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

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.

1. Introduction

First, the variational iteration method was proposed by He [1] in 1998 and was successfully applied to autonomous ordinary differential equation, to nonlinear partial differential equations with variable coefficients. In recent times a good deal of attention has been devoted to the study of the method. The reliability of the method and the reduction in the size of the computational domain give this method a wide applicability. The VIM based on the use of restricted variations and correction functional which has found a wide application for the solution of nonlinear ordinary and partial differential equations, e.g., [2, 3, 4, 5, 6, 7, 8, 9, 10]. This method does not require the presence of small parameters in the differential equation, and provides the solution (or an approximation to it) as a sequence of iterates. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives and whereas the Adomian decomposition method was before the Nineteen Eighties, it was developed by Adomian [11, 12] for solving linear or nonlinear ordinary, partial and Delay differential equations. A large type of issues in mathematics, physics, engineering, biology, chemistry and other sciences have been solved using the ADM, as reported by many authors [13]. The Adomian decomposition method (ADM) [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] is well set systematic method for practical solution of linear or nonlinear and deterministic or stochastic operator equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), integral equations, integro-differential equations, etc. The ADM is considered as a powerful technique, which provides efficient algorithms for analytic approximate solutions and numeric simulations for real-world applications in the applied sciences and engineering. It allows us to solve both nonlinear initial value problems (IVPs) and boundary value problems (BVPs) [17, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46] without unphysical restrictive assumptions such as required by linearization, perturbation, ad hoc assumptions, guessing the initial term or a set of basic functions, and so forth. The accuracy of the analytic approximate solutions obtained can be verified by direct substitution. More advantages of the ADM over the variational iteration method is mentioned in Wazwaz [22, 28]. A key notion is the Adomian polynomials, which are tailored to the particular nonlinearity to solve nonlinear operator equations. A key concept of the Adomian decomposition series is that it is computationally advantageous rearrangement of the Banach-space analog of the Taylor expansion series about the initial solution component function, which permits solution by recursion. The selection behind choice of decomposition is nonunique, which provides a valuable advantage to the analyst, permitting the freedom to design modified recursion schemes for ease of computation in realistic systems.

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 [56] 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 $v_i=-v_n$, Our purpose is to determine the saturation of injected fluid $S_i\left(x,t\right)$ during oil recovery process which is a function of distance $\xi$ and time $\theta$, therefore saturation $S_i$ has been chosen as a dependent variable while $x\text{and}t$ 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 [78]); viz,

$$v_n=-v_i$$

Where $v_i\&v_n$ are the seepage velocities of injected & native liquids respectively.

The relation between relative permeability and phase-saturation,

$$k_i=S_i^3$$

$$k_n=1-\alpha S_n,\mathrm{\alpha }=1.11$$

Where $k_i\&k_n$ denotes fictitious relative permeability. $S_i\&S_n$ 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; [78]

$$v_i=-\left(\frac{k_i}{{\delta }_i}\right)K\frac{\partial p_i}{\partial x}$$E1

$$v_n=-\left(\frac{k_n}{{\delta }_n}\right)K\frac{\partial p_n}{\partial x}$$E2

$$v_i=-v_n$$E3

$$p_c=p_n-p_i$$E4

$$\phi \left(\frac{\partial S_i}{\partial t}\right)+\frac{\partial v_i}{\partial x}=0$$E5

$$\phi \left(\frac{\partial S_n}{\partial t}\right)+\frac{\partial v_n}{\partial x}=0$$E6

Where v${}_i$ and v${}_n$ are the seepage velocities, k${}_i$ and kare the relative permeabilities ${\delta }_i$ and ${\delta }_n$ are the kinematic viscosities (which are constants), p${}_i$ and p${}_n$ are pressure of the injected and native liquid respectively, $\phi \text{and}K$ are the porosity and the permeability of the homogeneous porous medium; S${}_i$ is the saturation of the injected liquid; p${}_c$ 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,

p${}_c$=$\beta S_i$ [70], we get,

$$\phi \left(\frac{\partial S_i}{\partial t}\right)+\frac{\partial }{\partial x}\left[\mathit{KD}\left(S_i\right)\beta \left(\frac{\partial S_i}{\partial x}\right)\right]=0$$E7

Where D(S${}_i$) = $\frac{k_i{k}_n}{{\delta }_n{k}_i+{\delta }_i{k}_n}$ and β being small capillary pressure coefficient.

