Water film rupture in blocked oil recovery by gas injection: Experimental and modeling study

Accepted Manuscript Water Film Rupture in Blocked Oil Recovery by Gas Injection: Experimental and Modeling Study Seyedamir Mirazimi, Behzad Rostami, Mohammad-Hossein Ghazanfari, Maryam Khosravi PII: DOI: Reference:

S0009-2509(16)30704-7 http://dx.doi.org/10.1016/j.ces.2016.12.043 CES 13315

To appear in:

Chemical Engineering Science

Received Date: Revised Date: Accepted Date:

20 August 2016 28 November 2016 16 December 2016

Please cite this article as: S. Mirazimi, B. Rostami, M-H. Ghazanfari, M. Khosravi, Water Film Rupture in Blocked Oil Recovery by Gas Injection: Experimental and Modeling Study, Chemical Engineering Science (2016), doi: http://dx.doi.org/10.1016/j.ces.2016.12.043

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Water Film Rupture in Blocked Oil Recovery by Gas Injection: Experimental and Modeling Study Seyedamir Mirazimi a, Behzad Rostami a,*, Mohammad-Hossein Ghazanfari b, Maryam Khosravi c a

Institute of Petroleum Engineering, School of Chemical Engineering, College of Engineering, University of

Tehran, Tehran, Iran b

Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran

c

IOR Research Institute, National Iranian Oil Company, Tehran, Iran

*

Corresponding author: Behzad Rostami Postal address: Institute of Petroleum Engineering-College of Engineering, University of Tehran, North Kargar St., Tehran, Iran, P.O Box: 113654563) Email address: [email protected] Phone number: + 989125210473, +982161114736

1

Abstract Water shielding phenomenon generally occurs after waterflooding in water-wet rocks, and impedes direct contact between the oil and the injected gas in tertiary gas injection processes. In this work, a set of visualization experiments were performed on micromodel patterns including designed dead-end pores with a film of water on the surface of pore bodies, which is a more realistic representation of porous media. The experiments were conducted at different miscibility conditions, and the required time for water to be displaced from the throat by the swelling of oil was measured for first contact miscible (n-C5/CO2) and immiscible (n-C10/CO2) systems. In the next step, a model was proposed to simulate the results of the experiments, based on the work of Bijeljic et al. (2003). As the impact of nonideal mixing in this process has not been previously discussed in the available literature, the new model was developed by taking into account the changes in the partial molar volumes of oil and gas components using the PR and the SRK equations of state, and also by considering the mass transfer from the surrounding water on the pore body into the shielded oil. The rupture times predicted by the model were compared with the measured experimental data, as well as those reported by Campbell and Orr (1985). It was found that inclusion of partial molar volumes of components improves the accuracy of the model. The results also revealed the significant role of the water film on the pore body surfaces in mass transfer rate between the phases in water-wet media. The close agreement between the results of the model proposed in this study and the experimental data shows that it can be helpful for developing more accurate multiphase compositional models. Keywords: Water shielding, Non-ideal mixing, Mathematical modeling, Micromodel experiment, Dead-end pore

2

1. Introduction Waterflooding is the most commonly used secondary oil recovery method in the oil industry; but even in homogenous reservoirs with high sweep efficiency, a major part of oil remains behind in the form of immobilized pore-scale ganglia surrounded by water. In such cases, tertiary miscible gas injection is considered as an effective method to recover the remaining oil. Therefore, it seems essential to investigate the recovery mechanisms of the trapped oil by gas and to recognize the engaged parameters in the production process of hydrocarbons to gain a comprehensive knowledge about the behavior of oil reservoirs under tertiary gas injection. As a result, in this study it has been tried to perform a fundamental study on the recovery of the trapped oil in dead-end pores by gas injection after waterflooding, both theoretically and experimentally. In water-wet media, a thin continuous water film is present on the surface of the reservoir rock in pores and throats. After waterflooding, due to the increase of water saturation, this film coalesces into a water layer in some narrower throats, isolating the oil behind from the main oil bank in the reservoir in form of ganglia or dendrites. When gas is injected into the system, the water barrier formed previously impedes direct contact between the blocked oil and the injected gas on the other side, making the oil inaccessible by gas, and preventing mass transfer between the phases in order to attain miscibility. This phenomenon is generally referred to as “water shielding” or “water blocking”. However, the injected gas manages to reach the stagnant oil through dissolving in water and diffusing into oil. As gas diffuses into oil, the non-flowable oil swells gradually pushing the water, until it completely displaces the water barrier present in the throat, and unblocks itself. Water shielding has been observed in laboratory experiments on cores by various researchers. Raimondi et al. (1961) experimented miscible displacement in the presence of two phases. Their observations indicated that the displacement of oil, as non-wetting phase, becomes less 3

efficient as the water saturation is increased above the irreducible range. However, the trapped oil is not completely isolated by water, and is recoverable by miscible displacement. Thomas et al. (1963) studied miscible displacement in both the wetting and non-wetting phase in two-phase systems. The results showed that dispersion is a function of saturation, and a part of saturation effect is as a result of trapping of the displacing and displaced fluid in dendritic structures or dead-end pores due to the presence of a second phase. Raimondi and Torcaso (1964) investigated the distribution of the oil phase resulting from increasing and decreasing the water saturation by imbibition. They concluded that oil is trapped upon imbibition of water, and it is the trapped oil which seems to make tertiary recovery operations uneconomic. Stalkup (1970) performed displacements of laboratory oils by propane in sandstone cores in the presence of high water saturations. The experiments revealed that only part of the oil is flowable in the presence of high water saturation, while the other part remains motionless in locations blocked by water. Although, this trapped oil can be recovered by molecular diffusion into the flowing propane. He also observed that oil trapping for strongly water-wet sandstones is more severe than weakly water-wet reservoir rocks. Shelton and Schneider (1975) assessed the effect of mobile water saturation on oil recovery from sandstone cores by miscible displacement. They found out that in contrast to the water-wet case, trapping of hydrocarbon does not occur in oil-wet porous media. Nevertheless, diffusion contributes to the recovery of the trapped non-wetting phase, and the required contact time is directly related to the saturation of water. Similar results confirming the unfavorable effect of water saturation as well as the importance of rock wettability on the severity of oil trapping were obtained by Lin and Huang (1990), Tiffin et al. (1991), Wylie and Mohanty (1997 & 1999) and Cable et al. (2004) who also conducted laboratory core experiments to study the performance of tertiary miscible displacements.

