# Modeling Wettability Variation during Long-Term Water Flooding

Authors: Renyi Cao; Changwei Sun; Y. Zee Ma
Journal: Journal of Chemistry
 (2015)

---

## Abstract

Surface property of rock affects oil recovery during water flooding. Oil-wet polar substances adsorbed on the surface of the rock will gradually be desorbed during water flooding, and original reservoir wettability will change towards water-wet, and the change will reduce the residual oil saturation and improve the oil displacement efficiency. However there is a lack of an accurate description of wettability alternation model during long-term water flooding and it will lead to difficulties in history match and unreliable forecasts using reservoir simulators. This paper summarizes the mechanism of wettability variation and characterizes the adsorption of polar substance during long-term water flooding from injecting water or aquifer and relates the residual oil saturation and relative permeability to the polar substance adsorbed on clay and pore volumes of flooding water. A mathematical model is presented to simulate the long-term water flooding and the model is validated with experimental results. The simulation results of long-term water flooding are also discussed.


---

## Body

## 1. Introduction

Wettability is an important factor that affects oil recovery in water flooding. In the past, oil reservoirs were commonly interpreted to be strongly water-wet, because water phase is generally the initial fluid contact with the reservoir rock. However, some researchers found that many reservoir rocks were not strongly water-wet [1, 2]. The latter results were obtained without considering the natural surfactant in crude oil, such as asphalt and paraffin substances, which are easily absorbed on the solid-liquid interface and change reservoir rocks to oil-wet [3]. Kusakov [4] found that the rupture of water film could lead to crude oil directly contacting the quartz surface and the surface would become oil-wet instead of water-wet. Schmid [5] claimed that wettability could be changed and strong water-wet rock could become weak water-wet after contact with crude oil.For the sandstone reservoirs, the wettability changes during deposition process, accumulation process, and water flooding process (Figure1). During deposition process, the pores are saturated with water to form a layer of water film during the deposition process [6, 7], as shown in Figure 1(a). Thus, the reservoir rocks are water-wet. During the accumulation process, the partial polar substances in crude oil penetrate water film to adsorb on rock surface through van der Waals, electrostatic, hydrogen bonding, and acid-base [8–11], and the rock with the polar substances changes to oil-wet, as shown in Figure 1(b). Note, however, that if the partial polar substances do not penetrate water film, the rock will be water-wet. During long-term water flooding, the polar substances can be washed out and desorbed from the rock surface; the wettability may change to water-wet, as shown in Figure 1(c).Figure 1
Wettability variation during reservoir formation process.
(a)
Deposition process
(b)
Accumulation process
(c)
Water-flooding processWettability may affect almost all the variables in core analysis, including capillary pressure, relative permeability, oil recovery, and EOR. Morrow [12] confirmed that a higher water flooding recovery could be obtained in strongly water-wet core through one-dimensional water flooding experiments. The one-dimensional displacement experiments with high pore volume (PV) of injected water [13] showed that the changes of wettability and pore structure during long-term water flooding reduced the critical capillary number and the residual oil saturation. When the injected water was 5000 PV of rock core, the oil displacement efficiency of cores was about 57%; when the injected water was 1000 PV or even 10000 PV, the oil displacement efficiency was increased (near 80%) [13]. Therefore, under different water flooding conditions, oil displacement efficiency of the rock is not a constant.The wettability changes during long-term water flooding, and this could affect the residual oil saturation and oil recovery. A number of studies showed that, during low-salinity water flooding, the ions could exchange between injecting water and rock, and it could lead to adsorption of divalent ions and minerals dissolution, change the wettability to water-wet, and enhance oil recovery [14–20]; it could also lead to changes of relative permeability [21–25]. Recently, mechanism on wettability alternation, such as surfactant flooding, gas displacement, and low salinity water flooding has been studied [16, 20, 26–30].The main factor affecting wettability is the amount of polar substances adsorbed on the clay surface. In order to model the wettability variation, we should first evaluate the amount of polar substances adsorbed on the surface of clay. Five factors that impact the polar adsorption and desorption should be considered, and these include the concentration of polar substances in crude oil and on rock surface [31], salinity and pH of injecting water and formation water [32, 33], clay contents in rock [28], and flow rate of water through the pore [31]. Yet, an accurate description of wettability alternation model during long-term water flooding from injecting well or aquifer is lacking, which will lead to failure in history matching and unreliable forecasts using numerical simulation [34, 35].The purpose of this paper includes:(
1
) characterizing the adsorption of polar substance during long-term water flooding from injecting water or aquifer and then relate the residual oil saturation and relative permeability to the polar substance adsorbed on clay and pore volumes of flooding water and (
2
) building a mathematical model to simulate the long-term water flooding. The simulation results of long-term water flooding are also discussed in this paper.


## 2. Mathematical Models for Long-Term Water Flooding

### 2.1. Conditions

The following conditions are assumed in our mathematical modeling for long-term water flooding.(1)
Temperature of reservoir is constant.
(2)
Reservoir fluids are oil and water phases.
(3)
Components include water, oil, polar substance (AO), clay,N
a
+, C
a
2
+, M
g
2
+, F
e
3
+, O
H
-, and other anions.
(4)
No phase transition process is in reservoir.
(5)
Porosity and permeability don't change during long-term water flooding.
(6)
Gravity is not considered.


