SPE 153954

Analytical Study of Effects of Flow Rate, Capillarity and Gravity on CO2/Brine Multiphase Flow System in Horizontal Corefloods Chia-Wei Kuo and Sally M. Benson, Stanford University

Copyright 2012, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Western North American Regional Meeting held in Bakersfield, California, USA, 19–23 March 2012. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract This paper presents an approximate semi-analytical solution for predicting the average steady-state saturation during

multiphase core flood experiments over a wide range of capillary and gravity numbers. Recently, the influences of flow rate,

gravity, and sub-core heterogeneity on the brine displacement efficiency have been studied using the 3-D simulator TOUGH2

(Kuo et al. 2010). These studies have demonstrated that the average saturation depends on the capillary and gravity numbers

in a predictable way.

The purpose of this paper is to provide a simple approximate semi-analytical solution for predicting the average saturation

during core flood experiments, thus avoiding the need for 3-D simulations. A two dimensional analysis of the governing

equations for the CO2/brine multiphase flow system at steady-state is used to develop the approximate semi-analytical

solution. We have developed a new criterion to identify the viscous-dominated regime at core scale. Variations of interfacial

tension, core permeability, length of the core, and the effects of buoyancy, capillary and viscous forces are all accounted in

the theoretical solutions. We have also shown that three dimensionless numbers (NB, Ngv, Rl) and two critical gravity

numbers (Ngv,c1, Ngv,c2) are required to properly capture the balance of viscous, gravity, and capillary forces. There is good

agreement of the average saturations between the 3-D simulations and the model. This new model can be used to design and

interpret multiphase flow core-flood experiments.

1. Introduction

Recently, the influences of flowrate, gravity, and capillarity on immiscible CO2/brine two-phase flow system at core scale

have been studied in laboratory experiments (Perrin et al. 2009; Perrin and Benson 2010; Krevor et al. 2011) as well as using

a 3-D numerical simulation (Krause et al. 2009; Kuo et al. 2010; Shi et al. 2010). Although theoretical analysis of multiphase

flow systems has identified flow regions controlled by different types of flow behavior, most of them focused on oil/water

systems where the effect of gravity has been neglected (Lenormand et al. 1988; Pickup and Stephen 2000; Jonoud and

Jackson 2008; Hussain et al. 2011). Without considering gravity, the flow regime is often studied at steady-state for two

extreme limits: one is when the viscous force is large such that the capillary pressure is negligible. In this viscous limit,

fractional flow is uniform. The other limit is when the flow rate is very low and capillary forces dominate the fractional flow

distribution. In the capillary limit, capillary equilibrium is assumed. Characterizing different flow regimes is important and

very useful for upscaling. Dimensionless numbers are often used to analyze different regimes since they combine the effects

of flow rate, interfacial tension, the permeability of the core, and gravity segregation (Fulcher et al. 1985; Avraam and

Payatakes 1995; Skauge et al. 1997; Skauge et al. 2000; Virnovsky et al. 2004; Cinar et al. 2006). Motivated by the need to

effectively design and accurately interpret multiphase flow coreflood experiments and by the previous multiphase flow

literature, issues such as the influences of flow rate, gravity as well as permeability and interfacial tension on CO2/brine are

addressed and systematically investigated by analyzing different dimensionless numbers in this paper.

However, the boundaries between viscous and capillary limits are ambiguous in the literature. Jonoud and Jackson (2008)

develop a new dimensionless group using three dimensionless numbers (transverse and longitudinal Peclet numbers and end-

point mobility ratio: PeT, Pe

L, M

e) to characterize the balance of viscous and capillary forces and use numerical simulations to

determine empirically the threshold values of these two limits. Their results show that the balance between viscous forces

along the direction of flow and capillary forces perpendicular to the flow control the boundaries. The viscous-dominated

regime is valid for the PeT greater than the minimum values of Pe

T while the capillary-dominated regime is valid for the Pe

T

2 SPE 153954

smaller than the maximum values of PeT.

For gas/oil or gas/water systems, the density difference between two fluids is large, and the effect of gravity becomes

important (Rossen and Duijn 2004; Nordbotten et al. 2005; Kopp et al. 2009). Strong gravitational forces could lead to

distinct gravity segregation between the injected nonwetting phase and wetting phase. In addition, when flow rates are low,

gravity-capillary equilibrium is established. Zhou et al (1994) provided a 2D comprehensive analysis of the combined effect

of viscous, gravity and capillary forces on the fluid moving in the transverse direction. They identify six different flow

regions, the corresponding simplified equations, and conditions based on three dimensionless groups: gravity to viscous ratio

NgvM/(1+M), capillary to viscous ratio NcvM/(1+M), and the shape factor Rl2. They establish approximate bounds for the

transitions between regions by examining a wide range of experimental data in the literature. However, most of the data

sources are focused on imbibition and miscible flow types. In addition, their model only characterizes different flow regions

without solving the simplified equations.

