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Fluid flow equations in two and three dimensions can be compactly represented using concepts from vector analysis. This page reviews the mathematics involved.
Fluid flow equations in two and three dimensions can be compactly represented using concepts from vector analysis. This page reviews the mathematics involved.


==Continuity equation==
== Continuity equation ==
 
The continuity equation in three space dimensions for the Cartesian coordinate system, shown in '''Fig. 1''', is
The continuity equation in three space dimensions for the Cartesian coordinate system, shown in '''Fig. 1''', is


[[File:Vol1 page 0051 eq 001.png]]....................(1)
[[File:Vol1 page 0051 eq 001.png|RTENOTITLE]]....................(1)


<gallery widths="300px" heights="200px">
<gallery widths="300px" heights="200px">
Line 10: Line 11:
</gallery>
</gallery>


The flux terms (''J''<sub>''y''</sub>) and (''J''<sub>''z''</sub>) have meanings analogous to (''J''<sub>''x''</sub>) for flux in the ''y'' and ''z'' directions, respectively. If we write the components of flux as the flux vector [[File:Vol1 page 0051 inline 001.png]] = {''J''<sub>''x''</sub>, ''J''<sub>''y''</sub>, ''J''<sub>''z''</sub>}, '''Eq. 1''' can be written in vector notation as
The flux terms (''J''<sub>''y''</sub>) and (''J''<sub>''z''</sub>) have meanings analogous to (''J''<sub>''x''</sub>) for flux in the ''y'' and ''z'' directions, respectively. If we write the components of flux as the flux vector [[File:Vol1 page 0051 inline 001.png|RTENOTITLE]] = {''J''<sub>''x''</sub>, ''J''<sub>''y''</sub>, ''J''<sub>''z''</sub>}, '''Eq. 1''' can be written in vector notation as


[[File:Vol1 page 0051 eq 002.png]]....................(2)
[[File:Vol1 page 0051 eq 002.png|RTENOTITLE]]....................(2)


where the divergence of vector [[File:Vol1 page 0051 inline 001.png]] = {''J''<sub>''x''</sub>, ''J''<sub>''y''</sub>, ''J''<sub>''z''</sub>}, in Cartesian coordinates, is
where the divergence of vector [[File:Vol1 page 0051 inline 001.png|RTENOTITLE]] = {''J''<sub>''x''</sub>, ''J''<sub>''y''</sub>, ''J''<sub>''z''</sub>}, in Cartesian coordinates, is


[[File:Vol1 page 0051 eq 003.png]]....................(3)
[[File:Vol1 page 0051 eq 003.png|RTENOTITLE]]....................(3)


The divergence operator ∇• is an example of an operator from vector analysis that determines the spatial variation of a vector or scalar field. Following Fanchi, <ref name="r1" /> we first review the concepts of scalar and vector fields and then define gradient (grad), divergence (div), and curl operators.  
The divergence operator ∇• is an example of an operator from vector analysis that determines the spatial variation of a vector or scalar field. Following Fanchi, <ref name="r1">Fanchi, J.R. 2006. Math Refresher for Scientists and Engineers, third edition. New York: Wiley Interscience.</ref> we first review the concepts of scalar and vector fields and then define gradient (grad), divergence (div), and curl operators.
 
== Scalar and vector fields ==


==Scalar and vector fields==
We define scalar and vector fields in a Cartesian coordinate system with position vector
We define scalar and vector fields in a Cartesian coordinate system with position vector


[[File:Vol1 page 0051 eq 004.png]]....................(4)
[[File:Vol1 page 0051 eq 004.png|RTENOTITLE]]....................(4)
 
where [[File:Vol1 page 0051 inline 002.png|RTENOTITLE]] are unit vectors defined along the orthogonal {''x,y'',''z''} coordinate axes. If we can associate a scalar function (''f'') with every point in a region (''R''), then the scalar field may be written as


where [[File:Vol1 page 0051 inline 002.png]] are unit vectors defined along the orthogonal {''x,y'',''z''} coordinate axes. If we can associate a scalar function (''f'') with every point in a region (''R''), then the scalar field may be written as
[[File:Vol1 page 0051 eq 005.png|RTENOTITLE]]....................(5)


[[File:Vol1 page 0051 eq 005.png]]....................(5)
Examples of scalar fields include pressure, temperature, and saturation.


