SPE 164333-MS Enhanced Oil Recovery by

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injection in an oil reservoir may modify the rheology, mobility, wettability, and ... changing the wettability of reservoir rock through their adsorption on porous ...
SPE 164333-MS Enhanced Oil Recovery by Nanoparticles Injection: Modeling and Simulation Mohamed F El-Amin, SPE, Shuyu Sun, SPE, Amgad Salama; King Abdullah University of Science and Technology (KAUST), Saudi Arabia Copyright 2013, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Middle East Oil and Gas Show and Exhibition held in Manama, Bahrain, 10–13 March 2013. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract In the present paper, a mathematical model and numerical simulation to describe the nanoparticles-water suspension imbibes into a water-oil two-phase flow in a porous medium is introduced. We extend the model to include the negative capillary pressure and mixed relative permeabilities correlations to fit with the mixed-wet system. Also, buoyancy and capillary forces as well as Brownian diffusion are considered. Throughout this investigation, we monitor the changing of the fluids and solid properties due to addition of the nanoparticles and check for possible enhancement of the oil recovery process using numerical experiments. Introduction The applications of nanoparticles in oil/gas exploration/production become a promising field of research. The nanoparticles injection in an oil reservoir may modify the rheology, mobility, wettability, and other properties of the fluids and therefore need comprehensive investigations. For example, certain types of nanoparticles can be used as tracers for oil and gas exploration and others may be used in oilfields. Nanoparticles used in the oilfields to enhance water injection by virtue of changing the wettability of reservoir rock through their adsorption on porous walls. There are two types of polysillicon nanoparticles can be used in oil fields to improve oil recovery and enhance water injection, respectively [1]. The polysilicon nanoparticles are classified based on wettability of the surface of the PN. The first type is called lipophobic and hydrophilic polysilicon nanoparticles and exists in water phase only, while the second type is called hydrophobic and lipophilic polysilicon nanoparticles and exists in the oil phase only. The nanometer (nm) unit is equal to one billionth of a meter (nm=10-9 m). Particles are classified based on the diameter size, coarse particles (10,000 nm - 2,500 nm), fine particles (2,500 nm - 100 nm), nanoparticles (100 nm - 1 nm), nanoclusters (10 nm - 1 nm) and a narrow size distribution. Nanopowders are agglomerates of ultrafine particles, nanoparticles, or nanoclusters. One kind of polysilicon nanopowder has the range between 500 and 10 nm was used in oilfields to enhance water injection by changing wettability of the porous media. If a particle larger than a pore throat may block at the pore throat during nanoparticles transport with flow in the porous medium. Experimental results [1] illustrated that the sizes of polysilicon particles are in the range of 10 - 500 nm, while pore radii of a porous medium (sandstone) are from 6 to 6.3!104 nm. A few numbers of nanoparticles types have sizes slightly less than a pore throat may bridge at the pore throat. However, the nanoparticles can be adhered to the pore walls if the nanoparticles sizes are much less than the pore sizes. Ju et al. [2] reported that when the suspension of the polysilicon nanoparticles of one nanosize is injected into an oil reservoir, it could change the wettability of porous surfaces of sandstone and consequently have effects on water and oil flows. Ju et al. [2] and Ju and Dai [1] have founded their mathematical model of nanoparticles transport in two-phase flow in porous media based on the formulation of fine particles transport in two-phase flow in porous media provided in Refs. [3-5]. Improvements in the recovered volumes by injecting hydrophobic nanoparticles which enhance or reverse the initial reservoir wettability favoring an increase in the relative permeability of the oil phase and the capillary pressure drop between phase pressures have been reported in Ref. [6]. The authors [7, 8] introduced modeling and simulations of nanoparticles transport and two-phase flow in porous media. In the current work, we extend the model to include a negative capillary pressure and mixed relative permeabilities correlations to fit with the mixed-wet system.

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Mathematical Formulation Flow Model The govern equations of the two-phase water-oil flow in porous media are mass conservation equation, and constitutive equation. Assuming that there is no mass transfer between the two phases; the governing equations may be written as, ! !!! !! !!

!! ! !

! !! ! !! !! !!!!!!! ! !! !!!!!!! !!!! !!

!!! ! !! !!! !!!!!!! ! !! !!!!!!!

