spe 1578 pa boberg lantz

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Calculation of the Production

Rate of a Thermally Stimulated

Well

ABSTRACT

T

C BOBERG

R B LANTZ

JUNIOR

MEMBERS

AIME

This paper presents a

method

for calculating the pro

ducing rate

of

a well as a function

of

time following steam

stimulation. The calculations have proved valuable in both

selecting wells for stimulation alld

ill

determining

optimum

treatment sizes.

The

heat transfer

model

accounts for cooling

of

the oil

sand by both vertical alld radial conduction. Heat losses

for any

number

of productive sands separated by unpro

ductive rock are calculated for the injection. shut-in and

production phases

of

the cycle. The oil rate increase caused

by viscosity reduction due to heating is calculated by

steady-state radial flow equations. The response

of

Sllcces

sive cycles of steam injection call also he calculated with

this method.

Excellent agreement

is

shown betweell calculated and

actual field results. lso included are the results

of

several

reservoir

and

process variable studies. The method is best

suited for wells producing from a multiplicity

of

thin sands

where the bulk of the stimulated production comes from

the unheated reservoir. The flow equations used neglect

gravity drainage and sall/ration changes within the heated

region.

INTRODUCTION

This paper presents a calculation method which can be

used to predict the field performance

of

the cyclic steam

stimulation process.

The

calculation method enables the

engineer to select reservoirs that have favorable charac

teristics for steam stimulation and permits him to deter

mine how much steam must be injected to achieve fav

orable stimulation. While the calculation represents a con

siderable simplification

of

physical reality and the results

are subject to numerous assumptions which must be made

about the reservoir, it has been found that realistic calcu

lations

can

be made

of

individual well performance fol

lowing steam injection.

The

duration

of

the stimulation effect will depend pri

marily on the rate

at

which the heated oil sand cools which,

in turn,

is

determined

by

the rate at which energy

is

re

moved from the formation with the produced fluids and

conducted

from

the heated oil sand to unproductive rock.

A complete mathematical solution to this problem is a

formidable task,

and

finite difference techniques would un

doubtedly have

to

be used.

The

calculation method pre-

Original manuscript received

n

Society of Petroleum Engineers office

July 8, 1966. Revised manuscript received Oct. 31, .1966. a ~ e r SPE

1578)

was presented

a t SPE

41st

Annual Fall Meetmg

held

m

Dallas.

Tex., Oct. 2-5, 1966.

©

Copyright 1966 American

Institute

of Mining,

Metallurgical,

and Petroleum Engineers, Inc.

DECEMBER 966

ESSO

PRODUCTION

RESEARCH

CO.

HOUSTON

TEX

sented here utilizes analytic solutions

of

simple related

heat transfer and fluid flow problems.

The

method is suf

ficiently simplified that it can be used as a hand calcula

tion, although the calculations are somewhat lengthy and

laborious.

For

that reason, the analysis was programmed

for an IBM 7044 digital computer.

Well responses observed at the Quiriquire field in east

ern Venezuela have been matched using this program after

making suitable approximations for reservoir and well bore

conditions. One

of

the most valuable uses

of

this calcu

lation method

is

to assess the effect

of

reservoir and proc

cess variables on the stimulation response. This paper con

tains results of several studies made

of

key reservoir and

process parameters. Among the most important

of

these is

the influence

of

prior well bore permeability damage. I f a

well

is

severely damaged prior to stimulation, a higher

stimulation response will be observed than

if it

is undam

aged. I f a portion

of

this damage

is

removed. a permanent

rate improvement will occur.

THEORY

DESCRIPTION OF

CALCULATION

METHOD

The

process of cyclic steam stimulation

is

essentially one

of reducing oil viscosity around the well bore by heating

for a limited distance out into the formation through the

injection

of

steam. Suitable modifications

of

the calculation

technique presented here can be made so

that

stimula

tion of wells by hot gas injection

or

in situ combustion can

also be calculated.

A schematic drawing

of

the heat transfer and fluid flow

considerations included in the calculation method is shown

in Fig. 1

In

brief, the calculation assumes that the oil sand

is

uniformly and radially invaded by injected steam. For

wells producing from several sands, each sand

is

assumed

to be invaded to the same distance radially.

In

calculating

the radius heated r

h

energy losses from the wellbore and

conduction to impermeable rock adjacent

to

the producing

sands are taken into account. After steam injection is

stopped, heat conduction continues and oil sands with r

<

r

cool as previously unheated shale and oil sand at

r r

h

begin to warm.

The

effect

of

warming of oil sand

out

beyond

1'

has little effect on the oil production rate

compared to the effect

of

cooling

of

the oil sand nearer the

wellbore than r o Thus, in computing the oil production

rate, an idealized step function temperature distribution in

the reservoir is assumed where the original temperature

exists for r

> r

and where an average elevated tempera

ture exists for r r o

The

average temperature in the oil

sand for the region r

1'

is computed as a function

of

_ _

9References

given

a t end of paper.

1613



time following termination of steam injection by an energy

balance.

From

the average temperature, the oil viscosity in

this region is determined.

The

oil production rate is calcu

lated by a steady-state radial flow approximation which

accounts for the reduced oil viscosity in this region.

SIZE O F T H E H E A T E D REGIO . ;

Wellbore Heat Losses

To

calculate the size

of

the region heated by steam, it

is necessary to estimate the quantity of heat actually in

jected af ter well bore heat losses are taken into account .

