Study on fingero imbibition phenomena in inclined porous media by optimal homotopy analysis method

Ain Shams Engineering Journal (2016) xxx, xxx–xxx

Ain Shams University

Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com

ENGINEERING PHYSICS AND MATHEMATICS

Study on fingero imbibition phenomena in inclined porous media by optimal homotopy analysis method Shreekant Pathak *, Twinkle Singh Department of Applied Mathematics & Humanities, S. V. National Institute of Technology, Surat 395007, India Received 3 January 2016; revised 19 May 2016; accepted 20 June 2016

KEYWORDS Fingero imbibition; Homogeneous porous media; Optimal homotopy analysis method; Oil recovery process

Abstract The approximate solution of fingero imbibition phenomena governed by a nonlinear partial differential equation is discussed in the present paper. The primary oil recovery process was due to natural soil pressure, but in the secondary oil recovery process water flooding plays an important role. When water is injected through the injection well for recovering the reaming oil after primary oil recovery process, it comes to contact with the native oil and at that time the fingero imbibition phenomena occur due to different viscosity. For the mathematical modeling, we consider the homogeneous inclined porous medium and comparative study of without inclination and with inclination has been done, and also optimal homotopy analysis method has been used to solve the partial differential equation governed by the mathematical modeling. The graphical representation of the solution is given by MATHEMATICA and physically interpreted. Ó 2016 Faculty of Engineering, Ain Shams University Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction In this research paper, we have discussed the fingero imbibition phenomena in the horizontal direction with inclined homogeneous porous media [5–7]. When porous medium is already filled with some fluid and when it comes to contact with another fluid then there is a spontaneous flow of wetting fluid into the porous medium and counter flow of native fluid from the porous medium. This process is called imbibition. In our * Corresponding author. E-mail addresses: [email protected] (S. [email protected] (T. Singh). Peer review under responsibility of Ain Shams University.

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Pathak),

problem we consider the inclined homogenous porous media saturated by native fluid is oil and injected fluid is water. The non-wetting fluid (oil) comes to contact with wetting fluid (water), there is a spontaneous flow in porous media of water into oil formatted area and oil into water formatted area is known as imbibition. The fingero imbibition phenomena arise on account of simultaneous occurrence of two important phenomena imbibition and fingering. Shah and Verma [29] called this conjoint phenomenon as fingero- imbibition. During the secondary oil recovery process water is injected into the oil reservoir, instead of regular displacement of the whole front common interface, the protuberances will develop with irregular fingers in different shapes and sizes, because of the different viscosities of water and oil. Hence, these phenomena are known as fingero imbibition phenomena or instability phenomena. Water is inexpensive, easily findable, and it is usually more stable follow regress than all another approaches of secondary oil recovery, therefore injection of water is the ideal

http://dx.doi.org/10.1016/j.asej.2016.06.010 2090-4479 Ó 2016 Faculty of Engineering, Ain Shams University Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Pathak S, Singh T, Study on fingero imbibition phenomena in inclined porous media by optimal homotopy analysis method, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.06.010

2

S. Pathak, T. Singh

form of the secondary oil recovery. Water is injected through injection well and the oil produced at production well due to the oil displacement. The stability of injected water depends on the mobility ratio of oil and water, heterogeneity of porous medium and interaction forces [30,32]. Many researchers have discussed the fingero imbibition phenomena by finding its analytic as well as numerical solution. For example, Kataria and Mehta [18] had exploited an analytical solution of fingero imbibition phenomena in terms of infinite series. Most of earlier researcher, Scheidegger [28], Saffman and Taylor [27] had ignored the capillary pressure. Verma [33] had considered the heterogeneous porous medium and gain the statistical behavior of fingers. We had used an optimal homotopy analysis method [20,25] to obtain an approximate solution of the governed nonlinear partial differential equation. There are many analytic and numerical methods available in the literature to handle nonlinear partial differential equation. For example, adomin decomposition method (ADM) [1,8,10,12], homotopy analysis method (HAM) [20,24], homotopy perturbation method (HPM) [14,15], differential transform method (DTM) [2,9], variational iteration method (VIM) [11,16], homotopy perturbation sumudu transform method (HPSTM) [31], laplace decomposition method (LDM) [19], finite difference method (FDM) [26], finite element method (FEM) [13], finite volume method (FVM) [17]. 2. Basic idea of OHAM Optimal homotopy analysis method is semi analytic method based on the homotopy analysis method [20]. The homotopy analysis method was first time briefly introduced by Shijun Liao in his PhD thesis. Then many methods based on homotopy analysis method were introduced by researchers such as optimal homotopy analysis method, homotopy perturbation method, optimal homotopy asymptotic method, and spectral homotopy analysis method. Suppose the nonlinear partial differential equation in a general form N ½uðx; tÞ ¼ 0

