# Chebyshev wavelets collocation method for simulating a two-phase flow of immiscible fluids in a reservoir with different capillary effects

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## Abstract

At least in the last 10 years, considerable effort has been given to studying the dynamics of fluid flow in porous media. The phenomena is widely applicable in many areas of science and engineering. In many cases, the effect of capillary pressure and discontinuities in the two-phase flow dynamics is not fully clear, especially in petroleum reservoirs. In this paper, we introduce a new method based on the Chebyshev wavelets collocation method and the so-called operational matrices of integration. The method was implemented specifically for an oil–water-phase flow in heterogeneous reservoir using different capillary pressure treatments. Convergence and accuracy of this method were established and used to simulate the partial differential equations governing the two-phase model. The method incorporates the various conditions of the complex governing equations as a single system. The system is subsequently reduced into a simple set of algebraic equations making the problem easier to solve. Numerical results showed that the method is able to account for the expected discontinuities occurring in the flow process. It was also found that these discontinuities or jumps in the two-phase flow are caused by the capillary pressure as expected physically.

## Keywords

Two-phase flow Chebyshev wavelets Operational matrices## Introduction

In the past decade, considerable effort has been made in studying the dynamics of fluid flow in porous media due to its applicability in many areas of science and engineering (Soulaine et al. 2013; Pasquier et al. 2017; Ahammad and Alam 2017), like hydrology, ground water remediation, membrane separation, polymer filtration, and oil and gas recovery from reservoirs (Epshteyn and Riviere 2006; Peaceman 1977; Zhong et al. 2013; Al-Rbeawi 2017). In petroleum reservoirs, the process is studied to predict their future performance and also optimize recovery processes in the reservoirs (Ahammad and Alam 2017; Begum 2009). This has led to the modification and proposition of models and methods for numerical reservoir simulation. According to Ewing (1983), reservoir simulation is a standard tool for predicting the flow of fluids through porous media under various operating conditions.

Oil industries, for instance, are interested in improving numerical methods to simulate the recovery of oil to exploit the reservoirs in an optimal way (Pasquier et al. 2017). Fluids such as water or gas are mostly injected into the oil reservoir to improve the recovery of oil from the medium, changing the dynamics of the flow in the reservoir. Detailed study of fluid flow in porous media is challenging even in the case of the single-phase flow (Mozolevski and Schuh 2013). Flow simulation in reservoirs has been studied using finite-difference methods, finite-element methods, and finite-volume methods among others over the past years (El-Amin et al. 2015; Foroozesh et al. 2008; Sun and Yuan 2015; Yuan et al. 2015). The challenge is that these methods are either chosen based on the nature of the problem and the problem dynamics, or in some cases, the methods for simulation are changed anytime the dynamics of the process changes. Particularly, in heterogeneous porous media, most methods cannot clearly capture the expected reservoir properties due to the consistent changing dynamics in heterogeneous media. It appears that methods that can adapt to such changes are well suited to solve flow problems in heterogeneous porous media.

The two-phase flow in a reservoir has been investigated by some authors (Amaziane and Jurak 2007; Hoteit and Firoozabadi 2008b; Zhong et al. 2013; Pasquier et al. 2017). The models describing flow of fluids in a reservoir are mostly nonlinear or coupled partial differential equations with its solution providing insight into the dynamics of the flow process. The simultaneous flow of fluids in a reservoir and many other porous media is a highly complex phenomena (Alam 2017; Szymkiewicz 2007). Numerical simulation of two-phase flow through a reservoir remains very challenging (Ahammad and Alam 2017; de la Cruz and Monsivais 2013). The effect of capillary pressure, permeabilities and heterogeneity significantly influences the flow path of the fluids (Kou and Sun 2010; El-Amin et al. 2015).

Wavelets have numerous applications and have been extensively used for numerical approximations in relevant literature over the past few decades. it is important to note that wavelet methods have gained great popularity for finding numerical solutions to many forms of differential and integral equations (Araghi et al. 2012; Epshteyn and Riviere 2006; Heydari et al. 2012; Hooshmandasl et al. 2012; Lepik 2011; Li 2010; Alam 2017) due to its high level of accuracy, speed, and efficiency in estimating the solutions. Different wavelet families were applied in various studies examples of which are Haar, Daubechies, Chebyshev, Legendre, and B-spline wavelets. The hallmark of wavelets is their ability to study the function at different scale features (Daubechies 1992).