Various methods are available for estimating wellbore heat

losses.

0,11

A simple method which assumes a constant, aver

age temperature

of

the injected steam and an average

initial geothermal gradient computes the cumulative energy

lost during injection Q" as:

.(

aD)

2r.DKr,

T T

r

2 I

Q -

a

(1)

where

I

is read from Fig. 2 as a function of dimensionless

time

at, 1

1', .

For

uninsulated tubing or the case where steam

is in direct contact with the casing wall, r, is the inside

FLOW OF OIL

WATER

G S

. /VVVv

HE T

CONDUCTION

OIL

S ND

SH LE

OIL SAND

SH LE

OIL SAND

FIG.

I-SCHEMATIC REPRESE] o;TATlOl i

OF HEAT TRANSFER

Al iD

FLUID

FLOW CALCULATED BY

MATHEMATICAL MODEL.

10

F

====:::;... :

FIG.

2-1 FACTOR FOR WELLBORE HEAT Loss DETERMINATION.

1 6 1 4

casing radius; for insulated tubing.

1',

may be roughly ap

proximated as the inside tubing radius.

The

average down-hole steam quality X for the entire

steam injection period is then

X, =

X 'f - MQ'J

.

,

If,

2)

Heated Radills

During steam injection the oil sand near the well bore

is at condensing steam temperature

L.

the temperature

of

saturated steam at the sand face injection pressure.

Pressure fall-off away from the well during injection is

neglected in this analysis. and T, is assumed to exist out

to a distance

r o

where the temperature falls sharply to

Tr,

the original reservoir temperature.

In

reality, the tem

perature falls more gradually to reservoir temperature be

cause of the presence of the hot water bank ahead of the

steam,

but

this is neglected to simplify the calculation.

The heated zone radius is calculated by the equation of

Marx and Langenheim: In the case

of

multisand reser

voirs, if it is assumed that each sand is invaded uniformly

as

though all sands had the same thickness

h

and were in

vaded by equal amounts

of

steam:

r.'

=

hM. (XiII , + h, - /z .) f, .

4Kr.(T, - T,.)t,N,

(3)

The

function

l.

is plotted as a function

of

dimension

less time T

=

4Kt, h'(pC)] in Fig. 3.

The

use of·this equa

tion for multisand reservoirs also assumes that injection

times are sufficiently short and that interbedded shale is

sufficiently thick that no heating occurs

at

the mid-plane

of the shale during the injection period. Eq. 3 further as

sumes that the value

of average density times heat ca

pacity pC for the barren strata is the same as that for the

oil sands

[(pC),h

Ie

= (pC),].

For

multisand reservoirs, a heated radius for each sand

r i could be calculated assuming equal steam injection

per

foot

of

net sand. An average heated radius

rh

could then

be computed from rio

= h,r,,,'/ h,.

Practically speaking.

the r' for reservoirs consisting of more than three sands

of

reasonably uniform thickness can be approximated by

using Eq.

3.

T E M P E R A T U R E H I S TO R Y O F

T H E

H E A T E D R E G I O N (r ,

<

r

<

r.)

The average temperature Tav.

of

the heated region (or

I O r ~

t

~ - - -

0.01

; ; ; - - - - - ~ - - _ ; : _ : - - - - - - - - _ _ : _ _ = _ - - - - - ~ ~ - - - - - - - - ~

0.01

0.1

1.0 10

FIG.

3--DIMENSIONLESS

PUSITlOl i OF STEAMI:D·OUT REGION.

JO t T R 'i A L O F P E T R O L E U M T E C H l O L O ( ; Y



regions in the case of a multisand reservoir)

after

termina

tion of steam injection is calculated from an approximate

energy balance

around

the region r <

I'

<

r

h

:

T,,

= T,

+

(T, - T..)

[ v ~

(1 - 8) - 0],

OF

4)

In Eq. 4, and ~ are unit solutions of component con

duction problems in the radial and vertical directions, re

spectively. 8 is a correction term which accounts for the

energy removed from the oil sand by the produced oil,

gas and water. I f little energy is removed by the produced

fluids (the case of a low rate well), 8

can

be neglected and

Eq. 4 reduces to a product solution for the average tem

perature of

a series of right circular cylinders of radius

r conducting three-dimensionally to initially unheated

rock. The development of Eq. 4 is given in the Appendix.

Values for

V,'

and ;;-, for the case of production from a

single sand may be read

from

Fig.

4.

Relations

from

which

values of

y,

and

Y may

be calculated are developed in

the Appendix (Eqs. A-8 and A-16).

ENERGY REMo\ED \\ITH

THE PRODUCED FLums

The quantity 8 in Eq. 4 accounts for the energy re

moved from the formation with the produced fluids and

is defined by:

I

If

Hfdx

=

2 ZTrr,,'(pC),(L _

T..)' dlll1enslOnless .

(5)

I,

Eq. 4 must be solved in

a

stepwise

manner

since

H

the

energy removal rate, is a function of T g. Because ap

proximations are used in deriving Eq. 4, as 8 approaches

unity

1',,,

calculated by Eq. 4 can hecome less than

T,

.

This is physically impossible. When it happens, T,, should

be taken as equal to

T,.