(1) (2)

where ![-] is the porosity of the medium,!!! [kg m3] is the phase ! density, !! is the phase ! saturation and !! [m/s] is the phase ! velocity. ! stands for the water (wetting) phase, and ! stands for the oil (nonwetting) phase. ![m2] is the absolute permeability, ! !! [-] is the phase ! relative permeability, !! [Pa] is the phase ! pressure, ! ! !!!!! !!!! is the gravitational acceleration, and !! [m2 s-1] is the fluid viscosity. The fluid saturations for the two-phase flow of water and oil are interrelated by, !! ! !! ! !

(3)

The capillary pressure is the difference in pressure across the interface between two immiscible fluids, water and oil and thus defined by, !! ! !! !!!

(4)

and the total velocity is, !! ! !! ! !! ,

(5)

Summing the saturation equation for water phase and the oil phase, one obtains, ! ! !! ! !!!!!!

(6)

Moreover, adding the constitutive equations for each phase, Eqs. (1)-(2), and substituting into Eq. (5), we end up, !! ! !! !! ! !!! ! !! ! !!! ! !! ! !!!

(7)

where !! ! ! ! !! ! !!! is the mobility and !! ! ! !! ! ! !! ! is the total mobility and ! ! !! ! !! ! !! ! !! . In order to derive the pressure equation, substitute Eq. (7) into Eq. (6), we obtain, ! ! !! ! !!! ! !! ! !!! ! ! ! ! !!! ! !!!!!

(8)

Substituting the constitutive equation of the water phase, Eq. (2), into Eq. (1) gives, ! !!! !!

! ! ! !! ! !

(9)

Therefore, water velocity may be written as, !! ! !! !! ! !! ! !! !!! ! !! !! !!!!!!

(10)

where !! ! !! ! !! . Mixed-wet capillary pressure The aim of injecting the water-nanoparticles suspension into the oil reservoir is to change the wettability of the medium from oil-wet to water-wet, so, the rock may be considered as a mixed-wet rock. Therefore, it is worth to include a more general correlation for the capillary pressure in the model to represent its variations due to the wettability change. The most popular capillary pressure correlations are positive and limited to a primary drainage. Huang et al. [9] have proposed a general correlation to include all four branches of the bounding hysteresis loop: spontaneous and forced imbibition, and spontaneous and forced secondary drainage. Skjaeveland et al. [10] introduced a general capillary pressure correlation for mixed-wet reservoir rock based on the simple power-law form of Brooks and Corey for primary drainage capillary pressure. Between the two limits, a completely water-wet system and a completely oil-wet system, there are some other cases of mixed-wet systems need a correlation that should be symmetrical with respect to the two fluids since neither dominates the wettability.

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The general correlation is achieved by summing the two limiting expressions that resulting in the general expression, !! !!!"

!! ! !!

!!!

!!!!"

! !!

!! !!!"

!!!

!!!!"

(11)

where !! and !! are constants represents the entry pressure for imbibition and drainage, respectively. The constants !!!! and !!!! are the pore size distribution index for imbibition and drainage, respectively. As the oil pressure slowly decreases the water imbibes into the rock and consequently the water saturation increases. When the oil pressure is equal to the water pressure (pc=0), the saturation reaches the spontaneous water imbibition saturation. Increasing the saturation from this point can only be accomplished by forcing the water in, hence by increasing the water pressure above the oil pressure. By definition, the capillary pressure becomes negative. Mixed-wet relative permeabilities In a similar manner to the pc generalization, the relative permeabilities correlations may be extended to the mixed-wet systems that require symmetrization with respect to the two fluids. The relative permeability may be changed due to nanoparticles retention in a porous media. In order to define the variation of relative permeabilities caused by the nanoparticles let us firstly define the following quantities. A specific area of a sand core can be calculated by the following empirical equation [11], !!" ! !!

!

(12)

!

where ! is a constant. Also, it is important to define the total surface area in contact with fluids for all the size intervals of the nanoparticles per unit bulk volume as follows [1, 2], !!"! ! !"

!

!" ! !!!

(13)

!! is the diameter of particle interval size i. When !!"! ! !!" the total surfaces per unit bulk volume of the porous medium are completely covered by the nanoparticles adsorbed on the pore surfaces or entrapped in pore throats, while if !!"! ! !!" the surfaces per unit bulk volume of the porous medium are partially covered by the nanoparticles. Therefore, the relative permeabilities of the water and gas phases can be expressed as a linear function of the surface covered by the nanoparticles, i.e. ! ! !!"! ! !!" , thus, ! !"!! ! ! !" !

!!"! !!"

! !"!! ! ! !"

