Effect of viscosity on oil recovery by spontaneous imbibition from partially water-covered matrix blocks

Abstract

Spontaneous imbibition is an important mechanism of oil recovery from fractured reservoirs and unconventional reservoirs. Oil is produced by combining co- and counter-current imbibition when the matrix blocks was partially covered by water. In this paper, we focused on the effect of viscosity ratios on oil production by spontaneous imbibition and established the numerical model for one-dimensional linear imbibition with TEO-OW boundary conditions, which was validated by the experimental data. The effect of viscosity ratio on co- and counter-current imbibition is investigated and scaling result of the imbibition recovery curve for wide range of viscosity ratio using the conventional scaling equation was tested, which indicates that the close correlation was achieved only when oil-water viscosity ratios are higher. Then, a modified scaling equation was developed based on the piston-like assumption for one-dimensional co-current imbibition and close correlation of imbibition recovery curves was achieved when viscosity ratios are lower. Finally, correlation of imbibition recovery curves was improved for wide range of viscosity ratios by combining conventional and modified scaling equation. Results show that since the shape of imbibition recovery curves is not similar for different viscosity ratios, it is difficult to obtain the perfect correlation using the constant viscosity term.

Introduction

Spontaneous imbibition plays an important role in oil recovery from fractured reservoirs (Morrow and Mason, 2001; Rangel-German and Kovscek, 2006) and unconventional reservoirs (Cheng et al., 2018; Jia et al., 2019; Roshan et al., 2016). Oil production by spontaneous imbibition can occur in co- and counter-current manners according to boundary conditions (Karpyn et al., 2009). When the matrix blocks are fully covered by water, oil can only be produced by counter-current imbibition and when the matrix blocks are partially covered by water, oil can be produced by combining co- and counter-current imbibition (Hatiboglu and Babadagli, 2010; Pooladi-Darvish and Firoozabadi, 2000). In the past few decades, many research group concentrated on the various aspects of spontaneous imbibition by experimental (Fernø et al., 2013; Unsal et al., 2009), analytical (Cai et al., 2014; Nooruddin and Blunt, 2016) and numerical methods (Cheng, 2012; Nasriani et al., 2018).

The counter-current imbibition attracted far more attention than co-current imbibition because experiments with the boundary condition of matrix block fully covered by water is easier to carry out (Mason and Morrow, 2013). In addition, the pattern of fluid flow is simpler and easier to be expressed by mathematical models (Mason et al., 2009). In contrast, the research on co-current imbibition is less even though oil recovery efficiency by co-current imbibition is generally higher than counter-current imbibition because it did not need to overcome the capillary back pressure (Meng et al., 2015; Unsal et al., 2007a, 2007b). Actually, the partially water-covered boundary condition is widely existed in fractured reservoirs during water injection and unconventional reservoirs after hydraulic fracturing. Understanding the mechanism of oil recovery by spontaneous imbibition with partially water-covered boundary conditions can make a contribution to predicting production performance in fractured (Chen and Mohanty, 2013; Cui et al., 2020; Zhong et al., 2020) and unconventional reservoirs (He et al., 2020; Jia et al., 2020; Wang et al., 2018).

The pattern of flow fluid is very complex for spontaneous imbibition with partially water-covered boundary conditions because oil can be by produced by combining co-and counter-current imbibition (Meng et al., 2019). In order to investigate the mechanism of imbibition with partially water-covered boundary conditions, researcher generally simplify the boundary conditions. The most widely used boundary condition is TEO-OW boundary conditions, i.e., cylindrical cores with all lateral faces sealed and one end face connected with water and the other face connected with oil (Haugen et al., 2014). The schematic illustration of fluid distribution and pressure profiles for spontaneous imbibition with TEO-OW boundary conditions is shown in Figure 1. In the early stage of imbibition, pressure in non-wetting phase at the imbibition front (p_nw,f) is higher than capillary back pressure, and oil is produced by co- and counter-current imbibition. With the imbibition front advancing, p_nw,f decreases. When p_nw,f is lower than capillary back pressure, oil production by counter-current imbibition ceased and oil can only be produced by counter-current imbibition (Haugen et al., 2014, 2015).