4

The Mechanism of shielded oil recovery during tertiary gas injection has been investigated using micromodel experiments in several studies. Campbell and Orr (1985) studied the recovery of oil from dead-end pores, with and without the presence of water barriers shielding the oil, by using a visual pore-scale micromodel. As a part of their work, they performed a visual experiment with micromodel, in which Soltrol™ 130 (a mixture of C9 through C13 branched alkanes) in a dead-end pore shielded by water was recovered by CO2. Displacement was conducted at 77 F (25 C) and 1 200 psia (8.3 MPa), in which CO2 is first contact miscible with the oil. Figure 1 illustrates the process of the displacement. As CO2 diffuses through the water, the oil swells, displacing the water barrier. Eventually, the water film is dislodged by the flowing CO2, and then the oil in the pore is recovered, just when no water is present after 26.5 hours. CO2

CO2

CO2

water

water water oil

oil

oil

Figure 1: Illustration of tertiary miscible displacement performed by Campbell and Orr (1985) - recovery of Soltrol shielded by water in a dead-end pore by CO2 - from left to right: start of CO2 injection; position of water barrier after 18 hours; and position of water barrier after 26.5 hours

Their visual observations implied that the efficiency of both first contact and multiple contact miscible displacements were much higher in the absence of water. Kantzas et al. (1988) conducted visualization experiments of displacing residual oil by gravity assisted inert gas injection in water-wet micromodels. They observed that when air is allowed to enter the model, gas-water interface passes through large pore throats and enters pore bodies. It then moves to occupy the pore body and pushes the oil blob out of the pore. The coalescence of oil blobs forms an oil bank which grows with time as it moves towards the production end. In 5

addition, oil films are formed behind the oil bank in the pores invaded by air which maintain hydraulic continuity and allow leakage of oil towards the oil bank. Oren et al. (1992) investigated mechanisms of mobilization and recovery of waterflood residual oil during tertiary gas injection in 2-D strongly water-wet glass micromodels for systems with positive and negative spreading coefficients. Their experiments showed that oil recovery was significantly higher for the positive spreading system, due to flow through thin but continuous oil films. Laroche (1998) studied secondary and tertiary gas injection in heterogeneous wettability micromodels. He concluded that most of the time gas invades the oil-wet pathways which have the least resistance, as water bridges in water-wet regions drastically increase the resistance to flow. He also observed that the efficiency of gas injection in tertiary experiments is less than the secondary ones in the presence of isolated oil-wet patches surrounded by water in strongly water-wet regions. Sohrabi et al. (2008a & 2008b) performed flow visualizations at high pressure conditions to explore pore scale events during residual oil recovery by near-miscible hydrocarbon gas injection. Their observations revealed that when the injected gas came in contact with the residual oil, the gas front was covered with a small amount of oil. As the gas advanced further, an oil bank was formed and moved ahead of the gas front, supplying oil to pores where residual oil was present. As a result of this pore-scale mechanism, oil is transferred from the bypassed pore to the main gas flow stream where it can be subsequently recovered. A combination of a very low gas–oil interfacial tension (IFT), perfect wetting (or spreading) conditions of the oil near the critical point, and also the ability of the low-IFT oil and gas to simultaneously flow in the main flow stream, facilitates this recovery mechanism. Sayegh and Fisher (2008) investigated the efficiency of tertiary miscible CO2 injection with and without simultaneous injection of water by using 2-D micromodels. They found that oil was recovered more efficiently when CO 2 was co-injected with water than when it was injected alone. This was due to a mechanism

6

present during water/CO2 co-injection where the water shield was repeatedly built-up, rearranged, then thinned out by water snap-off, which exposed part of the oil droplet to the CO2 at some point, allowing direct contact between the oil and gas. Riazi et al. (2009) experimented different CO2 injection scenarios in glass micromodels. They observed that in tertiary injection experiments, water barriers initially prevented the production of the oil, but swelling of oil ruptured the water layer at later times, which then led to shrinkage of the oil as a result of the extraction of oil components by the CO2 and displacement of oil by film flow mechanism. They reported higher pore-level sweep efficiency and ultimate oil recovery in the case where no water was present. Hamedi Shokrlu and Babadagli (2015) studied the injection of miscible solvent (n-heptane) to recover the residual oil left after waterflooding by use of micromodel visualization experiments. Their results illustrated that the presence of water in porous media creates capillary barriers preventing the direct contact between the oil and the solvent. They concluded that rock wettability and oil viscosity have a significant influence on the amount of oil recovery. All of the experimental studies mentioned above indicated the unfavorable effect of the presence of water by trapping the oil. However, given enough time, the gas would make the oil swell gradually and displace the water barrier. Various researchers have proposed mathematical models to estimate the time required for water-shielded oil to break the water barrier. Grogan and Pinczewski (1987) proposed a 1-D numerical model that simulated the swelling of residual oil blobs by CO2 diffusion through a blocking water phase. Their model resulted in a set of coupled highly nonlinear partial-differential equations that constituted a boundary value problem for which the position of the oil/water interface was determined by numerical solution. Muller and Lake (1991) developed another 1-D pore model that calculated the mass flux between flowing and stagnant regions separated by a water film. The model considered solvent diffusion and diffusional extraction of oil accompanied by swelling

7

or shrinking of the stagnant hydrocarbon phase. The various process parameters were represented by a set of dimensionless numbers, which allowed the problem to be scaled to any size. Do and Pinczewski (1993) suggested a 1-D model based on the assumption that the CO2 concentration profiles in both oil and water phases may be approximated by simple power functions where exponents are allowed to vary with time. While their method greatly reduced the mathematical complexity of the moving boundary problem from a system of coupled partial differential governing equations to a system of coupled ordinary differential equations, it neglected volume changes from mixing which would greatly complicate the analysis. Bijeljic et al. (2003) developed a 1-D diffusion-based pore model using first Fick’s law to simulate the volume change of the oil due to multicomponent mass transfer from or to the gas phase in the presence of water blocking. Their model assumed an ideal mixture for the oil phase, and generated a system of ordinary differential equations (ODE) which could be solved simultaneously using a numerical solver. Riazi et al. (2011) developed a different 1-D model based on the second Fick’s law, which leads to one partial and one ordinary differential equation together with two auxiliary equations. They used the moving mesh mode of COMSOL multi-physics mathematical modeling software to solve the equations, which is based on the finite element method. One of the assumptions considered in all of the abovementioned models was ideal mixing, i.e., the volume of the mixture is the simple sum of the volumes of the pure components. However, studies on the volumetric characteristics of different hydrocarbon binary mixtures have proven their mixing process to be non-ideal. Various researchers have focused on the volumetric behavior of CO2, as the most common gas used for secondar in ection, with alkanes Cullick and

athis (1

, essi res et al. (2001),

and tertiar iga-Moreno et

al. (2005), and Song et al. (2012) measured the densities and excess volumes of binary system of CO2/decane, which is widely used as a typical model fluid representative of oil

