JOGCE Journal of Oil, Gas and Coal Engineering Vol. 3(1), pp. 023-033, August, 2018. © www.premierpublishers.org. ISSN: 0767-0974
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The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Mustapha Abdullahi School of Chemical Engineering and Analytical Science, University of Manchester, Manchester M13 9PL, UK Email: [email protected], Tel: +447388119656
The Capillary number hypothesis is very empirical in Surfactant flooding Enhanced Oil Recovery (EOR) method. which is of modest experience on the North Sea and many other offshore platforms. The capillary number drives the force wetting processes, which is controlled by the balance between capillary and viscous forces. The mobilization of oil trapped in pores of water- wet rock is steered by capillary number that is typically within specific ranges (ퟏퟎ−ퟓ to ퟏퟎ−ퟒ). There is high uncertainty and confusion in the parameterization of capillary number formula, as every quantity is given on a macroscale level. As demonstrated herein, a new microscopic capillary number parameterization was proposed. This paper is written to improve the numerical formulation of capillary number in surfactant flooding model. The new formula for capillary number was derived based on existing equations as a function of residual oil saturation and tested. Thus, the proposed mobility mechanism easily accounts for a broader critical range of capillary number (ퟏퟎ−ퟔ to ퟏퟎ−ퟒ) in comparison with available models with a critical capillary number (ퟏퟎ−ퟓ to ퟏퟎ−ퟒ). We used an existing model to quantify the effect of capillary number on a miscible and immiscible relative permeability curves by computing the interpolation parameter 푭풌풓 as a tabulated function of the Logarithm (base 10) of the capillary number using a new capillary number formulation.
Keywords: Critical capillary number; Capillary number; capillary desaturation; miscibility; Interpolation parameter.
INTRODUCTION
Up to two-thirds of the crude oil remains trapped in the 푢 is Darcy’s velocity, µ is the viscosity of the displacing reservoirs after primary and secondary recovery in an fluid, 𝜎 is the interfacial tension between oil and the average oil reservoir (Yan and Qiu, 2016). EOR is then surfactant solution, ∆p is the density difference between required to optimise the depletion, as the remaining oil is the phases, g is acceleration due to gravity and d is a trapped in the pore structure of the reservoir. characteristic dimension taken here as the pore diameter.
The fluid flow in porous media is mainly governed by three The role of interfacial tension and capillary number in a forces: gravitational force, viscous force and capillary dynamic air-liquid interface during an immiscible fluid forces. These three forces are integrated through the displacement flow driven by pressure was investigated by Capillary number (푁푐) and the bond number (퐵표) defined Yan & Qiu (2016). The critical capillary number was by: studied analytically, and they derived the theory regarding the critical capillary number with the wettability effect in a 푢휇 pressure-driven capillary tube. Their results show that the 푁푐 = (1) critical capillary number is associated with the contact 𝜎 angle, slip length and capillary radius. Basante (2010) ∆𝜌푔푑2 conducted a laboratory experimental work using two 퐵 = 표 𝜎 (2) sandstone cores to demonstrate the effect of capillary
The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Abdullahi M. 024
number on mobility control. The result of the experiment where 휆푤 and 휆표are constants for water and oil relative shows that raising the flow rate by a factor of 10 permeability, respectively, for the same rock type. With the (0.03ml/min to 0.3 ml/min) will decrease the oil recovery sandstone cores associated with the fluids in Shen’s factor considerably. The water-oil relative permeability experiment, the curve fitting of the experimental data leads characterises two-phase flow, and it’s difficult to determine to 휆표=2.006 and 휆푤 =3.807 (Shen et al., 2006). the functional form in a reservoir study. The addition of various chemical agents (For