
,v
a
i
) with i ∈{1, 2,3},
– 3 fluid pressures P
=(p
w
, p
v
, p
a
),

´
udez
de Castro, 2005), p. 111, we can also introduce pressures
p
ε
:= η
ε
p
ε
.
We can evaluate also the “macroscopic” mixture density
ρ :
= ρ
s
+ ρ
w
+ ρ
v
+ ρ
a
.
Let us also remark that, from he point of view of solid phase, fluid pressures P are
accompanied by a (partial) Cauchy stress tensor compound from components τ
ij
with i, j ∈
{
1,2,3}, whose (indirect) relation to V will be discussed later.
The porosity can be evaluated from the finite strain analysis by (Sanavia et al., 2002), p. 139.
The multiphase medium at the macroscopic level can be described as the superposition of all
phases ε, whose material point with coordinates x

,t) denotes the displacement at chosen time and zero indices are related to a
reference configuration, here in time t
= 0. Thus
F
ε
ij
(x
ε0
,t) :=
∂x
ε
i
(x
ε0
)
∂x
ε0
j
= δ
ij
+
∂u
ε
i
(x
ε0
,t)
∂x
ε0
j