ð1Þ

where N is a nonlinear operator, x and t are independent variables in domain X and u is an unknown function. The zeroorder deformation equation is given by ð1  qÞL½/ðx; t; qÞ  u0 ðx; tÞ ¼ c0 qL½/ðx; t; qÞ

ð2Þ

in which, q 2 ½0; 1 is the embedding parameter, c0 is a convergence control parameter, L is a linear operator and u0 ðx; tÞ is an initial guess of uðx; tÞ. According to OHAM we have great freedom to choose a linear operator and an initial guess. The corresponding mth-order deformation equation is L½um ðx; tÞ  vm um1 ðx; tÞ ¼ c0 dm ½um1 ðx; tÞ where dm ½um1 ðx; tÞ ¼ and vm ¼



0; 1;


1 @ m1 N ½/ðx; t; qÞ


ðm  1Þ! @qm1 q¼0

m61 m>1

Applying inverse operator both side of Eq. (3), we have

ð3Þ

um ðx; tÞ ¼ vm um1 ðx; tÞ þ L1 c0 dm ½um1 ðx; tÞ

ð4Þ

The Convergence control parameter c0 is optimized by the ‘‘optimization method”. Yabushita et al. [34] applied the HAM to solve two coupled nonlinear ODEs. They suggested the ‘‘optimization method” to find out the optimal convergence-control parameters by means of the minimum of the squared residual of governing equation as follows )#2 Z Z " (X m Em ðc0 Þ ¼ N un ðx; tÞ dX ð5Þ X

n¼0

Using the optimal value of convergence control parameter c0 and initial guess value u0 ðx; tÞ in Eq. (4), we get the approximate value of uðx; tÞ according to different values of m uðx; tÞ ¼ u0 ðx; tÞ þ u1 ðx; tÞ þ u2 ðx; tÞ þ . . .

ð6Þ

3. Mathematical formation For the mathematical formulation of fingero imbibition phenomena in inclined homogenous porous media, we consider certain following assumptions. We take a cylindrical piece of porous matrix having length L and inclination c with horizontal x-axis whose three sides are impermeable and only one side is open to inject water The injected water comes to contact with oil at common interface x ¼ 0, the fingero imbibition phenomena starts. (Fig. 1(A)) [24]. Due to irregularity of fingers size and shape, schematic shape of fingers suggested by Schidegger [28] (Fig. 1(B)). But, still it is difficult to find out the portion saturated by water due to irregular length; therefore, we took the average cross-sectional area covered by rectangular fingers (Fig. 1 (C)). Also the initial saturation of water at common interface is very small. Our main aim in the present paper was to find the saturation of water into the oil formatted area. In secondary oil recovery process, during water flooding, let injected fluid (water) and native fluid (oil). The seepage velocities of the fluids water and oil are given by Darcy’s law as follows by bear [4]   ki @pi Vi ¼  K þ qi g sin c ð7Þ li @x Vn ¼ 

  kn @pn þ qn g sin c K ln @x

ð8Þ

where ki and kn are the relative permeabilities, K is permeability of homogeneous porous medium, c is the angle of inclination, li and ln are the constant kinematic viscosities of displacing fluids water and oil respectively and pi and pn are the pressure of water and oil. In order to prepare the mathematical model for fingero imbibition phenomena, we assume that the densities of both injected and native fluids are constant. The continuity equation for these two fluids is given as 

