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\begin{document}

\title{LT-SCM-SNI method for solving definite solution problems of
three-interval composite nonlinear partial differential equations and
its application}



\author[1]{Xiaoxu Dong}%
\author[1]{Shun Li}%
\author[1]{Hong Guo}%
\author[1]{Wei Li}%
\author[1]{Wenjing Li}%
\author[1]{Qun Liu}%
\affil[1]{Xihua University}%


\vspace{-1em}



  
  \date{May 07, 2024}


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\selectlanguage{english}
\begin{abstract}
The definite solution problems of three-interval composite nonlinear
partial differential equations (PDE) under different conditions is
studied in the paper. Then the definite solution problems are solved by
Laplace transformation - similar constructing method - Stehfest
numerical inversion equation (LT-SCM-SNI method). Firstly, the definite
solution problems of three-interval composite nonlinear PDE is
transformed into the boundary value problems of three-interval composite
linear of ordinary differential equation (ODE) by linearization and
Laplace transformation (LT). Secondly, the solution in Laplace space of
the boundary value problems of three-interval composite linear of ODE is
obtained through using the similar constructing method (SCM). The
solution in real space of the definite solution problems of
three-interval composite nonlinear PDE is obtained through using the
Stehfest numerical inversion equation (SNI) and linearization equation.
Finally, the nonlinear spherical seepage model of three-interval
composite reservoir under infinite outer boundary conditions is solved
through using the LT-SCM-SNI method. The influence of various parameters
on dimensionless bottom-hole transient pressure is studied. The example
shows the simplicity and practicability of the method presented in the
paper.%
\end{abstract}%



\sloppy


LT-SCM-SNI method for solving definite solution problems of
three-interval composite nonlinear partial differential equations and
its application11 Project supported by Research on the coupling model of
fracturing flowback in shale gas reservoir development, Talent
introduction project of Xihua University (Z202068).

Xiaoxu Dong\textsuperscript{1,}22Corresponding author. Email address:
dongxiaoxu1028@163.com (Xiaoxu Dong)., Shunchu Li\textsuperscript{1},
Hong Guo\textsuperscript{1}, Wei Li \textsuperscript{1}, Wenjing Li
\textsuperscript{1}, Qun Liu\textsuperscript{1}

\textsuperscript{1} School of Science, Xihua University, Chengdu,
610039, China

dongxiaoxu1028@163.com; lishunchu@163.com; 13541638376@163.com;
supinf2003@aliyun.com; lwj2986345062@163.com; liuqun6661211@163.com

\textbf{Abstract:}The definite solution problems of three-interval
composite nonlinear partial differential equations (PDE) under different
conditions is studied in the paper. Then the definite solution problems
are solved by Laplace transformation - similar constructing method -
Stehfest numerical inversion equation (LT-SCM-SNI method). Firstly, the
definite solution problems of three-interval composite nonlinear PDE is
transformed into the boundary value problems of three-interval composite
linear of ordinary differential equation (ODE) by linearization and
Laplace transformation (LT). Secondly, the solution in Laplace space of
the boundary value problems of three-interval composite linear of ODE is
obtained through using the similar constructing method (SCM). The
solution in real space of the definite solution problems of
three-interval composite nonlinear PDE is obtained through using the
Stehfest numerical inversion equation (SNI) and linearization equation.
Finally, the nonlinear spherical seepage model of three-interval
composite reservoir under infinite outer boundary conditions is solved
through using the LT-SCM-SNI method. The influence of various parameters
on dimensionless bottom-hole transient pressure is studied. The example
shows the simplicity and practicability of the method presented in the
paper.

\textbf{Keywords:}Definite solution problems; Nonlinear partial
differential equations; LT-SCM-SNI method; Three-interval composite
reservoir; Sensibility analysis

1 Introduction

Nonlinear partial differential equations and their systems as models for
describing real-world phenomena often appear in the field of physical
sciences and engineering. In practical applications, the solutions to
the partial differential equation that satisfies some additional
conditions, such as initial conditions and/or boundary conditions are a
common concern. Therefore, the study of solutions to definite solution
problems of nonlinear partial differential equations plays an important
role in the study of nonlinear physical phenomena. Domestic and foreign
scholars have done a lot of research in this area.

