Investigation of Pressure Drop of Trapped Oil in Capillaries with

Sep 11, 2018 - Institute of Mud Logging Technology and Engineering, Yangtze University , Jingzhou , Hubei 434000 , P.R. China. ‡ School of Petroleum...
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Cite This: Ind. Eng. Chem. Res. 2018, 57, 13866−13875

Investigation of Pressure Drop of Trapped Oil in Capillaries with Circular Cross-Sections Long Long,*,†,‡ Yajun Li,‡ Houjian Gong,‡ Long Xu,‡ Qian Sang,‡ and Mingzhe Dong§ †

Institute of Mud Logging Technology and Engineering, Yangtze University, Jingzhou, Hubei 434000, P.R. China School of Petroleum Engineering, China University of Petroleum, Qingdao, Shandong 266580, P.R. China § Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4 ‡

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S Supporting Information *

ABSTRACT: An investigation is performed to determine the cause of the pressure drop for mobilizing a trapped oil slug in capillaries with circular cross-sections. This research is focused on pressure drop across the tube at different stages of oil mobilization and the physics behind the rapid decrease during slug separation. Pressure drop decease in the hold-up stage, which is too small to be observed experimentally, and the severe change of pressure drop during the slug detachment process are observed and quantitatively analyzed using computational fluid dynamics. The method used for the analysis is the equivalent viscosity model. For different slug lengths and multislug cases, the role of the capillary pressure difference in the process of trapped oil migration is well-recognized. The influence of oil dispersion in porous media on pressure drop when saturation is constant is also discussed. Compared with multislug pressure drops, the relative error of the equivalent viscosity model and computational fluid dynamics is less than 2.33%, which proves the accuracy of the equivalent viscosity model.

1. INTRODUCTION In recent years, with the development of modern society, the demand for oil has rapidly grown. In oil bearing formation, a type of porous media, the fluid flow channel can be considered an ensemble of microscopic pores. Along with the complex structure of the porous media and the rough walls of pores and throats, the study of the two-phase displacement mechanism in capillary systems is extremely difficult. The pores are usually reduced to capillaries to discuss the fluid dynamics phenomena in them.1 In contrast to pipe flow, the effect of gravity on the flow in pore scale tubes is negligible, while surface tension and capillary pressure fundamentally influence flow behavior. For the interaction between fluids, the velocity and pressure distribution in two-phase flow in capillaries are entirely different from in single-phase flow. As the most prevalent pattern of multiphase flow inside tubes, slug flow has received considerable research interest because of the great developmental potential and numerous industrial applications, especially in enhanced oil recovery. When an oil slug in a capillary is driven from one end to the other, in the form of a round-ended column, the viscous liquid (water represents the viscous liquid in this study) originally occupying the capillary is forced out with a low Reynolds number and leaves a layer between the inner wall and slug. Based on experimental observations of the movement of a long bubble in a tube, Fairbrother and Stubbs2 concluded that the bubble moved faster than the average velocity of the liquid and that the velocity of the bubble and the mean velocity of the liquid can be expressed as vb and vm, respectively. They also developed a fraction W for the relative velocity between the bubble and the © 2018 American Chemical Society

liquid, W =

v b − vm . vb

When the slug moves as a circular cylinder

with round ends surrounded by a liquid film of thickness h, which is far less than the capillary radius r, a simple equation can be used to calculate the thickness of the layer of liquid near the 2h wall, W = r . Using this equation, scholars have obtained the relationship between W and Ca through experiments. For capillary numbers between 10−3 and 10−2, W is a function of the μv capillary number, Ca = σ b , where μ is the viscosity of the displaced fluid and σ is the liquid surface tension. The relationship can be expressed as W = Ca1/2

(1)

This result was extended by Taylor3 to much higher velocity, with Ca values between 0.015 and 2. For larger capillary numbers, W approaches a limiting value of approximately 0.56. Many experimental studies have been reported by other researchers on slug flow in a capillary.4−6 Based on the solution of Fairbrother and Stubbs,2 the thickness of the film can be calculated using the capillary number. Marchessault and Mason7 and Bretherton8 derived thicknesses for different ranges of capillary numbers and developed empirical correlations simultaneously. Received: Revised: Accepted: Published: 13866

July 13, 2018 August 30, 2018 September 11, 2018 September 11, 2018 DOI: 10.1021/acs.iecr.8b03189 Ind. Eng. Chem. Res. 2018, 57, 13866−13875