Examples of scalar fields include pressure, temperature, and saturation.
If, instead of a scalar function (''f''), we can associate a vector [[File:Vol1 page 0051 inline 003.png|RTENOTITLE]] with every point in the region (''R''), we can construct a vector field of the form


If, instead of a scalar function (''f''), we can associate a vector [[File:Vol1 page 0051 inline 003.png]] with every point in the region (''R''), we can construct a vector field of the form
[[File:Vol1 page 0051 eq 006.png|RTENOTITLE]]....................(6)


[[File:Vol1 page 0051 eq 006.png]]....................(6)
The vector field is a function that assigns a vector to every point in the region ''R''. Examples of vector fields include the Darcy velocity field and seismic velocities.


The vector field is a function that assigns a vector to every point in the region ''R''. Examples of vector fields include the Darcy velocity field and seismic velocities.
== Gradient, divergence, and curl ==


==Gradient, divergence, and curl==
The spatial variation of a scalar or vector field can be determined with the del operator ∇. The del operator, ∇, is defined in Cartesian coordinates as
The spatial variation of a scalar or vector field can be determined with the del operator ∇. The del operator, ∇, is defined in Cartesian coordinates as


[[File:Vol1 page 0052 eq 001.png]]....................(7)
[[File:Vol1 page 0052 eq 001.png|RTENOTITLE]]....................(7)


The gradient of a scalar field (''f'') is obtained by operating on the scalar field with the del operator, thus
The gradient of a scalar field (''f'') is obtained by operating on the scalar field with the del operator, thus


[[File:Vol1 page 0052 eq 002.png]]....................(8)
[[File:Vol1 page 0052 eq 002.png|RTENOTITLE]]....................(8)


The direction of the gradient of the scalar field (''f'') evaluated at a point is oriented in the direction of maximum increase of the scalar field. In addition, the vector field, ∇''f'', is perpendicular to a surface that corresponds to a constant value of the scalar field ('''Fig. 2''').  
The direction of the gradient of the scalar field (''f'') evaluated at a point is oriented in the direction of maximum increase of the scalar field. In addition, the vector field, ∇''f'', is perpendicular to a surface that corresponds to a constant value of the scalar field ('''Fig. 2''').


<gallery widths="300px" heights="200px">
<gallery widths="300px" heights="200px">
Line 52: Line 55:
</gallery>
</gallery>


Two outcomes are possible when the del operator is applied to a vector field. One outcome is to create a scalar, and the other is to create a vector. A scalar is obtained when we take the dot product of the del operator with a vector field [[File:Vol1 page 0052 inline 001.png]]. The result is the divergence of the vector field.
Two outcomes are possible when the del operator is applied to a vector field. One outcome is to create a scalar, and the other is to create a vector. A scalar is obtained when we take the dot product of the del operator with a vector field [[File:Vol1 page 0052 inline 001.png|RTENOTITLE]]. The result is the divergence of the vector field.


[[File:Vol1 page 0052 eq 003.png]]....................(9)
[[File:Vol1 page 0052 eq 003.png|RTENOTITLE]]....................(9)


A vector is obtained when we take the cross product of the del operator with a vector field [[File:Vol1 page 0052 inline 001.png]]. The result is the curl of the vector field [[File:Vol1 page 0051 inline 003.png]].
A vector is obtained when we take the cross product of the del operator with a vector field [[File:Vol1 page 0052 inline 001.png|RTENOTITLE]]. The result is the curl of the vector field [[File:Vol1 page 0051 inline 003.png|RTENOTITLE]].