Ford and Pal et al. \textsuperscript{{[}6{]}} constructed a reliable
numerical scheme for solving two-sided space-fractional partial
differential equations. The use of numerical schemes usually solves a
slightly perturbed problem. Yang and Deng\textsuperscript{{[}19{]}}put
forward Riccati-Bernoulli sub-ODE method to construct exact traveling
wave solutions, solitary wave solutions and peaked wave solutions for
nonlinear partial differential equations. Han and Wang et al.
\textsuperscript{{[}7{]}} studied the existence and asymptotic behavior
of solution for a class of boundary value problems of nonlinear higher
order elliptic equations with two parameters. Sahadevan and Prakash
\textsuperscript{{[}13{]}} showed how to use invariant subspace method
to derive the exact solution of a time fractional nonlinear partial
differential equation. This reference helps to derive multiple exact
solutions. Madenci and Dorduncu et al. \textsuperscript{{[}10{]}}
obtained the numerical solutions of linear and nonlinear partial
differential equations (PDEs) by using differential operators. The
boundary conditions of model they studied are Dirichlet and
Neumann-characteristic boundary conditions. Yang and Goodyer et al.
\textsuperscript{{[}18{]}} introduced a new software tool for solving
efficient solutions of parabolic linear and nonlinear partial
differential equations. It is suitable for two different sample
problems, which illustrate the flexibility and robustness of the tool.
Abraham-Shrauner\textsuperscript{{[}1{]}} used the Power Index Method to
identify the possible exact analytical nonlinear solutions of any of two
characteristics of hyperbolic functions or any of three characteristics
of Jacobian elliptic functions of nonlinear partial differential
equations, which are invariant under the independent variables
translations. Benamou and Froese et al. \textsuperscript{{[}2{]}}
applied the classical ``direct'' technique involving Lie symmetries of
partial differential equations to the boundary value problems of
generalized Burgers equations with time-dependent viscosity coefficient
and solved the particular sub-cases. However, this method has its
limitations. Polyanin and Zhurov \textsuperscript{{[}11{]}} have derived
the nonlinear differential-difference equations of motion for viscous
incompressible fluid with finite relaxation rate and given the
corresponding exact solutions to these equations. The results obtained
can be used to solve some hydrodynamic problems for the
differential-difference model of viscous fluid. Lee and Manteuffel
\textsuperscript{{[}9{]}}proposed a natural framework for combining a
Newton linearization and a FOSLL discretization approach for nonlinear
partial differential equations. Yang and Deng et al.
\textsuperscript{{[}19{]}} proposed the Riccati-Bernoulli sub-ODE method
to construct the exact traveling wave solutions, solitary wave solutions
and spike wave solutions of nonlinear partial differential equations and
given the Backlund transformation of the Riccati-Bernoulli equation.
This method provides a powerful and simple mathematical tool for solving
some nonlinear partial differential equations in mathematical physics.

The study of reservoir seepage model is an important application of
boundary value problem of differential equation. In recent years, the
seepage theory has been continuously improved, but the theoretical and
applied research on nonlinear seepage problems is still in primary
stage. The study of nonlinear seepage mechanics has received more and
more attention and has become a new direction in the development of
modern seepage mechanics. Poon\textsuperscript{{[}12{]}} proposed a
composite reservoir model of rock with fractal characteristics. And
numerical solutions of the dynamic pressure of formation pressure is
obtained by using the Laplace numerical inversion. Tong et al.
\textsuperscript{{[}15-16{]}} obtained the asymptotic solution of the
seepage model by inverse Laplace transformation. They discussed the
influence of quadratic pressure gradient on pressure and described the
difference between linear and nonlinear solutions. Zhang and Jia et
al.\textsuperscript{{[}20{]}}established the nonlinear percolation model
for three-zone composite reservoir. Using variable substitution and
Laplace transform, the analytical solution of Laplace space under the
infinite boundary was obtained. By Stehfest numerical inversion, the
real space solution was also obtained. However, they only suggested that
we can use the Gaussian iterative elimination method to solve the
equation to obtain the Laplace space solution of the bottom hole
pressure. But they did not give the concrete expression of the solution.
Dong et al.\textsuperscript{{[}3-4{]}} obtained the SCM for solving the
boundary value problem of composite second-order linear homogeneous ODE.
Dong and Liu et al. \textsuperscript{{[}5{]}} established the nonlinear
three-region composite fractal reservoir. Using variable substitution
and Laplace transform, the analytical solution of Laplace space under
three different boundary conditions was obtained. By Stehfest numerical
inversion, the real space solution was also obtained.