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Industrial & Engineering Chemistry Research

volume of fluid method in a commercial computational fluid dynamics (CFD) package, FLUENT, to simulate two-phase flow in capillaries. With the detailed velocity field around the bubble, they validated Taylor’s result for reversal flow patterns. Their work provides insight into the hydrodynamics of slug flow inside small-scale capillaries. FLUENT is selected to simulate trapped slug behavior inside tubes in this research, as used by Taha and Cui, although they did not discuss the detachment process of a trapped slug. Kreutzer et al.26 investigated the pressure drop of a train of bubbles in capillary channels by using a finite element method, concluding that more bubbles per unit channel length will results in a higher pressure drop. Fuerstman et al.27 found that the pressure drop along the tube also depends on the number of bubbles and the viscosity of the liquid in the channel. However, these studies did not investigate the development of the wetting phase film during the slug mobilization process. In the microscopic flow simulation study of porous media (such as the network model),28−30 the pores are usually simplified as circular, triangular and rectangular. In this research, numerical simulations are carried out to study trapped slug flow behavior and pressure drop patterns in a water filled circular capillary, while the purpose of this study is to provide a theoretical basis for future flow studies in noncircular capillary tubes. The pressure drop across the slug for different displacement processes and velocity distributions in a tube are discussed in detail using CFD software. The trend in pressure drop curves are in good agreement with Dong’s experiment literature.10 Furthermore, the reason for extreme changes in pressure drop during the slug detachment process are explained for the first time using our simulation results. An equivalent viscosity model is introduced for pressure drop at different displacement stages with varying lengths and slug numbers. The multiplied capillary pressure difference in multi-oil slug flow is verified to be the primary factor for pressure drop increase. Compared with the simulation results, the accuracy of the method is also verified.

Most of the early research focused on a slug in motion, finding that film thickness h increases with increasing slug velocity vb. For oil slugs, the wetting film thickness decreases with decreasing slug velocity and approaches a constant value of approximately 0.7 μm when the slug is stationary. Chen5 measured this result and indicated that the scenario in actual rocks is more complicated since the distribution of the wetting liquid depends on the pore roughness. Slug flow in the porous media of the petroleum industry has smaller scale channels and much lower capillary numbers,9 and the mechanism for mobilizing a trapped oil slug under low capillary number is of great importance for oil recovery. Dong et al.10 experimentally investigated the mobilization of trapped oil slugs in a capillary tube filled with water under low displacement velocity. Using a highly sensitive pressure transducer, they measured the pressure drop across the tube and revealed that the pressure drop for the motion of the trapped oil slug can be divided into three stages: the build-up stage, hold-up stage, and steady flow stage. Based on experimental observations, the stages of pressure drop correspond to different oil slug shapes. In the build-up stage, with the injection in the wetting phase, the pressure drop increases gradually, and the ends of the slug begin to deform where the convex rear end flattens and the front end is raised, yet the slug body does not move forward. When the differential pressure increases to a specific value, the contact line of the rear end and the inner wall begins to move, and the slug body also moves forward with the flow. This is the beginning of the hold-up stage. With a more convex meniscus at the front end, the contact line of the front end and the inner wall remains in place in this stage, and a wetting liquid film appears around the front end. As the slug continues to move forward, the slug body detaches from the inner wall when the rear contact line touches the front contact line. Then, the slug begins to form a capsule shape and flows with a film around it. Meanwhile, the pressure drop decreases rapidly to a lower value. As a type of retention, the wetting liquid film is determined by the capillary forces and a thickness that approaches a constant value when the capillary number is less than 10−4.11 Contrasting air bubble flow, there is an internal flow in the oil slug, and the influence of the interaction between the immiscible liquids on the flow cannot be neglected.12,13 Consequently, the different pressure drop stages reflect different oil slug movement patterns. The seepage curve is the relationship curve between seepage velocity and pressure difference in porous media.14 The curve is usually a straight line, which can be used to obtain the permeability of the porous medium. Additionally, permeability is an important parameter for an oil reservoir characterization and engineering. From the macroscopic level, understanding the migration mechanism of an oil slug in a capillary fill with water is of theoretical significance for studying multiphase flow in low permeability reservoirs. In development of low-permeability oil reservoirs, characteristics of non-Darcy flow and the initial pressure gradient greatly affect the recovery of residual oil.15−17 Experimental results show that the seepage curve is not a straight line and the velocity is zero until a specific pressure value is reached.18 Based on previous studies, the velocity hysteresis is assumed to be a macroscopic representation of mobilizing trapped oil slugs in pores. Other researchers have studied slug flow in tubes using numerical methods.19−21 Using the finite difference method to discuss the flow of liquid drops in a circular capillary, Hyman and Skalak22 concluded that the pressure drop across the slug decreases as slug velocity increases. Taha and Cui23−25 used the

2. ESTABLISHMENT AND ALGORITHM OF THE CFD MODEL 2.1. Governing Equations. The CFD software FLUENT 6.3 is used to simulate slug flow in a horizontal capillary. The volume of fluid method (VOF) is a commonly used technique for tracking the interface between fluids in the numerical simulation of multiphase fluid flow problems. The movement of the interface is tracked based on the distribution of α, the volume fraction of the liquid filled capillaries, which is defined as (1) α = 1 when the control volume is filled with phase 1; (2) α = 0 when the control volume is filled with phase 2; and (3) 0 < α < 1 when the control volume is on the interface. The governing equations are given as Continuity equation ∂ρ + ∇·(ρu) = 0 ∂t

(2)

Momentum equation ∂ (ρu) + u·∇u = ∇p + μ∇2 u + f ∂t

(3)