example, Surfactant) during Xu et al. (2011) conducted experiments to determine the chemical flooding in oil production will significantly change surfactant’s performance, such as the relationship the interfacial tension and increase the degree of difficulty between surfactant concentration and oil/water interfacial in measurement. Shen et al. developed an improved tension and the relationship between the surfactant method of measuring water-oil relative permeability concentration and the water viscosity. Their results show curves, shown that the relative permeabilities of both water that oil/water interfacial tension will decrease as the and oil phases will increase with decreasing interfacial surfactant concentration increase, surfactant flooding has tension (Shen et al., 2006). the capacity of enhancing oil recovery. Their results show that the optimum surfactant concentration is 2%, which A logarithm relationship exists between water-oil two- can improve oil recovery by the percentage of 0.22. Ren phase relative permeability and interfacial tension (Shen et et al. (2018) used both numerical and analytical method to al., 2006). characterise the migration, trapping and accumulation of Base on several experiments, relative permeability is 퐶푂2 in a saline aquifer during geological sequestration. considered as a function of saturation, interfacial tension, They used a 1D two-phased-flow model and solved the and properties of core pore only (Shen et al., 2006). model equations using the method of characteristics. Their results demonstrated that the 퐶푂2accumulated by (3) permeability hindrance is greater than that accumulated by 퐾 = 푚 (푆∗ )푛푤 푟푤 푤 푤 capillary trapping. ∗ 푛표 퐾푟표 = 푚표(1 − 푆푤) (4) Alquaimi et al. (2018) proposed a new capillary number where 푚푤 and 푚표 are coefficients of water and oil relative definition for fractures that depends on force balance and permeability functions respectively incorporates geometrical characteristics of the fracture ∗ Note that 푚푤 is the water relative permeability at 푆푤 = model. They conducted an experimental desaturation ∗ 1 and 푚표 is the oil relative permeability at 푆푤 = 0, procedure to test their capillary number definition and respectively. quantify the relationship between the pressure and By definition above, trapped ganglions. Bryan & Kantzas (2009) performed (5) core flooding experiments to investigate how Alkali- 푚 = (퐾∗ ) 푤 푟푤 푆표푟 Surfactant flooding can lead to improved heavy oil ∗ 푚표 = (퐾푟표)푆푤푐 (6) recovery. It was determined from their results that the performance of surfactant alone was not sufficient to Therefore the normalise formula can be derived from the emulsify oil, but can only increase the water-wetting of the two relative permeability equations above: glass, but the combination of alkali and surfactant can reduce the oil-water interfacial tension and oil/water (7) emulsions will be produced. Furthermore, their results 퐾∗ = (푆∗ )푛푤 푟푤 푤 show that the mechanism of emulsification and ∗ ∗ 푛표 퐾푟표 = (1 − 푆푤) (8) entrainment, which occurs during high rate flow in lower permeability cores is not as efficient in recovering The exponential indexes, 푛푤 푎푛푑 푛표 are found to be related additional oil. to interfacial tension and the pore size distribution parameters, 휆푤 and 휆표 (Brooks and Corey, 1964), as Laforce et al. (2008) used analytical solutions to study the development of multi-contact miscibility in simultaneous (9) water and gas (SWAG) injection into a reservoir, they 푛표 = 푛표(𝜎푤표,휆표) considered the application of 퐶푂2 storage in enhanced oil 푛푤 = 푛푤(𝜎푤표,휆푤) (10) recovery using a fully compositional one-dimensional and three-phase flow through porous media. Their results A particular relation between exponential constants and demonstrate that miscibility does not develop when the interfacial tension was derived, and the two-phase relative fraction of water in the injection mixture is sufficiently high permeability model from the relative permeability and define the minimum gas fraction necessary to achieve equations above has the form(Shen et al., 2006): miscibility and highlights the importance of improved relative permeability models.