Chebyshev wavelets are the types of wavelets constructed from Chebyshev polynomials as their basis functions. They have very excellent interpolating properties and gives better accuracy for numerical approximations (Heydari et al. 2014). Our purpose in this paper is to propose the Chebyshev wavelet collocation method for computing oil–water two-phase fluid flow in a reservoir. The present work is a continuation of the earlier work of same authors, Amoako-Yirenkyi et al. (2016) in which the Chebyshev wavelet collocation method was used for the numerical simulation of the single-phase flow in a reservoir.

In this paper, Chebyshev wavelet collocation method is used mutually with the operational matrix of integration to simulate the two-phase flow process in a reservoir with different capillary pressures. This paper is outlined as follows: in “Two-phase flow model”, the two-phase flow model is reviewed. In “First kind Chebyshev wavelets and its properties”, we describe the Chebyshev wavelets and its properties as the solution scheme. In “Chebyshev wavelet formulation of two-phase flow model”, we present the wavelet formulation of the two-phase flow model. The approximate solution to the flow problem is discussed in “Numerical results”. Finally, we conclude in “Conclusions”.

## Two-phase flow model

*w*and oil, the non-wetting phase denoted by

*o*), \(\phi\) is the porosity, \(q_\alpha\) is the injection or production rate per unit volume at phase \(\alpha\) (1/s), \(S_\alpha\) is the saturation at phase \(\alpha\), and \(u_\alpha\) represents the Darcy’s velocity (m/s). The Darcy’s law defines a linear relationship between the velocity of the fluid and the gradient of phase pressure \(P_\alpha\) (Pa):

*K*is the absolute permeability, \((\text {m}^2)\), \(k_{r\alpha }\) is the relative permeability (–) of phase \(\alpha\), \(\rho _\alpha\) \((\text {kg}/\text {m}^3)\), and \(\mu _\alpha\), \((\text {Pa}\,\text {s})\) are density and viscosity of phase \(\alpha\), respectively. Water and oil are assumed to fill the entire pore space in the medium. That is

## First kind Chebyshev wavelets and its properties

*a*, and translation parameter,

*b*, are allowed to vary continuously, we obtain a family of continuous wavelets as

*a*and

*b*to assume discrete values as \(a = a_0^{-n}\) and \(b=mb_0a_0^{-n}\), where \(a_0>1, b_0>0\), with

*n*and

*m*being positive integers, gives a family of discrete wavelets

*k*, the first kind Chebyshev wavelets family is defined on the interval [0, 1] as

*m*is the degree of the first kind Chebyshev polynomials,

*t*is the normalized time and

*m*defined on the interval \([-1,1]\). The coefficients in Eq. (19) are to ensure orthonormality of the constructed wavelets. The Chebyshev polynomials are generally calculated recursively from the set of equations:

*M*is a fixed positive integer greater than 2. The set of Chebyshev Polynomials are orthogonal with respect to the weight function (Heydari et al. 2014):

*w*(

*t*) must be dilated and translated (Araghi et al. 2012; Heydari et al. 2014) as

*f*(

*t*) defined on the interval [0, 1) which is squared integrable can be expanded by the Chebyshev wavelets family as

*C*, \(\Psi (t)\) are \(2^{k-1}M\) column vectors given by

*f*(

*x*,

*t*) defined on the square \([0,1)\times [0,1)\) may be expanded using the Chebyshev wavelets basis as

*f*(

*x*,

*t*) in Eq. (29) is truncated, then we have

*D*is a \(2^{k-1}M\times 2^{k-1}M\) matrix.

### Convergence analysis

*f*(

*t*).

### Theorem 3.1

*If the Chebyshev wavelet expansion of a continuous function* *f*(*t*) *converges uniformly*, *then the Chebyshev wavelet expansion converges to the function* *f*(*t*) (Adibi and Assari 2010).

### Proof

*p*and

*q*are fixed and evaluating the integral term wise which is justified by uniform convergence on [0, 1]

*f*and

*g*have the same Fourier expansion based on the Chebyshev wavelets. We, therefore, conclude that \(f(t) = g(t)\) for \(t\in [0,1]\).

### Theorem 3.2

*If a function*\(f(t) \in L^2([0,1]),\)

*with bounded second order derivative*\(|f''(t)|\le {\mathcal {K}}\)

*can be expanded as a sum of infinite Chebyshev wavelets*:

*then*

*which implies the Chebyshev wavelets expansion converges uniformly to*

*f*(

*t*).