@Si @vi þ ¼0 @t @x

ð9Þ



@Sn @vn þ ¼ 0: @t @x

ð10Þ

where  is the porosity and Si and Sn are the functions of saturations. Also, we consider the porous matrix is fully

Please cite this article in press as: Pathak S, Singh T, Study on fingero imbibition phenomena in inclined porous media by optimal homotopy analysis method, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.06.010

Study on fingero imbibition phenomena

3 

     @Si @ ki @pn @pc  K ¼  qi g sin c @x li @t @x @x

ð15Þ

   @Sn @ kn @pn ¼ þ qn g sin c K  @x ln @t @x

ð16Þ

Adding Eq. (15) and (16) and using the relation (11) and integrating w.r.t. x, we get        ki kn @pn ki @pc ki kn Kþ K K qi K þ qn K g sin c  þ li ln @x li @x li ln ¼ VðtÞ ð17Þ where VðtÞ is a constant of integration. Simplify Eq. (17)

ki c K @p l @x @pn VðtÞ

þ i

¼ ki ki @x K þ lknn K K þ lknn K li li

ki q K þ lknn qn K g sin c li i

 ki kn K þ K li ln

ð18Þ

Using Eq. (18) into Eq. (15), we get



8 93 2 kn ki @pc Kðqn  qi Þ gsinc= K @x ln @Si @ 4ki < VðtÞ li 5





 K ¼ ki ki ; @t @x li : ki K þ kn K K þ kn K K þ kn K li

ln

li

ln

li

ln

ð19Þ

Oroveanu [22] gave the following pressure relation Figure 1 (A) Representation of fingero imbibition in a cylindrical piece of inclined hemogenous porous matrix, (B) schematic representation of fingers at level x for fingero imbibition phenomena in inclined homogeneous porous matrix of length L, and (C) average cross sectional area occupied by small fingers as rectangle for fingero imbibition phenomena [24].

saturated. i.e. the saturation of injected water and native oil is unity. Si þ Sn ¼ 1:

ð11Þ

The standard relation between relative permeabilities and phase saturation [23] is given by  ki ¼ S i ð12Þ kn ¼ 1  aSi The relation between capillary pressure and pressure of the native and injected fluids [24] is given by pc ¼ pn  pi :

ð13Þ

Also it is observed the capillary pressure is linearly dependent on the saturation of injected water [21] pc ðSi Þ ¼ bSi ;

ð14Þ

where b is constant. Use, Eqs. (7) and (8) into continuity Eqs. (9) and (10) and using relation (13), we have

1 pn ¼ p þ pc ; 2

p ¼

pn þ pi 2

ð20Þ

where p is the constant mean pressure, hence @pn 1 @pc ¼ @x 2 @x

ð21Þ

Substituting the value of Eq. (21) into Eq. (18) gives value of VðtÞ as     1 kn ki @pc ki kn VðtÞ ¼ þ K K qn K þ qi K g sin c ð22Þ 2 ln li @x li ln Substituting this value of VðtÞ from Eq. (22) into Eq. (19) we get     @Si @ ki 1 @pc ki ¼ þ qi Kg sin c K  ð23Þ  @t li @x li 2 @x Doing some simplifications and using Eqs. (12) and (14)   @Si Kb @ @Si q Kg @Si  Si ¼ þ i ð24Þ sin c 2li @x li @t @x @x Use dimensionless parameters X¼

x ; L

T¼

Kb t 2li L2

ð25Þ

The advantage of using dimensionless parameter is it reduced the parameters, therefore Eq. (24) reduced as   @Si @ @Si @ 2Lqn g sin c ¼ þA ðSi Þ; A ¼ Si : ð26Þ @T @X @X @X b subject to the initial and boundary condition [24]

Please cite this article in press as: Pathak S, Singh T, Study on fingero imbibition phenomena in inclined porous media by optimal homotopy analysis method, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.06.010