∗ ∗ [표.9371.푙표푔(휎푤표)+휆푤] (11) 퐾푟푤 = (푆푤) Lohne et al. (2012) investigated the influence of capillary ∗ ∗ [0.1960.푙표푔(휎푤표)+휆표] forces on segregated flow behind the displacement front 퐾푟표 = (1 − 푆푤) (12) The New Capillary Number Parameterization for Simulation in Surfactant Flooding
J. Oil, Gas Coal Engin. 025 by numerical simulations of homogeneous and 퐹 = 1 − [(훼푁 )푛 + 1]−1 (13) 푘푟 푐 heterogeneous models. Their results show that the positive effect of gravity segregation is that the oil floats Where 푛 ≃ −0.75 seems to fit data they conducted by up, accumulates under low permeable rocks and thereby laboratory measurements. The scaling parameter α is increases the effective horizontal oil mobility. They found used to fit the measured data. the magnitude of the incremental oil production to increase with increasing curvature of oil relative permeability. Thus, by referring to the latest previously published Hence, the positive effect of decreasing IFT is larger in literature on the miscibility development in surfactant mixed-wet formation than in water-wet formations. flooding, it can be concluded that the capillary number and Keshtkar et al. (2016) developed an explicit composition the interfacial tension plays a vital role in accurately and explicit-saturation method to study surfactant flooding predicting miscibility in Surfactant flooding to enhance oil sensitivity analysis on an oil production reservoir. Their recovery. The previous literature has not critically results show that the addition of surfactant causes a evaluated the mobility components as a function of the reduction in IFT between water and oil phases and miscibility level between surfactant, water, and oil, most of subsequently will trigger the mobility of the trapped oil and the research and discussion conducted in the literature are increase the oil production level. focused on miscibility in gas and oil system. Therefore is need to come up with a numerical model that will Felix et al. (2015) carried out various experiments to accurately give a clear picture of the transport components implement surfactant polymer flooding. Different slugs as a function of miscibility and various formulations in were injected after water flooding, and their results showed surfactant-water and oil system. different displacement efficiencies based on the mechanism chosen for the implementation of the surfactant polymer flooding. These experiments revealed METHODOLOGY AND MODEL SETUP the importance of selecting the right tool for the surfactant flooding as to optimise recovery. Xavier. (2011) Reservoir simulation is a useful tool for estimating the experimentally determined the influence of surfactant future behaviour of petroleum fields. In some cases, it can concentration on hydrocarbon recovery. The interfacial also be used for identifying particular phenomena in a tension between brine and kerosene was studied with the specific task. In this study, the investigation was performed use of sodium dodecyl sulphate (SDS) as a means of using numerical simulation experiments. New formulation reducing the interfacial tension. His findings were that the for the capillary number was derived. And Eclipse Black IFT decreases as the surfactant concentration increases Oil model with surfactant option was used for simulating and reaches a point of critical micelle concentration the displacement process to see the effect of off surfactant (CMC). Thus was able to find a critical surfactant and to test the transport mechanism which was derived as concentration at 0.3 wt% of the surfactant. shown in Figure 1. Another simulation was conducted to Cheng et al. (2005) presented results on developed validate the implementation of surfactant flooding using miscibility by gas injection in petroleum reservoir with the Matlab Reservoir Simulation Tool (MRST). aim to define a minimum miscibility pressure (MMP) for the fluid system. He proposed a method for the optimum number of grid cells and time step size for numerical simulation models and verified 1D first-contact miscibility displacement. Khanamiri et al. (2015) conducted laboratory surfactant flooding experiments with aged sandstone cores, surfactant sodium dodecylbenzene sulfonate was used at a concentration of 0.05wt% and 0.2wt% to enhance oil recovery. The result shows the effect of surfactant concentration on Interfacial tension i.e. decrease in IFT with an increase in surfactant concentration. They discovered that the low salinity surfactant flooding with 0.2wt% surfactant concentration did not result in higher oil recovery that the flooding with 0.05wt% surfactant concentration in tertiary low salinity surfactant injection. This is because the tertiary low salinity surfactant injection after secondary low salinity water injection is more efficient than the tertiary surfactant injection where the surfactant is injected after high salinity (secondary) and low salinity water (tertiary) injection.