Volume fraction equation ρ = αρ1 + (1 − α)ρ2 13867

(4) DOI: 10.1021/acs.iecr.8b03189 Ind. Eng. Chem. Res. 2018, 57, 13866−13875

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Industrial & Engineering Chemistry Research μ=

layers was 10. According to previous studies,2,3,7,8 the width of the mesh is sufficient to simulate the formation of water film around the oil slug. In the initial condition of the simulation, there was a 4-mm cylindrical oil slug with flat ends in the capillary, and the slugside face overlapped with the inner wall. The distance between the slug rear end and the inlet was 4 mm. With a fixed contact angle, the simulation continues until a steady oil slug is established. Figure 1 shows an oil slug at steady conditions with a 0.5° contact angle. To the ensure that the interface forms and the slug reaches the steady condition, the inlet velocity was set to two-stage with a UDF function: in the first stage, the entire capillary system was in the initial stage, and the velocity remained zero; when the steady condition was reached, the velocity was increased to 0.01 m/s31,32 in the second stage, which is the mobilized stage. At this flow rate, the capillary number is 2.5 × 10−4 and the Reynolds number is 9.98, and the flow in the tube is laminar. At this time, the capillary force plays a major role compared to the viscous force. Otherwise, the pressure-outlet was set to zero; thus, the pressure at the inlet could represent a capillary pressure drop.

αρ1μ1 + (1 − α)ρ2 μ2 αρ1 + (1 − α)ρ2

(5)

where ρ, ρ1, and ρ2 are the average density, density of phase 1, and density of phase 2, respectively. In this study, phase 1 is the fluid filled tubes and phase 2 is the fluid in the slug. μ, μ1, and μ2 are the average viscosity, viscosity of phase 1, and viscosity of phase 2, respectively. p is pressure, u is velocity, and f is the body force per unit volume. 2.2. Modeling of Slug Flow. Proper establishment of the slug model is important for obtaining good numerical simulation results. The microchannels where the oil slug flows can be considered perfect cylindrical capillaries, whose radius at each cross-section is equal. To simplify calculations, the axial-symmetric property is assumed. Therefore, a three-dimensional model can be substituted by half of a two-dimensional model. The configuration of the slug in a tube model is shown in Figure 1.

3. RESULTS AND DISCUSSION 3.1. Trapped Oil Slug Transport and Pressure Drop. The simulation results demonstrate that the pressure drop profile is in agreement with the three-stage theory10 and that different stages correspond to different oil slug shapes. Figure 2

Figure 1. Modeling oil slug flow in a capillary tube.

At the beginning of the simulation, an oil slug is placed in a water-filled tube, whose length is 20 mm, with an inner diameter of 1 mm. The oil drops used in this paper are derived from kerosene in the Fluent Database Material, while the viscosity was revised for different oil cases. Boundary conditions of velocityinlet and pressure-outlet were used for water injection at a constant flow rate. Under the premise of two liquids, the no-slip wall condition was applied to the inner walls. This oil slug flow simulation includes the following assumptions: (1) The tube is long enough, and the position of the slug ensures that the influence of entrance effects on the slug can be neglected. (2) The ratio of gravity to capillary forces is very small; therefore, the effect of gravity on the fluid distribution is negligible. (3) With two incompressible Newtonian liquids and low injection velocity, the flow in the capillary can be assumed to be laminar. (4) The influence of menisci appearance on the length of the slug is negligible. 2.3. Fluid Properties. The fluids employed for oil and water phases in this research were based on the Fluent Database Material. In the simulation, the interfacial tension and contact angle between oil and water were set as 0.04 N/m and 0.5°, respectively. The density and viscosity of these two fluids are shown in Table 1.

Figure 2. Pressure drop vs time for flow rate is 0.01 m/s.

is the pressure drop across the capillary during the trapped oil slug mobilization. When the slug starts to mobilize at a constant rate injection, similar to the experiment result, the pressure profile can be divided into the following stages: the build-up stage, hold-up stage, and steady flow stage. Figure 3 shows different oil slug profiles for four different stages. In the initial system, the inlet injection and the tube pressure drop are zero. Figure 3a shows the steady slug shape at the end of the initial stage. Due to the interfacial tension, the interfaces between oil and water (the red part is oil and the blue part is water) become arcs of circles. Moreover, the angles between the inner wall and the tangent lines at the intersection points of the interfaces and inner wall fit well with the set contact angle. In the mobilized stage, Figure 3b−d shows the oil slug shape in the build-up stage, hold-up stage, and steady flow stage. The simulation results indicate that the shape of the oil droplets changes accordingly when the trend of the pressure drop and

Table 1. Flow Parameters of Oil and Water fluid

density (kg·m−3)

viscosity (Pa·s)

water oil

998 780

0.001 0.1

2.4. Model Geometry. In a two-dimensional coordinate system, an axial-symmetric capillary was used in this paper. To control the calculation precision, a grid was used to simulate slug flow consisting of a 100 × 2000 quadrilateral control volume. To improve the resolution of the water film, which was initially built during slug deformation, the grid was refined near the wall. Using Boundary Layer tools, the widths of the first-row volumes were 1 μm, with a growth factor of 1.1, while the number of 13868

