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ApPENDIX A

DERIVATION OF THE MACROSCOPIC ~1ASS BALANCE EQUATION

Consider the four-phase system depicted in Figure A.l. It

will be assumed that within each phase, the classical balance laws of

continuum mechanics hold at any point. The microscopic mass balance

equation for a material species i within a phase, may be written

as:

where p

v

V. -1

.L (pw,) + V. (pw' v) - V • j. - p f. 3t 1 _ 1 _ _ -1 1

is the bulk density of the phase

is the mass concentration of species the total volume of the phase

is the mass fraction of species

is the local phase velocity

is the local velocity of species i

a

over

j. _ - pw. (v. -v) is the diffusive flux of species _1 1_1 -

(A. 1 )

f. is the external supply of species i to the phase. 1

189

FIGURE A.l: FOUR PHASE SYSTEM AVERAGING VOLUME

(after Hassanizadeh and Gray (1979a))

190

Here is has been assumed that there is no net production of species

within the phase.

The balance law (A.l) is not valid on the interface between two

phases a and S. At this interface, the following balance equation

holds (Eringen (1980)):

(p. (w - v) + j.) I . n as + (p. (w - v) + j.) I . n Sa 1 - - _1 a - 1 - - _1 S-

o (A.2)

where ~ is the velocity of the interface and ~as is the unit normal

vector pointing out of the a phase into the S phase, la indicates

the limit of a term as the interface is approached from the a phase

side. The right-hand side of (A.2) is zero because the interface does

not contribute mass to the system.

To obtain a macroscopic equation from equation (A.l) by an

averaging procedure, the following criteria should be satisfied (Has­

sanizadeh and Gray (1979 a)):

1) The characteristic length, D, of the averaging volume or area must satisfy the inequality:

£'«D«L

where £, is the microscopic scale of the medium and L is the scale of gross inhomogeneities. This criteria implies that a meaningful REV (representative elementary volume) is definable.

191

2} The total volume of interest may contain no macroscopic surface of discontinuity.

3} Macroscopic quantities must be consistent with microscopic quantities. i.e. macroscopic quantities must account exactly for the total amount of microscopic quantities.

4} Definitions of macroscopic variables should correspond to observable and measurable functions.

5} Certain smoothness conditions must be satisfied so that the final integral equation may be localized.

Averaging operators may be defined as follows:

VoLume average operator:

<f> (x.t) 0. -

Mass average operator:

(A.3)

<P>o.(~.!)dV f Ii~ + ~.t} f(~+~.t} Yo.(~ + ~.t}dv(~} dV (A.4)

where dV is the averaging volume

~ is a microscopic spatial position vector relative to the centroid of the averaging volume (see Figure A.l)

192

dv(~) is the microscopic differential volume

x is the position vector of the centroid of the averaging volume with respect to an inertial frame of reference

--- tlo Y (x+ ~,t) 0.--

if x+~ lies within phase a

otherwise

Yo. is called the phase distribution function.

The averaging procedure is as follows. First, the microscopic

equation is integrated over the a phase portions of the averaging volume

and divided by dV. Next, the resultant equation is integrated over all

the averaging volumes which encompass the space of interest. Finally, the

equation is localized to achieve the macroscopic equation. In the averag-

ing process, various theorems are employed which relate the integral of

a derivative to the derivative of an integral. For a discussion of these

theorems and their proofs, see Hassanizadeh and Gray (1979a) and Gray and

Lee (1977).

The averaged form of the microscopic equation is:

a ( -a) ( -a-a) a -a -at <P>", w,. + II' <P> v w· - v·J. - <P> f. '-" - a -, - -, a'

(A.5)

where

and Ie: 1

<p> dV a

193

(A.6)

(A.7)

Here, dAaS is the area of as-interfaces inside the averaging volume

dV and da is an infinitesimal element of area in the microscopic domain,

ea(pw;} represents the exchange of mass of species i due to phase change, Aa a I; represents the exchange of mass due to interphase diffusion. ~; is

an average flux vector which represents the macroscopically non-convective

flux of species i. Its existence is established by a tetrahedron argu­

ment (Hassanizadeh and Gray (1979a)). The sum of definitions (A.6) and

(A.7) (the right-hand side of (A.5)) represents the total flux of species

to the a phase from all other phases. This flux is due to both con-

vection and diffusion/dispersion and may be a positive or negative quan­

tity, Using the definition of je: and equations (A,6) and (A.7), the -1

right-hand side of (A.5) may be expressed as:

(A.8)

194

Equation (A.5) is subject to the following constraints:

1. The sum of the mass fractions of all species in a given phase should equal unity (from the definition of mass fraction):

L~~ = 1 i 1

(A.9)

2. The total flux of all species to (or from) the a phase is equal to the mass gained (or lost) by that phase (conservation principle):

(A 010)

3. The mass of the total system is conserved:

o

40 The mass of each species is conserved over the entire system:

L (ea(PWi) + I~) o (A.12) a

Constraint (4) is derived from the averaging of (A02) over the interfacial

area and is applicable to a conservative (non-reacting) species.

ApPENDIX B,I

PROPERTIES OF THE DIFFERENCE OPERATOR

AND ITS SOLUTIONS

Consider the differential operator A acting on some function u.

This operator may be approximated at a point i by a finite difference

operator L such that

L u· + R. 1 1

(B.1)

where Ri is a remainder term known as the truncation or local discre­

tization error. Let h be the measure of the size of one mesh element.

Then the difference operator L is defined as a consistent approxima-

tion to the differential operator A if II~II -+- 0 as h-+-O, where

II ~ II is some norm of the vector ~ conta i n i ng elements Ri 0

The remainder term Ri is most often found by manipulating Taylor

series expansions for a function to produce a specific finite difference

approximation plus some truncation error. The Taylor's series expansion

for u(x. + h) 1

about u(x i ) may be written as:

u(x. + h) 1

196

a h2 a2u u(x.) + h ..J! + 1 ax 2T-2 ax

(B.2)

The quantity O[h4] is an asymptotic expression for the truncation

error of this expansion. (B.2) is said to be accurate to the "order

h4." In a more rigorous sense, the notation f = O[hn] means that

a positive number, B, exists independent of h such that:

for all h.

Now consider the error in the approximate solution to the

differential equation. This error at a point is given by:

(B.3)

where ui is the true solution and ui is the approximate solution.

Then a solution to a finite difference equation is said to aonverge to

the solution of a differential equation if JJeJJ-+ 0 as h-+O. Note

that consistency is a property of the difference operator while conver­

gence is a property of the solution.

Another property of the difference operator is known as stability.

A numerical scheme is called stabZe if any errors introduced into the

197

computation (either by round-off or discretization) do not amplify

during subsequent computations. The stability of a specific numerical

scheme may be investigated by numerical experimentation (heuristic sta­

bility analysis), Fourier analysis (Von Neumann stability analysis), or

matrix stability analysis. The latter two methods may be applied only

to linear, constant coefficient finite difference approximations. For

the examination of the stability of nonlinear difference equations,

local linearization is necessary. See Lapidus and Pinder (1982) for a

discussion of these stability analysis techniques.

The relationship between consistency, stability, and convergence

for linear difference equations is expressed by Lax's Equivalence

Theorem:

Given a properly posed initial boundary value problem and a finite difference approxi­mation to this problem which is consistent, then stability is the necessary and sufficient condition for convergence.

For a more mathematical and detailed discussion of the above concepts

and of what constitutes a "properly posed" problem, see Richtmyer and

Morton (1967).

ApPENDIX B,2

ANALYSIS OF TRUNCATION TERMS

The value of a function f at the node i+1 (see Figure 3.1)

may be expressed in terms of the function's properties at node i by

use of a Taylor's series expansion (B.2):

i i

Similarly, the value of f at i-1 may be written as:

f. 1 1-

2 af {t.x J a2f

fi - t.x_ ax + -2-:Z ax

i

Subtracting {B.4b} from {B.4a} yields the expression:

i

i

(B.4b)

(B.5)

199

If 1[1 is approximated by the left hand side of (B.5), then the ax i leading truncation term is of order /::"x. Note that for the case

/::,.x+ = /::,.x_ (equal nodal spacing), this error term vanishes and the

approximation becomes accurate to (/::"x)2.