4

S. Pathak, T. Singh 

Sic ð1  eX Þ; 0 < X 6 1 Sic ; X¼0 Si ð0; TÞ ¼ Sic ; X ¼ 0 & T > 0 Si ð1; TÞ ¼ fðTÞ; X ¼ 1 & T > 0

E5 c 0

Si ðX; 0Þ ¼

0.0321973 0.0321973 0.0321973

4. Solution by OHAM

0.0321973

According to OHAM [20,25] first we construct the zeroth order deformation equation as ð1  qÞL½/ðX; T; qÞ  Si0 ðX; TÞ ¼ c0 qN ½/ðX; T; qÞ;

ð27Þ

where q 2 ½0; 1 is the embedding parameter, c0 is an auxiliary parameter also know as convergence control parameter. We @ choose auxiliary linear operator L ¼ @T , [3] and Si0 ðX; TÞ ¼ ð0:1e X þ 0:5X  TÞ [24] is an initial guess of Si ðX; TÞ. According to OHAM we have great freedom to choose linear operator and initial guess. Expanding /ðX; T; qÞ in Maclaurin series with respect to q, then the corresponding mth-order deformation equation is given by L½SiðmÞ ðX; TÞ  vm Siðm1Þ ðX; TÞ ¼ c0 dm ½Siðm1Þ ðX; TÞ;

ð28Þ

where m1 X 2 dm ½Sm1 ðX;TÞ ¼ ðSm1 ÞT  Sr ðSm1r ÞXX  ðSm1 ÞX  AðSm1 ÞX : r¼0

and vm ¼



0.0321973 0.0321973 0.0321973

c0 0.033946

Figure 2

0.033945

0.033944

0.033943

0.033942

Square residual error at 5th-order approximation.

Si1 ðX; TÞ ¼ 0:034ð0:076e X T  0:034e2X T þ 0:19T2  0:085e X T2  0:142T3 þ 0:5TX  0:0425e X T2 XÞ

ð31Þ

Si2 ðX; TÞ ¼ 0:034ð0:076e X T  0:034e2X T þ 0:19T2  0:085e X T2  0:142T3 þ 0:5TX  0:0425e X T2 XÞ  0:034ð0:0023e X T þ 0:001e2X T  0:01T2 þ 0:002e X T2

0; 1;

m61 m>1

þ 0:001e2X T2  0:00053X T2 þ 0:004T3 þ 0:001e X T3  0:0004e2X T3

Applying inverse operator to both sides on Eq. (28) Sim ðX; TÞ ¼ vm Siðm1Þ ðX; TÞ þ c0 L1 ½dm ½Siðm1Þ ðX; TÞ:

þ 0:0000067e3X T3  0:000003e4X T3 ð29Þ

The Convergence control parameter c0 plays an important role in the OHAM. One can gain convergent series solution simply by choosing a proper auxiliary parameter c0 . This is the reason why we call c0 as the convergence-control parameter. In 2007, Yabushita et al. [34] applied the HAM to solve two coupled nonlinear ODEs. They suggested the so-called ‘‘optimization method” to find out the optimal convergence-control parameters by means of the minimum of the squared residual of governing equation as follows 8 " 9  #2 = M X N < m X X 1 i j Em ðc0 Þ ¼ ; N Sin : ; ðM þ 1ÞðN þ 1Þ i¼0 j¼0 : M N n¼0 ð30Þ Eq. (30) gives the square residual error at mth-order approximation. It is obvious that we can find the optimal value of convergence-control parameter c0 at the any order of approximation. If there exists convergence-control parameter, c0 for which we get the minimum of the squared residua Em , is so called the optimal convergence-control parameter c0 . The optimal value of convergence control parameter c0 ¼ 0:0339 with minimum square residual error E5 = 3.2E02 at fifth order approximation, which can be noticed by Fig. 2. Substituting the optimal value of convergence control parameter and solving Eq. (29) for different values of m we have following approximations