DOI: 10.1021/acs.iecr.8b03189 Ind. Eng. Chem. Res. 2018, 57, 13866−13875

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Industrial & Engineering Chemistry Research

Figure 3. Deformation profile of oil droplets.

time curve change. In Figure 3b, at the initial displacement, both ends of the slug deform: the convex rear end meniscus gradually flattened and became concave under certain conditions; meanwhile, the convex front end meniscus was raised and moved forward. Accordingly, in this stage, the pressure drop across the tube increased with the development of menisci deformations. When the pressure drop reached a maximum at the end of the stage, the slug flow entered the hold-up stage, and the pressure began to change. Different from the pressure stability results in Dong’s conclusion,10 the pressure drop is essentially linearly decreasing over time in this stage, which was also observed by other scholars.31,32 In the hold-up stage, the contact line of the rear end and inner wall gradually moved toward the flow direction; conversely, the contact line of the front end and inner wall remained motionless. However, as the front end meniscus move forward, with an approximately hemispherical profile, an annular water film appeared between the front end and the tube wall, as shown in Figure 3c. Along with the development of this stage, the water film would grow longer, while the surface of the slug in contact with the inner wall continued to decrease. This stage lasted until the rear end traveled an entire slug length. At the end of the hold-up stage, two contact lines eventually make contact. This means the oil droplet is completely separated from the inner wall and is enclosed by the water film. Then, the system enters the steady flow stage, and the pressure drop curve decreases rapidly and remains at a lower value. As shown in Figure 3d, the curvature of the rear end meniscus increases, and the oil slug moves forward in capsule form. According to the relationship between the pressure drop curve and oil droplet deformation, the variation of pressure drop is a hydrodynamic reflection for the resistance of the inner wall to the slug during the transport. 3.2. Equivalent Viscosity Model. As mentioned above, during the hold-up stage, with the increased amount of slug enclosed by the water film and the decreased contact area between the slug and inner wall as the slug moves forward, the pressure drop curve becomes linearly downward-sloping. It is reasonable to assume that the variation in the pressure drop is related to the development of the water film. In this section, the pressure drop across the entire tube will be divided into several parts to discuss the influence of the water film and menisci deformation. As shown in Figure 4, as a residual oil slug in the capillary, the pressure drop at the two ends of the capillary can be expressed as

Figure 4. Pressure drop across the capillary tube with an oil slug trapped inside during build-up stage.

ΔP = ΔPw1 + ΔPw2 + ΔPo + ΔPc

(6)

where ΔP is the pressure drop across the entire capillary, ΔPw1, ΔPw2, and ΔPo are the Poiseuille pressure loss caused by the right side water slug, left side water slug, and oil slug, respectively, and ΔPc is the differential pressure caused by two menisci. According to the Hagen-Poiseuille theorem, the pressure drop of each slug can be calculated as follows: ΔPw1 = ΔPw2 = ΔPo =

8μw vL1 r2

(7)

8μw vL 2 r2

(8)

8μo vLo r2

(9)

where μw and μo are the viscosities of water and oil, respectively, r is the radius of the capillary, v is the average velocity of the slug flow, and L1, L2, and Lo are the length of the right-side water slug, left side water slug, and oil slug, respectively. Additionally, ΔPc can be written as the difference between the capillary pressures of the front and rear interfaces: ΔPc = ΔPc1 − ΔPc2

(10)

where ΔPc1 and ΔPc2 designate, respectively, the capillary pressure of the front and rear ends, which can be expressed as ΔPc1 =

2σow cos θc1 r

(11)

ΔPc2 =

2σow cos θc2 r

(12)

where σow is the interfacial tension between oil and water and θc1 and θc2 are the contact angles at front and rear ends, respectively. 13869

DOI: 10.1021/acs.iecr.8b03189 Ind. Eng. Chem. Res. 2018, 57, 13866−13875

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disappearance of the capillary pressure difference, which is a reason for the sharp decrease in the pressure drop. The pressure drop across the tube consists of two parts: the Poiseuille pressure losses caused by the water slugs and the capsule shape oil slug. The relationship between the pressure drops in this stage can be written as

By substituting eqs 7−12 into eq 6, the pressure drop across the entire tube can be obtained: ΔP =

8μw Lw v r

2

+

8μo Lov r

2

2σow(cos θc1 − cos θc2) r

+

(13)

where Lw = Lw1 + Lw2 denotes the total length of the water regime. Because of the hemispherical meniscus of the front-end during the hold-up stage, the contact angle θc1 = 0°. Therefore, eq 6 can be expressed as ΔP =

8μw Lw v r

2

+

8μo Lov r

2

2σow(1 − cos θc2) r

+

ΔP =

(14)

Figure 5. Pressure drop across a capillary tube with an oil slug trapped inside during the hold-up stage.