In this paper, we are interested in the drainage flow behavior in the CO2/brine two-phase flow in the horizontal core. We

identify different flow behavior in homogeneous and isotropic systems based on the similar dimensionless groups used in

Zhou et al. (1994). A two dimensional analysis of the governing equations accounting for viscous, gravity, and capillary

forces at steady-state is used to develop an approximate semi-analytical solution to predict non-wetting phase saturations

during core-flood experiments. A systematic parametric study of the flow regimes is performed numerically to help identify

flow regions. For example, a wide range of flow rates, permeability, interfacial tension, and three different lengths of the

core are investigated. The semi-analytical solution is validated by comparing the results under a wide range of input

parameters, including different fractional flows and relative permeability curves. This new model can be used to design and

interpret multiphase flow core-flood experiments as well as to gain greater insight into field-scale behavior.

2. Methodology The overall methodology for this study contains two parts. First, we conduct 3D high resolution simulations of core scale

CO2/brine flow over a wide range of injection flowrates. We simulate the injection of a constant ratio of CO2/brine at a given

flowrate into a horizontal and initially brine saturated core. The corresponding brine displacement efficiency is assessed

when the system reaches steady-state (defined as the time when the saturation is no longer changing and the fractional flows

of CO2 and brine are equal at the inlet and outlet of the core). Sensitivity studies incorporating a wide range of permeability,

core lengths and interfacial tension values have also been studied to generalize these results. Second, a theoretical analysis of

the multiphase flow equations is used to develop an approximate semi-analytical solution for predicting the average

saturation in the core as a function of gravity number, Bond number and several dimensionless parameters. Finally, we

compare the results of simulations and approximate semi-analytical solutions to test its validity over a wide range of

parameters.

2.1 Simulation Method

TOUGH2 MP with ECO2N (Pruss et al. 1999; Zhang et al. 2003; Pruess 2005) has been used in this paper to simulate a

series of CO2/brine core flood experiments. For the beginning part of the paper, we focus on 95% fractional flow of CO2 and

5% brine, which are injected simultaneously into a simulated core at a wide range of flowrates. Later we consider a wide

range of fractional flows. The simulated core is saturated with brine as the initial condition while the capillary pressure

gradient between the last slice of the core and the outlet slice is set to zero to minimize end effects. The dimensions of

simulated core are 5.08 cm in diameter and 15.24 cm long, and the core is modeled by 25 25 31 grid blocks with

dimensions 1.884mm 1.884mm 5.08mm. The reservoir condition is 50˚C and 12.4 MPa. Here we assume a

homogeneous core with petrophysical properties of Berea Sandstone (mean porosity of 0.202 and mean permeability of 430

md). Results for a wide range of permeability and interfacial tension values are provided later in the paper. The relative

permeability relations and the capillary pressure for each grid element are as follows:

CO w2

2

n n

w w wr

r,CO r,w

wr wr

1-S S -Sk = , k =

1-S 1-S

(1)

c w w

φp (S )=σ J(S )

k

(2)

2 2

1

λ 1/λ w P

w 1 1 * *λ

P*

S -S1J(S )=A ( -1)+B (1-S ) , S =

1-SS

(3)

where Sw is the average brine saturation, Swr is the residual brine saturation, nw

and nCO2 are the functional exponents for the

SPE 153954 3

brine and CO2 curves respectively, pc is capillary pressure and σ is the interfacial tension between CO2 and brine. φ and k are

the porosity and permeability; and J is a modified J-function (Silin et al. 2009) with five free parameters A1, B1, λ1, λ2, Sp.

The unknown parameters in relative permeability (Eq. 1) and capillary pressure (Eq. 2) are chosen to match the experimental

data provided by Perrin et al. (2009). Figure 1 illustrates the input curves for the capillary pressure and relative permeability

used in this paper.

The details regarding the simulation can be found in Kuo et al. (2010). The significance of flow rate, interfacial tension,

and core permeability on the CO2/brine flow systems is studied: flow rate is changed by several orders, from 0.0001 ml/min

up to 60 ml/min; interfacial tension and core permeability are varied by two orders of magnitudes to study a range of values.

Additionally, a range of core lengths is also included.

Figure 1: (LHS) Input relative permeability curves for CO2 and brine with Swr =0.15, nw =7, and nCO2=3; (RHS) The capillary pressure curve used in the simulations. The values of input parameters are A1=0.007734, B1=0.307601, λ1=2.881, λ2=2.255, Sp=0 and σ =22.47 dynes/cm

Figures 2, 3 and 4 show the core average CO2 saturations for the homogeneous cores as a function of traditional capillary

number Ca, alternative capillary number Ncv, and gravity number Ngv for two sensitivity studies using the definitions shown

in Eq. 4-6 (Fulcher et al. 1985; Zhou et al. 1994).

2t coCa= u μ σ (4)

2

*

c

cv 2

co t

kLpN =

H μ u (5)

2

gv

co t

ΔρgkLN =

Hμ u (6)

where ut is the total average Darcy flow velocity, μCO2 is CO2 viscosity, pc* is a characteristic capillary pressure, L the length

of the core, H the height of the core, Δρ=ρw-ρg is the density difference between CO2 and brine, and g is the acceleration of

gravity. The values of Ca, Ncv and Ngv are controlled by varying injection rates, interfacial tension, and permeability where

the other parameters are kept as constants. In general, higher flowrates, lower permeability, smaller density differences and

lower capillary pressure have higher Ca, or lower Ncv and Ngv. This high flowrate regime is representative of the near well

region. On the other hand, lower flowrates, hence the lower Ca or the higher Ncv and Ngv, are representative of the leading

edge of the plume and during fluid redistribution in the post-injection period.