(14)

where ! !"!! is the relative permeabilities of water/oil phase when the surfaces per unit bulk volume of the porous media is completely occupied by the nanoparticles. Ju and Fan [1] reported that the effective permeability ! !" ! !! !"!! of water after nanoparticles treating is improved 1.627 - 2.136 times the effective permeability before nanoparticles treating. However, the absolute permeability decreases about 10%. So, one may write ! !"!! ! !! ! !" , such that !! is the ratio of the water relative permeability due to nanoparticles adhering. Therefore, one may write Eq. (14) as follows, ! !"!! ! ! ! !! !! ! ! ! !" where !! !

!!"! !!"

(15)

. Similarly, for the oil phase, one may write,

! !"!! ! ! ! !! !! ! ! ! !"

(16)

where !! is the ratio of the oil relative permeability due to nanoparticles adhering. Nanoparticles Transport Model In this work, we assume only one interval size of the nanoparticles in the water suspension, the transport equation for the size interval i of the nanoparticles may be written as, ! !!! !! !!

! !! ! !!! ! ! ! !!! !! !!! ! ! ! ! ! !

(17)

where i=1,2,…,m. !! is the volume concentrations of nanoparticles in the interval size i. ! ! is the rate of change of particle

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volume belonging to a source/sink term. ! ! is the net rate of loss of nanoparticles in size interval i. Nanoparticles that have sizes smaller than microns have strong Brownian motion, and can bring nanoparticles very close to the pore wall. Therefore, nanoparticle may retention to decrease as flow velocity increases. The diffusion coefficient Di of the nanoparticle can be calculated using the Stokes-Einstein equation, !! !

!! !

(18)

!"#!!

where kB is the Boltzmann constant, T absolute temperature, !! is the particle diameter, and µ is the fluid viscosity. In Eq. (18), particle diffusion constant is inversely proportional to particle diameter !! , which increases with decreasing particle size. For instance, for water at 20°C the Brownian diffusivity for a 1-µm-diameter particle is 4.3!10-9 cm2/s, which is small compared to solute diffusion but potentially significant over the small distances within pore spaces [12]. The net rate of loss of nanoparticles may be written as [1, 2, 13-15], !! !

! !" ! !!

(19)

where !" ! ! !! ! !!! is the porosity variation due to release or retention of nanoparticles of interval i. !! is the volume of the nanoparticles of interval i in available on the pore surfaces per unit bulk volume of the porous medium. !!! is the volume of the nanoparticles of interval i entrapped in pore throats from the suspension per unit bulk volume of porous medium due to plugging and bridging. Alternately, !! and !!! may be defined in terms of the mass of particles per unit fluid volume deposited at the pore bodies !! and pore throats !!! of the porous medium as, !! ! !! !!! , !!! ! !!! !!!

(20)

where !! is the density of the particulate suspension. At the critical velocity of the surface deposition only particle retention occurs while above it retention and entrainment of the nanoparticles take place simultaneously (Gruesbeck and Collins[16]). A modified Gruesbeck and Collins's model for the surface deposition is expressed by [1], !!! !!

!

!!!! !! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! ! !! !!!! !! !! !!!!! !! !! !!! !! !!!!!!!!!!!!!! !! ! !!

(21)

where !!!! is the rate coefficients for surface retention of the nanoparticles in the interval i, !!!! is the rate coefficients for entrainment of the nanoparticles in the interval i, and !! is the critical velocity. Similarly, the rate of the entrapment of the nanoparticles in interval i is, !!!! !!

! !!"!! !! !!

(22)

where !!"!! is the pore throat blocking constants. Porosity may be changed because nanoparticles deposition on the pore surfaces or blocking of pore throats. The porosity variation may be written as [1, 13], ! ! !! !

!

!"

(23)

!

where !! is the initial porosity. Also, the permeability variation due to nanoparticles deposition on the pore surfaces or blocking of pore throats may be expressed as [2], ! ! ! ! !!! ! ! ! !!!!!

!

(24)

where ! ! is the initial permeability, !! is a constant for fluid seepage allowed by the plugged pores. The flow efficiency factor expressing the fraction of unplugged pores available for flow is given by, !!!!

! ! !!!! !!