direction becomes shorter as the water film develops. Since the water film thickness is much less than the tube radius r, the contact length between the slug and wall Lo′ and the water film length Lf meet the relationship:

ΔP =

(15)

ΔP =

r2

+

8μo Lo′v r2

+

8μf Lf v r2

ΔP =

where μf is the equivalent viscosity of the segment enclosed by the water film and is called the hold-up viscosity. It is reasonable to assume the relationship μf < μo for the lubricating effect of the water film. 2σ (1 − cos θ )

r2

(18)

8μw Lw v r

2

+

8μo Lov r

2

+

2σow(1 − cos θc2) r

(19)

8μw Lw v r

2

+

8μf Lov r

2

+

2σow(1 − cos θc2) r

(20)

o

can be calculated, and then the strip viscosity can be obtained by eq 18. According to the pressure relationship above, the pressure drops of different stages can be detected with the graph in Figure 2. The contact angle and equivalent viscosity results are 8μ L v 8μ L v listed in Table 3, where ΔPf = f 2 f and ΔPof = ofr 2 o o are the r pressure drops across the segment enclosed by water film in the hold-up stage and the steady flow stage, respectively. Due to the lubricating effect of the water film, the hold-up viscosity μf is less than the oil viscosity μo. Additionally, the decrease in pressure drop ΔP and the decrease in oil segment pressure drop between

8(μf − μo )v

Letting A = w w 2 o o + ow r c2 and B = r2 r for constant capillary tube geometry, fluid properties, and flow rates, eq 16 can be simplified as: ΔP = A + BLf

8μof Lovo

In the steady flow stage, the velocity of the capsule shape oil slug vo is greater than the average velocity. The phenomenon of the slug moving faster than the average velocity of the other phase has been discussed by scholars,2,7,8 and the relationship between the fraction W and capillary number Ca has been established, as shown in Table 2. Using these equations and the v −v definition of W, W = o v , the velocity of the capsule shape slug

2σ (1 − cos θc2) + ow r (16)

8(μ L + μ L )v

+

On the other hand, at the end of the hold-up stage, the oil slug is in a critical state and is about to be stripped from the inner wall. The differential pressure caused by capillary force still exists at this point, and eq 16 can be written as

The water film has a lubricating effect on the oil slug when the oil phase viscosity is larger.12 Otherwise, for the experimental and numerical research by other scholars,10,31,32 with the shape of the rear end meniscus unchanged, the contact angle θc2 is constant in the hold-up stage. Thus, eq 14 can be written as 8μw Lw v

r

2

where vo is the velocity of the capsule shape oil slug, μof is the equivalent viscosity of the segment enclosed by the water film in the steady flow stage is called the strip viscosity. Additionally, the two equivalent viscosity values are different because the velocity of the oil slug increases after separation. Different from eq 16, the pressure drop ΔP in the steady flow stage is independent of the length of the water film Lf. This is the reason the curve in this stage does not vary with time. The slope of the pressure drop curve decreases at the beginning of this stage due to the recovery of the rear-end interface. With regard to understanding the pressure drop mechanism in each stage, the equivalent viscosities can be obtained with the pressure drop curve. At the preliminary period of the hold-up stage, the water film is in a critical state and is about to spread and the water film length is Lf = 0. Equation 16 can be expressed as

As previously described, in the hold-up stage, shown as Figure 5, the contact length between the oil droplet and wall in the axial

Lo = Lo′ + Lf

8μw Lw v

(17)

The negative value of B is consistent with the linear drop in the pressure drop curve. At the start of the steady flow stage, the oil droplet is stripped from the inner wall by the water film and begins moving forward in capsule form. The separation of the oil slug causes the Table 2. Formula Table of W author Fairbrother and Stubbs2 Marchessault and Mason7 Bretherton8

equation W = Ca1/2 W = −0.1(μ/σ)1/2 + 1.78Ca1/2 W = 2.674Ca2/3 13870

7.5 × 10−5 < Ca < 0.014 7 × 10−6 < Ca < 2 × 10−4 10−5 < Ca < 10−2 DOI: 10.1021/acs.iecr.8b03189 Ind. Eng. Chem. Res. 2018, 57, 13866−13875

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Industrial & Engineering Chemistry Research Table 3. Pressure Drops and Equivalent Viscosities preliminary period of hold-up stage ΔP (Pa) ΔPw (Pa) ΔPo (Pa) ΔPc (Pa) θc2 (deg)

183.34 5.12 128.00 50.22 46.78

end period of hold-up stage ΔP (Pa) ΔPw (Pa) ΔPf (Pa) ΔPc (Pa) μf (mPa·s)

123.97 5.12 68.63 50.22 53.62

steady flow stage ΔP (Pa) ΔPw (Pa) ΔPof (Pa) μof (mPa·s)