The methods given in the above literature are not suitable for
complicated mathematical models and seepage models. Based on the above
research, this paper studies the definite solution problems of
three-interval composite nonlinear PDE under Dirichlet and
Neumann-characteristic boundary conditions. In section 2, we propose the
definite solution problems of three-interval composite nonlinear PDE and
solve definite solution problem by linearization equation, Laplace
transformation, similar constructing method and Stehfest numerical
inversion equation. In section 3, the nonlinear spherical seepage model
of three-interval composite reservoir under three kinds of outer
boundary conditions (infinite boundary, constant pressure boundary and
closed boundary) is solved by using the method obtained. In order to
compare the influence of nonlinear term on bottom-hole pressure, the
linear spherical seepage model is also solved in this section. In
section 4, the influence of various parameters on dimensionless
bottom-hole pressure is studied. The influence of nonlinear terms and
various parameters on the bottom-hole pressure is given in the form of
characteristic graphs and tables. The results show that the nonlinear
term and various parameters have great influence on the bottom-hole
pressure. The example shows the simplicity and practicability of the
method obtained in this paper.

2 The solution of the definite solution problem of three-interval
composite nonlinear PDE

In this section, we first propose the following definite solution
problem of three-interval composite nonlinear PDE:

\par\null

where, , , ,, , , , , , , are constants, and , .

Next, we study the solution to the definite solution problem . In order
to linearize the governing equation, the following variable
substitutions are proposed:

\par\null

Substituting the equation (13) into the solution problem (12), the
definite solution problem of nonlinear PDF (12) is transformed into the
following the definite solution problem of linear PDF:

\par\null

The Laplace transform is applied to the above definite solution problem
(14) of dimensionless PDE, which will be transformed into the following
boundary value problem of ODE:

\par\null

Where is Laplace space variable.

Combining the outer boundary conditions and of the boundary value
problem (15), the boundary value problem (15) is transformed into the
following boundary value problem:

where .

If and are two linearly independent solutions to second-order linear
homogeneous differential equations. According to equations (2)-(5), the
following functions of guide solutions are define by and :

\par\null

According to the similar constructing
method\textsuperscript{{[}17-18{]}} for solving the boundary value
problem of ODE, the solutions for inner, middle and outer intervals of
the boundary value problem (16) are expressed respectively as follow:

,

\par\null

where is called the similar kernel function of outer interval. Its
expression is as below:

\par\null

is called the similar kernel function of middle interval. Its expression
is as below:

\par\null

is called the similar kernel function of inner interval. Its expression
is as below:

\par\null

Using the Stehfest numerical inversion
equation\textsuperscript{{[}20{]}} to the equations (21)-(23), the
solutions for three intervals of the definite solution problem (14) in
Real space are

\par\null\par\null\par\null

where

\par\null

According to the equation (13), the solutions for three intervals of the
definite solution problem (1) in Real space are

3 The application of IT-SCM-SNI method

\textbf{3.1Nonlinear Seepage flow model of three-interval composite
reservoir}

This paper studies the physical model of nonlinear Seepage flow model of
three-interval composite reservoir as Figure 1.

To formulate the nonlinear seepage flow model, the main assumptions are
as follows:

2) Single-phase micro compressible fluid obeys Darcy's law and
isothermal curve;

3) Reservoir has equal thick, each direction is horizontal and has a
same nature;

4) Neglecting the capillary single-phase horizontal flow without gravity
effect;

5) There is no additional pressure drop at the interface of two seepage
interval;

6) Considering wellbore storage effects (in the beginning of opening
well, the fluid stored in the wellbore starts to flow, the oil in the
formation does not flow);

7) Formation pressure is initial reservoir pressure before producing;

8) The production well will be put into production with a fixed output
of .

