 Open access peer-reviewed chapter

# Study on Approximate Analytical Method with Its Application Arising in Fluid Flow

Written By

Twinkle R. Singh

Submitted: September 9th, 2020 Reviewed: April 2nd, 2021 Published: May 6th, 2021

DOI: 10.5772/intechopen.97548

From the Edited Volume

## Porous Fluids

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## Abstract

This chapter is about the, Variational iteration method (VIM); Adomian decomposition method and its modification has been applied to solve nonlinear partial differential equation of imbibition phenomenon in oil recovery process. The important condition of counter-current imbibition phenomenon as vi=−vn, has been considered here main aim, here is to determine the saturation of injected fluid Sixt during oil recovery process which is a function of distance ξ and time θ, therefore saturation Si is chosen as a dependent variable while xandt are chosen as independent variable. The solution of the phenomenon has been found by VIM, ADM and Laplace Adomian decomposition method (LADM). The effectiveness of our method is illustrated by different numerical.

### Keywords

• Variational Iteration method (VIM)
• nonlinear partial differential equations

## 1. Introduction

Same way Laplace Adomian’s Decomposition Method (LADM) was first introduced by Khuri [47, 48]. The Laplace Adomian Decomposition Method (LADM) is formed with combination of the Adomian Decomposition Method (ADM) Adomian [29, 49] and Laplace transforms. LADM is a promising method and has been applied in solving various nonlinear systems of differential equations [36, 50, 51, 52, 53, 54, 55, 56]. In a variety of applied sciences, systems of partial differential equations have attracted much attention e.g. [50, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75]. The general ideas and the essentiality of these systems are of wide applicability. Agadjanov  solved Duffing equation with the help of LDM. Elgazery [51, 76] had applied Laplace decomposition method for the solution of Falkner-Skan equation.

In the solution procedure of VIM; many repeated computations and computations of the unneeded forms, which take more time and effort beyond it, so a modification has been shown to reduce these unneeded forms.

On the other hand, few researchers have been discussed imbibition phenomenon in homogenous porous media with different point of view for example, researchers taking different perspectives for this phenomenon; [77, 78] and some others have analyzed it for homogeneous porous medium.

In this Present investigated model, Imbibition takes place over a small part of a large oil formatted region taken as a cylindrical piece of homogeneous porous medium. In this model, we have considered the important condition of counter-current imbibition phenomenon as vi=vn, Our purpose is to determine the saturation of injected fluid Sixt during oil recovery process which is a function of distance ξ and time θ, therefore saturation Si has been chosen as a dependent variable while xandt are chosen as independent variable.

## 2. Imbibition phenomenon

It is the process by which a wetting fluid displaces a non-wetting fluid the initially saturates a porous sample, by capillary forces alone. Suppose a sample is completely saturated with a non-wetting fluid, and same wetting fluid is introduced on its surface. There will be spontaneous flow of wetting fluid into the medium, causing displacement of the non-wetting fluid. This is called imbibition phenomenon. The rate of imbibition is greater if the wettability of the porous medium, by the imbibed fluid, is higher.

The mathematical condition for imbibition phenomenon is given by Scheidegger ); viz,

vn=vi

Where vi&vn are the seepage velocities of injected & native liquids respectively.

The relation between relative permeability and phase-saturation,

ki=Si3
kn=1αSn,α=1.11

Where ki&kn denotes fictitious relative permeability. Si&Sn denotes saturations of injected and native liquids respectively.

## 3. Mathematical structure of the model

According to the Darcy’s law, the basic equations of the phenomenon as; 

vi=kiδiKpixE1
vn=knδnKpnxE2
vi=vnE3
pc=pnpiE4
φSit+vix=0E5
φSnt+vnx=0E6

Where vi and vn are the seepage velocities, ki and kare the relative permeabilities δi and δn are the kinematic viscosities (which are constants), pi and pn are pressure of the injected and native liquid respectively, φandK are the porosity and the permeability of the homogeneous porous medium; Si is the saturation of the injected liquid; pc is the capillary pressure and t is the time. The co-ordinate x is measured along the axis of the cylindrical medium, the origin being located at the imbibition face x=0.

