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Simulation of cocurrent and countercurrent imbibition in water-wet fractured porous media.

INTRODUCTION

Oil recovery from fractured reservoirs poses several challenges compared to oil production from conventional reservoirs. Water injection is one of the most popularly known methods for recovering oil from fractured reservoirs. The process of oil recovery by water injection is highly influenced by spontaneous capillary imbibition in water-wet reservoirs. Reis and Cil (1993) maintain that spontaneous capillary imbibition has long been recognized as an important oil recovery mechanism in water-wet naturally fractured (heterogeneous) reservoirs. Spontaneous imbibition of water is a direct function of the capillary and gravity forces, and depends on the pore system, the wettability (Zhou et al., 2000), matrix block sizes and shape (Zhang et al., 1996), the interfacial tensions (Karimaie and Torsaeter, 2007) and initial water saturation (Viksund et al., 1998). Matrix capillary pressure is important in fractured reservoirs since it controls the rate of water uptake from the fracture network into the matrix and the efficiency of oil displacement by spontaneous imbibition.

In the reservoir, spontaneous imbibition can take place both in cocurrent and countercurrent flowing modes (Bourbiaux and Kalaydjian, 1988; Zhou et al., 2001). The study of spontaneous water imbibition is essential in the prediction of production performance in these reservoirs where the amount and rate of mass transfer between the matrix and the fracture influence the recovery and the production rate. The purpose of this paper, therefore, was to investigate the cocurrent and countercurrent water injection into fractured porous media. ECLIPSE-100 simulator was used to simulate oil recoveries from the two processes of spontaneous imbibition mechanisms.

MATERIALS AND METHODS

The reservoir description/model used in this study is based on Kazemi and Merrill (1979). The grid model which comprised of two matrix blocks with ten grid blocks surrounded by fractures is presented in Figure 1.

[FIGURE 1 OMITTED]

Water is injected at the bottom of the matrix block while oil is produced at the top of the matrix block. The relative permeability ([k.sub.r]) of the different phases in the matrix and fracture are evaluated using the following relationships:

Relative permeability in the matrix is evaluated from the equations:

[K.sub.rw] = [S.sub.w.sup.3]. (1)

[K.sub.ro] = ([S.sub.o] - [S.sub.or]. (2)

While the relative permeability in the fracture is evaluated using the following equations:

[K.sub.rw] = [S.sub.w]. (3)

[K.sub.ro] = 1 - [S.sub.w]. (4)

The relative permeability of the different phases in the fracture is assumed as a linear function of saturation with no irreducible or residual saturation and the capillary pressure in the fracture is assumed to be zero. These results are presented in Table 1.

A linear capillary pressure curve with end point of 1.30psi and 0.90psi at water saturation of [S.sub.w] = 0.43 and [S.sub.w] = 0.66 respectively is applied on the matrix block in this study. The reservoir model basic data extracted from Kazemi and Merrill (1979) as well as assumed parameter values are presented in Table 2.

In cocurrent imbibition, the injected water and the displaced oil flow through different faces in the same direction. Thus, the rate of injected water ([q.sub.w]) plus the rate of produced oil ([q.sub.o]) equal the total flow rate ([q.sub.t]), and is given as:

[q.sub.w] + [q.sub.o] = [q.sub.t]. (5)

The analytical equation for the cocurrent imbibition can be derived from Darcy's equation and capillary pressure, and this is expressed as:

[partial derivation]/[partial derivation]x (D([S.sub.w]) [partial derivation][S.sub.w]/[partial derivation]x - [q.sub.t]f([S.sub.w]) = [partial derivation][S.sub.w]/[partial derivation]t. (6)

Where:

D([S.sub.w]) and f([S.sub.w]) are expanded in equations 9 and 10 respectively.

In countercurrent flow, the water and oil flow through the same face in opposite direction. In other words, the rate of water injected ([q.sub.w]) equals the rate of oil produced ([q.sub.o]) which is expressed mathematically as:

[q.sub.w] = [q.sub.o]. (7)

An analytical equation for countercurrent imbibition can be derived using Darcy's equation, capillary pressure relationship between water and oil, as well as expression of the capillary pressure as a function of saturation (water). Thus, a one- dimensional (1D) countercurrent imbibition process can be described by a non-linear diffusion equation of the form:

[partial derivation]/[partial derivation]x (D([S.subw])[partial derivation][S.sub.w]/[partial derivation]x) = [partial derivation][S.sub.w]/[partial derivation]t. (8)

Where:

D([S.sub.w]) = K/[phi] [K.sub.ro]/[[mu].sub.0] f([S.sub.w]) d[P.sub.c]/d[S.sub.w]. (9)

f([S.sub.w]) = 1 / 1 + [K.sub.ro][[mu].sub.w]/[K.sub.rw][[mu].sub.o]. (10)

As earlier alluded to, the reservoir was modeled based on the description obtained from the work of Kazemi and Merrill (1979), and these reservoir parameters were incorporated in ECLIPSE input files in order to carry out the simulations for this research work. It should be noted that some input parameters such as rock compressibility, reservoir pressure, oil formation volume factor ([B.sub.o]), water formation volume factor ([B.sub.w]), etc. were assumed in order to complete the ECLIPSE input file.