### 2.2. Model of Variation for Polar Substances on the Rock Surface

The amount of polar substances adsorbed on the rock surface mainly is determined by the polar substances adsorbed on the clay and the content of clay in rock. The content of polar substances adsorbed on the rock can be described by(1)
C
rock
AO
=
C
clay
AO
·
C
rock
clay
,where C
rock
AO—concentration of polar substances (AO) adsorbed on the rock; C
clay
AO—concentration of polar substances (AO) adsorbed on the clay; and C
rock
clay—concentration of clay in the rock.The clay could fall off from the rock surface by the water flushing under long-term water flooding, and this process can be described by(2)
C
rock
clay
→
DESORPTION
C
water
clay
,where C
water
clay—concentration of clay in the aqueous phase (water).Assuming that the process for the clay desorption is nonlinear and the desorption rate is a function of the aqueous phase flow rateυ
w, the Langmuir adsorption law can be used to characterize it:(3)
d
C
rock
clay
d
t
=
K
1
·
υ
w
·
c
rock
clay
1
+
b
c
rock
clay
,where d
C
rock
clay
/
d
t—desorption rate; K
1, b—clay desorption constant; and υ
w—flow rate of water phase.Equation (3) implies the influence of total volume of water flushing the rock surface during the long-term water flooding, because the rate of clay desorption in rock is related to the flow rate of water phase.The equilibrium between the concentration of polar substances in the oil phase and the concentration of polar substances in clay mineral surface is expressed by(4)
C
clay
AO
↔
EQUILIBRIUM
C
oil
AO
,where C
oil
AO—concentration of the polar substance (AO) in the oil phase.This equilibrium can be represented by the exchange equilibrium equation with a coefficient,κ
exm, which is the equilibrium constant for the exchange between polar substances on the clay surface and the oil phase:(5)
κ
exm
=
C
clay
AO
C
oil
AO
.Formation water salinity and pH influence adsorption of polar substances on the rock surface [18, 28]. The equilibrium constant κ
exm can be characterized as the relationship between the total concentration of the metal cations in formation water and the concentration of hydroxide ions:(6)
κ
exm
=
K
2
∑
N
a
+
C
a
2
+
M
g
2
+
F
e
3
+
⋯
O
H
-
,where K
2—balance factor; N
a
+
C
a
2
+
M
g
2
+
F
e
3
+—the concentration of sodium ion in the aqueous phase.


### 2.3. Mass Conservation Equations and Pressure Equations

The continuity of mass for componentk in association with Darcy’s law is expressed as a function of the overall volume of component per unit pore volume (C
¯
k):(7)
∂
∂
t
ϕ
C
¯
k
·
ρ
k
+
∇
∑
l
=
1
n
p
ρ
k
C
k
l
·
U
¯
l
-
D
~
k
l
=
Q
k
+
Q
k
T
+
Q
k
F
S
+
Q
k
T
S
,where n
cv—total number of volume-occupying components (including water, oil, and polar substances); n
p—number of phases; C
¯
k—adsorbed concentration of component k, C
¯
k
=
1
-
∑
k
=
1
n
cv
C
^
k
∑
l
=
1
n
p
S
l
×
C
k
l
+
C
^
k, k
=
1,2
,
…
,
n
cv; ρ
k—density of pure component k; Q
k—injection/output rate of component k with Unit volume; and Q
k
T—output rate of component k by reaction in phase l with unit volume. Q
k
F
S is output rate of component k solid-phase under mechanical force (long-term erosion) conditions with unit volume. For clay component, it can be determined by (3). Q
k
T
S is output rate of ion balance for component k on rock surface with unit volume.The output rate of polar substances and clay components can be determined by(8)
Q
k
T
=
∑
l
=
1
n
p
S
l
·
K
k
l
exm
C
rock
k
,
(9)
Q
k
F
S
=
K
1
·
υ
w
·
c
rock
clay
1
+
b
c
rock
clay
,
(10)
Q
k
T
S
=
K
exm
C
rock
k
,where K
k
l
exm
=
d
C
l
k
/
d
t is the output rate of component k in the liquid phase l and K
1 is deposition rate of component k on the rock surface; consider (11)
K
exm
=
k
2
∑
N
a
+
C
a
2
+
M
g
2
+
F
e
3
+
O
H
-
.The aqueous-phase pressure is obtained by an overall mass balance on volume components [36]. Because of the following condition(12)
∑
k
=
1
n
cv
C
k
l
=
1
,the pressure equation in terms of the reference phase pressure (phase 1) is(13)
ϕ
C
t
·
∂
p
1
∂
t
+
∇
K
-
-
·
λ
r
T
c
·
∇
p
1
=
-
∇
∑
l
=
1
n
p
K
-
-
·
λ
rec
·
∇
h
+
∇
·
∑
l
=
1
n
p
K
-
-
·
λ
r
T
c
·
∇
p
c
l
1
+
∑
l
=
1
n
p
Q
k
+
Q
k
T
+
Q
k
F
S
+
Q
k
T
S
,where λ
rec
=
(
k
r
l
/
μ
l
)
·
∑
k
=
1
n
cv
·
ρ
k
·
C
k
l; λ
r
T
c
=
∑
l
=
1
n
p
λ
rec is the total relative mobility with the correction for fluid compressibility.The total compressibilityC
t is the volume-weighted sum of the rock or soil matrix (C
r) and component compressibilities (C
k
0) are(14)
C
t
=
C
r
+
∑
k
=
1
n
cv
C
k
0
·
C
~
k
,whereϕ
=
ϕ
R
·
1
+
C
r
p
ref
-
p
ref
0.


### 2.4. Relative Permeability Model

During long-term water flooding, the wettability changes with the reduction of polar substances on the rock surface, and the residual oil saturation and displacement oil efficiency will change as well. Since the polar substances on the rock surface is not constant, which depends on the time of displacement and flow rate of water phase, the residual oil saturation is a function of flow rate and time of water flooding (the cumulative volumes of water displacement).Since the variation of wettability is reflected by the relative permeabilities and residual oil saturation, different relative permeabilities (which could be measured with cores after different volumes of water displacement, Figure2) can be used to calculate the residual oil saturation and displacement oil efficiency [28].Figure 2
Relative permeabilities determined at different pore volumes of water injected.
(a)
(b)
(c)The relative permeability curve at low PV (initial wettability, Figure2(a)) of injected water reflects the initial wettability of rock with maximum contents (C
rock
AO
max
⁡ represented by C
max
⁡) of polar substances adsorbed on rock; the relative permeability curve at high PV (high water-wet at ultimate water displacement, Figure 2(c)) of injected water reflects the initial wettability of rock with minimum contents (C
min
⁡) of polar substances adsorbed on rock. We can use C
rock
clay to interpolate the relative permeabilities and residual oil saturation (Figure 2(b)) at different PV of injected water.Given relative permeability curves atC
max
⁡ and C
min
⁡, we can determine the maximum and minimum residual oil saturation, and the residual oil saturation S
or can be calculated with the linear interpolation methods [27]:(15)
S
or
=
S
or
max
⁡
-
C
-
C
min
⁡
C
max
⁡
-
C
min
⁡
S
or
max
⁡
-
S
or
min
⁡
,where S
or
max
⁡—the residual oil saturation at C
max
⁡ (low PV, initial wettability); S
or
max
⁡—the residual oil saturation at C
min
⁡ (high PV, high water-wet wettability).The endpoints of relative permeabilities are computed with a linear interpolation between the given input values atC
max
⁡ and C
min
⁡,(16)
K
r
l
end
=
K
r
l
end
max
⁡
-
S
or
-
S
or
min
⁡
S
or
max
⁡
-
S
or
min
⁡
K
r
l
end
max
⁡
-
K
r
l
end
min
⁡
,where K
r
l
end—the endpoint of relative permeability for phase l at S
or; K
r
l
end
max
⁡—the endpoint of relative permeability for phase l at S
or
max
⁡; and K
r
l
end
min
⁡—the endpoint of relative permeability for phase l at S
or
min
⁡.The relative permeabilities for phase l are assumed to be unique functions of their respective saturations only [37], described as(17)
K
r
l
=
K
r
l
end
S
l
∗
n
,where S
l
∗—the normalized saturations for phase l, S
w
∗
=
(
S
w
-
S
w
c
)
/
(
1
-
S
w
c
-
S
or
), S
o
∗
=
1
-
S
w
∗; and n—the exponent determined by fitting the data.