The nonlinear spherical seepage equations under cylindrical coordinates
of the three-interval composite reservoir are obtained through the mass
conservation equation (continuity equation), the equation of motion, the
equation of state for liquid and rock \textsuperscript{{[}21{]}}:

The basic equation of seepage in inner interval:

The basic equation of seepage in middle interval:

The basic equation of seepage in outer interval:

where is reservoir pressure in interval, ; is liquid isothermal
compression factor of interval, ; is rock isothermal compression factor
of interval, ; is rock porosity in interval, dimensionless; is liquid
viscosity of interval, ; is reservoir permeability in interval, ; is
effective wellbore radius, ; is reservoir radius of interval, ; is
well-bore radius, .

The reservoir pressures of three intervals are equal at of nonlinear
spherical seepage model of the three-interval composite reservoir. So
the initial condition is

Inner boundary condition:

where is formation volume factor, dimensionless; is well-bore storage
factor, ; is bottom-hole transient pressure, .

Convergence conditions:

Infinite outer boundary conditions:

where is initial reservoir pressure.

\textbf{3.2 The solution of nonlinear Seepage flow model of
three-interval composite reservoir}

In order to facilitate the research and description, reference yield is
introduced and dimensionless variables are defined as follows:

,,,,,,,,, , .

Substituting above equations into nonlinear spherical seepage model of
the three-interval composite reservoir, the definite solution problem of
dimensionless PDE is obtained as below:

\par\null

In order to linearize the seepage equation, the following variable
substitutions are proposed:

\par\null

Substituting equation into definite solution problem, The definite
solution problem of nonlinear PDF is transformed into the boundary value
problem of linear PDF as below:

\par\null

The Laplace transform .is applied to the above definite solution problem
(32) of dimensionless PDE, which will be transformed into the following
boundary value problem of ODE:

and are two linearly independent solutions of differential equations.
and are two linearly independent solutions of differential equations.
and are two linearly independent solutions of differential equations.
Here and are respectively the first and the second class of modified
Bessel functions of order .

We define the function as below:

According to equations - (20), functions of guide solution are as
follows:

According to equation, similar kernel functions of outer interval is as
follows:

\par\null

According to equation, similar kernel functions of middle interval is as
follows:

According to equation, similar kernel functions of inner interval is as
follows:

According to equations -, solutions of the dimensionless reservoir
transient pressure of inner, middle and outer intervals in the Laplace
space , and are :

\par\null

Using the Stehfest numerical inversion equation to the equation , the
solution of inner interval the definite solution problem in Real space
is

\par\null

where

According to inner boundary condition of the definite solution problem,
the expression of dimensionless bottom-hole transient pressure of the
nonlinear model in Real space is:

4 Characteristics of the nonlinear seepage

\textbf{4.1The transient pressure characteristic curve}

The dimensionless bottom-hole transient pressure dynamic response curves
of and of the nonlinear model of three-interval composite reservoir
under infinite outer boundary condition are drawn in Figure 2 by
compiling the corresponding program. Characteristic curve of bottom-hole
transient pressure and transient pressure derivative under infinite
outer boundary condition of three-interval composite reservoir can be
divided into six seepage stages.