44.40 5.12 39.28 30.20

different stages, Pof < Pf, indicate that the decrease in the equivalent viscosity is another reason for the sharp decrease of pressure drop. Compared with the calculated values using Fluent, which is 47.2°, the contact angle of the rear end θc2 is especially accurate, with a relative error less than 1%. Contrasting the hold-up stage and steady flow stage, the pressure drop rapidly decreases when slug separation occurs due to two reasons: the disappearance of the capillary pressure difference caused by the menisci in the front and rear ends and the decrease in the equivalent viscosity at the point of separation. 3.3. Different Lengths of Oil Slug. In the previous parts of this section, the equivalent viscosity model and reasons for the rapidly decrease of pressure drop were discussed. The influence of oil slug length will be studied in detail in this section. Three different lengths for the oil slug were selected: 3 mm, 5 mm, and 6 mm; the capillary geometry, fluid properties, boundary condition, and mesh properties are consistent with the 4 mm case. The distance between the slug rear end and the inlet was 4 mm. Pressure drop curves across the capillaries for different length slugs are shown in Figure 6. The 4 mm case was also considered

Figure 7. Hold-up stage deformation profile of oil droplets with different lengths.

Table 4. Pressure Drop and the Equivalent Viscosity of Oil Slugs with Different Lengths steady flow stage

hold-up stage

3 mm 5 mm 6 mm

ΔPc (Pa)

ΔPf (Pa)

θc2 (deg)

μf (mPa·s)

ΔPof (Pa)

μof (mPa·s)

50.45 50.57 49.85

50.19 84.88 100.02

46.79 46.85 46.49

52.28 53.05 52.09

29.68 48.98 58.74

30.43 30.13 30.11

μf is less than the oil viscosity μo and greater than the strip viscosity μof in the other cases. It is remarkable that the two types of equivalent viscosities in different cases are essentially identical. Consequently, the hold-up viscosity and strip viscosity are independent of slug length. 3.4. Multi-Oil Slugs Transport and Pressure Drop. In modern industry such as low permeability reservoirs during the intermediary and later development stage, multioil slugs will occur in real porous media.33−37 The results of the previous study indicate the migration process of an oil drop slug in a capillary is affected by the capillary pressure.10,31,32 The capillary pressure difference caused by two menisci is 38% of the total pressure drop in the hold-up stage for the 4-mm case. With the increase of menisci, the capillary pressure difference will also increase. It is reasonable to assume that the capillary pressure difference will multiply when one slug is divided into several sections. For the oil phase in porous media, this means that the oil saturation remains constant. Then, the additional pressure drop will affect the movement of the slug. In this section, multioil slug transport and pressure drop are simulated and discussed in detail. Setting a consistent simulation condition as the single slug case, in order to compare with single oil slug cases, there are five multioil slug cases designed and calculated to study the law of pressure drop curves. The cases are listed in Table 5. Meanwhile, the phased separation of two different length slugs in the same capillary will be studied. In the multislug cases, based on assumption 4, the Poiseuille pressure loss caused by water and oil segments is still defined as ΔPw and ΔPo. Because the shape of the rear-end meniscus is independent of slug length, the capillary pressure difference of

Figure 6. Pressure drop vs time for Ca = 2.5 × 10−4.

for comparison. The curves in Figure 6 have similar trends. Moreover, the curves can be divided into three stages similar to the experiment results, and the slopes of the curves in the holdup stage are similar. This phenomenon can be resolved by combining the hold-up stage slug profiles of the different length cases. As shown in Figure 7a−c, in the hold-up stage, there is a distinct observation that the 3 mm, 5 mm, and 6 mm slug cases have similar rear-end menisci to the 4 mm case, which was discussed in Figure 3c. This demonstrates that the contact angles θc2 for different lengths are the same. Therefore, under the same conditions, the capillary pressure difference in the hold-up stage is independent of slug length. Using the previous derivation, the contact angles and equivalent viscosities in these cases can be calculated and are listed in Table 4. Similar to the 4 mm case, the hold-up viscosity 13871

DOI: 10.1021/acs.iecr.8b03189 Ind. Eng. Chem. Res. 2018, 57, 13866−13875

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Industrial & Engineering Chemistry Research ΔP = ΔPw + ΔPo1 − 2 + 2ΔPf2 + 2ΔPc

Table 5. Distributions of Multislugs Lo (mm)

Lo1 (mm)

Lo2 (mm)

Lo3 (mm)

3 4 5 6 6

1.5 2.5 3 4 2

1.5 1.5 2 2 2

/ / / / 2

8μ (L − L )v

ΔP = ΔPw + ΔPo1 − 2 + ΔPf2 + ΔPof2 + ΔPc 8μ L v

(21)

ΔP = ΔPw + ΔPf1 + ΔPof2 + ΔPc 8μ L v

(23)

ΔP = ΔPw + ΔPof1 + ΔPof2

Similarly, at the end period of the hold-up stage, the oil viscosities in eqs 22 and 23 should be replaced by the hold-up viscosity μf, and the total pressure drop of the 1.5 mm−1.5 mm and 2 mm−2 mm−2 mm cases can be written as ΔP = ΔPw + ΔPf + 2ΔPc

(24)

ΔP = ΔPw + ΔPf + 3ΔPc

(25)