8

based on its similar physical properties such as viscosity and surface tension

lso,

edina-

erm de et al. (2013) performed similar studies on binary mixtures of CO 2/heptane and CO2/tridecane. In all of the abovementioned studies, non-ideal volumetric behavior was observed for the mixtures at different pressure and temperature conditions. In the present study, a number of micromodel experiments have been conducted at different pressures and temperatures, providing different miscibility conditions, to investigate the recovery of water-shielded oil in a dead-end pore by gas injection. A film of water was present on the surface of the pore structure in the tests, to be a better representation of waterwet media. The movement of water/oil interface due to swelling of oil by the dissolved gas in water was monitored in the experiments, and the water rupture time has been measured. In the next step, a mathematical model is developed based on the work of Bijeljic et al. (2003), to simulate the swelling process of oil displacing the water barrier. We show that the inclusion of partial molar volumes of oil and gas components in the model increases the accuracy of the results. In addition, it is demonstrated that considering the molecular diffusion between phases across the boundary of the trapped oil ganglion and the surrounding water film in the pore is necessary to achieve a realistic model of water-wet media. 2. Experimental Work In this section, a series of tertiary CO2 injection tests were conducted on glass-type micromodels including dead-end pores at both first contact miscible and immiscible conditions. The tests were performed when a film of water was present on the surfaces of pore bodies, surrounding the shielded oil, to be a better representative of the status of deadend pores at water-wet systems. Micromodel system allowed us to visualize the production process of the water-shielded oil, and to monitor the arrangement of the fluids, specially the position of the water/oil interface during the experiments. Figure 2 illustrates a schematic

9

diagram of the experimental setup used in this study. It consists of an air oven, in which the glass micromodel and three transfer cells containing water, oil and gas along with the connecting lines are kept at the test’s pressure and temperature, a back pressure regulator (BPR), and positive displacement pumps for injection of the fluids and maintaining the back pressure. The experiments were monitored by a digital microscope connected to a computer, which could take pictures and record videos from different stages of the experiments.

Digital Microscope

Micromodel

Back Pressure Regulator

Thermometer

Transfer Cells

Oven

PC

Injection Pump

Back Pressure Pump

Figure 2: A schematic diagram of the micromodel setup

The glass micromodel was made up of a pair of 10-mm-thick glass plates fused together by heating up to 700 C. A pattern has been etched onto the surface of one of the plates before the fusion, which created a void space between the plates representative of the porous media. Two holes were drilled at the inlet and outlet of the pore structure, allowing the injection and production of fluids into and from the enclosed pore space. The fluids used in the experiments were distilled water, carbon dioxide, and n-pentane or ndecane as the oil phase. The water and the oil were dyed blue and red using Methylene blue 10

and Sudan red respectively, to have visually distinguishable phases, followed by filtration using fine filter papers to separate any remaining particles. In order to minimize the gravitational effects, the orientation of the micromodel was horizontal in all of the tests. It was initially vacuumed, followed by injecting oil at atmospheric pressure and the desired temperature. After the system was completely filled with n-pentane or n-decane, the valve at the inlet of the micromodel was closed, and water was pumped into the system. The injection of water would increase the pressure of the system behind the valve, while the micromodel on the other side was still at atmospheric pressure. Therefore, when the valve was opened, water would invade the oil in the throats and displace it. This process could be repeated until the ideal oil-water configuration was achieved. Then, the back pressure was set at the desired value and water injection was continued until the overall pressure of the system was stabilized. Increasing the pressure of the system caused the oil ganglion to slightly shrink, which led to gradual formation of a thin film of water on the pore walls. Finally, the injection of gas was started and the experiment was recorded by digital microscope for further analysis. The gas injection continued after the oil displaced the water in the throat to investigate the production mechanism of the oil. 3. Mathematical Modeling All of the proposed models for estimating the rupture time of water film mentioned in section 1 used empirical correlations to estimate molecular diffusion coefficients and hydrocarbons solubility values in water at given temperature and pressure as input parameters. While each model had some advantages over the others, they all suffered from at least one of the following disadvantages: - The mathematical model required physical parameters such as partition coefficient, which can only be measured by laboratory experiments. 11

- Oil/gas mixing was assumed to be ideal, and volume changes owing to non-ideal mixing was neglected, which resulted in deviation from reality. - The compositions of in-situ oil and injected gas were ignored in the formulations, and the model treated oil and gas as two separate black-oil phases. - The model was one-dimensional, i.e., the gas/oil and water/oil mass transfer areas are equal, which is not true in the presence of water film on the surface of pore walls in water-wet systems. We have tried to develop an improved mathematical model based on that proposed by Bijeljic et al. (2003) by removing the abovementioned deficiencies. 3.1.One-Dimensional Model Bijeljic et al. (2003) developed a simple 1-D numerical model to investigate multicomponent mass transfer in water-blocking at the pore scale. They assumed an ideal mixture (i.e., the total volume is the summation of the volumes of oil and gas) for oil phase and neglected the convection term in their equations. They also assumed a uniform gas concentration in the oil phase at any time and that the time required for the gas in the water phase to reach steady state conditions is negligible, i.e. applying the first Fick's law for this part of their model. They investigated the impact of swelling and/or shrinkage of the oil phase created by mass transfer from or to the gas phase in the presence of water blocking.

12

Lo,in Lo (t)

gas

Lw,in Lw (t)

water

oil

water

gas

oil

Figure 3: (a) Representation of water-shielded oil in a dead-end pore; (b) simplified mathematical description of the system

As depicted in Figure 3, components diffuse through the water barrier from gas to the oil phase and vice versa. The water/oil boundary moves as the oil film thickness Lo(t) and the water film thickness Lw(t) vary in time by a length L(t). The final from of the Bijeljic’s model can be written as a system of ordinary differential equations, which can be solved simultaneously using a numerical solver. nc Lw (t ) M Di ,w S i ,w C i , g  C i ,o (t )  ,   i t Lw (t ) i 1 i

C i ,o (t ) t



D

i ,w

(1)

Lw (t )  S i ,w C i , g  C i ,o (t )   C i ,o (t )  Lw (t ) t  Lo (t )

,

(2)

where Di,w (m2/s) is the diffusion coefficient of component i in water and Si,w (mol of component i/mol of solution) is the solubility of hydrocarbon component i in water from the gas or oil phase. Ci,g (mol/m3) and Ci,o(t) (mol/m3 ) are the concentrations of component i in the gas and oil phase, respectively. Mi (kg/mol) is the molar mass of component i and ρi (kg/m3) is the density of pure component i. i el ic’s method has two ma or advantages that make it applicable for s stems in which none of the other proposed models can be used:

13

- It takes into account the composition of oil and gas, and calculates the diffusion flux of each component separately, while the other models treat the in-situ oil as a black-oil phase and assume the injected gas to be pure CO2. This makes these models inapplicable for cases where a multicomponent gas is injected into the system. - i el ic’s model, Contrar to all the other ones, does not require partition coefficient, defined as the ratio of the equilibrium concentrations of a solute in two immiscible solvents (water and oil in this case). It takes pressure, temperature and fluid compositions as the input parameters to estimate the water rupture time, while in other studies, CO 2 partition coefficient had to be interpolated between the values measured by Balint (1971) and Daniel and Gaddy (1972) for water and a crude oil. It means that for these models, experimental partition coefficient values have to be available for the desired system of fluids. Despite the abovementioned advantages, Bijeljic et al. (2003) assumed the mixing process of oil and gas to be ideal. This is in contrast with the results of various studies on the volumetric behavior of CO2 and alkanes, all of which demonstrated some degree of non-ideality for the mixtures at different pressure and temperature conditions. This implies that neglecting this phenomenon would lead to unreliable results for estimation of water rupture time in blocked oil recovery process. In this work, the previously described model has been improved, in order to obtain more realistic results by considering the effect of non-ideal mixing in miscible gas injection processes. The improved model can be expressed as the following system of ordinary differential equations: nc  (t )D S  (t )  Lw (t ) i ,o i ,w i ,w C i , g  C i ,o (t )      C i ,o (t )Lo (t ) i ,o  , t Lw (t ) t  i 1   

14

(3)

C i ,o (t ) t



D i ,w S i ,w C i , g  C i ,o (t )  Lw (t )  C i ,o (t )  Lw (t) t 

,

(4)

Lo (t )

where υi,o denotes partial molar volume of component i in oil phase which can be estimated from the Peng-Robinson (PR) equation of state using equation (5) (Danesh, 1998), or from the Soave-Redlich-Kwong (SRK) equation of state using equation (6) (Lin and Daubert, 1980):

 RT i ,o 

nc    bi P  o2  2bo  b 2    2bi RT  2 x j aij  2bi P o  b   o  b   bi a j 1    ci , 2 2 P o  2bo  b   2P o  b o  b   2RT o  b   a

(5)

nc

abi o  b bi  RT  j 1 1     b  o  b  o o  b   o  ci , a  2o  b  RT  2 2 o  b  o2 o  b  2 x j aij 

i ,o

(6)

where a and b are the attractive and repulsive term parameter of equation of state, respectively. R is the universal gas constant, and T and P are temperature and pressure of the system. ci is volume translation parameter which is calculated from correlations proposed by Jhaveri and Youngren (1988) and Peneloux et al. (1982) for the PR and the SRK equations of state, respectively. The details of the derivation of equations (3) and (4) have been described in Appendix A. 3.2.Two-Dimensional Model In water-wet systems, a continuous film of water covers the whole surface of the porous media. In case of tertiary gas injection, this water layer is present on the pore bodies, surrounding the trapped oil in dead-end pores, as well as on the main flow channel walls, in contact with the gas. In such a system, mass transfer occurs across both the water/oil and water/gas interfaces, causing the injected gas to dissolve in the water and to diffuse into the

15

trapped oil. The contact area between water and gas is considerably larger than that of oil and water, because the continuous water layer has spread over the whole pore space as a result of prior waterflooding, while the mass transfer area between oil and water is limited to the oil blob circumference (Figure 4). Accordingly, the oil/water interface controls the mass transfer rate between the injected gas and the trapped oil. On the other hand, the swollen oil cannot displace the water in the throat completely, on account of the high tendency of water to cover the surface of the throat walls. Instead, it channels through the water barrier and builds a narrow path in the center of the throat to reach the gas on the other side (Figure 5). gas water

oil

Figure 4: Representation of water-shielded oil in a dead-end pore: the injected gas dissolves in the water film spread on the surface of the porous media and diffuses into oil from the water surrounding the trapped oil ganglion gas water

oil

Figure 5: Schematic of water rupture process by swelling of the shielded oil in a dead-end pore during gas injection from the start of injection until oil/gas direct contact (left to right)

In order to estimate the water rupture time for such cases, simplifying the problem into a 1-D model cannot be correct, as the cross section area of the swelling oil channel in the throat differs significantly from that of water/oil mass transfer interface. Therefore, a 2-D form of 16

the model was derived to consider the effect of the water film present on the surface of the porous media. The 2-D form of the model has been presented in equations (7) and (8). nc  d  (t )D S  (t )  Lw (t ) o ,w i ,o i ,w i ,w C i , g  C i ,o (t )     C i ,o (t )Ao (t ) i ,o  d o ,th , Lw (t ) t  t i 1   

C i ,o (t ) t



d o ,w Di ,w S i ,w C i , g  C i ,o (t )  Lw (t )  C i ,o (t )d o ,th  Lw (t) t 

(7)

,

(8)

Ao (t )

where do,w (m) is the perimeter of the oil blob, do,th (m) is the width of the oil channel in the throat, and Ao(t) is the area of the oil blob. 4. Results and Discussion 4.1.Experimental Results A number of micromodel experiments were designed and conducted, four of which have been selected to be presented in this study based on the quality of initialization conditions and visual results: Two first contact miscible (FCM) tests using n-pentane and two immiscible tests using n-decane as the oil phase. Carbon dioxide was used as the injection gas in all of the experiments with an injection rate of 6 cm3/h, and the water rupture time was measured. Table 1 summarizes the details of the experiments. Table 1: Details of the experiments conducted in this study Fluid

Pressure

Temperature

System

(MPa)

(C)

1

n-C5/water/CO2

4.41

9

FCM

208

2

n-C5/water/CO2

4.41

9

FCM

85

3

n-C10/water/CO2

2.76

25

Immiscible

216

4

n-C10/water/CO2

2.76

25

Immiscible

1023

No.