All the simulation results shown in Fig. 2-4 are for a fractional flow of 95% CO2. The sensitivity studies of interfacial

tension are illustrated in the left hand side of the figures and permeability on the right. When comparing the average

saturation in terms of traditional capillary number Ca, alternative capillary number Ncv and gravity number Ngv, the efficiency

of brine displacement clearly falls into three separate regimes: a viscous-dominated regime where the saturation is

independent of Ca, Ngv and Ncv; a gravity-dominated regime where the average saturation is strongly dependent on the

dimensionless numbers; and a capillary-dominated regime characterized by low saturations with a small dependence on the

dimensionless variables. The transitions from the viscous- to gravity-dominated regime and from the gravity- to capillary-

dominated regime are dependent on the interfacial tension and permeability in Fig. 2 and 3. Results show that the transition

points occur earlier when gravity has stronger influence, for example, the lower interfacial tension and higher permeability

cases.

4 SPE 153954

However, plotting the same data in terms of gravity number results in the same transitions for different interfacial tensions

or core permeabilities (Fig. 4). Defining the first transition, from viscous- to gravity dominated regime, as a critical number 1

and the second transition, from gravity- to capillary-dominated regime, as a critical number 2, we observe that viscous forces

dominate when Ngv ≤ Ngv,c1. This regime is known as viscous-dominated regime where the brine displacement efficiency is

independent of interfacial tension and permeability. Gravity starts to have a significant effect on the multiphase flow

behavior when decreasing flowrates reduce the viscous forces. In this gravity-dominated regime (Ngv,c1 < Ngv < Ngv,c2), the

brine displacement efficiency is highly flowrate dependent and insensitive to the interfacial tension and permeability (Fig. 4).

The capillary forces dominate when flowrates are reduced further. In this capillary-dominated regime (Ngv,c2 ≤ Ngv), the brine

displacement efficiency is weakly dependent on the gravity number.

Figure 2: Average CO2 saturation as a function of traditional capillary number Ca

Figure 3: Average CO2 saturation as a function of alternative capillary number Ncv

Figure 4: Average CO2 saturation as a function of alternative gravity number Ngv

From Fig. 4, the viscous-dominated regime is clearly defined by Ngv ≤ 20 while the capillary-dominated regime begins when

Ngv ≥ 220. Therefore, Ngv,c1 = 20 and Ngv,c2 = 220. These two critical gravity numbers are independent of Bond number and

will be used in the analytical solution in the following sections.

SPE 153954 5

2.2 Development of a 2-D Semi-Analytical Solution

We first simplify our 3-dimensional problem into 2 dimensions (x-z direction). The properties of this 2D porous medium are

homogeneous and isotropic. Porosity and permeability are constant. Mass conservation equations and pressure equation for

incompressible flow are:

j j j,x j,z

j

S S u uφ u =φ =0

t t x z

(7)

tu 0 (8)

with the condition Sw+

2COS =1. uj,x and uj,z are Darcy velocity of phase j in the x and z direction, respectively. Darcy flow

velocities for both phases are given by

j j rj

j j j j,x j,z j

j

p p ku =-λ k , +ρ g u , u where λ =

x z μ

(9)

where uj, λj, pj, ρj and μj are Darcy flow velocity, relative mobility, pressure, density, and viscosity of phase j, respectively.

Boundary conditions used in the simulation are given as followings. At the top and the bottom boundaries,

uw,z=ug,z=0 (10)

In the flow direction x, total volumetric flow rate is sum of water and gas flow:

uw,x+ug,x=ut (11)

Using Darcy’s equation and the boundary conditions, the mass conservation of gas phase now becomes:

g g gt c c

w

S Mkλ Mkλu p p1 1- + Δρg- =0

t φ kλ x 1+M φ z z 1+Mx

(12)

M= λw/λg is mobility ratio. To non-dimensionlise the equation, we define xD=x/L, zD=z/H, tD=tut/φL, and pc(Sw) =pc*J (Sw).

Substituting all the defined terms into Eq. 12 yields the dimensionless mass conservation equation:

2

g rg rg rg

gv cv

D D D l D D D D

S Mk Mk Mk1 1 J J+ +N -N + =0

t x 1+M z 1+M R x x 1+M z z 1+M

(13)

where Ncv and Ngv are shown in Eq. 5 and 6, and Rl =L/H is the shape factor or called the aspect ratio. We can also define

Bond number NB as a ratio of gravity to capillary numbers:

gv

B *

cv c

N ΔρgHN =

N p (14)

Assuming steady-state (Eq. 15) and using separation of variables, we can obtain 2D time independent CO2 saturation SG (Eq.