Now consider the Taylor's series expansion of a function at

the spatial location i +t :

(B.6a)

Here, partial derivatives have been represented by prime notation for

convenience. An analogous expression for fi_~ is easily obtained:

f. 1.. 1-'2

Subtraction of (B.6b) from (B.6a) gives an alternative difference ex­

pression for the derivative of f at i:

(B.7)

Expressions for f.+L. and f. 1.. are also readily obtained from 1 '2 1-'2

Taylor's expansions:

200

(B.8a)

(B.8b)

Using the above expansions, a difference analogue to the differen-

a af I tial operator ax (n ax) i' may be developed. From (Bo 7), this dif-

ferential operator may be approximated as:

~ (,af) ax .. ax

The first term in the numerator of the right hand side of (B,9)

will now be examined:

(n af) ax i+l;;

(B. 1 0)

Here, (B.8a) has been employed. Two alternative ways of expressing

a function's derivative at the i+l;; level are given by:

201

[;,x+ 2

af f i+l - fi (2) "' 3 (Bo 11) ax = + fi+1.2 + O([;,x+)

l'lx+ 3! i+1.2

and

afl , t,x+ " 2 (Bo12)

ax = fi + -2- fi + O(t.x+)

i+~

These equations have been obtained by manipulation of Taylor's Series

approximations Equation (B,lO) may now be expanded further as:

(n af) ax i+~

(B.13)

II II

Here, (8,12) has been applied to both ni+1.2 and f i+1.2. Additional

( ) d . f fill manipulation of B,13 an expanSlon 0 i+~ about f l." . 1 d th 1 Yle s e

following result:

202

(n af) ax i+~

~x+ 3 iv t lIt

(T) l f; n; III I "" n;fi ~ + 2 -3-- n. f .- f .n·+-3-1 1 1 1

(B.14)

Incorporation of (B.14) and a similar expression obtained for

(n af) into (B.9) produces the final result: ax l' 1,

-'2

2 =

+ 0 (~x) 3 (B. 1 5 )

Note that the leading error term is of order ~x. This term will vanish

for equal nodal spacing. Under this condition, the difference expression

(first term in (B.15» becomes second-order accurate.

ApPENDIX B.3

THE NEWTON-RAPHSON ITERATION METHOD

Consider the nonlinear algebraic equation given by:

f(u) = 0 (B.16)

-An approximation for the root u of this equation may be obtained by

the following procedure. A Taylor's series expansion (see equation (B.2))

Of the function f may be written about an initial guess u(O):

(B.l7)

where 0(0) = Ii - u(O) is the error term. Note that terms in 0(0)

greater than order 1 have been truncated. Neglecting these terms and

solving for 0(0) yields:

(B.18)

An improved estimate for u may now be calculated from this error term:

204

(B.19)

At this point. another Taylor's series expansion may be written

about the new estimate. Continuation of this process leads to the

general algorithm:

v=O.l •••• (B.20)

Here, (v) refers to the iteration level. The iteration sequence given

by (B.20) is known as the Newton-Raphson method.

The process (B.20) may be extended to a system of N simultan-

eous equations in N variables. A truncated multivariable Taylor's

expression is given by:

f. (u.) 1 J

N af. (0) f.(u~O)) + ~ (1) (u. _ u~O))

1 J L au. J J j=l J

Let F be the Jacobi an of the vector functi on f:

af. (v) - (au ~ )

J

and let 6 be a correction vector:

i=l,N (B.2l)

(B.22)

205

(B.23)

Then Newton's method for a system of equations may be simply represented

as:

_ f~V) 1

(B.24)

By solving this matrix equation for o~V) and using equation (B.23), an

updated estimate for the unknown vector, u(v+l), may be obtained.

ApPEND! X C.1

MATRIX COEFFICIENTS FOR THE 1-D MODEL

This Appendix contains components of A and B for the mass

balance equations at node i. Let u be an ordered vector of 9 unknowns:

[P ,P ,wol ,P ,PWg.,w~,p ,P ,wol ]. Agiven oW i_l wg i _l i-l oWi 1 i oWi+l wgi +l i+l

mass balance equation at node may then be written as:

j 1, 9

where k is the equation number (k = 1, 3). (Summation notation applies.)