[[File:Vol1 page 0052 eq 004.png]]....................(10)
[[File:Vol1 page 0052 eq 004.png|RTENOTITLE]]....................(10)


The curl of the vector field [[File:Vol1 page 0051 inline 003.png]] is called the rotation of the vector field. It is a vector that is normal to the plane containing the vector field [[File:Vol1 page 0051 inline 003.png]]. The divergence of the gradient of a scalar field ( ''f'' ) is
The curl of the vector field [[File:Vol1 page 0051 inline 003.png|RTENOTITLE]] is called the rotation of the vector field. It is a vector that is normal to the plane containing the vector field [[File:Vol1 page 0051 inline 003.png|RTENOTITLE]]. The divergence of the gradient of a scalar field ( ''f'' ) is


[[File:Vol1 page 0053 eq 001.png]]....................(11)
[[File:Vol1 page 0053 eq 001.png|RTENOTITLE]]....................(11)


where we introduce the Laplacian operator,  
where we introduce the Laplacian operator,  


[[File:Vol1 page 0053 eq 002.png]]....................(12)
[[File:Vol1 page 0053 eq 002.png|RTENOTITLE]]....................(12)
 
in Cartesian coordinates.


in Cartesian coordinates.  
The gradient, divergence, curl, and Laplacian operators arise in many PDEs that affect petroleum engineering. For example, a vector field [[File:Vol1 page 0051 inline 003.png|RTENOTITLE]] is said to be irrotational if curl [[File:Vol1 page 0051 inline 003.png|RTENOTITLE]] = 0, and it is said to be solenoidal if div [[File:Vol1 page 0051 inline 003.png|RTENOTITLE]] = 0. These properties of the vector field are useful for analyzing the propagation of seismic waves. Another useful application of vector analysis is to the mathematical representation of fluid flow in two or three spatial dimensions. Two examples are presented next.


The gradient, divergence, curl, and Laplacian operators arise in many PDEs that affect petroleum engineering. For example, a vector field [[File:Vol1 page 0051 inline 003.png]] is said to be irrotational if curl [[File:Vol1 page 0051 inline 003.png]] = 0, and it is said to be solenoidal if div [[File:Vol1 page 0051 inline 003.png]] = 0. These properties of the vector field are useful for analyzing the propagation of seismic waves. Another useful application of vector analysis is to the mathematical representation of fluid flow in two or three spatial dimensions. Two examples are presented next.
== Incompressible flow ==


==Incompressible flow==
Incompressible flow occurs when the density of a fluid is constant. In this case, the continuity equation for flow of a fluid with density (''ρ'') and velocity [[File:Vol1 page 0052 inline 001.png|RTENOTITLE]] has concentration (''C'') and flux (J→) given by
Incompressible flow occurs when the density of a fluid is constant. In this case, the continuity equation for flow of a fluid with density (''ρ'') and velocity [[File:Vol1 page 0052 inline 001.png]] has concentration (''C'') and flux (J→) given by


[[File:Vol1 page 0053 eq 003.png]]....................(13)
[[File:Vol1 page 0053 eq 003.png|RTENOTITLE]]....................(13)


The concentration and density are scalar fields, and the velocity and flux are vector fields. The continuity equation without source or sink terms becomes
The concentration and density are scalar fields, and the velocity and flux are vector fields. The continuity equation without source or sink terms becomes


[[File:Vol1 page 0053 eq 004.png]]....................(14)
[[File:Vol1 page 0053 eq 004.png|RTENOTITLE]]....................(14)


A more suitable form of the continuity equation for describing incompressible fluid flow is obtained by substituting the differential operator,
A more suitable form of the continuity equation for describing incompressible fluid flow is obtained by substituting the differential operator,


[[File:Vol1 page 0053 eq 005.png]]....................(15)
[[File:Vol1 page 0053 eq 005.png|RTENOTITLE]]....................(15)


into '''Eq. 14''' to obtain
into '''Eq. 14''' to obtain


[[File:Vol1 page 0053 eq 006.png]]....................(16)
[[File:Vol1 page 0053 eq 006.png|RTENOTITLE]]....................(16)


In the case of incompressible fluid flow, density is constant and '''Eq. 16''' reduces to
In the case of incompressible fluid flow, density is constant and '''Eq. 16''' reduces to


[[File:Vol1 page 0053 eq 007.png]]....................(17)
[[File:Vol1 page 0053 eq 007.png|RTENOTITLE]]....................(17)


'''Eq. 17''' shows that the divergence of the velocity of a flowing, incompressible fluid is zero.  
'''Eq. 17''' shows that the divergence of the velocity of a flowing, incompressible fluid is zero.