Figure 1.

Schematic illustration of fluid distribution and pressure profiles in the cores with TEO-OW boundary condition (p_cb is the capillary back pressure).

Scaling of imbibition data with numerous variables can provide a method for the prediction of oil recovery from reservoirs from laboratory imbibition tests (Morrow and Mason, 2001). The basic scaling rules for imbibition was proposed by (Rapoport, 1955) and it was extended by defining a dimensionless time with consideration of core size and shape, viscosity ratio and capillary pressure by Mattax and Kyte (1962). Based on the work of Mattax and Kyte, a most widely used scaling equation was defined by Ma et al. (1997) and Zhang et al. (1996) as follows $$t_{\text{D,MMZ}}=\frac{1}{L_{\text{C}}{\hspace{0.17em}}^2}\sqrt{\frac{K}{\varphi }}\frac{\sigma }{\sqrt{{\mu }_{\text{w}}{\mu }_{\text{nw}}}}t$$ (1)where t_D,MMZ is the dimensionless time, dimensionless; L_C is the characteristic length, m; K is absolute permeability, m²; $\varphi$ is porosity, dimensionless; σ is interfacial tension, mN/m; μ_w is wetting phase viscosity, mPa · s; μ_nw is the non-wetting phase viscosity, mPa·s. The most parameters in this scaling equation were obtained by theoretical methods and have been validated by experimental and numerical results. This scaling equation has been widely validated by experimental and numerical results (Fischer et al., 2008; Mason et al., 2010). However, the geometrical mean of viscosities to compensate for the difference in viscosity was obtained by empirical method (Zhang et al., 1996) and some experimental results showed that it may only be applicable for a limited range of viscosity ratios (Fischer and Morrow, 2006; Mason et al., 2010). In addition, the viscosity term was proposed from the imbibition with fully covered by water. Whether it is applicable for spontaneous imbibition with partially water-covered needs to be investigated.

In this work, we focused on spontaneous imbibition with TEO-OW boundary conditions. The numerical model for spontaneous imbibition with TEO-OW boundary conditions was established and the basic model was validated by experimental data. The effect of viscosity ratio on imbibition was mainly investigated. Dimensionless time was tested for scaling of imbibition data for a wide range of viscosity ratios and a modified dimensionless time was proposed. The scaling results of imbibition data was improved by combining the dimensionless time proposed by Zhang et al. (1996) and the modified dimensionless time.

Numerical model

Seepage equation

In this work, we focused on one-dimensional spontaneous imbibition under strongly water-wet conditions with TEO-OW boundary conditions. The effect of gravity and the compressibility of fluids and rocks are neglected. The two-phase seepage equation for spontaneous imbibition has been widely used (Li et al., 2003; Meng et al., 2017) $$\frac{\partial }{\partial x}(\frac{K{k}_{\text{rw}}}{{\mu }_{\text{w}}}\frac{\partial p_{\text{w}}}{\partial x})=\varphi A\frac{\partial S_{\text{w}}}{\partial t}$$ (2) $$\frac{\partial }{\partial x}(\frac{K{k}_{\text{rnw}}}{{\mu }_{\text{nw}}}\frac{\partial p_{\text{nw}}}{\partial x})=\varphi A\frac{\partial S_{\text{nw}}}{\partial t}$$ (3)where, k_rw is the relative permeability to wetting phase, m²; k_rnw is the relative permeability to non-wetting phase, m²; A is the area of the cross section, m²; μ_nw is the non-wetting phase viscosity, mPa·s; p_w is the wetting phase pressure, MPa; p_nw is the non-wetting phase pressure, MPa; S_w is the wetting phase saturation, dimensionless; S_nw is the non-wetting phase saturation, dimensionless.