To study the effect of cocurrent and countercurrent imbibition on the reservoir model, the rate of water injection was varied. Three simulation runs for each mechanism, for a total of six scenarios were performed and these scenarios are presented in Tables 3.

RESULTS AND DISCUSSION

Figures 2 through 4 along with Table 4 present the results obtained from the cocurrent flow in the model. Figure 2 shows the oil recovery (FOE) at different injection rates. A comparison of the base-case and the cocurrent runs shows that the rate of oil recovery decreased as the injection rate was reduced. As depicted in Table 4, the recovery rate of the base-case was higher (0.00072 [Sm.sup.3]/day) when compared to RUNS1, 2 and 3 at 0.00049 [Sm.sup.3]/day, 0.00032 [Sm.sup.3]/day, and 0.00024 [Sm.sup.3]/day respectively. This is as a result of the imbibition rate decreasing as the injection rate was reduced. Hence, water is imbibed very slowly by the matrix as the injection rate decreased, resulting in a decrease in the rate of recovery. However, the ultimate recovery remained the same in all cases as this is generally governed by the residual oil saturation ([S.sub.or]) in the strongly water-wet systems (Behbahani et al., 2006). It is observable in the result (Figure 2) that the recovery rate in all cases varied between 0-300days but later on had about the same recovery (ultimate recovery) as the injection continued.

[FIGURE 2 OMITTED]

Figure 3 presents the oil production rate (OPR) from the cocurrent flow simulation runs. A comparison of the results depicts that early breakthrough occurred at high injection rate. Thus, the plot indicates that RUN3 had a delayed breakthrough compared to RUN1, RUN2 and the base-case. Mannon and Chilingar (1972) maintain that the rate of imbibition is proportional to the rate of injection. This explains the early breakthrough observed in the base-case as well as in RUN1 of this simulation study.

[FIGURE 3 OMITTED]

As earlier alluded to in this paper, it is generally assumed that the rate of water injected equals the rate of oil produced. Therefore, if water is imbibed rapidly as the injection rate increases, invariably oil will be produced proportional to the imbibition rate. Thus, this will result in rapid recovery rate as well as early breakthrough as the imbibed water displaces oil and increases rapidly in the matrix. Conversely, in the case of RUN2 and RUN3, the injected rate was very low such that the imbibed water level in the matrix increased slowly, resulting in a delayed breakthrough as depicted in Table 4.

Figure 4 shows the oil in place (OIP) in the matrix. It was observed that as water imbibed into the matrix, the volume of oil in the matrix reduced. Additionally, it was observed that the rate of the declination in oil volume depended on the injection rate. A comparison of the base-case with other cocurrent runs showed that the base- case scenario had a low declination rate when compared to other cases. This is as a result of the rate of recovery from the matrix which depended on the injection rate. However, all the cases had about the same OIP after 300days as observed in the FOE which depicted recovery from the matrix block.

[FIGURE 4 OMITTED]

Figures 5 through 7 along with Table 5 present the results obtained from the countercurrent flow in the simulation model. Figure 5 depicts the recovery from countercurrent flow at different injection rates. A comparison of the base-case with the various countercurrent runs showed that, while there was a slight difference in the rate of recovery among the different scenarios, the difference was not significant.

[FIGURE 5 OMITTED]

As depicted in Table 5, the recovery rate of the base-case was 0.00072 [Sm.sup.3]/day when compared to RUNS1, 2 and 3 at 0.00105 [Sm.sup.3]/day, 0.00131 [Sm.sup.3]/day, and 0.00144 [Sm.sup.3]/day respectively. This is as a result of the matrix block height and the injection rate. Since water was injected at a high rate, the water flowed through the fractures and surrounded the matrix block. As a result of capillary difference in the matrix-fracture, water imbibed into to the matrix to displace the oil from the matrix. The imbibition process in these cases did not depend on the injection rate since the matrix was surrounded by water. Thus, resulting in about the same rate of recovery as observed in the countercurrent flow in the model.

Figure 6 shows the oil production rate (OPR) from the countercurrent flow. The result depicts that early breakthrough occurred in all cases. This observation is as a result of the fact that water was injected at a very high rate. Thus, the injected water flowed through the fracture network and some of the water imbibed in the matrix to displace oil while others were produced through the fracture (breakthrough).

