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Vector analysis of fluid flow

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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.

Continuity equation

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

Vol1 page 0051 eq 001.png....................(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 Vol1 page 0051 inline 001.png = {Jx, Jy, Jz}, Eq. 1 can be written in vector notation as

Vol1 page 0051 eq 002.png....................(2)

where the divergence of vector Vol1 page 0051 inline 001.png = {Jx, Jy, Jz}, in Cartesian coordinates, is

Vol1 page 0051 eq 003.png....................(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

Vol1 page 0051 eq 004.png....................(4)

where 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

Vol1 page 0051 eq 005.png....................(5)

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

If, instead of a scalar function (f), we can associate a vector Vol1 page 0051 inline 003.png with every point in the region (R), we can construct a vector field of the form

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.

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

Vol1 page 0052 eq 001.png....................(7)

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

Vol1 page 0052 eq 002.png....................(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 Vol1 page 0052 inline 001.png. The result is the divergence of the vector field.

Vol1 page 0052 eq 003.png....................(9)

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

Vol1 page 0052 eq 004.png....................(10)

The curl of the vector field 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 Vol1 page 0051 inline 003.png. The divergence of the gradient of a scalar field ( f ) is

Vol1 page 0053 eq 001.png....................(11)

where we introduce the Laplacian operator,  

Vol1 page 0053 eq 002.png....................(12)

in Cartesian coordinates.

The gradient, divergence, curl, and Laplacian operators arise in many PDEs that affect petroleum engineering. For example, a vector field Vol1 page 0051 inline 003.png is said to be irrotational if curl Vol1 page 0051 inline 003.png = 0, and it is said to be solenoidal if div 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 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 Vol1 page 0052 inline 001.png has concentration (C) and flux (J→) given by

Vol1 page 0053 eq 003.png....................(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

Vol1 page 0053 eq 004.png....................(14)

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

Vol1 page 0053 eq 005.png....................(15)

into Eq. 14 to obtain

Vol1 page 0053 eq 006.png....................(16)

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

Vol1 page 0053 eq 007.png....................(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 Vol1 page 0053 inline 001.png in the multidimensional form

Vol1 page 0054 eq 001.png....................(18)

Substituting Eq. 18 into the 3D continuity equation gives

Vol1 page 0054 eq 002.png....................(19)

If we assume that Vol1 page 0051 inline 003.png and D are constant, we can simplify Eq. 19 to the form of

Vol1 page 0054 eq 003.png....................(20)

Eq. 20 is the 3D convection/dispersion equation. The term D2C is the dispersion term, and the term Vol1 page 0054 inline 002.png is the convection term.

Nomenclature

f = scalar function, Eq. 5
Vol1 page 0051 inline 002.png = unit vectors in Cartesian coordinates, Eq. 4
Jx, Jy, Jz = fluid flux in x-, y-, z-directions
Vol1 page 0051 inline 001.png = 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
Vol1 page 0051 inline 003.png = vector field, Eq. 6
Vol1 page 0075 inline 001.png = 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

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.

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