H the

heat

removal rate with the produced fluids, is

given by:

H

f

=

q (H..,,

+

H,,),

Btu/D

(6)

where

H

= [5.61(pC) +

R c ](T,, -

T,.),

Btu/STB oil

(7)

H = 5.61

p,,[R,,(/lr

- h

r

,. + R ,. h

f

. , ] , Btu/STB oil

(8)

wv • 1356 . . ,

=

0000

(P )R

bblliquid water

at

60F

£I

...

-

P ..

·

bbl stock-tank 011

(9)

when

P

>

p .. . and R .1' < R ... , R ... . = R when p ,

>

p,,;

if

R .,

calculated by the above formula is greater than

R,,,

then

R .,

=

R .

1.0

;:::-

0,8

0.6

0.4

0.2

o

0.01

r--.

~ I I I Il( -

tj

l

IIIII

I I

III

41l(t-t

j

)

.........,

Vr: 8= - - 2 -

V

: 8 = - - - 2 -

rh

IIIII

I

r

-

I

'-

VZ - I I ~ I ~ G L ~

~ A I ~ ~

,-

Vr

I

'

-.....

.......

0.1

1.0

10

10C

8 - DIMENSIONLESS TIME

FIG.

4- -S0LUTION FOR

;;r AN[);;z (SINGLE

SA;,\[).

DEeEMI lER,

1966

C L C U L T l O ~ OF

THE OIL

RATE

For conventional heavy oil reservoirs which have suf

ficient reservoir energy to produce oil under cold condi

tions, the use of steady-state radial flow equations appears

to be adequate for predicting the oil production rate re

sponse to steam injection.

These

equations are inadequate

for tar sands and pressure-depleted reservoirs where the

bulk of the stimulated production must come from oil sand

which is actually heated

rather

than from the portion

of

the reservoir that is still cold. The equations outlined in

the following section for this latter case will predict an un

realistically low rate response until the heated region itself

has become depleted or reduced in oil saturation.

Steady-State

A pproxill1ation

for

the Productivity

Index

For a given stage of reservoir depletion, the ratio

of

the

oil productivity index

(q,J

6..P) in the stimulated condition

i to that in the unstimulated condition

J

e

can be esti

mated by an equation of the form:

- _ i _ 1

-

J,

, dimensionless

(10)

{t c,

+

c,

p.,,,

The

geometric factors c, and c, include the pattern geo

metry and the skin factor of the well. For preliminary en

gineering calculations if no change in permeability occurs,

suitable values can be calculated from the formulas given

in Table I, which assume that r

« r,

and r r «

r,.

The effect of prior wellbore permeability damage is ac

counted for by using the effective well radius r in the re

lations (Table 1).

The

effective well radius is related to the

actual well radius r and skin factor S by the equation

(11)

Implicit in the assumptions used to derive Eq. 10 is the

assumption

that

heating and fluid injection have a negligi

ble effect on the permeability to oil.

To

adequately pre

dict changes in the permeability to oil during the injec

tion and production phases, prediction

of

two-dimensional,

three-phase behavior is required. The complexity of adding

these equations would undoubtedly necessitate the use

of

finite difference techniques to arrive at a solution

and

are

beyond the scope

of

the simplified model proposed in this

paper. When swelling clays are present, steam might re

duce the permeability, although cases

of

plugging attribu

table to steam injection in permeable unconsolidated sands

are apparently rare.

Influence and

Significance

of the Skin Factor

The

skin factor S has a large effect

on

the stimulation

response. The greater the skin factor, the greater c, will be

relative to

c,;

hence, the greater will be

the

influence of

the heated to cold viscosity ratio {tu'./P-., on the

ratio

of

stimulated to unstimulated productivity indices I. This can

be seen by referring to Eqs. 10 and 11 together with

the

defining relations for

c,

and

c,

given in Table

1.

TABLE

I-EXPRESSIONS FOR c, AND c, USED IN

EQ. 10

FOR

CASE

OF

NO

ALTERATION

OF

PERMEABILITY

BY

STIMULATION

System

c,

Radial-Pe onstant I (rhl rw J/ In(r,,/r Ie)

I (rhl

TtIJ

)/ ln(re/ru:)

(

'I, )

'I,'

In -

--

r

U'

2r,--

) ,

n - - V, - .

Th

2

re

:J

Radial-pi Declining

In ( I , · ) - 'h

1615



Another

way

to

understand the effect of S is to consider

k,j

and

rd, the permeability and radius

the region

around

the well bore where permeability

k:

(

k ) r

S =

k J

In-=-.

d rtl

12)

f

kd

is small relative to k and

r

> r ., then

S

will have a

Alteration oj Skin Factor hy Heating

In some wells asphaltene precipitation may be a cause

of

f this

and

similar damage can be allevi

by

heating, suitable modification

of c,

and c, can be

For

the

constant p, case where the skin factor S is re

to S,

following stimulation

and

r

> rd:

c, =

S, +

In rh/r .

S + In

relr ,

(13)

1n

r,/r

h

;

c: =

S

+

In

rJf ,

p, is de

Determination

of the Oil RaIl'

To

obtain the oil rate as a function of time.