[FIGURE 6 OMITTED]

Figure 7 depicts the oil in place (OIP) in the matrix block. A comparison of the base-case with the various countercurrent runs showed that, while there was a slight difference in the rate of declination of the oil volume in the matrix among the different scenarios, the difference was not significant. The reason that can be advanced for this observation is similar to an earlier discussion of Figure 5 in this paper. From the foregoing discussion, it is evidenced that countercurrent imbibition can be an important mechanism by which oil is displaced from the rock matrix. Kazemi et al. (1992) maintain that the function that relates the fluid exchange between the rock matrix and fractures is an important variable in modeling displacement in fractured reservoirs.

[FIGURE 7 OMITTED]

CONCLUSION

Water injection is an efficient recovery method in water-wet fractured porous media. In fractured porous media, the imbibition recovery mechanisms - cocurrent and countercurrent flow modes, which depend on the injection rate, can contribute to the recovery. Initial water saturation, wettability and interfacial tension (IFT) have a significant role in the recovery from fractured reservoirs. Based on the results of the simulation study of cocurrent and countercurrent imbibition in water-wet fractured porous media, the following conclusions are drawn:

* In cocurrent imbibition, the recovery rate decreased as the injection rate decreased.

* In countercurrent imbibition, there was no significant difference in the rate of recovery as the injection rate increased.

* The rate of recovery in countercurrent imbibition was higher than that of cocurrent imbibition.

* The ultimate oil recovery in both flow modes was the same in water-wet porous media as a result of the residual oil saturation in the matrix.

* Early breakthrough was experienced in countercurrent imbibition when compared to cocurrent imbibition.

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Dr. Francis D. Udoh & Anietie N. Okon

Department of Chemical & Petroleum Engineering University of Uyo, Uyo - Akwa Ibom State, Nigeria fdudoh@yahoo. com
Table 1: Fluid Saturation, Relative Permeability and Capillary
Pressure Values

Saturation Relative Permeability Capillary
 Pressure
[S.sub.w] [S.sub.o] [K.sub.rw] [K.sub.ro] [P.sub.c]
 (Bar)

0.43 0.34 0.000 0.000 0.08963
0.45 0.37 0.091 0.000 0.08722
0.48 0.40 0.111 0.000216 0.08384
0.50 0.45 0.125 0.001331 0.08122
0.53 0.47 0.149 0.002197 0.07764
0.55 0.50 0.166 0.004096 0.07522
0.60 0.52 0.216 0.005832 0.06922
0.63 0.55 0.250 0.009261 0.06564
0.66 0.57 0.287 0.012167 0.06205

Table 2: Reservoir Model Data

Property Value

Matrix Porosity 0.176
 ([[phi].sub.m])
Matrix Permeability 56mD
 ([k.sub.m])
Matrix Block 0.0508m
Fracture Width 0.003048m
Fractured Porosity 1.0
 ([[phi].sub.f])
Fractured Permeability 21000mD
 ([k.sub.f])
Oil Viscosity 4.60cP
 ([[mu].sub.o])
Water Viscosity 1.0cP
 ([[mu].sub.w])
Oil Density 820kg/[m.sup.3]
 ([[rho].sub.o])
Water Density 1000kg/[m.sup.3]
 ([[rho].sub.w])
Rock Compressibility 4.35 x [10.sup.-5] [Pa.sup.-1]
 ([c.sub.r])
Water Compressibility 5.29 x [10.sup.-5] [Pa.sup.-1]
 ([c.sub.w])
Reference Pressure 276 Bar
 ([P.sub.ref])
Initial Water Saturation 0.43
 ([S.sub.wi])
Oil Formation Volume Factor 1.52
 ([B.sub.o])
Water Formation Volume Factor 1.0
 ([B.sub.w])

Table 3: Injection Rates

Simulation Cocurrent Imbibition Countercurrent Imbibition
Runs Rate ([m.sup.3]/s) Rate ([m.sup.3]/s)

RUN1 0.000725 0.00290
RUN2 0.000483 0.00435
RUN3 0.0003625 0.00580

Table 4: Breakthrough and Oil Production Rate (cocurrent)

Simulation Run Breakthrough Oil Production Rate
Base-case (days) ([Sm.sup.3]/day)

 2.20 0.00072
RUN1 6.60 0.00049
RUN2 13.19 0.00032
RUN3 21.42 0.00024

Table 5: Breakthrough and Oil Production rate (countercurrent)

Simulation Run Breakthrough Oil Production Rate
 (days) ([Sm.sup.3]/day)

Base-case 2.20 0.00072
RUN1 1.65 0.00105
RUN2 1.10 0.00131
RUN3 1.10 0.00144
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Author:Udoh, Francis D.; Okon, Anietie N.
Publication:International Journal of Petroleum Science and Technology
Date:Sep 1, 2012
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