 0:0001e X T4 þ 0:0000091e2X T4  0:000008e3X T4  0:000006e2X T5  0:012TX þ 0:0029e X T2 X þ 0:001e X T3 X  0:001e2X T3 X  0:0012e X T4 X þ 0:000003e2X T4 X  0:000002e3X T4 X  0:000004e2X T5 X  0:0003e X T4 X2  7:07  107 e2X T5 X2 Þ

ð32Þ

and so on. Adding the initial guess value Si0 & first approximation Si1 to fifth approximation Si5 , we get the approximate solution of fingero imbibition phenomenon as Si ðX; TÞ ¼ ð0:1e X þ 0:5X  TÞ  0:034ð0:07e X T  0:034e2X T þ 0:19T2  0:085e X T2  0:14T3 þ 0:5TX  0:04e X T2 XÞ  0:034ð0:07e X T  0:034e2X T þ 0:19T2  0:085e X T2  0:14T3 þ 0:5TX  0:04e X T2 XÞ  0:034ð0:07e X T  0:034e2X T þ 0:19T2  0:085e X T2  0:1T3 þ 0:5TX  0:04e X T2 XÞ þ 0:034ð0:003e X T þ 0:001e2X T  0:01T2 þ 0:002e X T2 þ 0:001e2X T2  0:0005e3X T2 þ . . .

ð33Þ

Please cite this article in press as: Pathak S, Singh T, Study on fingero imbibition phenomena in inclined porous media by optimal homotopy analysis method, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.06.010

Study on fingero imbibition phenomena Table 1

5

Numerical value for the fingero imbibition phenomenon through homogeneous porous media with A ¼ 0:76091.

X

T¼0

T ¼ 0:1

T ¼ 0:2

T ¼ 0:3

T ¼ 0:4

T ¼ 0:5

T ¼ 0:6

T ¼ 0:7

T ¼ 0:8

T ¼ 0:9

T ¼ 1:0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.1000 0.1105 0.1221 0.1350 0.1492 0.1649 0.1822 0.2014 0.2226 0.2460 0.2718

0.0987 0.1129 0.1282 0.1448 0.1628 0.1823 0.2036 0.2267 0.2521 0.2798 0.3101

0.0972 0.1151 0.1342 0.1546 0.1765 0.2000 0.2253 0.2527 0.2823 0.3145 0.3496

0.0956 0.1174 0.1404 0.1647 0.1906 0.2182 0.2477 0.2794 0.3136 0.3505 0.3906

0.0944 0.1200 0.1469 0.1753 0.2053 0.2371 0.2710 0.3072 0.3461 0.3880 0.4334

0.0935 0.1232 0.1541 0.1866 0.2208 0.2570 0.2954 0.3363 0.3802 0.4274 0.4784

0.0934 0.1271 0.1621 0.1989 0.2374 0.2781 0.3212 0.3670 0.4161 0.4688 0.5259

0.0942 0.1320 0.1713 0.2123 0.2553 0.3007 0.3486 0.3996 0.4542 0.5128 0.5762

0.0961 0.1381 0.1817 0.2272 0.2748 0.3250 0.3780 0.4344 0.4947 0.5595 0.6298

0.0994 0.1457 0.1937 0.2437 0.2961 0.3512 0.4095 0.4716 0.5379 0.6095 0.6871

0.1043 0.1549 0.2074 0.2621 0.3194 0.3797 0.4436 0.5115 0.5844 0.6630 0.7486

Table 2 X

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X

Comparative numerical value of saturation of injected water Si ðX; TÞ for h ¼ 15 , 0 & 15 . T=0