Components of A and B for the water balance equation (equation - --number 1) are as follows:

1 n+ 1 A = - -- [T n+ 1 1 [T S Y ] 12 (lIx)2 w. 1 ] -l:IX w wwi

- 1- 2

Sn+l 1 a

A14 = lit [£ ~ + K as]. ow W 1

aSn+ l A - 1 [~w_+S(a+Qc)].+_l_[ ]n+l

15 - lit £ ap W ~w~ 1 2 TW wg (lIX)+ i+l

2

follows:

207

A = ____ 1 ___ [, ]n+1 1 [n+1 ] 18 2 W + AX 'W SWYW 1·

(!J.X) + i+1 L.I

2

Components for the species one equation (equation 2) are as

= (0 )n+1 [_yaP _1 + 1 aP (pn+1 W1'0 1· O~O AX 2 ~ 09

i L.I 4(!J.x) °i i+1 pn+1 ) 09i_1

A23 = - [L] _1_ [poE: Sn+1 00] _ y [, ]n+1 1 ° . (A ) 2 ° . 1 ° . ° i - 1 2!J.x p 1 L.lX _ 1-"2 1

208

A 1 [S (oSl + l)J + [LJ. 1_1_ [po~Sn+1DoJ 26 = At OE W1 0" \ 2 ~ 1

Ll po , '(l!.x)+ 0 ;+2

[ 1 J 1 [0 n+1 0 [ In+1 1 A29 = - 0; --2 p ES 0 0] . 1 + Yo. 'To; + 1 2l!.x

p (l!.x)+ '+2'

209

Components for species two equation (equation 3) are as follows:

o ( ) n+ 1 1 1 {[ 0 n+ 1 0 + YoPi TO i-l 211x + --2 PESO 0 J. 1

(llX) 1--- 2

210

e: {. wOS 0sP + w[ w asw In+1 ° ° aso n+1 A35 = At 2 oP o. p. w2 P . + P.[w2 -p---J.

1 1 a wg 1 1 a wg 1

o { 1 [0 In+ 1 1 ° n+ 1 } + Pi --2 w2'o . 1 + --2 [w2'oJ. 1 (Ax) 1- - (Ax) 1 + -

- 2 + 2

- S (gKgwKwo )n.+1[1 9 J} P 2 2 + w2Sg 1· gi 1

+ [K~OK~WJ~+1[(pgsg)n+1e:Dg]i_l r 2

211

212

83 (continued)

£ J[ ° 0' OS] on [pWSwKW20.](Wo,n. _ ') + ~t 1 w2P S0 80 . - P 0. w,. -1 1 1 1 1

213

All variables are evaluated at the n time level unless otherwise

indicated. The subscripts i-l, i, i+l indicate the spatial point at

which the variable is evaluated. Note that these coefficients have

been written for non-uniform nodal spacing. Definitions of all terms

dealing with ~x are as follows (refer to Figure 3.1):

2 (~x) = (~x) (~x)

For uniform nodal spacing (~x)2 = (~X)! = (~X):

ApPENDIX C,2

NEWTON-RAPHSON r~ATRIX COEFFICIENTS FOR THE I-D MODEL

This appendix contains components of F, the Newton-Raphson matrix,

for equations at a node i. The ordered vector ~f unknowns o(v+l) is equi­

valent to u(v+l) - u(v), where the superscript indicates th~ iteration - -

level of evaluation and u is the vector described in Appendix C.l.

Fll = _1_ a 2 aP (.6.x) _ oWi _l

- __ 1 __ y aP a (,(v)) + All 2.6.x w,. w oWi _l i-l

F =_1_ a 12 (.6.x)2 apwg .

