Volume 04 Issue 10-2024
43
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
10
Pages:
43-47
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
Filtration processes of liquid solutions are fundamental in many natural and industrial applications, such as
environmental protection, chemical engineering, water purification, and petroleum extraction. This article develops a
mathematical model for describing filtration processes and explores various numerical methods for solving the
governing equations. The model is based on Darcy’s law, continuity equation, and constitutive relations of liquid
solutions in porous media. Numerical methods, including finite difference, finite element, and finite volume
approaches, are discussed with applications to various filtration scenarios. We also provide analysis of the stability,
convergence, and efficiency of these methods.
KEYWORDS
Filtration
processes, liquid solutions, mathematical model, Darcy’s law, numerical methods, finite difference, finite
element, finite volume, porous media.
INTRODUCTION
Filtration processes are critical for understanding fluid
transport through porous media. These processes
have broad applications, from the extraction of
hydrocarbons from oil reservoirs to the treatment of
polluted groundwater and industrial filtration systems.
Understanding how liquids move through porous
structures requires robust mathematical models and
efficient numerical methods to solve them.
Research Article
MATHEMATICAL MODEL AND NUMERICAL METHODS OF FILTRATION
PROCESSES OF LIQUID SOLUTIONS
Submission Date:
October 02, 2024,
Accepted Date:
October 07, 2024,
Published Date:
October 12, 2024
Crossref doi:
https://doi.org/10.37547/ajast/Volume04Issue10-07
Alisherov Akramboy Alisherovich
Samarkand City Specialized Boarding School No. 1., Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 04 Issue 10-2024
44
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
10
Pages:
43-47
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
This article aims to formulate a mathematical model for
the filtration of liquid solutions through porous media,
develop appropriate boundary and initial conditions,
and implement numerical methods to solve the model.
The focus is on filtration in homogeneous and
heterogeneous media, considering both isotropic and
anisotropic cases.
The fundamental equation governing the filtration of
liquids in porous media is Darcy’s Law, which describes
the flow of a fluid through a porous material. For
incompressible, Newtonian
fluids, Darcy’s law is given by:
where:
The mass conservation of an incompressible fluid in a
porous medium is expressed by the continuity
equation:
For incompressible flows, this reduces to:
Volume 04 Issue 10-2024
45
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
10
Pages:
43-47
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
where
ρ\rhoρ
is the density of the fluid.
The governing equations for the filtration process are
obtained by combining Darcy’s law and the continuity
equation. For a single-phase, incompressible fluid in a
porous medium, the resulting partial differential
equation (PDE) for pressure PPP is:
This equation is supplemented by appropriate
boundary conditions, which may include Dirichlet,
Neumann, or mixed conditions depending on the
specific problem being modeled.
To solve the governing PDEs for filtration, various
numerical methods are employed. The choice of
method depends on the complexity of the porous
structure, the nature of the boundary conditions, and
the required accuracy.
The finite difference method approximates derivatives
using discrete differences on a grid. For example, the
second-order
accurate
central
difference
approximation for the spatial derivative in one
dimension is:
In multidimensional problems, finite difference
schemes are applied to each direction independently.
FDM is suitable for structured grids and simple
geometries.
The finite element method divides the computational
domain into smaller elements and uses interpolation
functions (usually polynomials) to approximate the
solution within each element. FEM is particularly
advantageous
for
complex
geometries
and
heterogeneous materials. The weak form of the
governing equations is derived, and the solution is
obtained by minimizing the residual over the entire
domain.
The variational form of the pressure equation in FEM is:
where
is the test function, and Ω represents the computational domain
Volume 04 Issue 10-2024
46
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
10
Pages:
43-47
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
The finite volume method is widely used in engineering
applications because it ensures local conservation of
mass and other conserved quantities. In FVM, the
domain is divided into control volumes, and fluxes are
calculated across the boundaries of these volumes.
The discretized form of the governing equation for
each control volume
Numerical methods for solving PDEs must be stable
and convergent to ensure that the solutions are
physically meaningful. Stability analysis often involves
the Courant
–
Friedrichs
–
Lewy (CFL) condition, which
provides a criterion for the time step in explicit time-
stepping schemes.
For implicit methods, stability is generally guaranteed,
but they require solving large linear systems at each
time step. Convergence refers to the behavior of the
numerical solution as the grid is refined. A method is
said to converge if the solution approaches the true
solution as the grid spacing decreases.
To demonstrate the application of the developed
mathematical model and numerical methods, we
consider a case study of fluid filtration through a
heterogeneous porous medium. The permeability field
is assumed to vary spatially according to:
is the reference permeability, and
is the
characteristic length scale of the heterogeneity. The
finite element method is used to solve the pressure
distribution in the domain, and the results are
compared with experimental data.
The mathematical model and numerical methods
developed in this article provide a framework for
simulating filtration processes of liquid solutions in
porous media. The choice of numerical method
depends on the geometry of the domain, material
properties, and desired accuracy. While finite
difference methods are efficient for simple
geometries, finite element and finite volume methods
offer greater flexibility and accuracy for complex
systems.
Future research may focus on extending the model to
account for multi-phase flows, reactive transport, and
the coupling between mechanical deformation and
fluid flow in deformable porous media.
REFERENCES
Volume 04 Issue 10-2024
47
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
04
ISSUE
10
Pages:
43-47
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
1.
Bear, J. (1972). Dynamics of Fluids in Porous Media.
Elsevier.
2.
Peaceman, D. W. (1977). Fundamentals of
Numerical Reservoir Simulation. Elsevier.
3.
Aziz, K., & Settari, A. (1979). Petroleum Reservoir
Simulation. Elsevier.
4.
Versteeg, H. K., & Malalasekera, W. (2007). An
Introduction to Computational Fluid Dynamics: The
Finite Volume Method. Pearson Education.