(25)

where !!!! is the coefficient of flow efficiency for particles i. The value of the exponent l has range from 2.5 to 3.5. For the nanoparticles transport carried by fluid stream in the porous media, deposition on pore surfaces and blockage in pore throats may occur. The retained particles on pore surfaces may desorb for hydrodynamic forces, and then possibly adsorb on other

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sites of the pore bodies or get entrapped at other pore throats. Results and Discussion In order to get physical insights for the problem under consideration, we consider a flow of type countercurrent imbibition [17-20]. In countercurrent imbibition both water and oil phases flow through one inflow-outflow boundary. Therefore, the total velocity becomes zero, !! ! !. The flow equations may be written as, !!! !!

! !!! !

!!! ! !!

!! ! !

(26)

and the water velocity becomes, !!!

!! ! !!! !!

!!

! !!!

(27)

Therefore, the flow equation for the water phase becomes, ! !!! !!

!!

! !!

!!!

! ! !!

!!

! !!!

!!

(28)

The corresponding transport equation for the interval i of the nanoparticles is given by, ! !!! !! !!

! !!

!!! !!

!

! !!

!!! !!

!!! !!

! !!

(29)

where !! !

!!!! ! !!"!!

!! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! ! !!

!!!! ! !!"!!

!! !! ! ! ! !!!! !! !! ! !! !! !!!!!!!!!!!!!!!!!! !! ! !!

(30)

The surface deposition of the nanoparticles in the interval i is, !!! !!

!

!!!! !! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !! ! !! !!!! !! !! ! !!!! !! !! ! !! !! !!!!!!!!!!!!!! !! ! !!

(31)

The rate of entrapment of the nanoparticles in interval i is, !!!! !!

! !!"!! !! !!

(32)

The initial conditions are, !! ! !!" ! !! ! !! ! !!! ! !!!!!!!!!!!!!!!!!!! ! !!!!!! ! ! ! !

(33)

where ! is the rock depth and !!" is the initial water saturation. The boundary conditions are, !! ! ! ! !!" ! !! ! !!!! ! !! ! !!! ! !!!!!!!!!!!! ! !!!!!! ! ! !"! !"

!

!"! !"

!

!"! !"

!

!!!! !"

! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! !!!!!!!!!! ! !

(34) (35)

where !!!! is the concentration of nanoparticeles in the nanoparticles suspension at the inlet boundary. The governing equations (28) – (32) are solved numerically along with the initial and boundary conditions, (33) - (35). An efficient algorithm is used to solve the above high-nonlinear parabolic partial differential equation in one space variable ! and time !. The Galerkin method is used for spatial discretization [21], while the time integration for the resulting ordinary differential equation is done with an adaptive time step. 100 points of the spatial grid were used during calculations and were enough to provide an acceptable accuracy. Now, we consider one size nanoparticles suspension in the water phase at the inlet, with the following parameter [1], !!!! ! !"!! !! , !!"!! ! !!!"!! !! , !!!! ! !"!! !! , !! ! !!!!!"!! !!!!, and !! ! !!!!!"!! !! ! !!. The nanoparticles diameter is taken as 40 nm and concentration !!!! ! !!! (without nanoparticles), 0.0009, 0.004, and 0.01. The remaining model parameters are; !!" ! !!" ! !!!!", !! ! !! ! !!!, !! ! !"", !! ! !!"", !! ! !, ! ! !, !! ! !!!, !! ! !!!", !!"! ! !!"! ! !, ! ! ! ! !, ! ! !!!, !! ! !!!"!!"!!" , !! ! !!!, !! ! !. The normalized nanoparticles

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concentration is plotted in Fig. 1 against the dimensionless distance with various inlet concentrations and imbibition times. From this figure it can be seen that the concentration of nanoparticles increases as the concentration at the inlet increases. Also, an interesting phenomenon can be observed from Fig. 1 that the concentration increases with time but at after a certain time of imbibition concentration of nanoparticles start to decrease with time. This may be interpreted by increasing the rate of nanoparticles precipitation on the pore surface of the porous medium. 1 0.9 2 hour, Ci,0=0.0009

0.8

01 day, Ci,0=0.0009 0.7

10 day, Ci,0=0.0009 80 day, Ci,0=0.0009

Ci/Ci,0

0.6

2 hour, Ci,0=0.004 01 day, Ci,0=0.004

0.5

10 day, Ci,0=0.004 0.4

80 day, Ci,0=0.004

0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5 z/H

0.6

0.7

0.8

0.9

1

Fig. 1: Normalized nanoparticles concentration against dimensionless distance with various inlet concentrations and imbibition times.