16). The derivation is detailed in Appendix A for reference.

g

D

S=0

t

at steady state (15)

2

lD

cvB D B D

R- ax

N-N bz -N bz

1 2 3 4SG C e +C e +C e +C (16)

Unknown variables a and b are functions of saturation and mobility ratio M (see Appendix B). C1, C2, C3 and C4 are

functions of xD and zD. Based on this solution, the time independent CO2 saturation SG depends only on its position xD and

6 SPE 153954

zD, M, as well as the dimensionless numbers Rl, Ncv and NB. Since it is difficult to integrate Eq. 16 to obtain the average core

saturation, we assume that the saturation at a particular point (x0,D, z0,D) is representative the core average saturation2COS :

2CO 0,D 0,DS SG x ,z (17)

Now C1, C2, C3 and C4 become some constants and the two terms in the exponent ax0,D and bz0,D are unknown variables

which will be determined later by fitting the simulation results. To eliminate some unknown parameters, we use several

assumptions observed from the simulation results. First, Fig. 4 shows that average CO2 saturations are independent of gravity

number Ngv when Ngv ≤ Ngv,c1: 2

lB 0,D

2 gv,c1

R- N ax

CO N

gv gv,c1

gv

S 0 when N N e 0

N

(18)

Therefore, in the viscous-dominated regime, Ngv ≤ Ngv,cl, the core average saturation becomes the Buckley-Leverett solution

SBL:

B 0,D

2

-N bz

CO 3 4 BLS C e +C S . (19)

The average saturation in this regime is independent of Bond number, which implies C3=0 and C4= SBL. Second, saturations

are observed (Kuo et al. 2010) to be a constant SBL when Bond number equals to zero (g=0), which results in C2= -C1. Based

on all restrictions above, Eq. 16 can be rewritten as the following form:

2

l0,D

B 0,D cv

2

R- ax

-N bz N

CO 1 BLS C e -1 e +S (20)

Let the term ax0,DNB/Ngv,cl≡d1, and then we need to satisfy2

l 1-R de 0 based on Eq. 18. Once we choose an appropriate value

of d1 to satisfy this condition, we can replace the unknown term ax0,D in terms of d1, NB, and the critical gravity number Ngv,c1.

Similarly, Figure 4 also illustrates that saturations are not sensitive to Bond number when Ngv ≤ Ngv,c2, which leads to

2 B 0,DCO -N bz

gv gv,c2

B

S0 when N N e 0

N

(21)

Defining the exponent term bz0,D≡d2/Ngv,c2, choosing an appropriate value of d2 to satisfy 2 B-d N Ngv,c2e 0 , and the saturations

are now determined by Bond number and gravity number, shown in Eq. 22.

2l gv,c1B

2 1gv,c2 gv

2

R NN-d -d

N N

CO 1 BLS =C e -1 e +S

(22)

C1, d1, and d2 are constant parameters which can be determined by simulation results or experimental data. From the

previous definitions, d1 and d2 are proportional to a and b, respectively. Based on Appendix B, the parameter a depends on

fCO2/krg while parameter b depends on fCO2. Therefore we can rewrite d1 and d2 in terms of two known variables:

d1=d11SBL/krg(SBL) and d2=d22SBL. On the other hand, the values of Bond number we are interested in are small (0.021-0.21)

in this paper, hence we can combine C1NB into another constant C11, and obtain C1=C11/NB. Substituting above assumptions

back into Eq. 22, our final form of saturation equation becomes:

2l gv,clB BL

22 BL 11gv,c2 rg BL gv

2

R NN S-d S -d

N k S N11

CO BL

B

CS = e -1 e +S

N

(23)

where C11 , d11 and d22 are constants determined by curve matching the semi-analytical solution with the simulation results.

Therefore the average saturation is controlled by Bond number NB, gravity number Ngv, the Buckley-Leverett solution SBL,

and relative permeability krg once the critical gravity numbers, the aspect ratio Rl as well as C11, d11, and d22 are known.

In the high flowrate regime or for Ngv ≤ Ngv,c1, the average saturation (and hence, SBL) is determined solely from the

fractional flow curve based on Buckley-Leverett theory (1942) which neglect gravity and capillary pressure:

SPE 153954 7

SBL = SBL(2COf ) (24)

where

2COf =1 1+M (25)

Once we know the input relative permeability curves and the CO2 fractional flow, we can determine the corresponding SBL.

