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# Vector analysis of fluid flow

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 ....................(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 = {Jx, Jy, Jz}, Eq. 1 can be written in vector notation as ....................(2)

where the divergence of vector = {Jx, Jy, Jz}, in Cartesian coordinates, is ....................(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,  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 ....................(4)

where 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 ....................(5)

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

If, instead of a scalar function (f), we can associate a vector with every point in the region (R), we can construct a vector field of the form ....................(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 spatial variation of a scalar or vector field can be determined with the del operator ∇. The del operator, ∇, is defined in Cartesian coordinates as ....................(7)

The gradient of a scalar field (f) is obtained by operating on the scalar field with the del operator, thus ....................(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 . The result is the divergence of the vector field. ....................(9)

A vector is obtained when we take the cross product of the del operator with a vector field . The result is the curl of the vector field . ....................(10)

The curl of the vector field is called the rotation of the vector field. It is a vector that is normal to the plane containing the vector field . The divergence of the gradient of a scalar field ( f ) is ....................(11)

where we introduce the Laplacian operator, ....................(12)

in Cartesian coordinates.

The gradient, divergence, curl, and Laplacian operators arise in many PDEs that affect petroleum engineering. For example, a vector field is said to be irrotational if curl = 0, and it is said to be solenoidal if div = 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 has concentration (C) and flux (J→) given by ....................(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 ....................(14)

A more suitable form of the continuity equation for describing incompressible fluid flow is obtained by substituting the differential operator, ....................(15)

into Eq. 14 to obtain ....................(16)

In the case of incompressible fluid flow, density is constant and Eq. 16 reduces to ....................(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 in the multidimensional form ....................(18)

Substituting Eq. 18 into the 3D continuity equation gives ....................(19)

If we assume that and D are constant, we can simplify Eq. 19 to the form of ....................(20)

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

## Nomenclature

 f = scalar function, Eq. 5 = unit vectors in Cartesian coordinates, Eq. 4 Jx, Jy, Jz = fluid flux in x-, y-, z-directions = 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 = vector field, Eq. 6 = position vector, Eq. 4 x,y,z = space dimensions