(28)

where ΔPf1 = f 2 1 . r Ultimately, the longer slug separates from the wall, and the two capsule shape slugs move forward together, as shown in Figure 8d. The total pressure drop can be written as

(22)

For the 2 mm−2 mm−2 mm case, the total pressure drop can be expressed as ΔP = ΔPw + ΔPo + 3ΔPc

(27)

where ΔPof2 = of 2 2 o . r As shown in Figure 8c, when the rear end meniscus of the longer slug travels the entire slug length, the total pressure drop can be expressed as

where n is the number of oil slugs. For the two-slug case in the preliminary period of the hold-up stage, 1.5 mm−1.5 mm, 2.5 mm−1.5 mm, 3 mm−2 mm, and 4 mm−2 mm, the total pressure drop can be written as ΔP = ΔPw + ΔPo + 2ΔPc

8μ L v

where ΔPo1 − 2 = o 1 2 2 and ΔPf2 = f 2 2 . r r When the shorter slug enters the steady flow stage and the capillary pressure difference disappears, the profile of the slugs can be shown as in Figure 8b. The total pressure drop can be written as

multiple slugs ΔPcn is a multiple of the single slug ΔPc with the same total length: ΔPcn = nΔPc

(26)

(29)

8μ L v

where ΔPof1 = of 2 1 o . r According to the derivation above, the total pressure drops for the five cases in different stages can be obtained with the equivalent viscosities and rear-end contact angle θc2 derived from the single oil slug case. The results are listed in Table S1. Maintaining the same capillary geometry, fluid properties, boundary conditions, and mesh properties as the 4 mm case, Fluent is used to simulate multioil slug flow. Figure 9a,d is the simulation results of pressure drop across the capillary versus time for multiple slugs and an equal length single slug. In Figure 9a, the pressure drop for multiple slugs (1.5 mm−1.5 mm case) is greater than that of an equal length single slug (3 mm case) in the hold-up stage. Since the length of a single slug in the 1.5 mm−1.5 mm case is shorter, the multislug case completes the hold-up stage earlier, and its curve rapidly decreases initially. Nevertheless, the pressure drop between the two cases is consistent in the steady flow stage. With such an equal multislug case, the 2 mm−2 mm−2 mm case has a similar trend to the 1.5 mm−1.5 mm case in Figure 9d. For the 2.5 mm−1.5 mm case in Figure 9b, which has two different length slugs in a same capillary, the pressure drop curve at the hold-up stage can be divided into two parts because of the slugs’ separation. According to the previous derivation and analysis, the first slope on the curve represents the process in which the two oil droplets undergo the hold-up stage at the same time, and then the pressure drop rapidly decreases into the second part, since the shorter slug detaches from the wall. Similarly, the pressure drop declines gradually in the second slope. Moreover, the value of the second part is smaller than that of first part. It is due to a decrease in pressure drop caused by hold-up viscosity, which also can be reflected by the ΔPf2 in eqs 26 and 27. In Figure 9d, by eqs 22 and 23, the pressure drop of the trislug case (2 mm−2 mm−2 mm) is greater than that of two slugs (4 mm−2 mm) or one slug (6 mm) with the same length. This means that the pressure drop in the hold-up stage is much larger in multislug systems due to the increased capillary pressure drop caused by dispersive oil slugs. Therefore, it is reasonable to conclude that, for constant oil saturation, when the oil

However, for two different lengths of slugs in the same capillary, 2.5 mm−1.5 mm, 3 mm−2 mm, and 4 mm−2 mm, phased separation occurs in the hold-up stage. The shorter oil slug will complete the hold-up stage earlier and be stripped from the inner wall, while the longer segment remains in this stage. Based on the analysis of the oil slug migration process, the hold-up stage can be divided into three substages: stage I, stage II, and stage III, which are shown in Figure 8. The lengths of the

Figure 8. Diagram of the process of the oil slug from the inner wall.

two slugs are set as Lo1 and Lo2, while Lo1 is the length of the longer slug. When the shorter slug is at the end period of the hold-up stage, the total pressure drop can be expressed as 13872

DOI: 10.1021/acs.iecr.8b03189 Ind. Eng. Chem. Res. 2018, 57, 13866−13875

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Industrial & Engineering Chemistry Research

Figure 9. Pressure drop vs time for multislug case.