It IS

neces

to

know the unstimulated productivity index J, and

p,

as a function of the cumu

Then

the stimulated oil rate q h

qoh = J J,. : P. STB/D

14)

T is determined from Eq. 10;

J,

is obtained from

extrapolated plot of the productivity index history of

prior

to stimulation; and : P the pressure draw

Pe

- p ,

during the period

of

stimulated produc

The

static pressure p,

can

be estimated from an extra

of a plot of p, vs cumulative oil produced prior

enance benefits of the injected fluids can be neglected.

PERFORMANCE

SUCCEEDING CYCLES

To calculate the performance

of

cycles following the

made

for the residual heat left in

The

energy remain

by:

Heat

remaining =

u,,'(pC)J1N,

(T,,, -

T,.)

(15)

An

approximate method

of

taking this energy into ac

is to add it

to that

injected during the succeeding

that

injection takes place into a res

that

is at original temperature

T,.

This additional

The major

assumption involved in this approximation is

at

the beginning

of

is a conservative assumption since calcu

heat

losses for cycles after the first will be higher

han

those actually observed. A more optimistic analysis

ould assume

that

all of the energy remaining, including

in

the

shale,

is

added to the energy injected

on

the suc

TABLE 2-STEAM STIMULATION TEST DATA FOR WelL

Q-594

Reservoir Characteristics

Depth. It

Section thickness ft

Net

sand thickness ft

Number of

sands

Oil

viscosity. cp

Pre-stimulation

Oil

rate, BID

WOR, bbl/bbl

GOR, scflbbl

Stimulation

Steam injected MM b

Wellhead inie ctio n pressure psig

Injection time including

shut-in

days

Durat ion of cycle including

injection

days

Stimulated

producing

time. days

Actual oil production bbl

Calculated oil

production

bbl

Estimated

cold

production bbl

RESULTS

First

Cycle

18.1

770

46

487

378

80,803

79,140

47,600

MATCH OF FIELD

TEST

PERFORMANCE

WITH CALCULATIONS

4,050

470

183

16

133

135

0.83

985

Second Cycle

19.2

800

55

354

288

47,813

53,700

30,400

A stimulation field test conducted in the Quiriquire field

of eastern Venezuela provides an excellent check

of

the

utility of the calculation method. Comparisons of actual to

calculated results are presented for the first two cycles of

steam stimulation

of

Well Q-594 which produces from 16

sands in a 470-ft interval (Table

2).

A comparison be

tween calculated and actual oil production rates after

stimulation for Well Q-594 is given in Fig. 5. It should be

noted that the calculated curve is a match of observed data

rather than a true prediction. The observed pressure draw

down. down time and water-oil ratio were used as input

data for the calculations.

The only skin effect used in the match of the production

history of the well was an effective skin due to the perfor

ated casing. The effective well radius used in the calcula

tion was 0.00176 ft compared to the actual casing radius

of 0.292 ft to account for a perforation density of four

holes per foot. A plot

of

the effective radius for various

perforation densities based on Muska ' is illustrated in Fig.

6. The

external drainage radius

r,

was estimated as 570 ft.

The

estimated decline in static pressure p, and cold pro

ductivity index

J

as a function of cumulative oil produced

following steam injection is given in Table 3.

The

water

cut for this well following stimulation is given in Ref. 9.

The agreement of the calculated and observed rate his

tory for the first cycle is excellent. Calculated cumulative

oil produced at the end

of

the first cycle differs by less than

-ACTUAL

---CALCULATED

DATA: SEE

TABLE

2

400 ----------------------------------

>-

300

III

: 100-

5

OLi

__

~ _ i l W l l

__

__________

__

o 200 400 600 800 1000

TIME

SINCE

START

OF

FIRST STEAM INJECTION -

DAYS

FIG. 5-C0Il1PARISOl \

OF CALCULATED AND ACTUAL OIL RATES

AFTEft STEAII1

INJECTlOl \

FOR

QUlRIQUlRE WELL

Q-594.

JO l iR : \A L o F P E T I IO L E U M

TECHl \Ol .OGY



3 per cent from the cumulative oil actually produced from

the well. Temperature profiles obtained some time after

production had been initiated indicated reasonably uniform

heating of all 16 sands. Temperatures of highest and lowest

sands were cooler than the middle sands; however, the

average temperature across the producing interval agreed

closely with the value of T,,, calculated by this method.

The

difference between the observed and calculated oil

rate for the second cycle of the Quiriquire well is greater.

Observed cumulative oil

at

the end of the second cycle

differs by about 10 per cent from that calculated. How

ever, agreement between calculated and observed perform

ance is still adequate for engineering estimates.

P R O C E S S

Y

A R I A B L E S T U D I E S

The calculation method described here has been used to

study the effects of several reservoir and process variables

on the steam stimulation process. Studies of this type pro

vide preliminary estimates

of

the energy required to

achieve desired stimulation levels and additional insight

into which variables have the greatest influence on the

production response of a steam-stimulated well (Figs . 7

through 10).

The

incremental oil-steam ratio has been se

lected as the primary dependent variable for these studies

since it can be directly related to the economics

of

the

process. The incremental oil-steam ratio is defined as the

ratio of the stimulated less primary oil production to the

amount of steam injected expressed as barrels of conden

sate.

The

values

of

this ratio are maximum values which

occur when the computed oil production rate has returned

to the extrapolated value

of

the cold rate. The influences of

I-

...

...

I

V I

::::>

o

ct

ar::

IO-J

-

-

w

~

w

>

~

u

w

...

w

,

_.-

_.