3. Results

3.1 Approximate Semi-analytical Solution

To compare our theoretical predictions with the simulation results shown earlier, we need to determine variables C11, d11, d22,

SBL and two critical numbers Ngv,c1 and Ngv,c2. Figure 5 shows that the average CO2 saturation of the base case as a function

of capillary number Ncv (LHS) and the CO2 fractional flow curve (RHS). The critical numbers Ngv,c1 and Ngv,c2 are

determined to be 20 and 220 respectively by examining the simulation results shown in Fig. 4. Table 1 provides the values of

dimensionless parameters used in the curve fitting to the base case with interfacial tension σ and core permeability k (see

Table 2 for parameter values). Substituting these values back into Eq. 23, we get the general form of core average CO2

saturation:

2BL l

BBL

rg BL gv

2

S RN -3.4-S k S N70.968CO BL

B

60S = e -1 e +S

N

(26)

For the base case, Rl is equal to 3, SBL is 0.324 when the fraction flow of CO2 equals to 0.95 (Fig. 5), and the value of krg(SBL)

is 0.05538. Therefore, the core average CO2 saturation for the base case is

B

gv

2

180N -- N220CO

B

60S = e -1 e +0.324

N

(27)

Eq. 27 is valid for the cases with the same aspect ratio, input relative permeability curves and the same Buckley-Leverett

solution SBL. It can be applied to the base case and the sensitivity cases with only permeability or interfacial tension

changing since the Buckley-Leverett solution SBL and relative permeability krg are independent of those variables. The only

variable parameters in Eq. 27 are Bond number NB and gravity number Ngv.

Figure 5: (LHS) Comparison of average CO2 saturation as a function of capillary number Ncv between theoretical values and simulation results for the homogeneous or base case (σ, k); (RHS) fractional flow curve based on our input relative permeability curves (Eq.1)

Table 1: Values of unknown variables used to match the base case. Sensitivity studies also share the same values

Ngv,c1 Ngv,c2 C11 d11 d22

20 220 60 0.17 3.1

Table 2: Berea core properties and fluid properties used in the homogeneous cores for the base case

mean φ mean k (md) σ (mN/m) L (cm) H (cm) μbrine (cp) μCO2 (cp)

0.202 430 22.47 15.24 5.08 0.558 0.046

8 SPE 153954

Figure 6 and 7 show the sensitivity studies for permeability (43md and 4300md) and interfacial tension (7.49 mN/m and

67.41 mN/m) respectively. The Figures compare the simulation results and the predicted values based on Eq. 27. As shown,

we can replicate the simulation results quite well, especially the transition from the viscous- to gravity-dominated regime.

However, we have a slight mismatch in the capillary-dominated regime for the cases when capillary force is strong (0.1k and

3σ). For the cases where capillary force is relatively small such as 10k (4300md) and σ/3 (7.49 mN/m), the semi-analytical

solution matches best in the transition from viscous- to gravity-dominated regime but deviates slightly in the gravity- and

capillary-dominated regime.

Figure 6: Comparison of average CO2 saturation as a function of capillary number Ncv between theoretical values and simulation results for the sensitivity cases of core permeability. LHS: (σ, 0.1k), RHS: (σ, 10k)

Figure 7: Comparison of average CO2 saturation as a function of capillary number Ncv between theoretical values and simulation results for the sensitivity cases of interfacial tension. LHS: (3σ, k), RHS: (σ/3, k)

3.2 Model Validation

Here we test the validity of Eq. 26 by using different core lengths (Fig. 8), different fractional flows of CO2 (Fig. 9), and

different input relative permeability curves (Fig. 10).

3.2.1 Effects of Core Length (15.24-45.72 cm)

The average CO2 saturations as a function of capillary numbers for the two different core lengths (2L: 30.48cm and 3L: 45.72

cm) are shown in Fig. 8. Since the input relative permeability curves and the fractional flow of CO2 (fCO2=0.95) for these two

sensitivity cases are the same as the base case, we can expect the Buckley-Leverett solution SBL is still 0.324. The aspect

ratios Rl now become 6 and 9, respectively. As shown, the theoretical values can predict simulation results quite well. The

longer core lengths encounter the stronger gravity effect, and hence result in a slightly earlier transition from viscous- to

gravity-dominated regime.

SPE 153954 9

Figure 8: Comparison of average CO2 saturation as a function of capillary number Ncv between theoretical values and simulation results for the sensitivity cases of length of the core. LHS: 2L =30.48cm, RHS: 3L=45.72 cm

3.2.2 Different Fractional Flows of CO2

Fig. 9 shows the average CO2 saturations as a function of capillary numbers for CO2 fractional flows of 0.79, 0.51 and 0.34.

Although the same input relative permeability curves are used, different fractional flow of CO2 results in different Buckley-

Leverett solution SBL and hence different krg(SBL) values. Again, the semi-analytical model predicts average saturations very

well, especially for the high fractional flows.

Figure 9: Comparison of average CO2 saturation as a function of capillary number Ncv between theoretical values and simulation results for the sensitivity cases of different fractional flows of CO2.

10 SPE 153954

3.2.3 Different Relative Permeability Curves

The final cases to test our theoretical model use different relative permeability curves with 0.95 fractional flow of CO2. The

other parameters such as those in Table 1 and 2 are the same. For these cases, the Buckley-Leverett solution SBL and

krg(SBL) are different for each case. Fig. 10 shows the average CO2 saturations as a function of capillary numbers for six

different input relative permeability curves. The first four cases (the 1st and 2

nd rows of Fig. 10) use the same form of relative

permeability functions shown in Eq. 1. The base cases use parameters nw=7, nCO2=3, and Swr=0.15 (IRP1) while the other

three cases change one or two parameters at a time. One has a lower residual brine saturation (IRP2), one has a higher

exponent of brine nw (IRP3), and the last one has a higher exponent of nCO2 as well as a lower residual brine saturation

(IRP5). The 2D model predicts the lower residual brine saturation (IRP2) and the higher nw (IRP3) very well while the

predictions for the higher nCO2 and lower residual brine saturation (IRP5) have slight deviations in the gravity- and capillary-

dominated regime. However, the semi-analytical solution still captures the transition from viscous- to gravity-dominated

regime quite accurately. The last two cases (the 3rd

row of Fig. 10) use Corey’s equation with residual saturation 0.05 and

0.15, respectively. The results show that even with different relative permeability functions, the semi-analytical solutions

still match the simulation results quite well.