## 2.1. Conditions

The following conditions are assumed in our mathematical modeling for long-term water flooding.(1)
Temperature of reservoir is constant.
(2)
Reservoir fluids are oil and water phases.
(3)
Components include water, oil, polar substance (AO), clay,N
a
+, C
a
2
+, M
g
2
+, F
e
3
+, O
H
-, and other anions.
(4)
No phase transition process is in reservoir.
(5)
Porosity and permeability don't change during long-term water flooding.
(6)
Gravity is not considered.


## 2.2. Model of Variation for Polar Substances on the Rock Surface

The amount of polar substances adsorbed on the rock surface mainly is determined by the polar substances adsorbed on the clay and the content of clay in rock. The content of polar substances adsorbed on the rock can be described by(1)
C
rock
AO
=
C
clay
AO
·
C
rock
clay
,where C
rock
AO—concentration of polar substances (AO) adsorbed on the rock; C
clay
AO—concentration of polar substances (AO) adsorbed on the clay; and C
rock
clay—concentration of clay in the rock.The clay could fall off from the rock surface by the water flushing under long-term water flooding, and this process can be described by(2)
C
rock
clay
→
DESORPTION
C
water
clay
,where C
water
clay—concentration of clay in the aqueous phase (water).Assuming that the process for the clay desorption is nonlinear and the desorption rate is a function of the aqueous phase flow rateυ
w, the Langmuir adsorption law can be used to characterize it:(3)
d
C
rock
clay
d
t
=
K
1
·
υ
w
·
c
rock
clay
1
+
b
c
rock
clay
,where d
C
rock
clay
/
d
t—desorption rate; K
1, b—clay desorption constant; and υ
w—flow rate of water phase.Equation (3) implies the influence of total volume of water flushing the rock surface during the long-term water flooding, because the rate of clay desorption in rock is related to the flow rate of water phase.The equilibrium between the concentration of polar substances in the oil phase and the concentration of polar substances in clay mineral surface is expressed by(4)
C
clay
AO
↔
EQUILIBRIUM
C
oil
AO
,where C
oil
AO—concentration of the polar substance (AO) in the oil phase.This equilibrium can be represented by the exchange equilibrium equation with a coefficient,κ
exm, which is the equilibrium constant for the exchange between polar substances on the clay surface and the oil phase:(5)
κ
exm
=
C
clay
AO
C
oil
AO
.Formation water salinity and pH influence adsorption of polar substances on the rock surface [18, 28]. The equilibrium constant κ
exm can be characterized as the relationship between the total concentration of the metal cations in formation water and the concentration of hydroxide ions:(6)
κ
exm
=
K
2
∑
N
a
+
C
a
2
+
M
g
2
+
F
e
3
+
⋯
O
H
-
,where K
2—balance factor; N
a
+
C
a
2
+
M
g
2
+
F
e
3
+—the concentration of sodium ion in the aqueous phase.


## 2.3. Mass Conservation Equations and Pressure Equations

The continuity of mass for componentk in association with Darcy’s law is expressed as a function of the overall volume of component per unit pore volume (C
¯
k):(7)
∂
∂
t
ϕ
C
¯
k
·
ρ
k
+
∇
∑
l
=
1
n
p
ρ
k
C
k
l
·
U
¯
l
-
D
~
k
l
=
Q
k
+
Q
k
T
+
Q
k
F
S
+
Q
k
T
S
,where n
cv—total number of volume-occupying components (including water, oil, and polar substances); n
p—number of phases; C
¯
k—adsorbed concentration of component k, C
¯
k
=
1
-
∑
k
=
1
n
cv
C
^
k
∑
l
=
1
n
p
S
l
×
C
k
l
+
C
^
k, k
=
1,2
,
…
,
n
cv; ρ
k—density of pure component k; Q
k—injection/output rate of component k with Unit volume; and Q
k
T—output rate of component k by reaction in phase l with unit volume. Q
k
F
S is output rate of component k solid-phase under mechanical force (long-term erosion) conditions with unit volume. For clay component, it can be determined by (3). Q
k
T
S is output rate of ion balance for component k on rock surface with unit volume.The output rate of polar substances and clay components can be determined by(8)
Q
k
T
=
∑
l
=
1
n
p
S
l
·
K
k
l
exm
C
rock
k
,
(9)
Q
k
F
S
=
K
1
·
υ
w
·
c
rock
clay
1
+
b
c
rock
clay
,
(10)
Q
k
T
S
=
K
exm
C
rock
k
,where K
k
l
exm
=
d
C
l
k
/
d
t is the output rate of component k in the liquid phase l and K
1 is deposition rate of component k on the rock surface; consider (11)
K
exm
=
k
2
∑
N
a
+
C
a
2
+
M
g
2
+
F
e
3
+
O
H
-
.The aqueous-phase pressure is obtained by an overall mass balance on volume components [36]. Because of the following condition(12)
∑
k
=
1
n
cv
C
k
l
=
1
,the pressure equation in terms of the reference phase pressure (phase 1) is(13)
ϕ
C
t
·
∂
p
1
∂
t
+
∇
K
-
-
·
λ
r
T
c
·
∇
p
1
=
-
∇
∑
l
=
1
n
p
K
-
-
·
λ
rec
·
∇
h
+
∇
·
∑
l
=
1
n
p
K
-
-
·
λ
r
T
c
·
∇
p
c
l
1
+
∑
l
=
1
n
p
Q
k
+
Q
k
T
+
Q
k
F
S
+
Q
k
T
S
,where λ
rec
=
(
k
r
l
/
μ
l
)
·
∑
k
=
1
n
cv
·
ρ
k
·
C
k
l; λ
r
T
c
=
∑
l
=
1
n
p
λ
rec is the total relative mobility with the correction for fluid compressibility.The total compressibilityC
t is the volume-weighted sum of the rock or soil matrix (C
r) and component compressibilities (C
k
0) are(14)
C
t
=
C
r
+
∑
k
=
1
n
cv
C
k
0
·
C
~
k
,whereϕ
=
ϕ
R
·
1
+
C
r
p
ref
-
p
ref
0.