Whitson et al. (1997) proposed a mathematical formulation for relative permeability and the interpolation function for a Gas-oil system as a function of the capillary number: Figure 1: Simulation process
The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Abdullahi M. 026
1. Proposed new capillary number equation The first term is the applied pressure gradient and the second contains gravity and other acceleration terms. Capillary pressure relationship is expressed by a single The LHS of inequality is comprised of the pressure dimensionless function known as the Leverett Number gradient and length of the alternative flow path around the (Leverett, 1941): trap and can be defined as a dimensionless length regarding the pore entry radius: 푃 퐾 푁 = 푐 √ 퐿푒 𝜎푇(휃) 휙 (14) 푚 = ∆퐿/푟푛 (20) by defining 푇(휃) as: 푃 (푆)푟 The geometrical factor ψ for the interface front with an 푇(휃) = 푐 푛 (15) advancing angle α and receding angle β is similar to that 2𝜎 given by Melrose and Brandner (Melrose, 1974): In a cylindrical oil mass of average radius a, the average length as proposed by (Stegemeier, 1974) is: 푇(훽)
휓 = 푇(ά) − 푟 (21) 2휎 푏⁄ 퐿 = (16) 푟푛 푎훻휙 푟 푟 푏⁄ is the pore body/ pore neck radii, 푏⁄ is comparible In the case of wetting phase trapping, residual fluid is held 푟푛 푟푛 in rings, interconnected with only thin water layers as to the Difficulty index by (Dullien et al., 1972) and its described by Basante (2010). The geometrical factor for defined as: 0 0 the contact angle interface passing through the pore is: 1 1 훼{푟 , 푟 }. 푑푟 . 푑푟 퐷 = [ − ] ∫ ∫ 푏 푛 푛 푏 (22) 푐표푠(휃 − 휂) 푟 푟 푇(휃) = (16) 푛 푏 ∞ 푟푏
1 + (푟푡/푟푛)(1 − 푐표푠휂) 푇(휃) is the pore shape wetting factor, The maximum The difficulty index D is an index measuring the difficulty of recovering waterflood residuals in tertiary surfactant interface curvature exists at 푛푚 Basante (2010) is given as: flooding. 푠푖푛 휃 For special cases of 훼 = 훽 = 0, 휓 = 푟푛[(1/푟푛) − (1/푟푏)]. 푛 = 휃 − 푎푟푐 푠푖푛 [ ] (17) 푚 ( ) 1 + 푟푛/푟푡 The fluid-rock only partially describes the structure of the rock k; the pore inlet size distribution can be express by combining Eq (14) and eq (15): 2√퐾/휙 푟푛(푆) = 푁퐿푒(푆) (23)
Substituting equation Equations (21), (22) and (24) into equation (19) gives
1 1 퐾 훻훷 [ ] . [ ] . [2푚(푆)] . [ ] ≥ 1 푁2 (푆) 휓(푆) 휙 𝜎 (24) 퐿푒
m= dimensionless alternative part length (a dimensionless length regarding the pore entry radius). We can define the dimensionless length of the entire multi- Figure 2: MultiPore Model pore trapped oil mass as: ∆퐿 푓 = For a nonwetting immobile (Oil) phase such as that shown 푎 (25) in figure 2 above, a force balance demonstrating the oil will be displaced if the applied pressure exceeds the net where a is the average radius of trapped oil mass. restraining capillary pressure as proposed by (Stegemeier, When equation (26) is combined with equation (16) an 1974): expression for the alternative flow path regarding interfacial tension and the pressure gradient is: 2𝜎 ∆푃퐴 = 훻훷 . ∆퐿 ˃ . 휓 = ∆푃푐 푟푛 (18) 2𝜎푓 ∆퐿 = √ (26) The applied potential gradient is defined as: 훻훷 By using equation (21), (24) and (27), The dimensionless 푑푝 flow path will be given as: 훻훷 = + ∆𝜌푔(1 + 퐺) (19)
푑퐿 푓 1/2 휙𝜎 1/2 (27) 푚(푆) = [푁퐿푒(푆)] . [ ] . [ ] 2 퐾훻훷
The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Trapping at a higher ratio of 훻훷 will result from the locations. The new pattern will have smaller, but still 휎 closely fitting pieces, so that in the low viscous/capillary separation at both original and at weaker filament J. Oil, Gas Coal Engin. 027