Boundary condition

In the numerical model, the boundary conditions of water-covered-face could be described as (Pooladi-Darvish and Firoozabadi, 2000) $$p_{\text{w}}=0,t=0^{+},x=0$$ (4) $$p_{\text{nw}}=p_{\text{cb}},t=0^{+},x=0$$ (5)where p_cb is the capillary back pressure, MPa. In the oil-covered-face, the pressure in non-wetting is equal to zero and water could not be expelled. Therefore, the boundary conditions in the numerical model be expressed as (Meng et al., 2019) $$q_{\text{w}}=0,t=0^{+},x=L$$ (6) $$p_{\text{nw}}=0,t=0^{+},x=L$$ (7)

Relative permeability and capillary pressure functions

The relative permeability to oil and water can be expressed by modified Corey-functions (Meng et al., 2019; Pooladi-Darvish and Firoozabadi, 2000) $$k_{\text{rw}}=k_{\text{rwe}}{\left(\frac{S_{\text{w}}-S_{\text{wi}}}{1-S_{\text{or}}-S_{\text{wi}}}\right)}^{n_{\text{w}}}$$ (8) $$k_{\text{ro}}=k_{\text{roe}}{\left(\frac{1-S_{\text{or}}-S_{\text{w}}}{1-S_{\text{or}}-S_{\text{wi}}}\right)}^{n_o}$$ (9)

The capillary pressure is expressed as follows (Meng et al., 2019; Pooladi-Darvish and Firoozabadi, 2000) $$p_c=-B\ln \left(\frac{S_{\text{w}}-S_{\text{wi}}}{1-S_{\text{or}}-S_{\text{wi}}}\right)$$ (10)where B is the capillary pressure parameter (MPa), which could be expressed as follows (Brown, 1951) $$B=\sigma \sqrt{\varphi /K}$$ (11)

Validation of the numerical model

In this work, the numerical model was solved by IMPES method. The numerical results of the base case was validated against the experimental results (CHP2 Core) with TEO-OW boundary condition by Haugen et al. (2014). The value of parameters in the base case are listed in Table 1. The numerical calculations were carried with different grid blocks. Numerical results show that when the grid blocks exceed 50, the variation of numerical results is very small. Therefore, all numerical calculations were carried out with 50 grid blocks. The numerical results were used to compare with the experimental results and a close match was obtained (Figure 2). This indicated that the numerical model can be used to simulate spontaneous imbibition with TEO-OW boundary conditions.

Table 1.

Parameter values for the base numerical model.

Parameter	Value
L (core length), cm	16.6
A (cross-sectional area), cm2	11.35
K (permeability), μm2	4.5 × 10−3
f (porosity), dimensionless	0.46
Swi (initial water saturation), dimensionless	0.02
μw (water viscosity), mPa · s	1.09
μnw (oil viscosity), mPa · s	1.47
σ (interfacial tension), mN/m	31.5
krwe (end-point of the water relative permeability curve), m2	0.25
kroe (end-point of the oil relative permeability curve), m2	0.65
nw (Corey-exponent for the water relative permeability curve)	6
no (Corey-exponent for the oil relative permeability curve)	2

Figure 2.

Comparison of numerical results and experimental results.

Numerical results

In order to study the effect of viscosity ratio on oil recovery by spontaneous imbibition with TEO-OW boundary conditions, numerical calculations were carried out for a wide range of viscosity ratios and oil recovery from oil face and water face were calculated, respectively. It is shown that oil production was mainly by co-current imbibition. However, it is noticed that oil production by counter-current imbibition increases with increasing oil-water viscosity ratios (Figure 3), which coincides with the experimental results by Haugen et al. (2015). The plots of total oil recovery versus imbibition time are shown in Figure 4. It is noticed that the shape of oil recovery curves was not similar, which makes it very difficult for scaling of imbibition data.

Figure 3.

Oil recovery by co-current (from oil face) and counter-current imbibition (from water face) for a wide range of viscosity ratios.