T = 0.1

T = 0.2

h ¼ 15

h ¼ 0

h ¼ 15

h ¼ 15

h ¼ 0

h ¼ 15

h ¼ 15

h ¼ 0

h ¼ 15

0.1000 0.1105 0.1221 0.1350 0.1492 0.1649 0.1822 0.2014 0.2226 0.2460 0.2718

0.1000 0.1105 0.1221 0.1350 0.1492 0.1649 0.1822 0.2014 0.2226 0.2460 0.2718

0.1000 0.1105 0.1221 0.1350 0.1492 0.1649 0.1822 0.2014 0.2226 0.2460 0.2718

0.0987 0.1129 0.1282 0.1448 0.1628 0.1823 0.2036 0.2267 0.2521 0.2798 0.3101

0.1000 0.1155 0.1321 0.1500 0.1692 0.1899 0.2122 0.2364 0.2626 0.2910 0.3219

0.1001 0.1156 0.1322 0.1500 0.1692 0.1899 0.2122 0.2363 0.2625 0.2909 0.3218

0.0972 0.1151 0.1342 0.1546 0.1765 0.2000 0.2253 0.2527 0.2823 0.3145 0.3496

0.1000 0.1205 0.1421 0.1650 0.1892 0.2149 0.2423 0.2714 0.3026 0.3360 0.3719

0.1003 0.1208 0.1423 0.1651 0.1893 0.2149 0.2423 0.2714 0.3026 0.3360 0.3719

T = 0.3

T = 0.4

T = 0.5

T = 0.6

h ¼ 15

h ¼ 0

h ¼ 15

h ¼ 15

h ¼ 0

h ¼ 15

h ¼ 15

h ¼ 0

h ¼ 15

h ¼ 15

h ¼ 0

h ¼ 15

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0956 0.1174 0.1404 0.1647 0.1906 0.2182 0.2477 0.2794 0.3136 0.3505 0.3906

0.1000 0.1255 0.1521 0.1800 0.2092 0.2399 0.2723 0.3064 0.3426 0.3810 0.4219

0.1005 0.1260 0.1525 0.1803 0.2094 0.2401 0.2724 0.3065 0.3427 0.3812 0.4221

0.0944 0.1200 0.1469 0.1753 0.2053 0.2371 0.2710 0.3072 0.3461 0.3880 0.4334

0.1000 0.1305 0.1621 0.1950 0.2292 0.2649 0.3023 0.3414 0.3826 0.4260 0.4719

0.1008 0.1312 0.1628 0.1955 0.2297 0.2653 0.3026 0.3418 0.3830 0.4265 0.4725

0.0935 0.1232 0.1541 0.1866 0.2208 0.2570 0.2954 0.3363 0.3802 0.4274 0.4784

0.0999 0.1355 0.1721 0.2100 0.2492 0.2899 0.3322 0.3764 0.4226 0.4710 0.5219

0.1012 0.1366 0.1731 0.2109 0.2500 0.2906 0.3330 0.3771 0.4234 0.4719 0.5229

0.0934 0.1271 0.1621 0.1989 0.2374 0.2781 0.3212 0.3670 0.4161 0.4688 0.5259

0.0999 0.1404 0.1821 0.2250 0.2692 0.3149 0.3622 0.4114 0.4626 0.5160 0.5718

0.1017 0.1421 0.1836 0.2263 0.2704 0.3161 0.3634 0.4126 0.4639 0.5174 0.5735

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0942 0.1320 0.1713 0.2123 0.2553 0.3007 0.3486 0.3996 0.4542 0.5128 0.5762