==Three-dimensional (3D) convection/dispersion equation==
== Three-dimensional (3D) convection/dispersion equation ==
The convection/dispersion equation in three dimensions is obtained by writing flux [[File:Vol1 page 0053 inline 001.png]] in the multidimensional form


[[File:Vol1 page 0054 eq 001.png]]....................(18)
The convection/dispersion equation in three dimensions is obtained by writing flux [[File:Vol1 page 0053 inline 001.png|RTENOTITLE]] in the multidimensional form
 
[[File:Vol1 page 0054 eq 001.png|RTENOTITLE]]....................(18)


Substituting '''Eq. 18''' into the 3D continuity equation gives
Substituting '''Eq. 18''' into the 3D continuity equation gives


[[File:Vol1 page 0054 eq 002.png]]....................(19)
[[File:Vol1 page 0054 eq 002.png|RTENOTITLE]]....................(19)


If we assume that [[File:Vol1 page 0051 inline 003.png]] and ''D'' are constant, we can simplify '''Eq. 19''' to the form of
If we assume that [[File:Vol1 page 0051 inline 003.png|RTENOTITLE]] and ''D'' are constant, we can simplify '''Eq. 19''' to the form of


[[File:Vol1 page 0054 eq 003.png]]....................(20)
[[File:Vol1 page 0054 eq 003.png|RTENOTITLE]]....................(20)


'''Eq. 20''' is the 3D convection/dispersion equation. The term ''D''∇<sup>2</sup>''C'' is the dispersion term, and the term [[File:Vol1 page 0054 inline 002.png]] is the convection term.  
'''Eq. 20''' is the 3D convection/dispersion equation. The term ''D''∇<sup>2</sup>''C'' is the dispersion term, and the term [[File:Vol1 page 0054 inline 002.png|RTENOTITLE]] is the convection term.
 
== Nomenclature ==


==Nomenclature==
{|
{|
|''f''
|=
|scalar function, '''Eq. 5'''
|-
|-
|[[File:Vol1 page 0051 inline 002.png]]
| ''f''
|=  
| =
|unit vectors in Cartesian coordinates, '''Eq. 4'''  
| scalar function, '''Eq. 5'''
|-
|-
|''J''<sub>''x''</sub>, ''J''<sub>''y''</sub>, ''J''<sub>''z''</sub>
| [[File:Vol1 page 0051 inline 002.png|RTENOTITLE]]
|=  
| =
|fluid flux in ''x''-, ''y''-, ''z''-directions
| unit vectors in Cartesian coordinates, '''Eq. 4'''
|-
|-
|[[File:Vol1 page 0051 inline 001.png]]
| ''J''<sub>''x''</sub>, ''J''<sub>''y''</sub>, ''J''<sub>''z''</sub>
|=  
| =
|fluid flux vector, '''Eq. 2'''  
| fluid flux in ''x''-, ''y''-, ''z''-directions
|-
|-
|(''J''<sub>''x''</sub>)<sub>''x''</sub>
| [[File:Vol1 page 0051 inline 001.png|RTENOTITLE]]
|=  
| =
|fluid flux in ''x''-direction at location ''x''  
| fluid flux vector, '''Eq. 2'''
|-
|-
|(''J''<sub>''y''</sub>)<sub>''y''</sub>  
| (''J''<sub>''x''</sub>)<sub>''x''</sub>
|=  
| =
|fluid flux in ''y''-direction at location ''y''  
| fluid flux in ''x''-direction at location ''x''
|-
|-
|(''J''<sub>''z''</sub>)<sub>''z''</sub>  
| (''J''<sub>''y''</sub>)<sub>''y''</sub>
|=  
| =
|fluid flux in ''z''-direction at location ''z''  
| fluid flux in ''y''-direction at location ''y''
|-
|-
|''q''  
| (''J''<sub>''z''</sub>)<sub>''z''</sub>
|=  
| =
|source term
| fluid flux in ''z''-direction at location ''z''
|-
|-
|''S''  
| ''q''
|=  
| =
|surface
| source term
|-
|-
|[[File:Vol1 page 0051 inline 003.png]]
| ''S''
|=  
| =
|vector field, '''Eq. 6'''
| surface
|-
|-
|[[File:Vol1 page 0075 inline 001.png]]  
| [[File:Vol1 page 0051 inline 003.png|RTENOTITLE]]
|=  
| =
|position vector, '''Eq. 4'''  
| vector field, '''Eq. 6'''
|-
|-
|''x,y'',''z''  
| [[File:Vol1 page 0075 inline 001.png|RTENOTITLE]]
|=
| =
|space dimensions
| position vector, '''Eq. 4'''
|-
|-
| ''x,y'',''z''
| =
| space dimensions
|}
|}