- ,-1

- __ 1 __ y aP a ( , ( v) ) + A12 2.6.x w,. w wg i _l i-l

215

+ _1 _ __ d_ (T(v) )[p(v) _ p(v) ] ( ) 2 dP w wg. wg

l'>x + wg; ;+1 , i+l 2

+ __ 1 __ y aP d (,(v)) + A17 2l'>X w,' w oW;+l ;+1

+ _1 _ --::-,-a_

( 2 aP LlX) oW;_l

+_l_L a 2 0 aP (Llx) p . ow. 1 - 1 1-

1 a - 2LlX Yo; aP

oW;_l

F = 1 22 4(LlX)2

[0 ](v) w1 To ;

216

( 0 DOS(v)) o(v) o(v) p E: 0 . 1 [w1. - w ]

1- 2 1 1;-1

( 0) (v) Tow1 ;-1 + A21

P [p(v) p(v) ] So. 09i+1 09;-1 1

1 a (0 )(v) [p(v) + --2 P w1 To . 1

(Llx) a W9;_1 1- 2 09;

+_l_L a ( )2 0 ap

Llx p. W9· 1 - 1 1-

( 0 DOS(v)) o(v) o(v) p E: 0 . 1 [w1 . - w1. 1]

1-2 1 1-

1 a - 2Llx Yo. aP

1 W9;_1

217

1 1 a 0 0 ( ) o(v) o(v) + -- - -- (p ED S v ) [wl - W ]

( ) 2 0 aP o· 1 1. 1 I1X p . ow . , - -2 i , --, ,

218

(\!) () wO J[P \! ,., og., 1- 1-

+ _, _ L _d _ ( 0 DOS ( \! ) ) 0 ( \! ) 0 ( \! )

2 0 aP P E: 0 • 1 [W1. - W ] (6X) p. wg. 1- -2 1 1 i - 1

- 1 1

219

as ( \J ) as hI ) + ~t {[E -;:;--pO + KS Cl]AD, + [ 0 + S ]AP } + A ~ a ~ E ~ OCl ~ wg. 26

OW; 0 OW; wg; ,

+_'_L a

( 0 ) ( \J ) [P ( \J ) P ( \J ) ] w, TO ,1 og ;-1 og,'+1 1+2'

( )2 0 ap t::.X + P; oW;+,

220

F28 (continued)

+_1_L a ( 2 0 ap

LlX)+ Pi W9i+1

( 0 DOS(v)) o(v) o(v) P E [w - W ] o .1 11• 11.+1 1+ 2

0 p. a (0 )(v) [p(v) _ p(v) ] F31

_ 1 ---2 aP w2'0 . 1 09i 09i_1 (LlX)_ OW i _1 1- 2

w p. a (w )(v) [p(v) _ p(V) +_1_ ] (LIX):

ap w2'W . 1 w9i w9i_1 oW i _1 1- 2

1 o(v) a ( ) a ( ) + _ [w 1 J[pOy (~ ). v + pWy (~ KWo).v ] 2L\X 1. - aP '01-1 w1.ap 'W21-1 1-1 °i OW i _1 OWi _1

221

F 31 (cont; nued)

+ a aP oW;_l

o p. F = _1_ a (W02TO) (.v), [p(v) - p(v) ]

32 (A)2 aP og og uX _ wg; -1 1- 2 ; ;-1

+ _ _ ,_ a (0 0 (,,) o(v) o(v) 2 ap P ED So v ). , [w1 - w ]

(~x) wg; _ 1 1- 2 ; -1 1 ;

+ _ _ ,_ o(v) " [ 1J[-=-..::...D _ (Kwo (pw ~DwS( v)) )

( )2 w1. - aP 2 ~ ~X 1-1 wg. . 1 w. , 1-1 1- 1- 2

F32 (cont;nued)

0

F =~ d

33 (l1X): 0 oWl

;-1 0

p. d +-'-

(l1X): 0 oWl

;-1

+ lOB 1 4(L'lx)2 p 0;

222

1 ] [ ( KW20 ) ,~ v ) a aPWg.