1 1 day, Ci,0=0.0 0.9

10 day, Ci,0=0.0 40 day, Ci,0=0.0

0.8

1 day, Ci,0=0.004 0.7

10 day, Ci,0=0.004 40 day, Ci,0=0.004

Sw

0.6

1 day, Ci,0=0.01 10 day, Ci,0=0.01

0.5

40 day, Ci,0=0.01 0.4 0.3 0.2 0.1 0 0

0.1

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0.4

0.5 z/H

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0.8

0.9

1

Fig. 2: Water saturation against the dimensionless distance with various imbibition times and inlet concentrations Fig. 2 shows the water saturation against the dimensionless distance with various imbibition times and inlet concentrations. As expected one may note that the saturation of water increases with the time imbibition. Also, the concentration of

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nanoparticles increases the water saturation in particular after significant time of imbibition. Ratios of the permeability and porosity are plotted against the dimensionless distance with various imbibition times and inlet concentrations in Figs. 3 and 4, respectively. A reduction in the permeability and the porosity based on equations (23) and (24) due to the precipitation of the nanoparticles on the pores walls. Of course both inlet concentration and imbibition time reduces the permeability and the porosity. 1 0.99 10 day, Ci,0=0.0009

0.98

20 day, Ci,0=0.0009 40 day, Ci,0=0.0009

K/K0

0.97

80 day, Ci,0=0.0009 10 day, Ci,0=0.004

0.96

20 day, Ci,0=0.004 40 day, Ci,0=0.004

0.95

80 day, Ci,0=0.004 10 day, Ci,0=0.01 20 day, Ci,0=0.01

0.94

40 day, Ci,0=0.01 80 day, Ci,0=0.01

0.93 0.92 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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1

z/H

Fig. 3: Ratio of the permeability against the dimensionless distance with various imbibition times and inlet concentrations

1

0.995

!/!0

0.99

10 day, Ci,0=0.0009 20 day, Ci,0=0.0009 40 day, Ci,0=0.0009 80 day, Ci,0=0.0009 10 day, Ci,0=0.004 20 day, Ci,0=0.004 40 day, Ci,0=0.004 80 day, Ci,0=0.004 10 day, Ci,0=0.01 20 day, Ci,0=0.01 40 day, Ci,0=0.01 80 day, Ci,0=0.01

0.985

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0.975

0.97 0

0.1

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1

z/H

Fig. 4: Ratio of the permeability against the dimensionless distance with various imbibition times and inlet concentrations

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0.5

40 day

0.45

Ci,0=0.0

0.4 0.35

Ci,0=0.004

0.3

Ci,0=0.01

0.25

krw

kro

0.2 0.15 0.1 0.05 0 0

0.1

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1

S

Fig. 5: Water and relative permeabilities against the normalized water saturation with various inlet concentrations after 40 day of imbibition 0.7 Ci,0=0.01 0.6

01 day 10 day

0.5

40 day

0.4

kro

krw

0.3

0.2

0.1

0 0

0.1

0.2

0.3

0.4

0.5 S

0.6

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1

Fig. 6: Water and relative permeabilities against the normalized water saturation with various imbibition times with 0.01 of inlet concentration Figs. 5 and 6 illustrate, respectively, the water and relative permeabilities against the normalized water saturation with various inlet concentrations and times of imbibition. It is noteworthy that as the nanoparticles concentration increases the water relative permeability decreases. In general, the relative permeability of a fluid is higher when it is the nonwetting fluid. Thus, the water relative permeability is higher in the oil-wet system than it would be at the system was water-wet. This occurs because the wetting fluid (water) tends to travel through the smaller, less permeable pores, while the non-wetting fluid (oil) travels more easily in the larger pores. This means increasing the nanoparticles aids changing the system from oil-wet to water-wet which in turn enhances the oil recovery. On the other hand, the oil relative permeability is high because the oil phase flows through the centers of the larger pores. Capillary pressure profiles are plotted in Fig. 7 against the normalized

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water saturation with various times of imbibition and inlet concentrations. From this figure one may note that at a high water saturation the capillary pressure becomes negative because the oil pressure is greater than the water pressure, which means the system is converted to a water-wet system. 6000 01 day, Ci,0=0.0 10 day, Ci,0=0.0 40 day, Ci,0=0.0 01 day, Ci,0=0.004 10 day, Ci,0=0.004 40 day, Ci,0=0.004 01 day, Ci,0=0.01 10 day, Ci,0=0.01 40 day, Ci,0=0.01