Miscibility

Rupture Time (min)

The tests with n-decane were performed at 25 C and 400 psi (2.76 MPa), in which the inplace and injected fluids are known to be completely immiscible (Liu et al., 2015). The 17

miscible tests were performed at proper conditions to make sure of the first contact miscibility of the n-pentane/CO2 system, i.e. 9 C and 640 psi (4.41 MPa). Both of these assumptions were confirmed by CMG WINPROP and by visual observations during the experiments, when the oil came into direct contact with the injected CO 2 after the displacement of the water from the throat. Figure 6 illustrates the configuration of the fluids right after the rupture of the water barrier. As it can be seen, there is a distinct interface between CO2 and n-decane, while no interface can be distinguished for the CO 2/n-pentane system, as CO2 is a liquid which is first contact miscible with n-pentane at the pressure and temperature conditions of the experiment.

immiscible interface

miscible interface

CO2

CO2

water film

oil

oil

Figure 6: Oil/gas configuration in dead-end pore right after the rupture of the water barrier: immiscible CO2/ndecane system in experiment 3 (left) and first contact miscible CO2/n-pentane system in experiment 1 (right)

In all the experiments, the observed volumetric behavior of the phases was similar. The configuration of oil and water at different times in experiment 1 has been depicted in Figure 7. When CO2 is introduced into the system, it replaces water in the center of the main flow channel, creating a conduit towards the outlet of the system, leaving behind a continuous film of water on the channel walls (Figure 7-1). The molecules of the injected CO2 dissolve and move in the water phase, pass through the water barrier in the throat and reach the oil in the 18

pore. Once the CO2 reaches the other side of the water barrier, it starts to dissolve in the oil ganglion and increases its volume as a result of swelling (Figure 7-2). As the injection of gas continues, the oil blob keeps expanding and starts to advance in the pore neck; however, it cannot completely displace the water and fill the whole width of the throat. Instead, it forms a finger-like structure and channels through the water barrier (Figure 7-3 to Figure 7-7). This is because the surface of the glass is strongly water-wet, which causes a film of water to be held on the throat walls. As the oil channel moves forward in the throat, the diffusion path of CO2 molecules heading towards the oil becomes shorter and the advancing speed of the oil front increases as a result of higher mass transfer from the gas phase. As it can be seen, it takes the oil 116 minutes to reach the half of the throat length (from Figure 7-2 to Figure 7-5), while passing the other half and reaching the gas on the other side (from Figure 7-5 to Figure 7-8) only takes 30 minutes. During the experiments, the position of the water/oil interface in the throat was monitored as the oil blob swelled by CO2 diffusion through water, and time from the start of gas injection into the system till it came to direct contact with the oil was measured.

19

CO2 water

1

2

3

4

6

7

8

oil

5

Figure 7: The position of the water (blue) and oil (red) interface in dead-end pore structure at different times for experiment 1: t1=4 min, t2=72 min, t3=124 min, t4=151 min, t5=188 min, t6=203 min, t7=207 min, t8=208 min

However, after breaking the water barrier, there was a great difference in the time required for the production of the oil in the dead-end pore between the miscible and immiscible displacements. Figure 8 compares the oil in the pore at rupture time and after 30 minutes for experiment 2. It can be inferred that the major part of the red n-pentane was recovered by the miscible gas, leaving behind a mixture mostly composed of the colorless liquid CO 2. Figure 9 demonstrates the oil in the pore at rupture time and after 15 hours for experiment 3. Contrary to experiment 2, the n-decane present in the dead-end pore remained almost untouched by the immiscible CO2.

20

CO2

CO2 miscible interface

water

dilute oil

oil

Figure 8: N-pentane in experiment 2 at rupture time (left) and after direct contact with miscible CO2 for 30 minutes (right)

immiscible interface

CO2

CO2

water

oil

oil

Figure 9: N-decane in experiment 3 at rupture time (left) and after direct contact with immiscible CO2 for 15 hours (right)

The great difference between the recovery times of the oil blob in direct contact with gas between the miscible and immiscible displacements can be due to high interfacial tension (IFT) of CO2-oil, higher oil viscosity and low solvent power of CO 2 at gaseous state in the immiscible experiment. For the oil ganglia directly in contact with the flowing CO 2, the predominant mechanism is the vaporization of the oil as a result of the extraction of oil 21

components by the CO2 (Riazi et al., 2009). In the immiscible experiment, CO2 is in gaseous state with low density, while it turns into a dense fluid with much higher density as a result of pressure elevation and temperature reduction in the miscible test. As the CO 2 density increases, the main intermolecular distance decreases and specific interactions between the gas and oil increase proportionally, giving rise to the rate of extraction (Luque de Castro et al., 1994). In addition, the presence of capillary forces in the immiscible experiment prevents the injected CO2 from entering the pore and displacing the oil, while in the miscible displacement, capillary forces are eliminated due to zero interfacial tension. Furthermore, the lower oil viscosity in miscible injection due to the mixing of CO 2 and the oil phase compared to the immiscible conditions makes it easier to flow in the porous medium and contributes to the production of oil. 4.2.Modeling Results In this part, the proposed 1-D model was utilized to predict the rupture time of water barrier using ODE23 function of MATLAB®, which utilizes second and third order Runge Kutta formulas. It was employed to estimate the water rupture time for a laboratory micromodel experiment conducted by Campbell and Orr (1985), in which water-shielded oil was recovered from a dead-end pore by miscible CO2 injection. This would help to compare the prediction ability and the precision of the new model with the original

i el ic’s model and

those proposed by other researchers. In order to estimate the water rupture time, the initial length of water and oil in the dead-end pore, as well as solubility and diffusion coefficients of hydrocarbon components in water are required. Figure 10 presents the configuration of oil and water at the beginning of the experiment. 22

0.4mm CO2 water

2mm

1mm

oil

4mm

Figure 10: Initial fluid configuration and dimensions of the experiment performed by Campbell and Orr (1985)

As CO2 dissolves in the water barrier, it passes the 2-mm throat and reaches the 1-mm water droplet. At this point, CO2 can pass different paths to reach oil. As shown in the above figure, these paths are in the form of chords of the circle with different lengths from 0 to 1 mm. The formula for calculating the length of chord c of a circle with a radius R making a central angle θ is c  2RSin  2  . Therefore, the average length of the chords which CO 2 can pass to 2

reach the oil can be calculated as c   2RSin  2  d  2  0.637mm . As a result, the initial 0 thickness of water in the model was set to 2.64 mm. As the volume of oil is 7.43 times bigger than water, the initial thickness of oil was set to 19.59 mm. For calculating diffusion coefficients of CO2 and methane in water, models proposed by Lu et al. (2013) and Guo et al. (2013) have been used, respectively. The general form of both models can be written as the following expression: m

T  D  D 0   1 , T S 

(9)

where D0, Ts and m for CO2 and methane are presented in Table 2.