distribution is more dispersed, the pressure drop across the circular pores in porous media will be larger in the initial displacement stage. When the oil droplets gradually enter the steady flow stage, the pressure drop decreases incrementally. Eventually, the pressure reaches a stable value as all the slugs enter the steady flow stage. Additionally, through the equivalent viscosity model and eqs 21−29, the pressure drop at different stages can be calculated, and the results have been marked with black lines in the diagrams in Figure 9. As shown in these graphs, the calculation results using the equivalent viscosity model are in agreement with the CFD simulation results; meanwhile, the relative error is less than 2.33%. 3.5. Effect of Capillary Number. In previous sections, the mobilization of trapped oil slugs have been studied and discussed in detail. The difference between the pressure drops of the hold-up stage and the steady flow stage has been the focus of this research, and the different length and multislug cases have been studied to reveal the detachment mechanism. The pressure drops at the hold-up stage and the steady flow stage, which are resulted from the capillary pressure difference and the equivalent viscosities, can be seen as an extra pressure for the trapped oil separating from the inner wall. As described above, the extra pressure consists of three parts: the internal friction produced by oil contact with the wall during the hold-up stage, the change of friction near the inner wall, and the disappearance of the

capillary pressure difference. The latter two parts occur at the moment of detachment. In eq 17, it can be seen, in the oil and water viscosity fixed case, the extra pressure caused by the internal friction is related to the average velocity and the length of the slug, which corresponds to capillary number and oil saturation in porous media. Similarly, for the moment of slug detachment, the change of friction is also related to the average velocity v, which also determines the slug velocity vo (Table 2). Therefore, it is necessary to study the influence of capillary number on extra pressure for slug flow at the pore level. Six different inlet velocities are chosen for slug flow: 0.03 m/s, 0.02 m/s, 0.015 m/s, 0.008 m/s, 0.005 m/s, and 0.002 m/s, while the capillary geometry, fluid properties, boundary condition, and mesh property are consistent with the 4 mm case in 3.1. Figure 10 is the pressure drop curves across the capillaries in cases with different capillary numbers (7.5 × 10−4, 5 × 10−4, 3.75 × 10−4, 2 × 10−4, 1.25 × 10−4, and 5 × 10−5). For these flow rates, the Reynolds number of the largest one is 30, and the flow in the tube is laminar. This indicates that the capillary force plays a major role compared to the viscous force. In these cases, the pressure drop curves also appear in three stages. When the capillary number increases, the pressure drops in the hold-up stage and the steady flow stage becomes larger, indicating that the resistance on the trapped oil increases. Additionally, the slope of the pressure drop in the hold-up stage becomes larger as 13873

DOI: 10.1021/acs.iecr.8b03189 Ind. Eng. Chem. Res. 2018, 57, 13866−13875

Industrial & Engineering Chemistry Research

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Article

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86 186 6148 3604. E-mail: [email protected] (L. Long). ORCID

Long Long: 0000-0002-2429-8521 Mingzhe Dong: 0000-0001-5093-0116 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial supports from Natural Science Foundation of China (No. 51204198, 51274225) and National Basic Research Program of China (No. 2014CB239103).

Figure 10. Pressure drop vs time for capillary numbers.

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NOMENCLATURE v = velocity of the liquid, m·s−1 h = thickness of liquid film, mm r = capillary radius, mm μ = viscosity, mPa·s σ = liquid surface tension, N·m−1 Ca = capillary number ρ = density, kg·m−3 p = pressure, Pa ΔP = pressure drop, Pa L = length, mm θ = contact angle, deg

the capillary number increases. This trend denotes that the increase in capillary number exacerbates the change in equivalent viscosity, and the extra pressure in higher velocity cases is much greater at the droplet detachment. On the other hand, it is noteworthy that the build-up stage and the hold-up stage would continue to be prolonged with smaller capillary number. Therefore, it is reasonable to obtain a hypothesis that, when Ca becomes small enough, the trapped oil would keep in contact with the inner wall in the displacement and stay in the build-up stage or the hold-up stage. Meanwhile, in comparison with the case of detached oil, the pressure drop crosses the tube in the case of staying in the build-up stage or the hold-up stage would be maintained at a higher value. And these areas are the subsequent focus of this paper.

Subscripts

b = bubble m = mean w = water o = oil c = capillary f = water film

4. CONCLUSION In this study, computational fluid dynamics with the VOF method is employed to simulate a series of oil slug flows in a capillary tube at the pore scale. The pressure drop in the driving process of oil slug flow in the capillary is discussed stage-bystage. An equivalent viscosity model is proposed to explain the change in pressure drop at different stages. Then, the causes of the rapid pressure drop decrease when slug separation that occurs is explored: the disappearance of the capillary pressure difference caused by the front-end and rear-end menisci and the decrease in the equivalent viscosity at the point of separation. To the best of our knowledge, this is the first time pressure drops for oil slug flows with different lengths and multi-oil slug flows with the same length have been discussed and researched. Different stages of multi-oil slug flow were analyzed. Furthermore, the multiplied capillary pressure difference in multi-oil slug flow is verified to be the primary factor for increased pressure drop. The calculation results carried out using the equivalent viscosity model proposed in this paper are in significant agreement with the simulation results. The equivalent viscosity model and pressure drop trends during slug flow can be used in dynamic pore-scale models of flow in porous media to predict multiphase flow functions.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.8b03189. Table of pressure drops at each stage (PDF) 13874

DOI: 10.1021/acs.iecr.8b03189 Ind. Eng. Chem. Res. 2018, 57, 13866−13875

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DOI: 10.1021/acs.iecr.8b03189 Ind. Eng. Chem. Res. 2018, 57, 13866−13875