Combing equations (1)-(5) and using the relation for capillary pressure as,

pc=βSi , we get,

φSit+xKDSiβSix=0E7

Where D(Si) = kiknδnki+δikn and β being small capillary pressure coefficient.

It is assumed is that an average value of D(Si) =D¯ (Si)

Using the transformation,

ξ=xL,θ=LtφL2,0xLSioB.E8

Eq. (7), becomes;

Siθ+βD¯Si2Siξ2=0

Siθ=βD¯SiSi2ξ2

Siθ=εSi2ξ2Whereε=βD¯SiE9

By the Hopf-Cole transformation [79, 80] equation (9) reduces to the Burger’s equation.

Siθ+SiSiξ=εSiξξE10

With the condition

Siξ0=Si0eξ at time θ = 0 and ξ > 0

### 3.1 Solution of the Burger’s equation by variational iteration method

To add the basic concepts of VIM, considering the below mentioned nonlinear partial differential equations:

Luxt+Ruxt+Nut=gxt,ux0=exE11

Where L=t,R is a linear operator which has partial derivatives with respect to x, Nu(x,t) is a nonlinear term and g(x,t) is an inhomogeneous term.

As per the VIM [6, 7];

Un+1xt=Unxt+0tλLUn+RUn¯+NUn¯gE12

Where λ is called a general Lagrange multiplier [81, 82] which can be identified optimally via vatiational theory, RUn¯ and NUn¯ are considered as restricted variations,

i.e. δRUn¯=0,δNUn¯=0 calculating variation with respect to Un;

λ'τ=01+λττ=t=0E13

The Lagrange multiplier, therefore, can be considered as λ=-1.

Now, substituting the multiplier in (12), then

Un+1xt=Un0tLUn+RUn+NUngE14
Siθ+SiSiξ=εSiξξE15

With the constrain

Siξ0=Si0eξ at time θ = 0 and ξ > 0

To solve equation (10) by VIM, substituting in equation (14) by

RUn=Unx2
NUn=UnUnxx

& g(x,t) = 0

And can obtain the following variational iteration formula:

Sin+1=Sin0θSinτ+SinSinξεSinξξE16

Using (14), the approximate solutions Unxtare obtained by substituting;

Siξ0=Si0eξE17

Approximate solutions are given below;

Si1=Si0eξβ10θ;whereβ10=Si02e2ξεSi0eξ
Si2=Si0eξβ10θ+β11θ22whereβ11=β10Si0eξ

Similarly,

Si3=Si0eξβ21θ22!+β22θ33!

And so on…..

Notes on VIM

From the analysis we can observed is this:

1. VIM can contain a series solution not exactly like ADM.

2. VIM needs many modifications to overcome the wasted time in the repeated calculations and unneeded terms.

Now applying ADM to equation (10); we get

Siξθ=Lθ1SiSiξεSiξξE18

And recursive relation is:

Siξ0=eξ

Then:

Si1ξθ=β10θ
Si2ξθ=β120θ33εβ21θ22
Si3ξθ=β1230θ44εβ31θ33

and so on…

Now, applying (LADM) Laplace transform with respect to t on both sides of (10);

Sixt=L11sLSi0Si0ξεSi0ξξ
Si1=β10eξθ
Si2=β102e2ξεβ10eξθ22!
Si3=β03εβ13θ33!

And so on…

## 4. Interpretation

It is concluded that for the non linear partial differential equation of imbibitions phenomenon in oil recovery process, through graphs, it has been observed that the saturation of injected water during imbibition, increases and it is noted that LADM gives faster accuracy compare to VIM and ADM (Figures 17).

## 5. Conclusions

The VIM, the ADM and the LADM are successfully applied to Burger’s equation. The results which are obtained by ADM are a powerful mathematical tool to solve nonlinear partial differential equation. It has been noted that this method is reliable and requires fewer computations; and scheme LADM gives better and very faster accuracy in comparison with VIM.