-

I

- - - + ~ ~ - - + -

/

/

i

:

/

/

/

:

/

I

I

~ - -

I

[7

I

/

,

- -

/

/

,

/

,

1 1

T

rw= 0.25 FT

I

PERF. RADIUS

=0.25

IN.

-_ .

I

-- ---

i

,

i

_-_._

I

T

I

i

i i

2

4

6

8

10

NO.

OF

HOlES/FT

FIG. 6--EFFECTIVE \VELL

RADIUS

FOR PERFORATED CASED HOLE

(AFTER

MUSKAT ).

D E C E M I I E I I 1 9 6 6

TABLE

3-ESTIMATED DECLINE IN STATIC

PRESSURE

AND COLD

PRODUCTIVITY WITH CUMULATIVE

OIL PRODUCED

FOLLOWING

STEAM

INJECTION fOR WELL

Q·59

Cold

Cumulative

Productivity

Ojl Production Static Index

M bbl) Pressure psio) B/D/psi)

0

490 0.312

100

410

0.281

200

330

0.256

300

250

0.231

400

170

0.206

skin damage, sand-shale ratio, stabilized water-oil ratio,

pre-stimulation oil production rate, gross sand thickness,

steam injection rate, total heat input and back-pressuring

of the well early in the production phase on the maximum

incremental oil-steam ratio are discussed

in

the following

sections.

Effect of kin Damage

The amount of skin damage present in a well prior to

stimulation can have a tremendous effect on the produc

tion response of the well when it is steam stimulated. This

is

true even if no well bore c leanup (i.e., damage removal)

is obtained although the stimulation benefits are greater

when well bore cleanup

is

achieved.

The effect of damage became evident when stimulated

production histories for some California wells became

available which showed better than 20-fold increases in

productivity following steam stimulation."'" This com

pared with three- to four-fold calculated increases which

would be predicted by steady-state flow formulas assum

ing no permeability damage. As an example, the effect of

different assumed skin factors on the stimulation response

of a hypothetical well (Well A) is shown in Fig. 7. Reser

voir and fluid properties used in the example are shown in

Table

4.

For all wells

in

Table

4,

a temporary increase

in

the water-oil ratio following steam injection was assumed.

This increase is shown in Fig. 8 as a function of the cumu

lative water produced divided by the cumulative water in

jected. This plot represents the average of actual data

from two Quiriquire wells. The calculated curves in Fig. 7

assume that the wells in question have no change in the

skin factor following steaming; thus, production returns

to the pre-stimulation rate. A permanent improvement

in

productivity would result if post-stimulation skin factors

were reduced.

Effect of Oil Viscosity

For a given temperature rise, the viscosity reduction of

a low viscosity oil is much less pronounced than for a high

viscosity crude. Thus, peak stimulated oil rates following

steam injection will be smaller, the lower the original oil

o l O O O r - - - , - - - , - - - , - - - - , - - - , - - - , - - - ,- - - - , - - ,

o

8 0 0 r - - - - - + - - - + - - - + - - - - ~ - -+--+-- - .+-

i

6 0 0 r - - - - + - - 1 - + - < - - + - - ~ - - - + - - - - L - - ~ - ..

- · -

z

o

5 4 0 0 f - - - + - - + + - ~ '

;:

o

o

,

DATA GIVEN IN

TABLE

4

'

....

2 0 0 r - - - - - ~ - - - ~ - - ~ - - ~ ~ ~ - - -

STEAM ' ~ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; : ; : ; : : : : : d

INJ

i

5

FIG. 7-EFFECT OF SKIN DAMAGE ON STEAM STIMULATION

RESPONSE.

1617



TABLE 4-STEAM STIMULATION

TEST

AND CALCULATION DATA

Reservoir Characteristics

Depth,

It

Section thickness,

ft

Net sand

thickness, ft

Number

of

sands

Reservoir temperature

of

Oil

viscosity,

cp:

at TI

at 300F

Skin

factor

Effective well

radius

ft

Pre-stimulation

Oil role,

BID

Oil productivity index,

B/D/psi

WOR, bbl/bbl

GOR, scf/bbl

Stimulation

Steam injected MM Ib

Wel lhead

injection

surface conditions

Pressure, psig

Well

A

(Fig. 7)

867

546

335

7

97

900

13

20

to

60

0.45

30

0.128

0.35

63

6.9

440

Well B

(Fig. 9)

3,000

Well C

(Fig. 10)

3,740

200

200 to

1 ,088

67

234

6

100

4010 1,000

1 to 6

o

0.00176

300

1.0

1.0

1,000

40

780

18

120

133

8

o

0.00176

99

0.3

0.57

600

16.6

770

Temperature, of

450 520 518

Steam

quality,

dimensionless

1.0 0.95

0.95

Injection

time, days 21 80 55

Shut-in time, days 13 4 5

*Includes effect of

well

completions, perforations, etc., jf any.

Well D

(Fig. 11)

4,000

1,400

467

36

125

70

3

o

0.25

230

0.5

0-1.0

s

42 to 126

1,500

600

0.95

18 to 54

3

viscosity. However, it

is

not obvious how much less incre

mental oil will be produced over the entire cycle length

since the lower heat removal rate with the produced fluids

will cause stimulated production to last longer for the light

er oil. I f there were no heat losses from the heated portions

of

the oil sands, the increased cycle length

for

the lower

viscosity oil should give the same incremental oil recovery

as for the heavier oil although at a reduced stimulated rate.

onsequently, heat losses to unproductive rock are im

portant

in evaluating the effect

of

oil viscosity

on

incre

mental oil recovery.