==References==
== References ==
<references>
<ref name="r1">Fanchi, J.R. 2006. ''Math Refresher for Scientists and Engineers'', third edition. New York: Wiley Interscience.</ref>
</references>


==Noteworthy papers in OnePetro==
<references />
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read


Kreyszig, E. O. 2011. Advanced Engineering Mathematics, 10th. John Wiley & Sons Inc. New York.  
== Noteworthy papers in OnePetro ==
 
== Noteworthy books ==
 
Kreyszig, E. O. (2011). Advanced Engineering Mathematics. (10th). New York, New York: John Wiley & Sons Inc. [http://faculties.sbu.ac.ir/~sadough/pdf/Advanced%20Engineering%20Mathematics%2010th%20Edition.pdf Online Resource] or [http://www.worldcat.org/oclc/314161 Worldcat]
 
== External links ==


==External links==
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro


==See also==
== See also ==
[[Mathematics of fluid flow]]
 
[[Mathematics_of_fluid_flow|Mathematics of fluid flow]]
 
[[Numerical_methods_analysis_of_fluid_flow|Numerical methods analysis of fluid flow]]


[[Numerical methods analysis of fluid flow]]
[[Diagonalizing_the_permeability_tensor|Diagonalizing the permeability tensor]]


[[Diagonalizing the permeability tensor]]
[[PEH:Mathematics_of_Fluid_Flow]]


[[PEH:Mathematics of Fluid Flow]]
[[Category:5.3 Reservoir fluid dynamics]]

Latest revision as of 20:22, 4 June 2015

Fluid flow equations in two and three dimensions can be compactly represented using concepts from vector analysis. This page reviews the mathematics involved.

Continuity equation

The continuity equation in three space dimensions for the Cartesian coordinate system, shown in Fig. 1, is

RTENOTITLE....................(1)

The flux terms (Jy) and (Jz) have meanings analogous to (Jx) for flux in the y and z directions, respectively. If we write the components of flux as the flux vector RTENOTITLE = {Jx, Jy, Jz}, Eq. 1 can be written in vector notation as

RTENOTITLE....................(2)

where the divergence of vector RTENOTITLE = {Jx, Jy, Jz}, in Cartesian coordinates, is

RTENOTITLE....................(3)

The divergence operator ∇• is an example of an operator from vector analysis that determines the spatial variation of a vector or scalar field. Following Fanchi, [1] we first review the concepts of scalar and vector fields and then define gradient (grad), divergence (div), and curl operators.

Scalar and vector fields

We define scalar and vector fields in a Cartesian coordinate system with position vector

RTENOTITLE....................(4)

where RTENOTITLE are unit vectors defined along the orthogonal {x,y,z} coordinate axes. If we can associate a scalar function (f) with every point in a region (R), then the scalar field may be written as

RTENOTITLE....................(5)

Examples of scalar fields include pressure, temperature, and saturation.

If, instead of a scalar function (f), we can associate a vector RTENOTITLE with every point in the region (R), we can construct a vector field of the form

RTENOTITLE....................(6)

The vector field is a function that assigns a vector to every point in the region R. Examples of vector fields include the Darcy velocity field and seismic velocities.