Phase I is the pure wellbore storage phase. The fluid seepage is
principally affected by wellbore storage. Wells open and produce. The
effusion seepage in the wellbore. The crude oil in the formation is at
rest. The double log curves bottom-hole transient pressure and transient
pressure derivative show a unit slope. The linear model is the same as
the nonlinear model, that is, the nonlinear seepage is the seepage of
the crude oil in the underground porous medium. Phase II is the
transition seepage phase. The derivative curves are influenced by the
skin factor and wellbore storage. Drilling, completion and other
construction work may cause formation plugging, so there is pollution in
the near well interval of oil well. Phase III is the seepage phase of
inner interval. It reflects the formation characteristics of the inner
interval. When the inner interval reaches the spherical flow stage, the
transient pressure derivative curves no longer follows the ''0.5 line''
rule, but is located below the ''0.5 line'' due to the non-linear effect
(The linear model follows the ``0.5 line'' rule). Phase IV is the
seepage phase of middle interval. It is a reflection of the formation
characteristics of the middle interval. Compared with the inner zone,
the transient pressure derivative curve of middle interval will
transform a step. If the physical properties of the middle interval
(permeability, porosity and rock compression coefficient, etc.) are
worse than the inner interval, the step of the transient pressure
derivative curve will rise. If the physical properties of the middle
interval are better than the inner interval, the step of the transient
pressure derivative curve of middle interval will decline. Phase V is
the seepage phase of outer interval. It is a reflection of the formation
characteristics of the outer interval. The change of the transient
pressure derivative curve of outer interval is similar to the phase IV.
Phase VI is the total system spherical seepage phase. The transient
pressure derivative curve converges to a horizontal line. It is
reflection of infinite outer boundary condition.

\textbf{4.2 Parameter sensibility analysis}

In this section, we will study the effects of , , , , , , and on
dimensionless bottom-hole transient pressure and transient pressure
derivative of nonlinear seepage flow model of three-interval composite
reservoir under infinite outer boundary condition. Transient pressure
and transient pressure derivative characteristic curves of nonlinear
seepage flow model affected by , , , , , , and are shown in Figures
3-10.

Firstly, we consider the effect of the quadratic pressure gradient on
dimensionless bottom-hole transient pressure and transient pressure
derivative by keeping the other parameters constant. Transient pressure
and transient pressure derivative characteristic curves of nonlinear
seepage flow model of three-interval composite reservoir under infinite
outer boundary condition affected by is shown in Figure 3.

Figure 3 exhibits the transient pressure characteristic curves of
nonlinear spherical seepage under different values of quadratic pressure
gradient by keeping the other parameters constant. It can be seen that
the quadratic pressure gradient has obviously impacted on the phase
II\textasciitilde{}VI of spherical seepage phase. Furthermore, the
bottom-hole transient pressure and transient pressure derivative
decrease with the increase of the quadratic pressure gradient .

Secondly, we consider the effect of the on dimensionless bottom-hole
transient pressure and transient pressure derivative by keeping the
other parameters constant. Transient pressure and transient pressure
derivative characteristic curves of nonlinear seepage flow model of
three-interval composite reservoir under infinite outer boundary
condition affected by is shown in Figure 4.

Figure 4 exhibits the transient pressure characteristic curves of
nonlinear spherical seepage under different values of the dimensionless
parameter group by keeping the other parameters constant. It can be seen
that the dimensionless parameter group has obviously impacted on the
phases II-III of spherical seepage phase. Furthermore, the bottom-hole
transient pressure and transient pressure derivative increase with the
increase of parameter group.

Thirdly, we consider the effect of the reservoir radius of inner
interval on dimensionless bottom-hole transient pressure and transient
pressure derivative by keeping the other parameters constant. Transient
pressure and transient pressure derivative characteristic curves of
nonlinear seepage flow model of three-interval composite reservoir under
infinite outer boundary condition affected by is shown in Figure 5.

Figure 5 exhibits the transient pressure characteristic curves of
nonlinear spherical seepage under different values of dimensionless
reservoir radius of inner interval by keeping the other parameters
constant. It can be seen that the dimensionless reservoir radius of
inner interval has obviously impacted on the phase III of spherical
seepage phase. Furthermore, the bottom-hole transient pressure and
transient pressure derivative decrease with the increase of reservoir
radius of inner interval .

Fourthly, we consider the effect of the reservoir radius of middle
interval on dimensionless bottom-hole transient pressure and transient
pressure derivative by keeping the other parameters constant. Transient
pressure and transient pressure derivative characteristic curves of
nonlinear seepage flow model of three-interval composite reservoir under
infinite outer boundary condition affected by is shown in Figure 6.