Figure 10: Comparison of average CO2 saturation as a function of capillary number Ncv between theoretical values and simulation results for six different input relative permeability curves. Those simulations share the same constant parameters in Table 1 and 2.

SPE 153954 11

4. Discussion of Results In previous sections, we have shown that simulation results and theoretical predictions agree very well. The average CO2

saturation is determined by three physical forces. The relative magnitudes of gravity to capillary forces and viscous to

gravity forces are characterized as Bond number NB and gravity number Ngv. The combine effect of these forces has shown

in Eq. 26. Similar to what we observed in Eq. 27, the average CO2 saturation can be simplified to Eq. 28 once we know the

exact bounds (Ngv,c1 and Ngv,c2) for the transitions between regions.

2l gv,clB

gv,c2 gv

2

R NN- -N N11

CO BL

B

CS = e -1 e +S

N

(28)

The semi-analytical model provides a simple tool to determine approximate values for the critical numbers. Comparing Eq.

28 and Eq. 26, we can estimate the critical numbers Ngv,c1 and Ngv,c2 in terms of the Buckley-Leverett solution SBL and relative

permeability to gas evaluated at SBL, krg (S BL):

BL

gv,c1 gv,c2

rg BL BL

S 70.97N 3.4 and N

k S S (29)

We can expect lower relative permeability to CO2 results in higher CO2 saturation and hence increases the value of SBL. Eq.

29 works well for all the cases tested in this paper.

Table 3 summarizes three different flow regimes and the corresponding core average CO2 saturation expressions based on

Eq. 28. In general, at high flow rates or so-called viscous-dominated regime (Ngv ≤ Ngv,c1), the brine displacement

efficiencies for homogeneous cores are flowrate independent at the given fractional flow of CO2. The value of SBL can be

predicted from Buckley-Leverett theory neglecting capillarity and gravity. When buoyancy forces begin to dominate

multiphase flow (Ngv,c1 < Ngv < Ngv,c2), average saturations are highly flow rate dependent. In this gravity-dominated regime,

the average saturation is mainly dependent on gravity number. When Ngv > Ngv,c2, saturation becomes less sensitive to the

gravity number. In the limit of capillary-gravity equilibrium, the average saturation asymptotically approaches to a constant

value.

Table 3: Summary of flow regions for general cases

Flow region Conditions Time independent CO2 saturation

Viscous-dominated regime Ngv ≤ Ngv,c1

2CO BLS S

Gravity-dominated regime Ngv,c1 < Ngv < Ngv,c2

2l gv,cl

gv

2

R N-

N11

CO BL

gv,c2

CS S - e

N

Capillary-dominated regime Ngv,c2 ≤ Ngv

2

2 2

l gv,cl l gv,cl11 11 11

CO BL BL

gv,c2 gv gv,c2 gv,c2 gv

R N R NC C CS S - 1- S -

N N N N N

5. Conclusions

A new semi-analytical solution has been developed that can be used to predict the influence of gravity and capillary numbers

on the average saturation expected during multiphase flow. Practical applications include helping to design core flood

experiments, including assuring that relative permeability measurements are made in the viscous dominated regime,

evaluating potential flow rate dependence, influence of core-length on a multiphase flow experiments, and influence of fluid

properties on the experiments.

The new semi-analytical solution can be used to calculate the average saturation in terms of several important

dimensionless numbers such as mobility ratio, relative permeability to gas evaluated at SBL, aspect ratio Rl, Bond number NB

and gravity number Ngv:

2BL l

BBL

rg BL gv

2

S RN -3.4-S k S N70.97CO BL

B

60S = e -1 e +S

N

(26)

The solution has been compared to 3D high resolution simulations to study the effects of flowrate, gravity, interfacial tension,

core-length, and core permeability on two-phase immiscible flow. The proposed 2D semi-analytical technique predicts the

12 SPE 153954

brine displacement efficiency of 3D homogeneous CO2/brine two-phase flow simulations very well when Bond number

ranging from 0.021-0.21. Theoretical predictions match the corresponding simulation results not only for the base case but

also for many sensitivity cases. It can also apply to various fractional flows of CO2 as well as different input relative

permeability curves.

6. Acknowledgments

The authors would like to gratefully acknowledge the financial support of the Global Climate and Energy Project (GCEP) at

Stanford University.

