The Mathematics of Fluid Flow Through Porous Media. Myron B. Allen, III

The Mathematics of Fluid Flow Through Porous Media

nabla dot left-parenthesis rho omega Subscript alpha Baseline bold v Subscript alpha Baseline right-parenthesis 2nd Column equals r Subscript alpha Baseline comma 3rd Column alpha 4th Column equals 1 comma 2 comma ellipsis comma upper N semicolon 3rd Row 1st Column ModifyingBelow StartFraction partial-differential Over partial-differential t EndFraction left-parenthesis rho omega Subscript alpha Baseline right-parenthesis With presentation form for vertical right-brace Underscript left-parenthesis upper I right-parenthesis Endscripts plus ModifyingBelow nabla dot left-parenthesis rho omega Subscript alpha Baseline bold v right-parenthesis With presentation form for vertical right-brace Underscript left-parenthesis II right-parenthesis Endscripts plus ModifyingBelow nabla dot bold j Subscript alpha With presentation form for vertical right-brace Underscript left-parenthesis III right-parenthesis Endscripts 2nd Column equals ModifyingBelow r Subscript alpha Baseline With presentation form for vertical right-brace Underscript left-parenthesis IV right-parenthesis Endscripts comma 3rd Column alpha 4th Column equals 1 comma 2 comma ellipsis comma upper N semicolon EndLayout"/>

where bold j Subscript alpha Baseline equals rho omega Subscript alpha Baseline bold-italic nu Subscript alpha is the diffusive flux of constituent alpha . In the last form, we refer to the terms labeled (I), (II), (III), and (IV) as the accumulation, advection, diffusion, and reaction terms, respectively.

The following exercise reassuringly shows that the multiconstituent mass balance reduces to the single‐constituent mass balance if we use the definitions of the mixture density rho and the barycentric velocity bold v and ignore the distinctions among constituents.

Exercise 2.15 Use the definitions of the multiconstituent density rho and the barycentric velocity bold v to show that Eq. (2.28) is equivalent to

StartFraction upper D rho Over upper D t EndFraction plus rho nabla dot bold v equals 0 period

2.5.4 Multiconstituent Momentum Balance

The differential momentum balance for multicomponent continua, in a form paralleling Eqs. (2.29) and (2.30), is

(2.31) (2.32)

Here, bold m Subscript alpha represents the rate of momentum exchange into alpha from other constituents, excluding momentum exchanges associated purely with the transfer of mass into alpha from other constituents. The term minus bold v Subscript alpha Baseline r Subscript alpha gives the rate of momentum exchange into alpha attributable to mass exchange from other constituents. Equation (2.31) plays a central role in modeling fluid velocities in porous media, as discussed in Sections 3.1 and 3.2.

As with the multiconstituent mass balance equation, one can retrieve the momentum balance for a simple continuum by summing over all constituents and ignoring the distinction among them. This derivation requires a bit of tensor notation encountered again in Section 5.1.

Exercise 2.16 For any two vectors bold a comma bold b , the dyadic product bold a circled-times bold b is a tensor having the following action on any vector bold u :

(2.33) left-parenthesis bold a circled-times bold b right-parenthesis bold u equals bold a left-parenthesis bold b dot bold u right-parenthesis period

Verify that the mapping bold u right-arrow from bar bold a left-parenthesis bold b dot bold u right-parenthesis is linear.

Exercise 2.17 Recall from Section 2.2 that the matrix representation of any tensor sans-serif upper A with respect to an orthonormal basis StartSet bold e 1 comma bold e 2 comma bold e 3 EndSet has entries bold e Subscript i Baseline dot sans-serif upper A bold e Subscript j . Compute the matrix representation of bold a circled-times bold b .

Exercise 2.18 Sum Eq. (2.31) and use Eq. (2.32), together with the definitions of multiconstituent density rho and barycentric velocity bold v , to get

rho StartFraction upper D bold v Over upper D t EndFraction minus nabla dot sans-serif upper T minus rho bold b equals 0 comma

where

bold b equals StartFraction 1 Over rho EndFraction sigma-summation Underscript alpha equals 1 Overscript upper N Endscripts rho Subscript alpha Baseline bold b Subscript alpha

gives the total body force per unit mass and

(2.34) sans-serif upper T equals sigma-summation Underscript alpha equals 1 Overscript upper N Endscripts left-parenthesis sans-serif upper T Subscript alpha Baseline minus rho bold-italic nu Subscript alpha Baseline circled-times bold-italic nu Subscript alpha Baseline right-parenthesis period

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