## 2.4. Relative Permeability Model

During long-term water flooding, the wettability changes with the reduction of polar substances on the rock surface, and the residual oil saturation and displacement oil efficiency will change as well. Since the polar substances on the rock surface is not constant, which depends on the time of displacement and flow rate of water phase, the residual oil saturation is a function of flow rate and time of water flooding (the cumulative volumes of water displacement).Since the variation of wettability is reflected by the relative permeabilities and residual oil saturation, different relative permeabilities (which could be measured with cores after different volumes of water displacement, Figure2) can be used to calculate the residual oil saturation and displacement oil efficiency [28].Figure 2
Relative permeabilities determined at different pore volumes of water injected.
(a)
(b)
(c)The relative permeability curve at low PV (initial wettability, Figure2(a)) of injected water reflects the initial wettability of rock with maximum contents (C
rock
AO
max
⁡ represented by C
max
⁡) of polar substances adsorbed on rock; the relative permeability curve at high PV (high water-wet at ultimate water displacement, Figure 2(c)) of injected water reflects the initial wettability of rock with minimum contents (C
min
⁡) of polar substances adsorbed on rock. We can use C
rock
clay to interpolate the relative permeabilities and residual oil saturation (Figure 2(b)) at different PV of injected water.Given relative permeability curves atC
max
⁡ and C
min
⁡, we can determine the maximum and minimum residual oil saturation, and the residual oil saturation S
or can be calculated with the linear interpolation methods [27]:(15)
S
or
=
S
or
max
⁡
-
C
-
C
min
⁡
C
max
⁡
-
C
min
⁡
S
or
max
⁡
-
S
or
min
⁡
,where S
or
max
⁡—the residual oil saturation at C
max
⁡ (low PV, initial wettability); S
or
max
⁡—the residual oil saturation at C
min
⁡ (high PV, high water-wet wettability).The endpoints of relative permeabilities are computed with a linear interpolation between the given input values atC
max
⁡ and C
min
⁡,(16)
K
r
l
end
=
K
r
l
end
max
⁡
-
S
or
-
S
or
min
⁡
S
or
max
⁡
-
S
or
min
⁡
K
r
l
end
max
⁡
-
K
r
l
end
min
⁡
,where K
r
l
end—the endpoint of relative permeability for phase l at S
or; K
r
l
end
max
⁡—the endpoint of relative permeability for phase l at S
or
max
⁡; and K
r
l
end
min
⁡—the endpoint of relative permeability for phase l at S
or
min
⁡.The relative permeabilities for phase l are assumed to be unique functions of their respective saturations only [37], described as(17)
K
r
l
=
K
r
l
end
S
l
∗
n
,where S
l
∗—the normalized saturations for phase l, S
w
∗
=
(
S
w
-
S
w
c
)
/
(
1
-
S
w
c
-
S
or
), S
o
∗
=
1
-
S
w
∗; and n—the exponent determined by fitting the data.