7. Nomenclature

2

1

1

CO 2

2

A =J-function fitting parameter

B =J-function fitting parameter

Ca=Traditonal Capillary number

f =Fractional flow of CO

g=Accelerataion of gravity, m/s

H=Height of the core, m

J=Leverett's J function

k=A

r,j

B

cv

gv

gv,c1

verage permeability, md

k =Relative permeability of phase j

L=Length of the core, m

M=Mobility ratio

N =Bond number

N =Alternative Capillary number

N =Gravity number

N =Critical gravity number of tra

gv,c2

j

c

*

c

l

nsition 1

N =Critical gravity number of transition 2

n =Functional exponent of relative permeability of phase j

p =Capillary pressure, Pa

p =Characteristic capillary pressure, Pa

R =Aspect ratio

2

BL

CO

P

w

wr

*

2

S =Buckley-Leverret saturation

S =Core average saturation

S =J-function fitting parameter

S =Average brine saturation

S =Residual brine saturation

S =Normalized brine saturation

SG=Steady-state CO s

D

j

t

D

D

aturation

t =Dimensionless time-coordinate

u =Flow velocity of fluid phase j, m/s

u =Total average Darcy flow velocity, m/s

x =Dimensionless x-coordinate

z =Dimensionless z-coordinate

Δρ=Density differenc 3

2

j

2

1,2

j

e between CO and brine, kg/m

μ =Viscosity of phase j, cp

=Porosity

σ =CO -brine interfacial tension, N/m

λ =J-function fitting parameter

λ =Relative mobility of phase j

8. References

1. Avraam, D. G., and A. C. Payatakes. 1995, Generalized relative permeability coefficients during steady-state two-phase flow in

porous media, and correlation with the flow mechanisms. Transport in Porous Media 20: 135-168

2. Buckley, SE. and Leverett, MC, 1942, Mechanism of fluid displacement in sands ,Trans. AIME 146,107

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crossflow effects in two-phase displacements, SPE Journal 11 (2): 216-226. SPE-90568-PA. DOI: 10.2118/90568-PA

4. Fulcher, R. A., Ertekin, T., and Stahl, C. D., 1985, Effect of capillary number and its constituents on two-phase relative

permeability curves, J. Petrol. Tech., (Feb.), 249-260.

5. Hussain, F., Cinar,Y., and Bedrikovetsky, P., 2011, A Semi-Analytical Model for Two Phase Immiscible Flow in Porous Media

Honouring Capillary Pressure, Transport in Porous Media

6. Jonoud S., Jackson M., 2008, New criteria for the validity of steady-state upscaling. Transport Porous Med. 71(1), 53–73

7. Kopp, A., Class, H., Helmig, R., 2009, Investigations on CO2 storage capacity in saline aquifers—Part 1: dimensional analysis of

flow processes and reservoir characteristics. Int. J. Greenhouse Gas Control 3(3), 263–276.

8. Krause, M., Perrin, J.-C., and Benson, S. M., 2009, Modeling Permeability Distributions in a Sandstone Core for

History Matching Coreflood Experiments, SPE126340 9. Krevor, S., Pini, R., Li, B., and Benson, S., 2011, Capillary heterogeneity trapping of CO2 in a sandstone rock at reservoir

conditions, Geophysical Research Letters 38, L15401

10. Kuo, C-W, Perrin, J-C., and Benson, S., 2010, “Effect of gravity, flow Rate, and small scale heterogeneity on multiphase flow of

CO2 and brine”, SPE132607, SPE Western Regional Meeting, 27-29 May 2010, Anaheim, California, USA8

SPE 153954 13

11. Lenormand, R., Touboul, E., Zarcone, C., 1988, Numerical models and experiments on immiscible displacements in porous

media, J. Fluid Mech., 189 (1988), pp. 165–187

12. Nordbotten, J.M., Celia, M.A., Bachu, S., 2005, Injection and storage of CO2 in deep saline aquifers: analytical solution for CO2

plume evolution during injection, Transport Porous Media, 58, pp. 339–360

13. Pickup, G.E., and Stephen, K.D., 2000, An assessment of steady-state scale-up for small-scale geological models, Petroleum

Geosci, 6 (2000), pp. 203–210

14. Perrin, J.-C., Krause, M., Kuo, C.-W., Miljkovic, L., Charob, E., Benson, S.M., 2009. Core-scale experimental study of relative

permeability properties of CO2 and brine in reservoir rocks. Energy Procedia 1, 3515–3522.

15. Perrin, J-C., and Benson, S., 2010, An experimental study on the influence of sub-core scale heterogeneities on CO2 distribution

in reservoir rocks, Transport in Porous Media

16. Pruess, K. 2005, “ECO2N: A TOUGH2 Fluid property module for mixtures of water, NaCl, and CO2,” Lawrence Berkeley

National Laboratory Report LBNL-57952, Berkeley, CA

17. Pruess, K., Oldenburg, C. and Moridis, G, 1999, “TOUGH2 User’s Guide, V2.,” Lawrence Berkeley National Laboratory Report

LBNL-43134, Berkeley, CA, Advances In Water Resources, 29,397-407.