It is assumed is that an average value of D(S${}_i$) =$\overline{D}$ (S${}_i$)

Using the transformation,

$$\xi =\frac{x}{L},\theta =\frac{\mathit{Lt}}{\phi L^2},0\le x\le \frac{{\mathit{LS}}_{\mathit{io}}}{B}.$$E8

Eq. (7), becomes;

$$\left(\frac{\partial S_i}{\partial \theta }\right)+\beta \overline{D}\left(S_i\right)\frac{{\partial }^2{S}_i}{\partial {\xi }^2}=0$$

$\frac{\partial S_i}{\partial \theta }=-\beta \overline{D}\left(S_i\right)\frac{\partial {S_i}^2}{\partial {\xi }^2}$

$$\frac{\partial S_i}{\partial \theta }=\epsilon \frac{\partial {S_i}^2}{\partial {\xi }^2}\text{Where}\epsilon =-\beta \overline{D}\left({\mathrm{S}}_i\right)$$E9

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

$$S_{i_{\theta }}^{\ast }+S_i^{\ast }{S}_{i_{\xi }}^{\ast }=\epsilon {S_i^{\ast }}_{\mathit{\xi \xi }}$$E10

With the condition

$S_i^{\ast }\left(\xi ,0\right)=S_{i_0}^{\ast }{\sout{e}}^{\xi }$ at time $\theta$ = 0 and $\xi$ > 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:

$$\begin{array}{l}\mathit{Lu}\left(x,t\right)+\mathit{Ru}\left(x,t\right)+\mathit{Nu}\left(,t\right)=g\left(x,t\right),\\ u\left(x,0\right)=e^x\end{array}$$E11

Where $L=\left(\frac{\partial }{\partial t}\right),R$ 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];

$$U_{n+1}\left(x,t\right)=U_n\left(x,t\right)+\underset{0}{\overset{t}{\int }}\lambda \left\{{\mathit{LU}}_n+\overline{{\mathit{RU}}_n}+\overline{{\mathit{NU}}_n}-g\right\}\mathit{d\tau }$$E12

Where $\lambda$ is called a general Lagrange multiplier [81, 82] which can be identified optimally via vatiational theory, $\overline{{\mathit{RU}}_n}$ and $\overline{{\mathit{NU}}_n}$ are considered as restricted variations,

i.e. $\delta \overline{{\mathit{RU}}_n}=0,\delta \overline{{\mathit{NU}}_n}=0$ calculating variation with respect to $U_n$;

$$\begin{array}{l}{\lambda }^{\text{'}}\left(\tau \right)=0\\ 1+\lambda {\left(\tau \right)}_{\tau =t}=0\end{array}$$E13

The Lagrange multiplier, therefore, can be considered as $\lambda$=-1.

Now, substituting the multiplier in (12), then

$$U_{n+1}\left(x,t\right)=U_n-\underset{0}{\overset{t}{\int }}\left\{L\left(U_n\right)+R\left(U_n\right)+N\left(U_n\right)-g\right\}\mathit{d\tau }$$E14

$$S_{i_{\theta }}^{\ast }+S_i^{\ast }{S}_{i_{\xi }}^{\ast }=\epsilon {S_i^{\ast }}_{\mathit{\xi \xi }}$$E15

With the constrain

$S_i^{\ast }\left(\xi ,0\right)=S_{i_0}^{\ast }{\sout{e}}^{\xi }$ at time $\theta$ = 0 and $\xi$ > 0

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

$${\mathit{RU}}_n=-U_{n_x}^2$$

$${\mathit{NU}}_n=U_n{\left(U_n\right)}_{\mathit{xx}}$$

& g(x,t) = 0

And can obtain the following variational iteration formula:

$${S_i^{\ast }}_{n+1}=S_{i_n}^{\ast }-\underset{0}{\overset{\theta }{\int }}\left\{S_{i_{n_{\tau }}}^{\ast }+S_{i{}_n{}^{}}^{\ast }{\left(S_{i{}_n}^{\ast }\right)}_{\xi }-\epsilon S_{i_{\mathit{n\xi \xi }}}^{\ast }\right\}\mathit{d\tau }$$E16