ratio region, residual saturation may be practically unchanged. The above is the Capillary number equation derived as a function of the residual saturation or initial saturation By combining equations (25) and (28), the ratio of the depending on availability of experimental data. viscous to capillary forces originally proposed by Brooks The parameters in the proposed capillary number equation and Corey (1964) is equated to three properties of the are given as follows: fluid-rock system: 푁퐿푒 Leverett Number ϕ Porosity 퐾훻훷 휓 Geometrical factor for curvature 푁 (푆) = ≥ [휙푁2 (푆)] . [휓2(푆)]. [1/2푓] 푐1 𝜎 퐿푒 (28) K Permeability 푆푤 Water saturation The first right-hand side of the inequality defines the 푁푐 Capillary number geometric of the rock pore network, the second term f Dimensionless length of the entire pore defines pore body/pore neck radii and its connection with a contact angle, and the third is a constant fluid geometric The Leverett Number is: property. 푃 퐾 푁 = 푐 √ (31) 퐿푒 𝜎 휙 Another dimensionless number 푁푉퐶 is obtained by substituting Darcy’s law into Equation (29) and placing The geometrical factor for curvature is: relative permeability in the rock-fluid property term, these results in the equation below: 푇(훽) 휓 = 푇(훼) − 푟 ⁄푟 (32) 푢휇 푏 푛 푁 (푆) = ≥ [휙푁2 (푆)] . [퐾 (푆)휓2(푆)]. [1/2푓] 푐 𝜎 퐿푒 푟푤 (29) where 훼 is the advancing contact angle of oil/water interface, 훽 is the receding contact angle of oil/water
interface, 푟 is pore body radii, 푟 is pore neck radii This number, which differs from 푁 by a factor of 퐾 , 푏 푛 푐1 푟푤 And the Dimensionless length of the entire pore is given segregates all rock properties to the Right-hand Side and as: thereby provides a good measure for comparative ease of recovery from different rocks. 퐿 푓 = 푎 (33 Although 푁퐿푒(S) in equation (30) was derived as a function of initial saturations, it can be expressed in terms of where 퐿 is Length of multi pore oil mass, 푎 is the average normalized residual oil,푆푅 = 푆표푟푐⁄푆표푟, because 푆표푟푐 itself is radius of a multi pore oil mass. a single function of initial saturation (S). 푆표푟 is defined as the maximum trapped saturation or the residual saturation The geometrical factor and the dimensionless length (f) at a small 푁푐. values can only be obtained experimentally. To test the proposed equation, various parameters may be used as 푆표푟 is a rock property only, the relative permeability to proposed by Melrose and Brandner (1975). Thereby given water, 퐾푟푤, can be expressed in terms of normalized the value of 휓 = 0.35 and 푓 = 2.7, the capillary number saturation , when we substitute this function into Equation and the interpolation parameters of various sample (30) relates normalized residual oil and capillary number: computed and given in 푢휇 푁 (푆 ) = ≥ [휙푁2 (푆 )] . [퐾 (푆 )휓2(푆 )] . [1/2푓] 퐶 푅 𝜎 퐿푒 푅 푟푤 푅 푅 (30)
Table 1: Calculations of Capillary number and the interpolation parameter using the proposed formulation
Calculations of Capillary number and the interpolation parameter 푭풌풓
2 2 ϕ NLe Krw(Sw) ψ f ϕN K ((S ))ψ log 10 (Nc) Nc interpolant N = Le rw w C 2f
The New Capillary Number Parameterization for Simulation in Surfactant Flooding
0.3 0.44 0.000002 0.35 18 4 X 10−8 -7.4 0
0.48 0.0049 1 X 10−6 -5.3 0
0.51 0.12 1.2X 10−4 -4.3 0.5
0.62 0.4 1.6 X 10−3 -3.7 1
0.77 0.8 2 X 10−2 -1.6 1
Abdullahi M. 028
2. Numerical Simulation model for surfactant Permeability X 퐾푥푥 100 mD flooding Permeability Y 퐾 100 mD The purpose of the numerical simulation is to verify the 푦푦 adaptability of the new formulation and the reliability of the Permeability Z 퐾 20 mD parameters used. Also to identify miscibility from capillary 푧푧 number dynamics and the effect of relative permeability interpolation to numerically replicate miscibility. To Initial pressure 푃푖 300 bar demonstrate the development effect of surfactant flooding, analyse the model-based calculated capillary number by Top depth 1000 m studying the dynamic changes in Velocity, interfacial tension, and local adsorption. Also to examine the various effect of transport component as a function of miscibility. Table 4: Reservoir Fluid parameters (Jørgensen, 2013) Initial water saturation 푆푤,푖 0.2 The surfactant flooding simulation was implemented on MRST. The surfactant model in Eclipse assumes black-oil 3 Surface Density of water 𝜌휔,푠푐 1080 Kg⁄sm fluid representation. However, in MRST the surfactant is assumed to be only dissolved in the aqueous phase and is Surface Density of oil 𝜌 800 Kg⁄sm3 added to the injected water as a mass per volume 표,푠푐 concentration (Kg/Sm3). The geological model used in the Reference viscosity of oil 휇 0.61 mPa s simulation is a modified 1-D surfactant data file from the 표,푟푒푓 MRST. The surfactant specific data consisting of tabulated values are obtained from experimental work done by Xu et Reference viscosity of water 휇푤,푟푒푓 5.0 mPa s al. (2011) as tabulated in Table 2. The reservoir physical properties and fluid properties used in the simulation are Reference pressure 푃푟푒푓 300 bar given in Table 3 and Table 4 respectively.