The

effect

of

initial oil viscosity is shown in Fig. 9 for

Well B of Table 4. These results indicate more than a 50

per

cent increase in incremental oil recovered for a 1,000-

cp oil over a 40-cp oil.

2.0

1.6

CI:

~ <

w

1.2

I:::l

..:c

I :>

z

0.8

~

-c(

W-A

C): : l

0.4

~

c(-

I:ti

uJJ

CI:

0.0

0..

I

I

I

I

\

I

I

I

i

i

I

\'

1

i

I

I

i

1

I

I

1

i

~

,

I

1

i

r--

I

I

1

I

I

I

I

i

i

I

-0.4

I

I

I

0.0

0.2

0.4 0.6 0.8 1.0

1.2

CUM.

WATER

PROD.jCUM.

WATER

INJ.

FIG. 8--EFFLCT

OF

STEAM STIMULATION ON WATER-OIL RATIO.

16111

Effect

of

Sand Shale Ratio

The effect of the ratio

of

sand to shale thickness on the

incremental oil-steam ratio for a single cycle

is

depicted

in

Fig. 10 for the conditions assumed for Well C in Table

4.

The

net sand thickness and number

of

sands were held

constant in these calculations. and the gross section thick

ness was varied to vary the sand-shale ratio. The decline

in the incremental oil-steam ratio as the sand-shale ratio

decreases (increasing gross section thickness)

is

the result

of

increased heat losses to the interbedded shales. These

greater heat losses result

in

an accelerated temperature de

cline; hence, an increase in the rate of decline of produc

tivity occurs. The above conclusions about sand-shale ra

tio hold only as long as the average individual sand thick·

ness is roughly constant. Thicker sands will cool more

slowly and the stimulation response will last longer for the

same sand-shale ratio than will be the case for thin sands.

Effect of Post Stimulation Water Oil Ratio

nd

Gross

Sand

Thickness

The

effects

of

post-stimulation water-oil ratio and level

of steam input per foot

of

gross thickness are shown in Fig.

11 for a well having the conditions indicated for Well D

in Table 4.

Fig. 11 indicates that the incremental oil-steam ratio de

pends markedly on the quantity

of

steam injected

per

foot

1 8

c:a

.....

~ 1.6

g

1.4

«

CI:

:E 1.2

«

t

' -

1.0

.....

o

4

0.8

Z

:E 0.6

CI:

U

~ 0.4

40

BASIC

DATA

GIVEN TABLE 4

I

1

1.

.1 .1

J.

100

1000

OIL

VISCOSITY - CP

FIG.

9 EFFECT

OF VISCOSITY 0:\

bCREMENTAL O I L / S T E A ~ 1

RATIO.

~

0.5

-

" 'a l

~ ~ 0 4

Oal

0.3

" ' 0

i=

0 2

~

« .

... co:

co:

I

-T- - t - - ' -

I

- --

L

; ~

- 1 1 F - - - + - - t - - - - + - ~ _ _ _ _ t _ _ - ~ - ~

DATA: m

~ B l ~

4

l

u 0.1

f--i-----j

z

FIG.

100CALCULATEU EFFECT

OF

SANU/SHALE RATIO 0:\

INCREMENTAL OIL/STEAM

RATIO.

JOl l IC\AL OF

I 'ETI tOLEl · \ t

T E C H N O L O G \ '



of gross section and on the stabilized water-oil ratio. The

detrimental effect

of

a high pre-stimulation water-oil ratio

is a consequence

of

the high heat capacity

of

the produced

water. Since the

heat

capacity

of

water

is

approximately

twice that

of

the crude oil. a high water-oil ratio results

in a high rate of energy removal as fluids are produced

from the formation.

High producing gas-oil ratios will also have a detrimental

effect on stimulation benefit. Both

of

these effects can be

seen from Eqs. 8 and 9 Under high temperature condi

tions a high gas-oil ratio can remove a large amount of

water in the vapor phase accompanied by a high latent

heat

of

vaporization. Consequently, both high water-oil

ratios and gas-oil ratios cause a more rapid decline in res

ervoir temperature.

Thicker

sections will require a proportionately greater

input

of

steam to achieve a given stimulation response, all

other

variables being equal.

The

curves plotted in Fig.

11

apply specifically only for the conditions of Table 4 (i.e.,

an

undamaged well having the indicated cold productivity

with negligible cold rate decline expected over the cycle

life, and a very thick section where heat loss rates for the

many individual sands can be assumed essentially equal).

Process COlltrol Variables

Three

variables relating to process control are the rate

of

steam injection, the total heat input and the degree

of

back-pressuring

of

a well early in the production phase of

a stimulation cycle.

Within the limits of

injection pressure considerations,

it

appears to be beneficial to inject the steam at as high a rate

as possible. The high injection rates provide two benefits:

(1) well bore heat losses as a percentage

of

total heat in

jected

are

reduced and (2) a given amount

of

energy can

be injected in a shorter period

of

time, thus reducing the

lost production while the well is being steamed.

It should be noted in Fig.