## 3. Numerical Models

Based on (7), the polar substances and clay only dissolve in water phase, and the governing equations can be described as follows:(18)
∂
φ
λ
k
ρ
k
C
~
k
,
w
∂
t
+
∂
∂
x
ρ
k
C
k
,
w
u
x
w
-
φ
S
w
K
x
x
k
w
∂
C
k
,
w
∂
x
1111111111111111111111111.11
+
K
x
y
k
w
∂
C
k
,
w
∂
y
111111111111111111111111111
+
K
x
z
k
w
∂
C
k
,
w
∂
z
+
∂
∂
y
ρ
k
C
k
,
w
u
y
w
-
φ
S
w
K
y
x
k
w
∂
C
k
,
w
∂
x
11111111111111111111111.1111
+
K
y
y
k
w
∂
C
k
,
w
∂
y
111111111111111111111111111
+
K
y
z
k
w
∂
C
k
,
w
∂
z
+
∂
∂
z
ρ
k
C
k
,
w
u
z
w
-
φ
S
w
K
z
x
k
w
∂
C
k
,
w
∂
x
11111111111111111111111.1111
+
K
z
y
k
w
∂
C
k
,
w
∂
y
111111111111111111111111111
+
K
z
z
k
w
∂
C
k
,
w
∂
z
=
R
k
k
=
1,2
,
…
,
n
cv
,where (19)
λ
k
=
1
-
∑
m
=
1
n
C
^
m
S
w
,
K
x
x
k
w
,
K
x
y
k
w
,
K
x
z
k
w
K
y
x
k
w
,
K
y
y
k
w
,
K
y
z
k
w
K
z
x
k
w
,
K
z
y
k
w
,
K
z
z
k
w
,is the diffusion tensor of component k, R
k
=
Q
k
+
Q
k
T
+
Q
k
F
S
+
Q
k
T
S.If the water phase is selected as the reference phase, with the assumption that the component of oil cannot be dissolved in water phase, the pressure equation in water phase is expressed as(20)
ϕ
C
t
∂
p
w
∂
t
-
∇
K
-
-
λ
r
t
c
∇
p
w
=
∑
k
=
1
n
cv
R
k
.The oil phase pressure equations can be computed by adding the capillary pressure between phases. The implicit form to process the pressure of water phaseP
w and the explicit form to process the coefficients can be used: (21)
φ
C
t
V
n
(
P
w
,
i
n
+
1
-
P
w
,
i
n
)
Δ
t
-
Δ
y
Δ
z
K
-
-
λ
r
t
c
i
+
1
/
2
n
P
w
,
i
+
1
n
+
1
-
P
w
,
i
n
+
1
Δ
x
i
+
1
/
2
-
K
-
-
λ
r
t
c
i
-
1
/
2
n
P
w
,
i
n
+
1
-
P
w
,
i
-
1
n
+
1
Δ
x
i
-
1
/
2
-
Δ
x
Δ
z
K
-
-
λ
r
t
c
j
+
1
/
2
n
P
w
,
j
+
1
n
+
1
-
P
w
,
j
n
+
1
Δ
y
j
+
1
/
2
-
K
-
-
λ
r
t
c
j
-
1
/
2
n
P
w
,
j
n
+
1
-
P
w
,
j
-
1
n
+
1
Δ
y
j
-
1
/
2
-
Δ
x
Δ
y
K
-
-
λ
r
t
c
m
+
1
/
2
n
P
w
,
m
+
1
n
+
1
-
P
w
,
m
n
+
1
Δ
z
m
+
1
/
2
-
K
-
-
λ
r
t
c
m
-
1
/
2
n
P
w
,
m
n
+
1
-
P
w
,
m
-
1
n
+
1
Δ
z
m
-
1
/
2
=
V
∑
k
=
1
N
cv
R
k
.Define (22)
T
l
,
i
±
1
/
2
=
Δ
y
Δ
z
K
-
-
λ
r
l
c
i
±
1
/
2
n
Δ
x
i
±
1
/
2
,
T
l
,
j
±
1
/
2
=
Δ
x
Δ
z
K
-
-
λ
r
l
c
j
±
1
/
2
n
Δ
y
j
±
1
/
2
,
T
l
,
m
±
1
/
2
=
Δ
y
Δ
z
K
-
-
λ
r
l
c
m
±
1
/
2
n
Δ
z
m
±
1
/
2
,
T
t
w
,
i
±
1
/
2
=
Δ
y
Δ
z
K
-
-
λ
r
t
c
i
±
1
/
2
n
Δ
x
i
±
1
/
2
,
T
t
w
,
j
±
1
/
2
=
Δ
x
Δ
z
K
-
-
λ
r
t
c
j
±
1
/
2
n
Δ
y
j
±
1
/
2
,
T
t
w
,
m
±
1
/
2
=
Δ
y
Δ
z
K
-
-
λ
r
t
c
m
±
1
/
2
n
Δ
z
m
±
1
/
2
.Thus, (21) can be reformulated to (23)
φ
C
t
V
n
Δ
t
P
w
,
i
n
+
1
-
P
w
,
i
n
-
T
t
w
,
i
+
1
/
2
P
w
,
i
+
1
n
+
1
-
P
w
,
i
n
+
1
+
T
t
w
,
i
-
1
/
2
P
w
,
i
-
1
n
+
1
-
P
w
,
i
n
+
1
+
T
t
w
,
j
+
1
/
2
P
w
,
j
+
1
n
+
1
-
P
w
,
j
n
+
1
+
T
t
w
,
j
-
1
/
2
P
w
,
j
-
1
n
+
1
-
P
w
,
j
n
+
1
+
T
t
w
,
m
+
1
/
2
P
w
,
m
+
1
n
+
1
-
P
w
,
m
n
+
1
+
T
t
w
,
m
-
1
/
2
P
w
,
m
-
1
n
+
1
-
P
w
,
m
n
+
1
=
V
∑
k
=
1
N
cv
R
k
.Equation (23) includes seven pairs of diagonal matrix for the three-dimensional space: (24)
a
i
,
j
,
k
P
i
,
j
,
k
-
1
+
b
i
,
j
,
k
P
i
,
j
-
1
,
k
+
c
i
,
j
,
k
P
i
-
1
,
j
,
k
+
d
i
,
j
,
k
P
i
,
j
,
k
+
e
i
,
j
,
k
P
i
-
1
,
j
,
k
+
f
i
,
j
,
k
P
i
,
j
+
1
,
k
+
g
i
,
j
,
k
+
1
P
i
,
j
,
k
-
1
=
h
i
,
j
,
k
.Based on the UTCHEM simulator (UTCHEM, Version 6.1, 1999), we use the incomplete LU decomposition method to precondition (24) and use the conjugate gradient method to solve (24). In order to improve the accuracy in the front of displacement, we choose the third-order format TVD method [38] as the difference scheme to process the coefficients.


## 4. Results and Discussion

We use the model to simulate a one-dimensional formation; the parameters are shown in Table1. The parameters of polar substances are referenced from the experiments [39].Table 1
Parameters of the numerical model.
Parameters
Values
Porosity
0.25
Permeability/mD
1000
Grid numbers
100
Length/cm
7.5


### 4.1. Experiments and Verification

#### 4.1.1. Experimental Materials and Conditions

The cores used in the experimental material were sealed cores from the reservoir formation (Table2).Table 2
Parameters of experimental cores.
Core number
Lengths (mm)
Diameter (mm)
Dry weight (g)
Porosity (%)
Gas-measuredpermeability (×10−3 μm2)
2-1
25.3
39.5
33.3
23.1
2535
2-2
25.1
42.5
35.7
25.7
2513Brine was prepared to simulate the formation water of 1080 mg/L salinity (Table3). Oil with 5.3 cp of viscosity at 80°C was a mixture of crude oil and light oil.Table 3
Ion content and salinity of brine.
Ions
Na+
Ca+
Mg2+
K+
Cl−
HCO
3
-
SO
4
2
-
Concentration (mg/L)
11.224
768
187
15.0
19.211
7.6
72The temperature was kept at 80°C for all the tests. The brine was injected by a constant-velocity pump of 1.0 mL/min.