Table 2: Surfactant properties used [5] SIMULATION RESULTS AND DISCUSSION Csurf IFT(N⁄m) 휇푤 (cP) Adsorption The simulation is carried out using MRST by comparing (Kg⁄sm3) (Kg⁄Kg) various parameters including the capillary number between cases of zero surfactants (퐶 = 0푘푔/푠푚3) and fifty 0 0.05 0.61 0 surfactant concentration (퐶 = 50푘푔/푠푚3) along a reservoir grid cell with size (x). The position of the reservoir grid is 30 1E-05 0.8 0.0005 donated with x in metres and the reservoir grid used range from 0 to 100 metres. 100 1E-06 1 0.0005
Table 3: Reservoir physical parameters (Jørgensen, 2013) Porosity Φ 0.3
The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Figure 3: Capillary number profile within the grid with surfactant concentration of 0풌품/풔풎ퟑ and 50풌품/풔풎ퟑ
J. Oil, Gas Coal Engin. 029
0 surfactant concentration 50 surfactant concentration 0.06 0.06 0.05 0.05 0.04 0.04 time= 1day time= 1day 0.03
0.03 IFT IFT 0.02 time= 240 0.02 time= day 240days 0.01 0.01 0.00 time= 480 0 time= day 480days 0 50 100 0 50 100 Position x (m) Position x(m)
Figure 4: Interfacial tension profile within the grid with surfactant concentration of 0풌품/풔풎ퟑ and 50풌품/풔풎ퟑ
The New Capillary Number Parameterization for Simulation in Surfactant Flooding
0 surfactant concentration 50 surfactant concentration
5.00E-09 5.00E-09
time= 1 day time= 1 day Vel
time= 240 Vel time= 240 day day
5.00E-10 time= 480 5.00E-10 time= 480 day day 0 50 100 0 50 100 Position x(m) Position x(m)
Figure 5: Absolute velocity profile within the grid with surfactant concentration of 0풌품/풔풎ퟑ and 50풌품/풔풎ퟑ
0 1 0 1 Figure 6: Oil saturation distribution map after flooding with surfactant concentration of 0풌품/풔풎ퟑ and 50풌품/풔풎ퟑ
The capillary numbers in each grid cell were calculated clearly indicates that the velocity of the phases does not 푢휇 using the equation 푁 = in MRST. The flow regime affect the miscibility development. It can be observed that 푐 휎 pattern in each grid cells is defined, and each cell has its both the capillary number and interfacial tension profiles shows a sharp front and smeared front as the simulation unique 푁푐 value. From Figure 3, The capillary number can be seen increasing from 10−8 to 10−2. The interfacial time increases at a certain point along the grid block which is caused by the presence of surfactant in the injecting fluid tension decreases along the grid cells from 0.05푁/푚 to added from the injector well located in the first grid. There 10−6푁/푚 as seen in Figure 4. Figure 4 shows the velocity is no transition zone on the profile because there is no profiles and it can be deduced that the velocity remain diffusion in the block and the permeability is constant. The constant within the grids cell and does not changes. This Abdullahi M. 030
sharp front on the figures indicates the miscible zone, and the smeared front is the immiscible zone.
Figure 6 show different distribution maps after flooding with a surfactant concentration of 0푘푔/푠푚3 and 50푘푔/푠푚3. The figure clearly show the influence of surfactant in EOR flooding as the model with surfactant concentration of 50푘푔/푠푚3 produces larger oil recovery after flooding.
The New Capillary Number Parameterization for Simulation in Surfactant Flooding
Figure 7: Interpolation parameter against the Log(Nc) to identify miscibility from capillary number dynamics
Miscibility is achieved by interpolation between the immiscible relative permeability curves and the miscible relative permeability curves. The Interpolation parameter 퐹푘푟 values must span between the range value of [0, 1]. Miscibility is the function of the interpolation parameter. The interpolation parameter is described by a function (log10 푁푐). This logarithmic function is defined with an equation: 푁표푠푢푟푓 log10 푁푐 − log10 푁푐 퐹푘푟 = 푠푢푟푓 푁표푠푢푟푓 (34) log10 푁푐 − log10 푁푐 푁표푠푢푟푓 where 푁푐 is the model-based capillary number, 푁푐 푠푢푟푓 is the minimal values of the capillary numbers, 푁푐 is the maximal values of the capillary numbers.