23

Table 2: Calculated parameters of equation (9) for CO2 and methane Parameter

CO2

Methane

D0 (m2/s)

13.942×10-9

15.95×10-9

Ts (K)

227

229.8

m

1.7094

1.8769

For calculating the solubility of CO2 and methane in water, algorithms proposed by Duan et al. (2006) and Duan and Mao (2006) were implemented, respectively. These are thermodynamic models, which calculate the solubility as a function of temperature, pressure and salt composition. In order to calculate the diffusion coefficient and solubility of heavier hydrocarbon components in water, a similar method to that used by Bijeljic et al. (2003) was used in this study. Table 3 summarizes the calculated diffusion coefficients and solubility values for CO 2 and hydrocarbon components. Table 3: Calculated diffusion coefficient and solubility values in water for Campbell’s experiment Solubility

Diffusion

(mol/mol)

Coefficient (m2/s)

CO2

2.51×10-02

1.92×10-09

C9

2.01×10-07

5.89×10-10

C10

7.36×10-08

5.54×10-10

C11–13

9.56×10-09

4.99×10-10

Component

The calculated values were fed into both models and water thickness was calculated as a function of injection time for Campbell and Orr’s experiment The results have been depicted in Figure 11. The model proposed by Bijeljic et al. predicts that after about 21 hours of CO2 injection, oil completely displaces the water barrier and comes into direct contact with CO 2, while this happens after 26.7 and 26.2 hours based on the new model incorporating the PR and the SRK equations of state, respectively. It is worth mentioning that this time had also

24

been estimated around 23.3 h by Riazi et al. (2011) and 21.4 h by Grogan and Pinczewski (1987). 3

Bijeljic This work by PR EOS This work by SRK EOS Riazi (Single Point) Grogan (Single Point) Experiment

Water Thickness (mm)

2.5 2 1.5 1 0.5 0 0

5

10

15

Time (hour)

20

25

30

Figure 11: Simulation of the experiment reported by Campbell and Orr (1985)

The results demonstrated that the estimated times calculated by the new 1-D model using both the PR and the SRK equations of state were in good agreement with that reported by Campbell and Orr (1985), which confirms that inclusion of partial molar volumes of components has led to improvement of the model precision and more realistic results; however, considering high uncertainties in the initial configuration of the fluids in this experiment, and the assumptions made in defining the corresponding 1-D model, a series of micromodel experiments have been designed and performed which allow for more accurate investigation of the model. It should be noted that the presence of water film on the surface of the porous media was not considered in Campbell and Orr’s experiment No water was reported to be present nor in the dead-end pore neither on the main flow channel surfaces. Also, the oil filled the whole width of the pore neck as it swelled with time. This might be attributed to the fact that the micromodel used in their studies had been oil-wet, which can be inferred from the figures in 25

their paper by considering the shape of the water/oil interface and the contact angle with the glass (Figure 1). As a result, this experiment could be simplified into a 1-D model. In this work, it was tried to simulate the formation of water film on the surfaces of pore walls in highly water-wet micromodels. For such systems, the 1-D model would not be sufficient for predicting the required time for unblocking the trapped oil, as the mass transfer rate between oil and gas is enhanced owing to the extension of water-oil contact area in the pore. As a result, the 2-D system was subsequently used to estimate the water rupture time for experiments 1 to 4. The solubilities and diffusion coefficients of the components were calculated using the same correlations and procedure explained before, which have been presented in Table 4. Table 4: Calculated diffusion coefficient and solubility values in water for experiments 1 to 4

Experiment 1 & 2 Component

Solubility

Experiment 3 & 4

Diffusion 2

Solubility

Diffusion

(mol/mol)

Coefficient (m /s)

(mol/mol)

Coefficient (m2/s)

CO2

2.92×10-02

1.28×10-09

1.40×10-02

1.92×10-09

n-C5 / n-C10

3.96×10-05

5.18×10-10

2.38×10-08

5.89×10-10

The initial dimensions of areas occupied by the phases in these experiments were measured using image analysis technique, and were fed into the 2-D model, to calculate the rupture time. The results have been summarized in Table 5. Table 5: Summary of the initial dimensions and the results from experiments and the 2-D mathematical model

No.

Lw

do,w

do,th

Ao 2

Texperiment

Tmodel-PR

Tmodel-SRK

(mm)

(mm)

(mm)

(mm )

(min)

(min)

(min)

1

1.66

16.90

0.57

19.49

208

152

155

2

1.26

18.44

0.81

22.89

85

114

116

3

0.97

17.21

0.84

20.45

216

270

258

4

1.12

20.53

1.76

27.50

1023

956

854

26

As it has been shown, the 2-D model was able to predict the water rupture time with acceptable precision for both miscible and immiscible conditions. It proves that in addition to the initial thickness of the water barrier, there are other dimensional aspects of the system which also affect the required time for water rupture: the contact area between the oil blob and the surrounding water film in the pore, and the cross section of the oil channel in the throat. It is clearly obvious that the initial thickness of water barrier impacts the rupture of water negatively, because the average diffusion path of gas molecules increases when the water barrier is thicker, taking a longer time to reach the shielded oil and making it swell. This is in agreement with equation (7), where the Lw (t ) t term on the left hand side that denotes the reduction rate of water barrier thickness has an inverse relationship with Lw on the other side. The same statement can be made about the cross section of the oil channel in the throat, as do,th is in the denominator of the right hand side of equation (7), implying that the shrinkage speed of the water barrier decreases with increase of do,th. This is true because do,th represents the diameter of the channel through which the oil blob swells as a result of gas diffusion, and the swollen oil advances more slowly when the channel is broader. Comparing the results of experiments 3 and 4 confirms these facts, where the increase in Lw and do,th values has resulted in a much higher rupture time for the same system of fluids in the same pressure and temperature conditions.

On the contrary, increase of the area of the oil blob and the contact between the oil and the surrounding water film in the pore (Ao and do,w in the numerator of right hand side of the equation) amplifies the diminishment of the water barrier by providing a larger area for the occurrence of mass transfer between the shielded oil and the dissolved gas in water.

Meanwhile, in addition to dimensional aspects of the system, physical parameters such as temperature, pressure and the type of the fluids are of the same importance, if not more, in the 27

recovery time of water shielded oil. Looking at the dimensional characteristics of experiments 2 and 3, one could expect the rupture time of experiment 2 to be higher than that of experiment 3, since Lw2 > Lw3 and considering the close initial values of do,th, do,w and Ao in these tests. However, the experimental results show that the rupture time of experiment 3 is about 2.5 times longer than that of experiment 2. This can be attributed to the alteration of pressure and temperature conditions of the tests, which makes the solubility of CO2 in water in experiment 3 to be less than half of that in experiment 2, while its diffusion coefficient in water is only 1.5 times bigger in experiment 3.