18. Rossen W. R., van Duijn C. J., 2004, Gravity segregation in steady-state horizontal flow in homogeneous reservoirs. J Pet Sci

Eng 43, 99–111

19. Shi, J.-Q., Xue, Z. and Durucan, S., 2010, Supercritical CO2 core flooding and imbibition in Tako sandstone–Influence of sub-

core scale heterogeneity, Int. J. Greenhouse Gas Control

20. Silin, D., Patzek, T., Benson, S., 2009, A model of buoyancy-driven two-phase countercurrent fluid flow. Transport in Porous

Media, 76, pp. 449–469

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experiments, SCA-9707 paper

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Porous Media 54: 167–192

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Reservoir Eng. 12(3): 173–178

Appendix A

The dimensionless mass conservation equation for gas phase is

2

g rg rg rg

gv cv

D D D l D D D D

S Mk Mk Mk1 1 J J+ +N -N + =0

t x 1+M z 1+M R x x 1+M z z 1+M

(13)

where Ncv and Ngv are shown in Eq. 5 and 6, and Rl =L/H is the shape factor or called the aspect ratio. Since M, J, and krg all

depend on saturations, we can rewrite the dimensionless equation as

3 31 2

g g rg g rg g rg gcv

gv cv2

D g D g D D w D D w Dl

F FF F

S S Mk S Mk S Mk SNd 1 d dJ dJ+ +N + +N =0

t dS 1+M x dS 1+M z x dS 1+M x z dS 1+M zR

(A-1)

Or a more compact form

g g g g gcv

1 3 gv 2 cv 32

D D D D D D Dl

S S S S SN+F + F +N F +N F =0

t x x x z z zR

(A-2)

where F1, F2, and F3 are functions of saturations. At steady-state,

g

g D D

D

S=0 S SG x ,z

t

(A-3)

Assume Ncv and Ngv are independent of xD and zD, and then we can rewrite equations in terms of xD and zD separately:

14 SPE 153954

cv 3 cv 3

1 2 gv2

D 1 D D gv 2 Dl

N F N FSG SGF SG+ +F N SG+ =0

x F x z N F zR

(A-4)

Since xD and zD are independent, we assume that the dependence of the steady-state solution SG on xD and zD is separated,

that is:

D D D DSG x ,z =X(x )Z(z ) (A-5)

Substituting saturation back into the equation, and defining a=F1/F3, b=F2/F3, and Bond number

gv

B *

cv c

N ΔρgHN =

N p (14)

Then Eq. (A-4) becomes

2 gvcv1

2

D D D B Dl

F NNF d 1 dX d 1 1 dZX+ + Z+ =0

X dx a dx Z dz N b dzR

(A-6)

To find a solution that satisfies Eq. (A-6), then

2l

Dcv

B D

cvR a1 D2 - xNDl

D 1 D

-N bz

D 2 D2 D

B D

N dΧΧ+ =const c z

dxR aΧ=A z e +c z

1 dΖΖ=B x e +c xΖ+ =const c x

N b dz

(A-7)

Substituting Eq. (A-7) back to Eq. (A-5) yields

2 2l l

D B D Dcv cv B D

R a R a- x -N bz - x

N N -N bz

1 D D 2 D D 3 D D 4 D DSG=C x , z e +C x , z e +C x , z e +C x , z (A-8)

where C1, C2, C3 and C4 are functions of xD and zD.

Appendix B

From Eq. (A-1), F1, F2, and F3 are defined as

1 g 2

g g

d 1 1 dMF S = =-

dS 1+M dS1+M

(B-1)

rg rg rg

2 g 2

g g g

Mk k dkd dM MF S = =

dS 1+M dS 1+M dS1+M

(B-2)

rg rgw

3 g

w

Mk MkdJ SF S = =J'

dS 1+M 1+M (B-3)

Assuming the derivative of relative permeability is proportional to itself,

rgrw

rw w rw rg g rg

w w

dkdkk '= =a k and k '= =-a k

dS dS (B-4)

SPE 153954 15

Using different forms of relative permeability results in different aw and ag. In this paper, aw and ag are as follows:

gw

w g

w wr wr

nna = and a =

S -S 1-S (B-5)

Since the mobility ratio M= λw/λg = μg krw/μw krg, the derivative of M with respect to Sw is

g -1 -2

rw rg rw rg rg w g

w w

μdMM'= = k 'k -k k k ' = a +a M

dS μ (B-6)

Substituting Eq. (B-4) and (B-6) back into (B-1) and (B-2) yields

1 g w g2 2

g g

d 1 1 dM MF S = =- = a +a

dS 1+M dS1+M 1+M

(B-7)

rg g w rg

2 g

g

Mk Ma -a MkdF S = =

dS 1+M 1+M 1+M

(B-8)

Therefore, we can derive the two variables a and b based on the definitions in Appendix A:

w g1

3 rg

a +aF 1a= =

F J'k 1+M (B-9)

g w2

3

a M-aF 1b= =

F J' 1+M (B-10)

Since fCO2=1/(1+M), then a and b becomes

2

w g CO

rg

a +a fa

J' k

(B-11)

2

g w

CO

a M-ab f

J'

(B-12)

Therefore a and b are proportional to fCO2/krg and fCO2, respectively.