Gradient, divergence, and curl

The spatial variation of a scalar or vector field can be determined with the del operator ∇. The del operator, ∇, is defined in Cartesian coordinates as

RTENOTITLE....................(7)

The gradient of a scalar field (f) is obtained by operating on the scalar field with the del operator, thus

RTENOTITLE....................(8)

The direction of the gradient of the scalar field (f) evaluated at a point is oriented in the direction of maximum increase of the scalar field. In addition, the vector field, ∇f, is perpendicular to a surface that corresponds to a constant value of the scalar field (Fig. 2).

Two outcomes are possible when the del operator is applied to a vector field. One outcome is to create a scalar, and the other is to create a vector. A scalar is obtained when we take the dot product of the del operator with a vector field RTENOTITLE. The result is the divergence of the vector field.

RTENOTITLE....................(9)

A vector is obtained when we take the cross product of the del operator with a vector field RTENOTITLE. The result is the curl of the vector field RTENOTITLE.

RTENOTITLE....................(10)

The curl of the vector field RTENOTITLE is called the rotation of the vector field. It is a vector that is normal to the plane containing the vector field RTENOTITLE. The divergence of the gradient of a scalar field ( f ) is

RTENOTITLE....................(11)

where we introduce the Laplacian operator,  

RTENOTITLE....................(12)

in Cartesian coordinates.

The gradient, divergence, curl, and Laplacian operators arise in many PDEs that affect petroleum engineering. For example, a vector field RTENOTITLE is said to be irrotational if curl RTENOTITLE = 0, and it is said to be solenoidal if div RTENOTITLE = 0. These properties of the vector field are useful for analyzing the propagation of seismic waves. Another useful application of vector analysis is to the mathematical representation of fluid flow in two or three spatial dimensions. Two examples are presented next.

Incompressible flow

Incompressible flow occurs when the density of a fluid is constant. In this case, the continuity equation for flow of a fluid with density (ρ) and velocity RTENOTITLE has concentration (C) and flux (J→) given by

RTENOTITLE....................(13)

The concentration and density are scalar fields, and the velocity and flux are vector fields. The continuity equation without source or sink terms becomes

RTENOTITLE....................(14)

A more suitable form of the continuity equation for describing incompressible fluid flow is obtained by substituting the differential operator,

RTENOTITLE....................(15)

into Eq. 14 to obtain

RTENOTITLE....................(16)

In the case of incompressible fluid flow, density is constant and Eq. 16 reduces to

RTENOTITLE....................(17)

Eq. 17 shows that the divergence of the velocity of a flowing, incompressible fluid is zero.

Three-dimensional (3D) convection/dispersion equation

The convection/dispersion equation in three dimensions is obtained by writing flux RTENOTITLE in the multidimensional form

RTENOTITLE....................(18)

Substituting Eq. 18 into the 3D continuity equation gives

RTENOTITLE....................(19)

If we assume that RTENOTITLE and D are constant, we can simplify Eq. 19 to the form of

RTENOTITLE....................(20)

Eq. 20 is the 3D convection/dispersion equation. The term D2C is the dispersion term, and the term RTENOTITLE is the convection term.

Nomenclature

f = scalar function, Eq. 5
RTENOTITLE = unit vectors in Cartesian coordinates, Eq. 4
Jx, Jy, Jz = fluid flux in x-, y-, z-directions
RTENOTITLE = fluid flux vector, Eq. 2
(Jx)x = fluid flux in x-direction at location x
(Jy)y = fluid flux in y-direction at location y
(Jz)z = fluid flux in z-direction at location z
q = source term
S = surface
RTENOTITLE = vector field, Eq. 6
RTENOTITLE = position vector, Eq. 4
x,y,z = space dimensions

References

  1. Fanchi, J.R. 2006. Math Refresher for Scientists and Engineers, third edition. New York: Wiley Interscience.

Noteworthy papers in OnePetro

Noteworthy books

Kreyszig, E. O. (2011). Advanced Engineering Mathematics. (10th). New York, New York: John Wiley & Sons Inc. Online Resource or Worldcat

External links

Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro

See also

Mathematics of fluid flow

Numerical methods analysis of fluid flow

Diagonalizing the permeability tensor

PEH:Mathematics_of_Fluid_Flow