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0 0 -1000

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S

Fig. 7: Capillary pressure against the normalized water saturation with various times of imbibition and inlet concentrations References [1] B. Ju and T. Fan, "Experimental study and mathematical model of nanoparticle transport in porous media," Powder Technology, vol. 192, pp. 195–202, 2009. [2] B. Ju, T. Fan, and X. Qiu, "A study of wettability and permeability change caused by adsorption of nanometer structured polysilicon on the surface of porous media, SPE-77938," presented at the SPE Asia Pacific Oil and Gas Conference and Exhibition, Melbourne, Australia, 2002. [3] X. H. Liu and F. Civian, "Characterization and prediction of formation damage in two-phase flow systems, SPE-25429," presented at the Production Operations Symposium, Oklahoma City, OK, U.S.A. [4] X. H. Liu and F. Civian, "A multiphase mud fluid infiltration and filter cake formation model, SPE-25215," presented at the SPE International Symposium on Oilfield Chemistry, New Orleans, LA, U.S.A. [5] X. H. Liu and C. Faruk, "Formation damage and skin factor due to filter cake formation and fines migration in the NearWellbore Region, SPE 27364," presented at the SPE Symposium on Formation Damage Control, Lafayette, Louisiana, 1994. [6] M. O. Onyekonwu and N. A. Ogolo, "Investigating the use of nanoparticles in enhancing oil recovery, SPE-140744," presented at the Annual International Conference and Exhibition, Tinapa-Calabar, Nigeria, 2010. [7] M. El-Amin, A. Salama, and S. Sun, "Modeling and simulation of nanoparticles transport in a two-phase flow in porous media," in SPE International Oilfield Nanotechnology Conference and Exhibition, Noordwijk, The Netherlands, 2012. [8] M. El-Amin, S. Sun, and A. Salama, "Modeling and Simulation of Nanoparticle Transport in Multiphase Flows in Porous Media: CO2 Sequestration," in Mathematical Methods in Fluid Dynamics and Simulation of Giant Oil and Gas Reservoirs, 2012. [9] D. D. Huang, M. M. Honarpour, and R. Al-Hussainy, "An improved model for relative permeability and capillary pressure incorporating wettability," in SCA, 1997, pp. 7-10. [10] S. Skjaeveland, L. Siqveland, A. Kjosavik, W. Hammervold, and G. Virnovsky, "Capillary pressure correlation for mixed-wet reservoirs," in SPE India Oil and Gas Conference and Exhibition, 1998. [11] J. S. Qin and A. F. Li, Physics of Oil Reservoir: Publishing Company, U. P. C, 2001. [12] T. Zhang, "Modeling of nanoparticle transport in porous media," 2012. [13] X. H. Liu and F. Civian, "Characterization and prediction of formation damage in two-phase flow systems, SPE-25429," presented at the Production Operations Symposium, Oklahoma City, OK, U.S.A, 1993. [14] X. H. Liu and F. Civian, "A multiphase mud fluid infiltration and filter cake formation model, SPE-25215," presented at the SPE International Symposium on Oilfield Chemistry, New Orleans, LA, U.S.A., 1993. [15] X. H. Liu and F. Civan, "Formation damage and skin factor due to filter cake formation and fines migration in the Near-

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Wellbore Region, SPE 27364," presented at the SPE Symposium on Formation Damage Control, Lafayette, Louisiana, 1994. [16] C. Gruesbeck and R. E. Collins, "Entrainment and deposition of fines particles in porous media," Soc. Pet. Eng. J., vol. 24, pp. 847–855, 1982. [17] M. El-Amin, A. Salama, and S. Sun, "Numerical and dimensional investigation of two-phase countercurrent imbibition in porous media," Journal of Computational and Applied Mathematics, 2012. [18] S. Sun, A. Salama, and M. El Amin, "Matrix-oriented implementation for the numerical solution of the partial differential equations governing flows and transport in porous media," Computers & Fluids, 2012. [19] M. El-Amin, A. Salama, and S. Sun, "Effects of Gravity and Inlet Location on a Two-Phase Countercurrent Imbibition in Porous Media," International Journal of Chemical Engineering, vol. 2012, 2012. [20] M. El-Amin and S. Sun, "Effects of Gravity and Inlet/Outlet Location on a Two-Phase Cocurrent Imbibition in Porous Media," Journal of Applied Mathematics, vol. 2011, 2011. [21] R. D. Skeel and M. Berzins, "A method for the spatial discretization of parabolic equations in one space variable," SIAM journal on scientific and statistical computing, vol. 11, pp. 1-32, 1990.