The interpolation parameter 퐹푘푟 value of 0 implies immiscible conditions and a value of 1 implies miscible conditions. The interpolation parameter 퐹푘푟 is computed to identify miscibility from capillary number dynamics and show the effect of relative permeability interpolation function to numerically replicate miscibility. From Figure 7 is can be clearly seen that the model is immiscible at the first timesteps in the grid cells during the surfactant flooding, partially miscible at median timesteps and fully miscible at the end of the grid. The gradual miscibility development of the model at every timestep is caused by the increasing level of the surfactant concentration during the flooding. This means miscibility develops gradually as we flood.
Figure 8: Water and Oil miscible/immiscible relative permeability curves
Figure 8 represents the two sets of relative permeability curves for water and oil, one curve for immiscible conditions (퐹푘푟=0) and one curve for fully conditions (퐹푘푟=1). Once the value for 퐹푘푟 is determined, the two relative permeability are scaled and averaged according to a specific method. The lower and upper end-points on the two relative permeability curves are used to calculate new end-point saturations by a weighted average with 퐹푘푟=1.
As 퐹푘푟 vary between 0 and 1, the relative permeability calculations vary. Since the relative permeability defined in the input deck consist of some discrete points and is not defined by a continuous function. Instead, new saturation variables will be created, and these two saturation values are used to calculate the relative permeability for both miscible and immiscible conditions at the target saturation, 푆푤.
Using these new saturation values, the relative permeability is interpolated in the miscible and immiscible table. J. Oil, Gas Coal Engin. 031
The New Capillary Number Parameterization for Simulation in Surfactant Flooding
The effective relative permeability in the grid cell with saturation 푆푤 and miscibility factor 퐹푘푟 is then the weighted average of the two curves (Equation below).
퐾 (푆 ) = 퐹 . 퐾 + (1 − 퐹 )퐾 푟푤 푤 푘푟 푟푤,푚푖푠푐 푘푟 푟푤,푖푚푚푖푠푐 (35) 퐾푟표(푆푤) = 퐹푘푟. 퐾푟표,푚푖푠푐 + (1 − 퐹푘푟 )퐾푟표,푖푚푚푖푠푐 (36)
Figure 10 Comparison between the new and standard capillary desaturation curve
Figure 10 shows the capillary desaturation curve, this shows the relation between the capillary number 푁푐 and the residual oil saturation of the model. The residual oil 1 saturation is deduced from the relation 푆 = 표푟 푁 훽 1+( 푐) Figure 9: Comparison between the influences of 휆 Interfacial tension on Capillary number computed using (Jørgensen, 2013). Given average values to be used in standard formulation in MRST and the newly proposed sandstone as λ = 0.0012 and 훽 = 1.25. From figure 9, the −6 formulation for capillary number oil mobility is observed to have begun at 푁푐 = 10 which is known as the critical value of 푁푐 when the newly Figure 9 shows a comparison between the newly derived proposed formulation is used. Using the standard formulation proposed and the standard formulation. The formulation or model-based calculated capillary number, −5 capillary number was plotted against interfacial tension the mobility is seen to have begun at 푁푐 = 10 . Literature computed from the new formulation proposed and the study has shown that critical value of 푁푐 in most capillary standard formulation. At first, the interfacial tension σ is desaturation curves is between 10−5 and 10−4. Thus the introduced at a pore level, the effect of surfactant is to new 푁푐 formulation provides a broader critical value range modify the interfacial tension to make σ a function of the than the standard formulation. surfactant concentration, C. At an upscale level, the The recovery factor of the simulations using MRST and change in relative permeability 퐾푟 will depend on the Eclipse@ is computed as: capillary number 푁푐 which measures the ratio between the viscous and capillary forces and is defined as 푟푒푐표푣푒푟푎푏푙푒 표푖푙 푅푒푐표푣푒푟푦 푓푎푐푡표푟 = 푢휇 푂푟푖푔푖푛푎푙 푖푛 − 푝푙푎푐푒 표푖푙 푁 = (37) 푐 𝜎 As the surfactant concentration is increased during the Table 5: Comparison between the recovery factor of flooding, the interfacial tension reduces, thus the residual Eclipse@ and MRST oil or trapped oil is forced to move. From the Proposed Eclipse® MRST formulation and standard formulation lines, the maximum OIL(recovery) 107.39 sm3 112.67 sm3 capillary number reached is greater using the newly Recovery factor 0.191 0.2 proposed formulation. By studying the total oil recovered in the two models above, the production data for oil is well matched. The small discrepancy may relate to errors by the simulator as the eclipse@ indicated few warnings during the simulation process, among was convergence issue, and also numerical dispersion issue may cause the small difference in the recovery factor. The oil bank seems to moves faster and has a higher oil recovery factor in MRST than in Eclipse@ as it gave a higher recovery factor as seen in . MRST is thus a vital and optimised tool for enhanced oil recovery simulation. Abdullahi M. 032