Figure 6 exhibits the transient pressure characteristic curves of
nonlinear spherical seepage under different values of dimensionless
reservoir radius of middle interval by keeping the other parameters
constant. It can be seen that the dimensionless reservoir radius of
middle interval has obviously impacted on the phase IV of spherical
seepage phase. Furthermore, the bottom-hole transient pressure and
transient pressure derivative decrease with the increase of reservoir
radius of middle interval . And the hump appears later with the increase
of reservoir radius of middle interval .

Fifthly, we consider the effect of dimensionless parameter on
dimensionless bottom-hole transient pressure and transient pressure
derivative by keeping the other parameters constant. Transient pressure
and transient pressure derivative characteristic curves of nonlinear
seepage flow model of three-interval composite reservoir under infinite
outer boundary condition affected by is shown in Figure 7.

Figure 7 exhibits the transient pressure characteristic curves of
nonlinear spherical seepage under different values of dimensionless
parameter by keeping the other parameters constant. It can be seen that
the dimensionless parameter has obviously impacted on the phase
II\textasciitilde{}VI of spherical seepage phase. Furthermore, the
bottom-hole transient pressure and transient pressure derivative
decrease with the increase of the parameter .

Sixthly, we consider the effect of dimensionless parameter on
dimensionless bottom-hole transient pressure and transient pressure
derivative by keeping the other parameters constant. Transient pressure
and transient pressure derivative characteristic curves of nonlinear
seepage flow model of three-interval composite reservoir under infinite
outer boundary condition affected by is shown in Figure 8.

Figure 8 exhibits the transient pressure characteristic curves of
nonlinear spherical seepage under different values of dimensionless
parameter by keeping the other parameters constant. It can be seen that
the dimensionless parameter has obviously impacted on the phase
IV\textasciitilde{}VI of spherical seepage phase. Furthermore, the
bottom-hole transient pressure and transient pressure derivative
decrease with the increase of the parameter .

Seventhly, we consider the effect of dimensionless parameter on
dimensionless bottom-hole transient pressure and transient pressure
derivative by keeping the other parameters constant. Transient pressure
and transient pressure derivative characteristic curves of nonlinear
seepage flow model of three-interval composite reservoir under infinite
outer boundary condition affected by is shown in Figure 9.

Figure 9 exhibits the transient pressure characteristic curves of
nonlinear spherical seepage under different values of dimensionless
parameter by keeping the other parameters constant. It can be seen that
the dimensionless parameter has obviously impacted on the phase
II\textasciitilde{}VI of spherical seepage phase. Furthermore, the
bottom-hole transient pressure and transient pressure derivative
decrease with the increase of the parameter on the phase
II\textasciitilde{}VI of spherical seepage phase. The bottom-hole
transient pressure and transient pressure derivative increase with the
increase of the parameter on the phase VI of spherical seepage phase..
And the hump appears later with the increase of reservoir radius of
middle interval .

Finally, we consider the effect of dimensionless parameter on
dimensionless bottom-hole transient pressure and transient pressure
derivative by keeping the other parameters constant. Transient pressure
and transient pressure derivative characteristic curves of nonlinear
seepage flow model of three-interval composite reservoir under infinite
outer boundary condition affected by is shown in Figure 10.

Figure 10 exhibits the transient pressure characteristic curves of
nonlinear spherical seepage under different values of dimensionless
parameter by keeping the other parameters constant. It can be seen that
the dimensionless parameter has obviously impacted on the phase IV of
spherical seepage phase. Furthermore, the bottom-hole transient pressure
and transient pressure derivative decrease with the increase of the
parameter .

5. Conclusions

The definite solution problems of three-interval composite nonlinear PDE
is transformed into the boundary value problems of three-interval
composite linear of ordinary differential equation (ODE) by
linearization and Laplace transformation. Then the solution in real
space of the boundary value problems of three-interval composite linear
of ordinary differential equation (ODE) is obtained by using similar
constructing method and the Stehfest numerical inversion equation. The
example shows the simplicity and practicability of the method obtained
for solving the class of definite solution problems.

The LI-SCM-SNI method is simple and effective for solving definite
solution problems of three-interval composite nonlinear partial
differential equations, nonlinear seepage flow model of three-interval
composite reservoir, etc.

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