### Proof

*f*(

*t*) uniformly supported by Theorem 3.1.

### Error analysis

*f*(

*t*) approximated by the Chebyshev wavelets series as

*P*vanishing moments, then the wavelets coefficients are bounded. That is

### Theorem 3.3

*Suppose*\(C^\mathrm{T}\psi (t) = \sum _{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}c_{nm}\psi _{nm}\)

*is the Chebyshev wavelets approximation to a function*

*f*(

*t*)

*in the function space*,

*then the error is bounded by the expression*:

### Proof

In the use of the Chebyshev wavelets, the interval \([0,\ 1]\) is divided into \(2^{k-1}\) subintervals \(I_n = [\frac{n-1}{2^{k-1}},\ \frac{n}{2^{k-1}}]\) on which the function *f*(*t*) is approximated.

*P*that interpolates

*f*(

*t*) on the subintervals with the error bound being

### Accuracy analysis

The evaluation of the accuracy of a numerical method is crucial to describing the performance and applicability to solving problems. Theorem 3.4 shows the accuracy of the Chebyshev wavelets representation of a function in the function space.

### Theorem 3.4

*Given the second-order derivative square-integrable function*

*f*(

*t*)

*defined on the interval*[0, 1)

*with bounded second-order derivative for some constant*\({\mathcal {K}},\)

*say*\(|f''(t)|\le {\mathcal {K}},\)

*then*

### Proof

## Chebyshev wavelet operational matrix of integration

*t*giving rise to the following results:

## Chebyshev wavelet formulation of two-phase flow model

*P*(

*x*,

*t*), is formulated from the pressure equation by letting

*x*for \(P_\mathrm{o}(x,t)\):

*t*to obtain the wavelet formulation of water saturation in the medium:

*Z*are also decomposed with the Chebyshev wavelets basis as

*A*,

*B*,

*E*, and

*H*are known, the two equations are solved simultaneously for

*C*and

*D*. The solution to \(P_\mathrm{o}(x,t)\) and \(S_\mathrm{w}(x,t)\) can be reconstructed from their wavelet coefficients

*C*and

*D*, respectively, using Eqs. (84) and (87).

## Numerical results

Relevant data for the oil–water reservoir simulation

Relevant data | |
---|---|

Porosity | \(\phi = 0.2\) |

Absolute permeability | \(k = 1\) |

Water viscosity | \(\mu _\mathrm{w} = 1~\text {cP}\) |

Oil viscosity | \(\mu _\mathrm{o} = 0.25~\text {cP}\) |

Density of water | \(\rho _\mathrm{w} = 1000~\text {kg}/\text {m}^3\) |

Density of oil | \(\rho _\mathrm{o} = 1000~\text {kg}/\text {m}^3\) |

Residual saturations | \(S_{\mathrm{{rw}}} = S_{\mathrm{{ro}}} = 0/S_{\mathrm{{rw}}} = 0, \ S_{\mathrm{{ro}}} = 0.2\) |

In Fig. 9, the density of the wetting phase was kept at \(1000~\text {kg}/\text {m}^3\) as in the previous cases, but that of the non-wetting phase is \(660~\text {kg}/\text {m}^3\). All the other parameter values are maintained including the force of gravity. Unlike the earlier results, where the distribution of the saturation had some discontinuity, varying the density of the fluids completely eliminates the discontinuity. Interestingly, the dynamics of the fluid dispersion in the medium is maintained.

## Conclusions

The proposed Chebyshev wavelets method has been successfully implemented in studying the two-phase slightly compressible flow in combination with their operational matrix of integration. This method was developed based on the first kind Chebyshev polynomial. The corresponding algorithm converges quite well, stable, and adapts to changing dynamics. Key to this study is the inclusion and treatment of the capillary pressure. The fluid dispersion in the reservoir displayed some discontinuities which in the literature may be due to the continuity of the capillary treatment. The study also revealed that the other properties of the process such as density could completely eliminate the discontinuity in saturation of the fluids.

## Notes

### Acknowledgements

We will like to acknowledge the support received from the National Institute for Mathematical Sciences, Ghana for this study

### Compliance with ethical standards

### Competing interests

The authors declare that they have no competing interests.

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