#### 4.1.2. Experimental Setup and Procedures

The displacement setup is shown in Figure3 and the procedures include the following.(1)
The core was saturated with brine and oil, and then core was socked to recover the initial wettability for 15 days.
(2)
The relative permeability was determined with 30 PV of brine injected with the nonsteady method, as shown in Figure5(a), and the pressure and flow rate of oil and water were recorded (Table 4).
(3)
After 30 PV of water flooding, 2000 PV of brine was continuously injected and the oil displacement efficiency was also continuously recorded (Figure4).
(4)
The core was saturated with oil again, and then the relative permeability was determined with 30 PV of brine, as shown in Figure5(b).Table 4
Displacement experiment data.
Core number
Injected rate(mL/min)
Viscosity(cp)
Interfacial tension(mN/m)
Sweep rate(10−5 m/s)
DisplacementEfficiency (%)
Residual oilsaturation (%)
Capillarynumber
2-1
1.00
5.3
1.1
E
-
01
2.15
65.4
25.5
1.0
E
-
03
2-2
1.00
5.3
2.3
E
-
02
2.72
74.9
18.8
6.3
E
-
03Figure 3
Setups for experimental displacement.Figure 4
Comparison of the result with experiment by Ji et al. [13]. In Ji’s experiment, the core with 2413 mD permeability was from Daqing Oilfield of China and the injected water reached 30,000 PV.Figure 5
Relative permeabilities determined at 30 PV and 2000 PV.
(a)
Relative permeabilities determined at 30 PV
(b)
Relative permeabilities determined at 2000 PV


#### 4.1.3. Experimental Results and Verification

In order to compare the result with another study, the displacement experiment was compared with results by Ji et al. [13] shown in Figure 4. Due to the different cores in experiments, only trends of increasing of oil displacement efficiency were compared, and the results show that the trends were the same for the experiment results of this paper and the results by Ji et al. [13].The relative permeability curves atC
max
⁡ and C
min
⁡ used in simulation are shown in Figures 5(a) and 5(b), and they represent the conditions with initial wettability and high water-wet wettability. The simulation result shows that the changing trend of oil displacement efficiency (E
d) and residual oil saturation (S
or) can match the experimental result well.In core number 2-1, brine was injected in the pore for 30 PV, the oil displacement was nearly 70%, and residual oil saturation had a value of 25%, shown in Figure6. From the experiment of core number 2-2, oil displacement efficiency could almost reach 80% while the brine was injected for more than 2000 PV as a high PV, shown in Figure 7. Based on the experiments of core number 2-1 and number 2-2, the oil displacement can be enhanced as the brine injected increases. The water displacement experiment data is shown in Table 4.Figure 6
Oil displacement efficiency, residual oil saturation, and water cut with semilogarithmic injected pore volume for 30 PV.Figure 7
Oil displacement efficiency, residual oil saturation, and water cut with injected pore volume for 2000 PV.Notice that many possible factors can influence the matching result between the simulation and experiment, and a better match can be obtained through adjusting the parameters of adsorption function (9) during the simulation (Figure 8).Figure 8
Oil displacement efficiency, residual oil saturation from experiment, and model during long-term water flooding.


### 4.2. Change of Polar Substances during Long-Term Water Flooding

The new model can simulate the ion exchange chemical reaction by defining different conditions of ion exchange and clay adsorption properties, and we can obtain the contents of polar substances (C
rock
AO) and clay on the rock (C
rock
clay) at different PV of injected brine (Figures 9 and 10). The variations of C
rock
AO and C
rock
clay are shown in Figure 11. C
rock
AO and C
rock
clay decrease with the increasing PV of injected brine, and the decreasing rate is high between 30 PV and 500 PV during water flooding. When above 500 PV, the C
rock
AO and C
rock
clay change slowly, which means that same reduction of polar substances needs more PV of brine injected at the high water cut stage. This is because the equilibrium of clay adsorption is relatively stable at low water cut, and great washing will break the equilibrium of polar substances adsorbed on clay under high water cut and water saturation above 30 PV.Figure 9
Adsorption contents of clay under the condition of different water flooding multiples (gmole/m3).
(a)
The initial condition
(b)
Water flooding multiples reach 30 PV
(c)
Water flooding multiples reach 1000 PV
(d)
Water flooding multiples reach 2000 PVFigure 10
Adsorption contents of AO under the condition of different water flooding multiples (gmole/m3).
(a)
The initial condition
(b)
Water flooding multiples reach 30 PV
(c)
Water flooding multiples reach 1000 PV
(d)
Water flooding multiples reach 2000 PVFigure 11
Oil displacement efficiency under the condition of different water flooding multiples.
(a)
When flooding multiples reach 50 PV
(b)
When flooding multiples reach 2000 PV


### 4.3. Change of Residual Oil Saturation during Long-Term Water Flooding

The polar substances adsorbed on rock surface play an important role in the wettability; the residual oil saturation (S
or) could be reduced with the wettability alternating to water-wet. The simulation results for S
or and E
d during a long-term water flooding are shown in Figure 12, in which one case considers and the other ignores the desorption of polar substances.Figure 12
Relationship between water flooding multiples and oil displacement efficiency within different adsorption conditions.When considering the desorption of polar substances,E
d at low PV (<50 PV) (Figure 11) can reach 60%; during 50–2000 PV E
d continues to increase to 70%, up to 10000 PV, and E
d could reach 75%. Therefore, improving water volume can significantly improve oil displacement efficiency.Most of conventional reservoir simulators ignore the desorption of polar substances. Here we setK
1 to be equal to 0 (3) to simulate the condition that the desorption is ignored, which implies that the desorption rate d
C
rock
clay
/
d
t is equal to zero. The predicted E
d while ignoring the desorption is lower. This is because S
or is a constant when the desorption of polar substances is ignored, and only one permeability curve is used in the simulation model. When the desorption of polar substances is taken into account, the wettability alternates to water-wet and the permeability curves under high PV of water injected are used in simulation model. Thus, the gradual reduction of S
or during long-term water flooding is reflected in simulation, and the calculated E
d is more accurate.


## 4.1. Experiments and Verification

### 4.1.1. Experimental Materials and Conditions

The cores used in the experimental material were sealed cores from the reservoir formation (Table2).Table 2
Parameters of experimental cores.
Core number
Lengths (mm)
Diameter (mm)
Dry weight (g)
Porosity (%)
Gas-measuredpermeability (×10−3 μm2)
2-1
25.3
39.5
33.3
23.1
2535
2-2
25.1
42.5
35.7
25.7
2513Brine was prepared to simulate the formation water of 1080 mg/L salinity (Table3). Oil with 5.3 cp of viscosity at 80°C was a mixture of crude oil and light oil.Table 3
Ion content and salinity of brine.
Ions
Na+
Ca+
Mg2+
K+
Cl−
HCO
3
-
SO
4
2
-
Concentration (mg/L)
11.224
768
187
15.0
19.211
7.6
72The temperature was kept at 80°C for all the tests. The brine was injected by a constant-velocity pump of 1.0 mL/min.