Besides, the volumetric behaviors of CO2/n-C5 and CO2/n-C10 binary mixtures are not the same. For binary systems of CO2 + alkane, the solubility of CO2 in the alkanes increases by increasing the pressure and reducing the temperature of the system. This is because in atmospheric conditions, the strength of the intermolecular forces operating within CO2 is far less than those of alkanes, due to the larger distance between CO 2 (gas) molecules compared to alkanes (liquid). By increasing the pressure, the distance between CO 2 molecules is reduced dramatically, causing the intermolecular forces operating within CO2 to increase much faster than those between alkane molecules. As the magnitude of the intermolecular forces of CO2 molecules becomes closer to alkane, the solubility of CO2 in the alkane increases, making the volume of the alkane bigger. Therefore, pressure plays a dominant role in squeezing CO2 molecules into the alkane phase.

On the other hand, elevation of temperature enhances the Brownian motion of molecules, which makes CO2 molecules get off the drag of alkane molecules by intermolecular forces and promotes the escape of CO2 molecules from the alkane phase. Therefore, the solubility of CO2 in the alkanes and the volume of CO2 + alkane systems are decreased with increasing temperature. 28

Moreover, for straight-chain alkanes, the volume expansion coefficient decreases as the chain of the alkane is increased under the same temperature and pressure. This is owing to the linear shape of the molecules which allows them to contact each other along the entirety of their length. Therefore, longer molecules have a larger surface area of contact, with correspondingly larger intermolecular forces. As a result, CO 2 molecules are more difficultly squeezed into the alkane phase, lowering the solubility of CO2 in the alkane and the volume of the CO2 + alkane system (Yang et al., 2012). As for the role of equation state in the calculations, it can be seen that the difference between rupture times calculated for each experiment using the PR and the SRK equations of state for estimation of partial molar volumes is not significant. It implies that the ability of both equations of state in predicting the partial molar volumes in this process is approximately the same, and neither of them has proven itself superior to the other one, as in experiments 1 and 3 SRK EOS has a better prediction, while for experiments 2 and 4 PR EOS has a closer estimation to the actual rupture time. 5. Conclusions In this work, a set of visualized miscible and immiscible tertiary gas injection experiments

were performed on micromodel apparatus to study the recovery process of trapped oil by water in dead-end pores. In all of the experiments, a water film was present on the surface of the pore walls, surrounding the oil ganglion in the dead-end pore, to represent the conditions existing in water-wet reservoirs. It was found that the presence of this water layer caused the cross section area of the swelling oil in the throat to be different from that of water/oil mass transfer interface. The configuration of oil and water were investigated during the experiments, and the water rupture times were measured.

29

In the next step, a mathematical model was developed to predict the rupture time, which included the impact of non-ideal mixing of oil and gas phases for the first time. It took into account the partial molar volumes of components calculated by the PR and the SRK equations of state. The model was verified b simulating the results of Campbell and Orr’s experiment (1985). The model predicted that after 26.7 h for the PR EOS and 26.2 h for the SRK EOS, the water barrier would be completely displaced from the throat. The actual reported time by Campbell and Orr was 26.5 hours, which proved that inclusion of non-ideal mixing effect in the calculations has led to higher accuracy of the model. Subsequently, the 2D model was employed to estimate the water rupture times for the experiments conducted in this study. It was found out that in the presence of water film on the surface of the water-wet pore walls, the thickness of the water barrier, as well as the oil/water boundary in the pore and the size of the oil channel in the throat affect the required time for rupture. Both the PR EOS and the SRK equations of state showed similar capability in the prediction of partial molar volumes. The results of the model were in close agreement with the experimental data for both miscible and immiscible displacements. This makes the new model attractive for possible applications in compositional simulation models where three phases of oil, water and gas are present in the system. References  Balint, V. (1971) Method de prévision du comportemet d'un gisement d'huile balayé par du gaz carboniqe. Revue de L'Institut Francais du Pétrol, 26 (6), 473–494. 

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 Bijeljic, B., Muggeridge, A. H. and Blunt, M. J. (2003) Multicomponent Mass Transfer across Water Films During Hydrocarbon Gas Injection. Chemical Engineering Science, 58, 2377-2388.

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 Cable, A., Spearing, M., Bahamaish, J., Dabbour, Y. and Kalam, Z. (2004) Gas Displacement Efficiency for a Low Permeability Carbonate Field. International Symposium of the Society of Core Analysts, Abu Dhabi, UAE.  Campbell, B. T. and Orr, F. M. (1985). Flow Visualisation for CO 2/Crude-Oil Displacements. SPE Journal, 25, 665–678.  Cullick, A. S. and Mathis, M. L. (1984) Densities and Viscosities of Mixtures of Carbon Dioxide and nDecane from 310 to 403 K and 7 to 30 MPa. Journal of Chemical & Engineering Data, 29, 3 3−3 6  Danesh, A. (1998) PVT and Phase Behaviour of Petroleum Reservoir Fluids. Elsevier, Amsterdam.  Daniel, R., Gaddy, V.L. (1972) Natural Barriers Formed in The Region of Hydrocarbon Reservoirs. (Hungarian) Banyasz, Kohasz, lapok, Koolaj Floidgaz, 5 (2), 37–45.  Do, H. D. and Pinczewski, W. V. (1993) Diffusion Controlled Swelling of Reservoir Oil by Indirect Contact with Injection Gas. Chemical Engineering Science, 48 (18), 3243-3252.  Duan, Z. and Mao, S. (2006) A Thermodynamic Model for Calculating Methane Solubility, Density and Gas Phase Composition of Methane-Bearing Aqueous Fluids from 273 to 523 K and from 1 to 2000 Bar. Geochimica et Cosmochimica Acta, 70, 3369-3386.  Duan, Z., Sun, R., Zhu, C. and Chou, I. (2006) An improved model for the calculation of CO 2 solubility in aqueous solutions containing Na+, K+, Ca2+, Mg2+, Cl-, and SO42-. Marine Chemistry, 98, 131-139.  Grogan, A. T. and Pinczewski, W. V. (1987) The Role of Molecular Diffusion Processes in Tertiary CO 2 Flooding. Journal of Petroleum Technology, 39, 591-602.  Guo H., Chen Y., Lu W., Li L. and M. Wang. (2013) In Situ Raman Spectroscopic Study of Diffusion Coefficients of Methane in Liquid Water under High Pressure and Wide Temperatures. Fluid Phase Equilibria, 360, 274-278.  Hamedi Shokrlu, Y. and Babadagli, T. (2015) Pore-Scale Investigation of Phase Distribution and Residual-Oil Development During Secondary and Tertiary Solvent Injection. SPE Reservoir Evaluation & Engineering, (18 (1), 39-52.  Jhaveri, B. S. and Youngren, G. K. (1988) Three-Parameters Modification of the Peng-Robinson Equation of State to Improve Volume Predictions. SPE Reservoir Engineering, 3 (3), 1033-1040.  Kantzas, A., Chatzis, I. and Dullien, F. A. L. (1988) Mechanisms of Capillary Displacement of Residual Oil by Gravity-Assisted Inert Gas Injection. SPE 17506, Rocky Mountain Regional Meeting, Casper, WY.