11 that

as the cumulative

steam input is increased, the incremental oil-steam ratio

curves pass through a maximum and begin to decline at

higher steam input. Thus, there appears to be an optimum

level

of

steam

input

for a given set

of

operating conditions,

and it should be possible to optimize the length

of

steam

injection to maximize the incremental oil-steam ratio.

The

- 2.4

CD

CD

-

CD

2.0

CD

0

i=

1.6

I:

IX

<I:

1.2

on

-

5

S

-

<I:

Z

IX

U

0

0

ms/hn

=

STEAM RATE LB/HR) /NET FT

COLD OIL RATE = .5 STB/D)/FT

- - ---

,

m

s

/h

n

=1500

m

s

/h

n

=1500

ms/h

n

= 500

t

I

I

50

100

150

200

250

300

mst;/h

- M

LB

STEAM/FT

OF GROSS

INTERVAL

FIG. I I -THEORETICAL PREDICTlO'i OF I l 'iCRDIE'iTAI. O l J j S T E ~ I

RATIO \ 5 STEA \OI }l'iJECTED.

UE( ;E I I I IER.

1 9 6 6

maximum in the incremental oil-steam ratio is the result

primarily of four factors:

(I)

increased heat losses associat

ed with the larger heated radius and longer cycle times

which resulted from higher energy inputs. (2) increased

lost production as the length

of

the injection period is in

creased, (3) a lower rate

of

increase

of

the heated radius

with energy injected as the heated radius becomes large.

and (4) a diminishing incremental benefit to the productiv

ity index by further increasing the heated radius. *

The

back-pressuring

of

a well early in the production

phase

of

a stimulation cycle can result in substantial in

creases in the calculated cumulative oil produced

at

cycle

end. Back-pressuring prevents

or

minimizes the flashing

of

produced water to steam which removes large quantities

of

energy which would otherwise be available to prolong

the stimulation effect. A pumping well can be back-pres

sured by one of two methods. First. the annulus pressure

can be controlled manually while the well

is pumped off.

Second, the well can be back-pressured more

or

less auto

matically by the column

of

liquid that will exist above the

pump when

pump

capacities are rate limiting. Table 5 illus

trates the effect of back-pressuring by the pump limiting

method on the computed results for Well Q-594.

Probably the optimum program

of

back-pressuring a

well

would be one in which the flowing bottom-hole pres

sure is maintained slightly above the saturation pressure

for steam at the existing bottom-hole temperature. This

would provide the maximum drawdown possible without

flashing a large fraction of the produced water to steam.

CONCLUSIONS

I.

The

simplified calculation method presented herein

can match history for conventional heavy oil wells and

can be used with confidence to make preliminary field se

lection and process variable studies for these cases. Steady

state radial flow equations, while adequate for predicting

steam stimulation response for most conventional heavy

oil wells, should not be used for tar sands and depleted res

ervoirs. For these cases the bulk

of

the oil production

comes from the region actually heated

rather

than from the

unheated region.

2 Process variables studies using the calculation show

that (a) wells having a high skin factor prior to stimulation

will respond most favorably to steam stimulation; a perma

nent rate improvement results if heating removes a portion

of the skin; (b) low produced gas-oil and water-oil ratios,

high steam injection rate, high sand-shale ratio, thick sands

and high original oil viscosity benefit the stimulation effect;

(c) thick sections require a proportionately greater energy

input to achieve a given incremental oil-steam ratio; (d)

flashing

of

produced water which causes rapid cooling

and deterioration

of

the stimulation response can be avoid

ed by back-pressuring the well early in the producing

cycle. This will permit more incremental oil to be produced

than would be attained if drawdowns were maximized

throughout the entire cycle.

'::This

is

true while

the heated radius

h is still

much smaller than

the drainage

radius

reo

As rh

becomes nearly equal

to rtJ productivity

increases

rapidly

with

increasing

it again.

Case

2

TABLE 5-EFFECT OF BACK·PRESSURE DURING THE

EARLY PART

OF

PRODUCT

ION

PHASE

Method of Back·Pressuring

Pump limited

No bock-pressure

Calculated Cumulative Inc.

Oil

ot Cycle End.

STB

Well Q·594)

35,000

17,000

1619



NOMENCLATURE

a = geothermal gradient, °F/ft

b

= square root

of

dimensionless time for radial

temperature decay (Eq. A-5, dimensionless)

B,

=

constant appearing

in

Eq. A-14, ft

c

C,

= constants appearing in Eq. 10, dimensionless

(Table I)

C

=

average specific

heat

of

gas, over temperature

interval T,. to T , ,

Btul

scf, 0 F

c C.' = average specific heats

of

oil and water over

the temperature interval T to

T

. .

Btu/lb

of

,

D = depth of the producing formation, ft

h = average thickness of the individual sand mem_

bers, ft

hi = enthalpy of liquid water at

T,,,

above 32F.

Btu/lb

hi.

=

specific enthalpy

of

vaporization

of

water at

T

DV

, Btu/lb

hiT = specific enthalpy of liquid water at T Btu/lb

hIs =

specific enthalpy

of

liquid water at T

Btu/lb

h, = individual sand thicknesses, ft

h, = artificially increased sand thickness used in Eq.

A-14, ft

HI

= rate at which energy is removed from the for

mation with the produced fluids at time I,

Btu/D

Ho.