Figure 4.

Total oil recovery for a wide range of viscosity ratios.

The correlation of imbibition recovery data by the use of the dimensionless time proposed by Zhang et al. (1996) is shown in Figure 5. The close correlation is observed when viscosity ratio is larger than 1. However, the dimensionless time exhibits an increasing trend when the viscosity ratio is lower than 1. Fischer et al. (2008) and Meng et al. (2017) reported the similar observation for pure counter-current imbibition. It is observed that the dimensionless time proposed by Zhang et al. (1996) may not be used to correlate wide range of viscosity ratio. Therefore, the scaling equation needs to be modified.

Figure 5.

Correlation of imbibition recovery data for various ratios using the dimensionless time proposed by Zhang et al. (1996).

Modification of viscosity term in scaling equation

A mathematical model for one dimensional co-current imbibition was developed by the assumption of piston-like displacement by Haugen et al. (2014) $${\left(\frac{x_{\text{f}}}{L}+D\right)}^2=D^2+Et$$ (12) $$D=\frac{1}{\left({\mu }_{\text{w}}/k_{\text{rw}}\right)\left(k_{\text{rnw}}/{\mu }_{\text{nw}}\right)-1}$$ (13) $$E=\frac{2K{p}_{\text{c,f}}}{\varphi L^2{S}_{\text{wf}}}\frac{1}{{\mu }_{\text{w}}/k_{\text{rw}}-{\mu }_{\text{nw}}/k_{\text{rnw}}}$$ (14)where x_f is the distance advanced by imbibition front, m; L is the length of core, m; p_c,f is the capillary pressure at the imbibition front MPa; S_wf is the average water saturation before the imbibition front, dimensionless.

It has been widely reported that saturation profiles are self-similar for spontaneous imbibition (Morrow and Mason, 2001; Ruth et al., 2007). The self-similarity displacement indicated that the average water saturation before the imbibition front is constant with the imbibition processing and the assumption of self-similarity has been validated by experiments as well (Dong et al., 1998; Li et al., 2006). According to assumption of the self-similarity, oil recovery by spontaneous imbibition could be expressed as follows $$R=\frac{x_{\text{f}}A\varphi \left(S_{\text{wf}}-S_{\text{wi}}\right)}{LA\varphi \left(1-S_{\text{wi}}\right)}=\frac{x_{\text{f}}\left(S_{\text{wf}}-S_{\text{wi}}\right)}{L\left(1-S_{\text{wi}}\right)}$$ (15)where R is oil recovery, dimensionless; S_wi is connate water saturation, dimensionless. In this paper, it is assumed that connate water is equal to zero, therefore $$R=\frac{x_{\text{f}}{S}_{\text{wf}}}{L}$$ (16)

Combining equations (12) and (16), yields $${\left(\frac{R}{S_{\text{wf}}}\right)}^2+2D\frac{R}{S_{\text{wf}}}=Et$$ (17)

Equation (17) can be written as $$t={\left(\frac{R}{S_{\text{wf}}}\right)}^2\cdot \frac{1}{E}+\frac{2D}{E}\frac{R}{S_{\text{wf}}}$$ (18)

Taking the derivative with respect to R of equation (18), yields $$\frac{dt}{dR}=\frac{2R}{E\cdot S_{{\hspace{0.17em}}_{\text{wf}}}^2}+\frac{2D}{E\cdot S_{\text{wf}}}$$ (19)

As can be seen from equation (19), dt/dR should be linearly related to R for one-dimensional linear co-current imbibition. For imbibition with TEO-OW boundary condition, oil is produced by one-dimensional linear co-current imbibition after the counter-current imbibition stopped. In the purely co-current imbibition stage for spontaneous imbibition with TEO-OW boundary condition, linear relationship may also be achieved between dt/dR and R. The relationship between dt/dR and R for TEO-OW imbibition with wide range of viscosity ratios is shown in Figure 6. It is shown that a linear relationship was achieved over a period of time. In Figure 6, only the linear period is provided. It is also shown that the linear period is reduced with increasing oil-water viscosity ratios because the purely co-current imbibition time decrease with increasing viscosity ratios.