0.0998 0.1454 0.1920 0.2399 0.2891 0.3398 0.3922 0.4464 0.5025 0.5609 0.6218

0.1023 0.1476 0.1941 0.2418 0.2910 0.3416 0.3940 0.4482 0.5045 0.5631 0.6242

0.0961 0.1381 0.1817 0.2272 0.2748 0.3250 0.3780 0.4344 0.4947 0.5595 0.6298

0.0998 0.1503 0.2020 0.2549 0.3091 0.3648 0.4221 0.4813 0.5425 0.6059 0.6717

0.1029 0.1533 0.2047 0.2575 0.3116 0.3673 0.4246 0.4839 0.5452 0.6089 0.6751

0.0994 0.1457 0.1937 0.2437 0.2961 0.3512 0.4095 0.4716 0.5379 0.6095 0.6871

0.0997 0.1553 0.2119 0.2698 0.3290 0.3897 0.4521 0.5162 0.5824 0.6508 0.7216

0.1037 0.1590 0.2155 0.2732 0.3324 0.3930 0.4554 0.5197 0.5861 0.6548 0.7262

0.1043 0.1549 0.2074 0.2621 0.3194 0.3797 0.4436 0.5115 0.5844 0.6630 0.7486

0.0996 0.1602 0.2219 0.2847 0.3490 0.4147 0.4820 0.5512 0.6223 0.6957 0.7715

0.1046 0.1649 0.2264 0.2891 0.3533 0.4190 0.4864 0.5557 0.6272 0.7010 0.7774

T = 0.7

T = 0.8

T = 0.9

T = 1.0

Please cite this article in press as: Pathak S, Singh T, Study on fingero imbibition phenomena in inclined porous media by optimal homotopy analysis method, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.06.010

6

S. Pathak, T. Singh Acknowledgments 0.7

Si Saturation

0.6 0.5 0.4 0.3

T

0.0

T

0.8

T

0.1

T

0.9

T

0.2

T

1.0

T

0.3

T

0.4

T

0.5

T

0.6

T

0.7

The authors are thankful to the Department of Applied Mathematics and Humanities, S.V. National Institute of Technology, Surat, for providing research facilities. References

0.2 0.1 0.0

0.2

0.4

0.6

0.8

1.0

X Distance

Figure 3 Saturation of injected water Si ðX; TÞ at different distance X with fix time level T ¼ 0:0; 0:1; 0:2; 0:3; 0:4; 0:5; 0:6; 0:7; 0:8; 0:9; 1:0.

5. Numerical and graphical representation The following Table 1 shows the numerical values of solution of saturation of injected water up to fifth order approximations by optimal homotopy analysis method using MATHEMATICA coding. For the numerical calculation we have chosen certain parameters as A ¼ 2Lqn g sin c=b ¼ 2  3  0:1  9:8  sinð15 Þ=2 ¼ 0:7609. Fig. 3 shows, the approximate solution of mathematical model of fingero imbibition phenomenon arising in secondary oil recovery process. The comparative values of saturation of injected water in inclined porous media and without inclination is sited in Table 2. From the angle of inclinations 15 ; 0 and 15 we can see that the saturation of injected water during the secondary oil recovery process is increasing with inclination angle 15 to 15 . 6. Conclusions In this paper, fingero imbibition phenomena is solved by optimal homotopy analysis method which is a well-known analytic method. The optimal value of convergence control parameter c0 ¼ 0:0339 is successfully obtained by reducing the square residual error. The comparative study of fingero imbibition phenomenon is sited in Table 2. The graph and tabular value shows that the saturation of injected water is increasing as distance X and time T are increasing. The comparison is made between with inclination and without inclination angle, shows that the saturation of injected water is decrees with inclined angle. If we consider the inclination angle in negative direction then the saturation of injected water is increased with inclined angle, which is consistent with physical fact. Hence we can say that the solution is physically consistent. Competing interests The authors declare that no competing interests exist.

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Shreekant Pathak is a Research Scholar at Department of Applied Mathematics, Sardar Vallabhbhai National Institute of Technology, Ichhanath, Surat – India. He passed M. Sc. in 2008 from Veer Narmad South Gujarat University, Surat – India. His field of interest is Fluid flow through porous media, Programming in mathematics, Mathematical Modeling.

Twinkle Singh is working as an Assistant Professor at Department of Applied Mathematics, Sardar Vallabhbhai National Institute of Technology, Ichhanath, Surat – India. She received PhD degree from Sardar Vallabhbhai National Institute of Technology, Ichhanath, Surat – India. She guided 4 PhD students and published many research articles in reputed journal. Her field of interest is Fluid flow through porous media, Groundwater flow, Mathematical Modeling.

Please cite this article in press as: Pathak S, Singh T, Study on fingero imbibition phenomena in inclined porous media by optimal homotopy analysis method, Ain Shams Eng J (2016), http://dx.doi.org/10.1016/j.asej.2016.06.010