Using (14), the approximate solutions $U_n\left(x,t\right)$are obtained by substituting;

$$S_i^{\ast }\left(\xi ,0\right)=S_{i_0}^{\ast }{\sout{e}}^{\xi }$$E17

Approximate solutions are given below;

$$S_{i_1}^{\ast }=S_{i_0}^{\ast }{e}^{\xi }-{\beta }_1^0\theta ;\mathit{\text{where}}{\beta }_1^0=\left(S_{i_0}^{{\ast }^2}{e}^{2\xi }-\epsilon S_{i_0}^{\ast }{e}^{\xi }\right)$$

$$S_{i_2}^{\ast }=S_{i_0}^{\ast }{e}^{\xi }-{\beta }_1^0\theta +{\beta }_1^1\frac{{\theta }^2}{2}\mathit{\text{where}}{\beta }_1^1={\beta }_1^0{S}_{i_0}^{\ast }{e}^{\xi }$$

Similarly,

$$S_{i_3}^{\ast }=S_{i_0}^{\ast }{e}^{\xi }-{\beta }_2^1\frac{{\theta }^2}{2!}+{\beta }_2^2\frac{{\theta }^3}{3!}$$

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.

To overcome these problems, following ADM and LADM is suggested.

Now applying ADM to equation (10); we get

$$S_i^{\ast }\left(\xi ,\theta \right)=L_{\theta }^{-1}\left[S_i^{\ast }{S}_{i_{\xi }}^{\ast }-\epsilon S_{i_{\mathit{\xi \xi }}}^{\ast }\right]$$E18

And recursive relation is:

$$S_i\left(\xi ,0\right)=e^{\xi }$$

Then:

$$S_{i_1}^{\ast }\left(\xi ,\theta \right)={\beta }_1^0\theta$$

$$S_{i_2}^{\ast }\left(\xi ,\theta \right)={\beta }_{12}^0\frac{{\theta }^3}{3}-{\mathit{\epsilon \beta }}_2^1\frac{{\theta }^2}{2}$$

$$S_{i_3}^{\ast }\left(\xi ,\theta \right)={\beta }_{123}^0\frac{{\theta }^4}{4}-{\mathit{\epsilon \beta }}_3^1\frac{{\theta }^3}{3}$$

and so on…

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

$$S_i^{\ast }\left(x,t\right)=L^{-1}\left[\frac{1}{s}L\left[S_{i_0}^{\ast }{S}_{i_{0_{\xi }}}^{\ast }-\epsilon S_{i_{0_{\mathit{\xi \xi }}}}^{\ast }\right]\right]$$

$$S_{i_1}^{\ast }={\beta }_1^0{e}^{\xi }\theta$$

$$S_{i_2}^{\ast }=\left({\beta }_1^{0^2}{e}^{2\xi }-{\mathit{\epsilon \beta }}_1^0{e}^{\xi }\right)\frac{{\theta }^2}{2!}$$

$$S_{i_3}^{\ast }=\left({\beta }_0^3-{\mathit{\epsilon \beta }}_1^3\right)\frac{{\theta }^3}{3!}$$

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 1–7).

Figure 1.

Plot of Saturation $S_i^{\ast }\left(\xi ,\theta \right)$ versus $\xi$ for VIM Solution.

Figure 2.

Plot of Saturation $S_i^{\ast }\left(\xi ,\theta \right)$ versus $\theta$ for VIM Solution.

Figure 3.

3-Dimensional VIM Solution.

Figure 4.

Plot of Saturation $S_i^{\ast }\left(\xi ,\theta \right)$ versus $\xi$ for ADM Solution.

Figure 5.

Plot of Saturation $S_i^{\ast }\left(\xi ,\theta \right)$ versus $\xi$ for LADM Solution.

Figure 6.

Plot of Saturation $S_i^{\ast }\left(\xi ,\theta \right)$ versus $\theta$ for LADM Solution.

Figure 7.

3-Dimensional LADM Solution

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.

Acknowledgments

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