Figure 6.

Plots of dt/dR versus R.

It is assumed that the slope of the straight line is k, then $$E=\frac{2}{k\cdot S_{{\hspace{0.17em}}_{\text{wf}}}^2}$$ (20)

Substituting equation (20) into equation (12), it can be obtained $${\left(\frac{x_{\text{f}}}{L}+D\right)}^2=D^2+\frac{2}{k\cdot S_{{\hspace{0.17em}}_{\text{wf}}}^2}t$$ (21)

Equation (21) can be written as $${\left(\frac{R}{S_{\text{wf}}}+D\right)}^2=D^2+\frac{2}{k\cdot S_{{\hspace{0.17em}}_{\text{wf}}}^2}F\left({\mu }_{\text{w}},{\mu }_{\text{o}}\right)\frac{t}{F\left({\mu }_{\text{w}},{\mu }_{\text{o}}\right)}$$ (22)where F (μ_w, μ_o) is the function of oil and water viscosity, and it is defined as $$k^{*}=\frac{2}{k\cdot S_{{\hspace{0.17em}}_{\text{wf}}}^2}F\left({\mu }_{\text{w}},{\mu }_{\text{o}}\right)$$ (23) $$t_{\text{D,NEW}}=t\sqrt{\frac{K}{\varphi }}\frac{\sigma }{F\left({\mu }_{\text{w}},{\mu }_{\text{o}}\right)}\frac{1}{L_C^2}$$ (24)where k* is scaling slope, dimensionless; t_D,NEW is the modified dimensionless time, dimensionless. In this paper, we focused on the viscosity term in the scaling equation. Therefore, the good scaling results mean that the scaling slope is close for wide range of viscosity ratios. If scaling slopes are equal for wide range of viscosity ratios, we can obtain $$k^{*}=\frac{2}{k\cdot S_{{\hspace{0.17em}}_{\text{wf}}}^2}F\left({\mu }_{\text{w}},{\mu }_{\text{o}}\right)=C$$ (25)where C is constant. Since good scaling results can be obtained by use of geometrical mean of the fluid viscosities when the viscosity ratios are equal (Meng et al., 2017), equation (25) can be written as $$k^{*}=\frac{2}{k\cdot S_{{}_{\text{wf}}\hspace{0.17em}}^2}\sqrt{{\mu }_{\text{w}}{\mu }_{\text{o}}}f\left(\frac{{\mu }_{\text{o}}}{{\mu }_{\text{w}}}\right)=C$$ (26) $$F\left({\mu }_{\text{w}},{\mu }_{\text{o}}\right)=\sqrt{{\mu }_{\text{w}}{\mu }_{\text{o}}}f\left(\frac{{\mu }_{\text{w}}}{{\mu }_{\text{o}}}\right)$$ (27)

Equation (26) can be written as $$f\left(\frac{{\mu }_{\text{o}}}{{\mu }_{\text{w}}}\right)=\frac{C\cdot k\cdot S_{{\hspace{0.17em}}_{\text{wf}}}^2}{2\sqrt{{\mu }_{\text{w}}{\mu }_{\text{o}}}}$$ (28)

The plots of $C·k{S}_{wf}/2\sqrt{{\mu }_w/{\mu }_o}$ versus μ_o/μ_w is shown in Figure 7. The relationship between $C·k{S}_{wf}/2\sqrt{{\mu }_w/{\mu }_o}$ and μ_o/μ_w can be expressed by the following function $$f\left(\frac{{\mu }_{\text{o}}}{{\mu }_{\text{w}}}\right)=a\cdot {\left({\mu }_{\text{o}}/{\mu }_{\text{w}}\right)}^b$$ (29)where a and b is constant and can be obtained by regression analysis. The regression result is shown in Figure 7 and the regression function gives $$f\left(\frac{{\mu }_{\text{o}}}{{\mu }_{\text{w}}}\right)=0.924{\left({\mu }_{\text{o}}/{\mu }_{\text{w}}\right)}^{-0.483}$$ (30)

Figure 7.