The New Capillary Number Parameterization for Simulation in Surfactant Flooding
CONCLUSIONS AND RECOMMENDATIONS NOMENCLATURE
CONCLUSIONS 푁퐿푒 Leverett Number 푁푉퐶 Capillary number A new equation for calculation of capillary number as a 푃푐 Capillary pressure major transport component and mechanism for oil P Pressure mobilisation was derived. The equation was tested and q Flow rate proven reasonable. The equation can be accepted as a r radius working hypothesis since the result of the capillary R Ratio of maximum pore radius/average pore radius numbers obtained all fell within the range of 푁푐 as determined by resistivity observed experimentally in various papers and models S Saturation of nonwetting phase available. Thus, the proposed mechanism easily 푆푅 Normalized residual oil saturation accounts for a broader critical range of capillary number T(θ) geometric factor for contact angle of interface passing (10−6 to 10−4) as compare with the standard models through toroidal pore with less broad critical capillary number ranges u Darcy’s velocity (10−5 to 10−4). α̂ Dullien pore volume distribution function α Advancing contact angle of oil/ water interface Miscibility was identified clearly from capillary number β Receding contact angle of oil/water interface dynamics, and the effect of relative permeability ∇Φ Potential gradient interpolation parameter was used to replicate ϕ Porosity miscibility. ᵑ pore angle interface ∆ρ density difference Absolute velocity of the phases was seen not to have σ Interfacial tension any influence on the capillary number distribution θ Advancing contact angle of fluid used to determine rock during the flooding. Therefore, the addition of surfactant property in flooding doesn’t change the velocities of the phases ψ The geometrical factor for curvature in the grid block. b pore body c cylinder An empirical performance of miscibility was certainly d drainage seen in the two relative permeability curves plotted, m maximum also seen in the plot of capillary number against the n pore neck interpolation parameter which shows how miscibility or Immobile nonwetting phase at maximum trapping develops and finally seen from the capillary orc immobile nonwetting phase below maximum trapping desaturation curve. These certainly illustrated the effect wr immobile wetting phase at maximum trapping of miscibility in the surfactant flooding in enhance wrc immobile wetting phase below maximum trapping recovery compared to the water flooding. 푉푏 Bulk volume f Dimensionless length of the entire pore The surfactant model in MRST produced significantly K Permeability higher oil (recovery) and a higher recovery factor in the μ Viscosity surfactant model implemented in MRST than Eclipse®. L Core length A area 휆 Mobility of phase RECOMMENDATIONS 퐾푟 Relative permeability 푆푛 Normalized saturation To investigate other methods that capture miscibility 푆푔푟 Residual gas saturation effects on relative permeability curves 푆푔푟 Initial gas saturation Experimental works should be carried out to describe C Trapping characteristic of the porous media the phase behaviour of oil/water system containing M Mobility ratio surfactant and published data validation. ω Todd-Longstaff mixing parameter Full-scale modeling of a 3-phase system (Oil-Water- 푁푐 Capillary number Gas) to study the effect of miscibility and hysteresis X Fractional distance of current saturation between simultaneously. drainage curve end point and hysteresis saturation
J. Oil, Gas Coal Engin. 033
The New Capillary Number Parameterization for Simulation in Surfactant Flooding
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Development of Gas/Oil Miscibility in Water and Gas Copyright: © 2018 Abdullahi M. This is an open-access Injection, in, pp. 21–24. Leverett MC (1941) Capillary Behavior in Porous Solids, article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted Transactions of the AIME, 142(01), pp. 152–169. doi: use, distribution, and reproduction in any medium, 10.2118/941152-G. Lohne A, Purwanto EY, Fjelde I (2012). SPE 154495 provided the original author and source are cited. Gravity Segregated Flow in Surfactant Flooding, (June), pp. 4–7.
The New Capillary Number Parameterization for Simulation in Surfactant Flooding