,-1

(0 )(v) [p(v) - p(v) ] W1 TO . 1 og; og;_l ,- 2"

(W)(v) [p(v) _ p(v) J W2TW . 1 Wg; Wg;_l ,- 2"

(wwT )(v) [p(v) J- p(v) J 20. og'+log'l , , ,-

223

w + Pi a (0 ltV) [p(v) _ p(v) ] -( )2 ap-- w2 T o ,log" wg,'_l

!::.X ow' ,---, 2

( ) 1 (v) (v) + loa (0 ) ( v ) J S P [P ( v) _ P v ] + S [w 0 _ w 0 ] l

AX P Yo, -;;----P W2T O ,' 1 0 og og 0 1 1 r Ll , 0 ow i i + 1 i - 1 it 1 i - 1

224

F34 (cont;nued)

+ ~ ( o)(v) 1 wSn _d_ (Kwo)~v) + Sn _d_ ( gKwoKgw)~v)} llt w2 ; ,P w. dP 2 1 9 dP P 2 2 1

lOW; OW;

+ _,_ [_d_ (( W OwS(v)) WO(V) 2 d P p e: W • 1 K2 . )

( llX ) + ow; 1 + 2" 1

wo(v) K 2;+1

225

F34 (continued)

_ 1 J 0 () (0 )(v) (v) (v) F35 - --2 lPi ~ w2'0 , 1 [p - P ]

C~x) w9, 1--2 09i 09i_1 - 1

1 0 () (0 )(v) [(v) (v) + --2 {P1' -:::--p w2'0 ,1 P - P ]

( A ) a 09,' 091'+1 uX + W9i 1+2"

226

F35 (continued)

1 0 '\ 0 () P () _ p(v) J + 1 o(v) o(v) l +-py -"-(WT).V{S[p V So[w1 -w1 Jr

6x 0i dPW9 . 2 0 1 0 09i+1 09i_1 i+1 i-1 1

+ .L ( o)(V) {[ W _d_ (dSW KWO)(V) + 0 _a_ (asO )(V) 6t W2 i Pi dPW9 dP 2 i P aP aPol i

i ow wg i v

227

1 O(v) ~ W W ( ) wo(v) + -- [w - 1] {_Q- (p e:D S v ) K

(IlX)! 1i+1 aPw9i w i+~ 2i+1

228

1 O(v) (v) () ( _ wO )(p v + 130 w1 . 1

1-1 i+1 09i_1

1 0 () P () ( ) 1 O(v) o(v) + P'f TV JS(pv _pv )+S(w -w )~

(f:XT 0 0i 1 0 09i_1 09i+1 0 1i _1 1i+1'

229

F36 (cont;nued)

. [~p + K~P ] + --'-- pWy s ,(V) __ d __ (woKwo)(~)[p(v) _ p(v) ] wg; OW; 2M W W w; (lW~ 2 2 , wgi+, wg;_,

230

F36 (continued)

0 Pi a (0 )(v) [p(v) p(v) ] F37 = --2 ap w2 To , 1

(llX)+ oWi+1 '+2 09 i 09i+1

W P; a (w )(v) [p(v) p(v) ] +---2 aP W2TW '+ 1 w9; w9i+ 1 (flX)+ oWi+1 ' 2

o 0 ( ) o(v) o(v) (p ED S v ) [w - w ]

o ,+1 1,'+1 1" , 2

1 o(v) 0 a ()(v) + W a - -2- [w1 - l][p Y -:"P=--- TO ,'+1 P Yw, ap

!;X ,'+1 0; a oWi+1 ' oW;+l

( Kwo ) (v)] 2 TO i+ 1

231

F37 (continued)

+ (K~OK~W)~v) aP a oWi+1

o - Pi 0 (W02To) (.v)l [p{v) _ p(v) ] F38 - ~( ) oP 091. 091.+1 D.X + W9i+1 1+ 2

W + _P_i_ -,,,-::...0_ (WOT lev) [p(v) _ p(v) ]

( )2 oP 2 W 1.+1 091. W9 1.+1 D.X + w9i+1 2

1 0 0 ( ) O(v) o(v) + -- 0 (p e:D Sov ). 1 [w1 - w1 ] ( D. )2 oP 1+- ;+1 ; x + W9i+1 2

1 o(v) 0 a ()(v) + W a - -2 - [w1 - l][p y .....,"P~- TO 1·+1 p YW. "P 6X 1.+1 01. a a W9i+1 1 W9i+l

1 O( v) ( ) + [ 1][ 0 KWO (pWe:DWSWV).1) --2 w1 - aP 2 (D.X) + i+ 1 wg i + 1 i + 1 1 + 2

+ a aP

W9i+1