Plots of $C·k{S}_{wf}/2\sqrt{{\mu }_w/{\mu }_o}$ versus viscosity ratio μ_o/μ_w (C = 0.001).

F (μ_o/μ_w) can be obtained by substituting equation (30) into equation (27). Then substituting F (μ_o/μ_w) into equation (24), yields $$t_{\text{D,NEW}}=t\sqrt{\frac{K}{\varphi }}\frac{\sigma }{0.924\sqrt{{\mu }_{\text{w}}{\mu }_{\text{o}}}{\left({\mu }_{\text{o}}/{\mu }_{\text{w}}\right)}^{-0.482}}\frac{1}{L_C^2}$$ (31)

Equation (31) is the modified dimensionless time based on the one-dimensional co-current imbibition. Oil recovery curves versus the modified dimensionless time is shown in Figure 8. It can be seen that the scaling results are not satisfying. In contrast to the dimensionless time proposed by Zhang et al. (1996), the correlation is close wen viscosity ratios are smaller. But when viscosity ratio is larger, the modified dimensionless time increases with the increasing viscosity ratio. This is because oil production is not by purely co-current imbibition for TEO-OW boundary condition. When viscosity ratio is lower, period of counter-current imbibition is very short and oil is mainly produced by co-current imbibition. However, when viscosity ratio is higher, period of counter-current imbibition is very long and even lasting the entire imbibition process. The modified dimensionless time is obtained by assuming purely co-current imbibition. Therefore, it may be not applicable for spontaneous imbibition with high viscosity ratios.

Figure 8.

Plots of oil recovery versus modified dimensionless time for various viscosity ratios.

It can be seen that close correlation can be obtained using the dimensionless time proposed by Zhang et al. (1996) when viscosity ratio is higher and using modified dimensionless time when viscosity ratio is lower. Therefore, the close correlation may be obtained by combining this two dimensionless time. In this work, the dimensionless time proposed by Zhang et al. (1996) was used when viscosity ratio is higher than 0.5 and modified dimensionless time was used when viscosity ratio is lower than 0.5. The correlation of imbibition recovery curves using this two dimensionless time was shown in Figure 9. It is shown the correlation is improved. However, since the shape of imbibition recovery curves are not similar for imbibition with TEO-OW boundary conditions, it is very difficult to obtain a fully coincident correlation.

Figure 9.

Plots of oil recovery versus dimensionless time for various viscosity ratios.

Conclusions

In this work, the numerical model for spontaneous imbibition with TEO-OW boundary condition is established and oil production by co- and counter-current imbibition is investigated for wide range of viscosity ratios. In addition, the correlation of imbibition recovery curves was studied by using the dimensionless time proposed by Zhang et al. (1996) and the novel scaling methods were proposed. The following conclusions can be obtained:

For spontaneous imbibition with partially water-covered boundary condition, the majority of oil is produced by co-current imbibition. However, the period of counter-current imbibition and the amount of oil produced by counter-current imbibition increase with the increasing oil-water viscosity ratios.

The shape of imbibition recovery curves is not similar for the different viscosity ratio, which makes it difficult to scale the imbibition curves. For purely linear co-current imbibition, a linear relationship cab be achieved between dt/dR and R.

Close correlation of imbibition recovery curves can be obtained only when oil-water viscosity ratio is high using the dimensionless time proposed by Zhang et al. (1996). A modified dimensionless time was proposed by assuming purely linear co-current imbibition and close correlation of imbibition curves was achieved when oil-water viscosity ratio is lower using the modified dimensionless time. The correlation of imbibition curves was improved by combining the dimensionless time proposed by Zhang et al. (1996) and the modified dimensionless time.