### 4.1.2. Experimental Setup and Procedures

The displacement setup is shown in Figure3 and the procedures include the following.(1)
The core was saturated with brine and oil, and then core was socked to recover the initial wettability for 15 days.
(2)
The relative permeability was determined with 30 PV of brine injected with the nonsteady method, as shown in Figure5(a), and the pressure and flow rate of oil and water were recorded (Table 4).
(3)
After 30 PV of water flooding, 2000 PV of brine was continuously injected and the oil displacement efficiency was also continuously recorded (Figure4).
(4)
The core was saturated with oil again, and then the relative permeability was determined with 30 PV of brine, as shown in Figure5(b).Table 4
Displacement experiment data.
Core number
Injected rate(mL/min)
Viscosity(cp)
Interfacial tension(mN/m)
Sweep rate(10−5 m/s)
DisplacementEfficiency (%)
Residual oilsaturation (%)
Capillarynumber
2-1
1.00
5.3
1.1
E
-
01
2.15
65.4
25.5
1.0
E
-
03
2-2
1.00
5.3
2.3
E
-
02
2.72
74.9
18.8
6.3
E
-
03Figure 3
Setups for experimental displacement.Figure 4
Comparison of the result with experiment by Ji et al. [13]. In Ji’s experiment, the core with 2413 mD permeability was from Daqing Oilfield of China and the injected water reached 30,000 PV.Figure 5
Relative permeabilities determined at 30 PV and 2000 PV.
(a)
Relative permeabilities determined at 30 PV
(b)
Relative permeabilities determined at 2000 PV


### 4.1.3. Experimental Results and Verification

In order to compare the result with another study, the displacement experiment was compared with results by Ji et al. [13] shown in Figure 4. Due to the different cores in experiments, only trends of increasing of oil displacement efficiency were compared, and the results show that the trends were the same for the experiment results of this paper and the results by Ji et al. [13].The relative permeability curves atC
max
⁡ and C
min
⁡ used in simulation are shown in Figures 5(a) and 5(b), and they represent the conditions with initial wettability and high water-wet wettability. The simulation result shows that the changing trend of oil displacement efficiency (E
d) and residual oil saturation (S
or) can match the experimental result well.In core number 2-1, brine was injected in the pore for 30 PV, the oil displacement was nearly 70%, and residual oil saturation had a value of 25%, shown in Figure6. From the experiment of core number 2-2, oil displacement efficiency could almost reach 80% while the brine was injected for more than 2000 PV as a high PV, shown in Figure 7. Based on the experiments of core number 2-1 and number 2-2, the oil displacement can be enhanced as the brine injected increases. The water displacement experiment data is shown in Table 4.Figure 6
Oil displacement efficiency, residual oil saturation, and water cut with semilogarithmic injected pore volume for 30 PV.Figure 7
Oil displacement efficiency, residual oil saturation, and water cut with injected pore volume for 2000 PV.Notice that many possible factors can influence the matching result between the simulation and experiment, and a better match can be obtained through adjusting the parameters of adsorption function (9) during the simulation (Figure 8).Figure 8
Oil displacement efficiency, residual oil saturation from experiment, and model during long-term water flooding.


## 4.1.1. Experimental Materials and Conditions

The cores used in the experimental material were sealed cores from the reservoir formation (Table2).Table 2
Parameters of experimental cores.
Core number
Lengths (mm)
Diameter (mm)
Dry weight (g)
Porosity (%)
Gas-measuredpermeability (×10−3 μm2)
2-1
25.3
39.5
33.3
23.1
2535
2-2
25.1
42.5
35.7
25.7
2513Brine was prepared to simulate the formation water of 1080 mg/L salinity (Table3). Oil with 5.3 cp of viscosity at 80°C was a mixture of crude oil and light oil.Table 3
Ion content and salinity of brine.
Ions
Na+
Ca+
Mg2+
K+
Cl−
HCO
3
-
SO
4
2
-
Concentration (mg/L)
11.224
768
187
15.0
19.211
7.6
72The temperature was kept at 80°C for all the tests. The brine was injected by a constant-velocity pump of 1.0 mL/min.


## 4.1.2. Experimental Setup and Procedures

The displacement setup is shown in Figure3 and the procedures include the following.(1)
The core was saturated with brine and oil, and then core was socked to recover the initial wettability for 15 days.
(2)
The relative permeability was determined with 30 PV of brine injected with the nonsteady method, as shown in Figure5(a), and the pressure and flow rate of oil and water were recorded (Table 4).
(3)
After 30 PV of water flooding, 2000 PV of brine was continuously injected and the oil displacement efficiency was also continuously recorded (Figure4).
(4)
The core was saturated with oil again, and then the relative permeability was determined with 30 PV of brine, as shown in Figure5(b).Table 4
Displacement experiment data.
Core number
Injected rate(mL/min)
Viscosity(cp)
Interfacial tension(mN/m)
Sweep rate(10−5 m/s)
DisplacementEfficiency (%)
Residual oilsaturation (%)
Capillarynumber
2-1
1.00
5.3
1.1
E
-
01
2.15
65.4
25.5
1.0
E
-
03
2-2
1.00
5.3
2.3
E
-
02
2.72
74.9
18.8
6.3
E
-
03Figure 3
Setups for experimental displacement.Figure 4
Comparison of the result with experiment by Ji et al. [13]. In Ji’s experiment, the core with 2413 mD permeability was from Daqing Oilfield of China and the injected water reached 30,000 PV.Figure 5
Relative permeabilities determined at 30 PV and 2000 PV.
(a)
Relative permeabilities determined at 30 PV
(b)
Relative permeabilities determined at 2000 PV