31

 Laroche, C. (1998) Secondary and Tertiary Gas Injection Experiments in Heterogeneous Wettability Micromodels. SPE 52067, SPE European Petroleum Conference, Hague, The Netherlands.  Lin, C. T. and Daubert, T. E. (1980) Estimation of Partial Molar Volume and Fugacity Coefficient of Components in Mixtures from the Soave and Peng-Robinson Equations of State. Industrial & Engineering Chemistry Process Design and Development, 19 (1), 51-59  Lin, E. C. and Huang, E. T. S. (1990) The Effect of Rock Wettability on Water Blocking During Miscible Displacement. SPE Reservoir Engineering, 5 (2), 205-212.  Liu, Y., Jiang, L., Tang, L., Song, Y., Zhao, J., Zhang, Y., Wang, D. and Yang, M. (2015) Minimum Miscibility Pressure Estimation for a CO2/n-Decane System in Porous Media by X‑Ray CT. Experiments in Fluids, 56 (7), Article 154.  Lu, W., Guo, H., Chou, I. M., Burruss, R. C. and Li, L. (2013) Determination of Diffusion Coefficients of Carbon Dioxide in Water between 268 and 473 K in a High-Pressure Capillary Optical Cell with in Situ Raman Spectroscopic Measurements. Geochimica et Cosmochimica Acta, 115, 183-204.  Luque de Castro, M. D., Valcarcel, M. and Tena, M. T. (1994) Analytical Supercritical Fluid Extraction. Springer, Berlin.  Muller, T. and Lake, L. W. (1991) Theoretical Study of Water Blocking in Miscible Flooding. SPE Reservoir Engineering, 6(4), 445-451.  Oren, P. E., Billiotte, J. and Pinczewski, W. V. (1992) Mobilization of Waterflood Residual Oil by Gas Injection for Water-Wet Conditions. SPE Formation Evaluation, 7 (1), 70-78.  Peneloux. A., Rauzy. E. and Freze, R. (1982) A Consistent Correction for Redlich-Kwong-Soave Volumes. Fluid Phase Equilibria, 8 (1), 7-23.  Raimondi, P. and Torcaso, M. A. (1964). Distribution of the Oil Phase Obtained upon Imbibition of Water. SPE Journal, 4 (1), 49-55.  Raimondi, P., Torcaso, M. A. and Henderson, J. H. (1961) The Effect of Interstitial Water on the Mixing of Hydrocarbons during a Miscible Displacement Process. Mineral Industries Experiment Station Circular, 61.  Riazi, M., Jamiolahmady, M. and Sohrabi, M. (2011) Theoretical Investigation of Pore-Scale Mechanisms of Carbonated Water Injection. Journal of Petroleum Science and Engineering, 75, 312-326.  Riazi, M., Sohrabi, M., Jamiolahmady, M., Irland, S. and Brown, C. (2009) Direct Observation of CO2 Transport and Oil Displacement Mechanisms in CO2/Water/Oil Systems. 15th European Symposium on Improved Oil Recovery, Paris, France.

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and Excess Volumes of CO2 + Decane Mixtures from 313 to 363 K and Pressures up to 25 MPa. Journal of Chemical & Engineering Data, 50, 1030−1037

Appendix A. Derivation of the Model in This Study ased on Fick’s first law of diffusion, the molar mass transfer rate of component i, Fi,w (mol/s) through a water film of length Lw(t) (m) is: Fi ,w (t )  

Aw D i ,w S i ,w (C i , g  C i ,o (t )) Lw (t )

,

(A.1)

33

where Aw (m2) is the water area perpendicular to the direction of mass transfer, Di,w (m2/s) is the diffusion coefficient of component i in water and Si,w (mol of component i/mol of solution) is the solubility of hydrocarbon component i in water from the gas or oil phase. Ci,g (mol/m3) and Ci,o (t) (mol/m3) are the concentrations of component i in the gas and oil phase, respectively. The molar mass transfer rate in the oil phase can be written as:  C i ,o (t )V o (t )  C i ,o (t ) V (t ) Fi ,o (t )    V o (t )  o C i ,o (t ) , t t t

(A.2)

The molar mass transfer rate in the oil phase is equal in absolute terms but takes place in the opposite direction to the molar mass transfer rate of the water phase: Fi,o(t)= -Fi,w(t) ,

(A.3)

Inserting equations (A.1) and (A.2) into equation (A.3) results in: C i ,o (t ) t



D i ,w S i ,w (C i , g  C i ,o (t )) / Lw (t )  C i ,o (t )(Lw (t) / t )

,

(A.4)

Lo (t )

The volume of component i in the oil phase, Vi,o(t) (m3), in the total volume of oil, Vo(t) (m3), can be written as: Vi,o(t)= υi,o(t) Ci,o(t) Vo(t),

(A.5)

where υi,o is partial molar volume of component i in oil phase. The rate of change of the volume of component i is: V i ,o (t ) t



i ,o (t ) t

 C i ,o (t ) Vo (t )  i ,o (t ) C i ,o (t ) Vo (t )  i ,o (t )   C i ,o (t ) Vo (t )  i ,o (t )Fi ,o (t ) , (A.6) t t

The swelling/shrinkage of the oil consisting of nc hydrocarbon components can be written as:

34

V o (t ) nc V i ,o (t ) ,  t t i 1

(A.7)

Combining equations (A.1), (A.6) and (A.7) we obtain: A D S (C  C (t ))  (t )  V o (t ) nc    i ,o (t ) w i ,w i ,w i , g i ,o  C i ,o (t )V o (t ) i ,o  , t Lw (t ) t  i 1 

(A.8)

Considering the fact that the rate of oil volume change is opposite to that of water, it follows that: nc  (t)   (t )Di ,w S i ,w (C i , g  C i ,o (t )) Lw (t )    i ,o  C i ,o (t )Lo (t ) i ,o  , t Lw (t ) t  i 1 

(A.9)

Equations (A.4) and (A.9) comprise a system of ordinary differential equations (ODE), which should be solved simultaneously using a numerical solver.

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Graphical abstract

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Highlights 

Micromodel experiments are performed on dead-end pores to study water film rupture.



A new mathematical model is proposed to estimate water rupture time.



The presence of water film on the pore walls of water-wet media is modeled.



Non-ideal mixing is considered in water rupture modeling for the first time.

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