## 4.1.3. Experimental Results and Verification

In order to compare the result with another study, the displacement experiment was compared with results by Ji et al. [13] shown in Figure 4. Due to the different cores in experiments, only trends of increasing of oil displacement efficiency were compared, and the results show that the trends were the same for the experiment results of this paper and the results by Ji et al. [13].The relative permeability curves atC
max
⁡ and C
min
⁡ used in simulation are shown in Figures 5(a) and 5(b), and they represent the conditions with initial wettability and high water-wet wettability. The simulation result shows that the changing trend of oil displacement efficiency (E
d) and residual oil saturation (S
or) can match the experimental result well.In core number 2-1, brine was injected in the pore for 30 PV, the oil displacement was nearly 70%, and residual oil saturation had a value of 25%, shown in Figure6. From the experiment of core number 2-2, oil displacement efficiency could almost reach 80% while the brine was injected for more than 2000 PV as a high PV, shown in Figure 7. Based on the experiments of core number 2-1 and number 2-2, the oil displacement can be enhanced as the brine injected increases. The water displacement experiment data is shown in Table 4.Figure 6
Oil displacement efficiency, residual oil saturation, and water cut with semilogarithmic injected pore volume for 30 PV.Figure 7
Oil displacement efficiency, residual oil saturation, and water cut with injected pore volume for 2000 PV.Notice that many possible factors can influence the matching result between the simulation and experiment, and a better match can be obtained through adjusting the parameters of adsorption function (9) during the simulation (Figure 8).Figure 8
Oil displacement efficiency, residual oil saturation from experiment, and model during long-term water flooding.


## 4.2. Change of Polar Substances during Long-Term Water Flooding

The new model can simulate the ion exchange chemical reaction by defining different conditions of ion exchange and clay adsorption properties, and we can obtain the contents of polar substances (C
rock
AO) and clay on the rock (C
rock
clay) at different PV of injected brine (Figures 9 and 10). The variations of C
rock
AO and C
rock
clay are shown in Figure 11. C
rock
AO and C
rock
clay decrease with the increasing PV of injected brine, and the decreasing rate is high between 30 PV and 500 PV during water flooding. When above 500 PV, the C
rock
AO and C
rock
clay change slowly, which means that same reduction of polar substances needs more PV of brine injected at the high water cut stage. This is because the equilibrium of clay adsorption is relatively stable at low water cut, and great washing will break the equilibrium of polar substances adsorbed on clay under high water cut and water saturation above 30 PV.Figure 9
Adsorption contents of clay under the condition of different water flooding multiples (gmole/m3).
(a)
The initial condition
(b)
Water flooding multiples reach 30 PV
(c)
Water flooding multiples reach 1000 PV
(d)
Water flooding multiples reach 2000 PVFigure 10
Adsorption contents of AO under the condition of different water flooding multiples (gmole/m3).
(a)
The initial condition
(b)
Water flooding multiples reach 30 PV
(c)
Water flooding multiples reach 1000 PV
(d)
Water flooding multiples reach 2000 PVFigure 11
Oil displacement efficiency under the condition of different water flooding multiples.
(a)
When flooding multiples reach 50 PV
(b)
When flooding multiples reach 2000 PV


## 4.3. Change of Residual Oil Saturation during Long-Term Water Flooding

The polar substances adsorbed on rock surface play an important role in the wettability; the residual oil saturation (S
or) could be reduced with the wettability alternating to water-wet. The simulation results for S
or and E
d during a long-term water flooding are shown in Figure 12, in which one case considers and the other ignores the desorption of polar substances.Figure 12
Relationship between water flooding multiples and oil displacement efficiency within different adsorption conditions.When considering the desorption of polar substances,E
d at low PV (<50 PV) (Figure 11) can reach 60%; during 50–2000 PV E
d continues to increase to 70%, up to 10000 PV, and E
d could reach 75%. Therefore, improving water volume can significantly improve oil displacement efficiency.Most of conventional reservoir simulators ignore the desorption of polar substances. Here we setK
1 to be equal to 0 (3) to simulate the condition that the desorption is ignored, which implies that the desorption rate d
C
rock
clay
/
d
t is equal to zero. The predicted E
d while ignoring the desorption is lower. This is because S
or is a constant when the desorption of polar substances is ignored, and only one permeability curve is used in the simulation model. When the desorption of polar substances is taken into account, the wettability alternates to water-wet and the permeability curves under high PV of water injected are used in simulation model. Thus, the gradual reduction of S
or during long-term water flooding is reflected in simulation, and the calculated E
d is more accurate.


## 5. Conclusions

We draw the following conclusions from our study.(1)
The wettability of oil reservoir changes to water-wet during long-term water flooding, and the residual oil saturation gradually reduces and oil displacement efficiency increases. Therefore, the alternation of wettability should be considered during simulation of long-term water flooding; otherwise it will lead to an incorrect history match and unreliable forecasting.
(2)
The desorption of polar substances from the rock during long-term water flooding is the mechanism of wettability variation. Since the wettability is reflected by the relative permeability curves in simulation model, the key point to characterize the process of long-term water flooding is to describe the relationship between relative permeability curves and adsorption of polar substances on rock. The factors for wettability variation include polar substances in crude oil, salinity, and pH of brine, clay in rocks, and flow rate during long-term water flooding.
(3)
An efficient approach for long-term water flooding simulation is achieved through building a water-oil two-phase multicomponent simulation model. This model describes most important physical phenomena during long-term water flooding, including polar substances desorption, residual oil saturation reduction, and variation of relative permeabilities. The model is solved through an implicit format to process the pressure and through an explicit format to process the coefficients; the third-order format TVD method was selected as the difference scheme to process the coefficients and to improve the accuracy in the front of displacement.
(4)
The simulation results have a good match to experimental results. The results have shown that the reduction of polar substances occurs under the condition of high water cut and high water saturation, the residual oil saturation can be gradually reduced by long-term water flooding, and high oil recovery can be obtained by increasing the volume of injected water.
(5)
For the field scale, before history match of simulation, a set of configuration parameters should be turned according to one-dimensional experiments, such as ion exchange, clay and polar substances adsorption-desorption, and relative permeability interpolation sets. All these turned parameters could be used in each grid of map brown field for numerical simulation. Then, the polar substances could be calculated for each grid and this value will be used in the relative permeability interpolation process. However, there are some limitations for residual oil reduction in this paper, such as the variation of porosity, and permeability should also be considered. We could define the relation of porosity and permeability with PV number to deal with this variation.



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*Source: 592951-2015-09-27.xml*