= [5.61(pC) + R,Ic',l(T,,, - T,.), Btu/STB oil

Hon

= 5.61

p,,[R (h

I

-

hI ) + R

,/zIgl, Btu/STB oil

I = dimensionless factor read from Fig.

2

as a

function

of aI,

I r,,'

J

=

ratio

of

stimulated to unstimulated productivity

indexes. dimensionless

J.,

J

= stimulated (hot) and unstimulated (cold) pro-

1620

ductivity indexes. respectively, STBI D/psi

J = zero order Bessel function of the first kind

J, = first order Bessel function of the first kind

k =

formation permeability, darcy

kd

=

damaged permeability, darcy

k

=

oil permeability, darcy

K = formation thermal conductivity. Btu/ f t /D/oF

I, = thickness of an individual interbedded shale.

ft

J = artificially reduced shale thickness used in Eq.

A-14. ft

M. = total mass of steam plus condensate injected

in the

current

cycle, Ib

N,

= number of individual sand members

P.

= static formation pressure existing at a distance

r, from the wellbore. psia

P

. = producing bottom-hole pressure, psi a

Pm, = saturated vapor pressure

of

water at T , , psia

q. = oil production rate, STB/D

qQ'

= oil production

rate

during stimulated produc

tion,

STB/D

Q ,

=

cumulative energy lost from wellbore during

steam

injection, Btu

r

=

radial distance from the well bore, ft

r, = well radius used in Eq. 1 for well bore heat

loss calculations, ft

rd = radius of the region of damaged permeability

k

d

, ft

r, = drainage radius of well, ft

r.

= radius of region originally heated, ft

r, = inside tubing radius, ft

r

w

=

effective well bore radius ,

ft

r

w = actual well bore radius, ft

R = total produced gas-oil ratio. scflbbl at stock

tank conditions

R w = total produced water-oil ratio, bbll bbl at stock

tank conditions

.l R

w

=

Rw -

Roo, bbllbbl

at stock-tank conditions

R o

=

normal (unstimulated> water-oil ratio, bbllbbl

at stock-tank conditions

R = water produced in the vapor state per stock

tank bbl oil produced, bbl water vapor (as

condensed liquid at

60F)/STB

Sk

=

kth

term in the series solution for

v

dimen

sionless

S

=

skin factor

of

well, dimensionless

S, = skin factor remaining after a stimulation treat

ment, dimensionless

I = time elapsed since start

of

injection for the cur

rent cycle, days

t, =

time of injection (current cycle), days

1',,, = average temperature

of

the originally heated

oil sand at any time

t,

OF

T, = original reservoir temperature, OF

T, =

condensing steam temperature at sand-face in

jection pressure. OF

v = temperature difference at r, Z and I above ini

tial reservoir temperature,

OF

v = T,,,, - T ,

of

v

v,

= unit solution for the component conduction

problems in the rand z direction, respec

tively. dimensionless

v V,

= integrated average

of

v

v,

for 0 < r < rio and

all

h

respectively, dimensionless

V

=

T,

-

T,.,

OF

w = constant appearing in Eq. A-16 equal to 4a

t

-

t,).

sq ft

W,

=

constant appearing in Eq. A-16, ft

Xi

=

average downhole steam quality during mJec

lion phase, Ib

vapor/lb

liquid plus vapor

X,,,,(

=

wellhead steam quality, Ib vapor/lb liquid plus

vapor

y

=

hypothetical thickness used in Eq. A-14, ft

Yo

= zero order Bessel function

of

the second kind

Y, = first order Bessel function of the second kind

z = vertical distance from bottom of lowest sand

in interval,

ft

_

Z = h,

;=1

a

=

overburden thermal diffusivity. sq ft/D

30

=

oil formation volume factor, STB

oill

res bbI

8 = quantity defined in Eq 5, dimensionless

JOUR:- AL O ,PETROLEUM TECJ\ : \ ,OLOGY



7i = i m e n s i o n l e ~ s time

_ _ 2 _

,

=

eTerfc

hiT)

;= \ I

T

-

v

(pC)1

= volumetric heat capacity

of

reservoir rock in

cluding interstitial fluids, Btu/cu

ft, OF

p ,

pw

= stock-tank fluid density of oil and water, Ib/

cu ft

T = 4Kt,/ h'(pC) dimensionless

ACKNOWLEDGMENTS

The authors wish to express their appreciation to Esso

Production Research Co, for permission

to

publish this

paper. Also, the comments and help

of A. R. Hagedorn

and A. G. Spillette are gratefully acknowledged.

REFERENCES

1.

Armstrong,

Ted

A.:

Steam

Injection Has Quick Payout in

Wilmington

Field , Oil Gas Jour. (March

21, 1966) 78.

2. Carslaw. H. S.

and Jaeger,

J. c.:

Condurtion

of

Heat

in

Sol·

ids,

2nd Ed., Oxford

U.

Prf'ss, London, England (1959) 346.

3. Carslaw, H. S.

and Jaeger, J.

c.:

ibid.,

5.1.

4. Long, Robf'rt T : Case History of Steam Soaking in Kern

River Field. California . JOUf. Pet.

Tech.

(Sept .

19(5)

91\9·

99.3.

5. Luk .. Y.

L.:

Intewals of

Ressel Functions.

McGraw Hill Book

Co . N. Y. (1962) 314.

6. Marx, J.

W. and

Langenheim. R. H.: Reservoir

Heating

by

Hot

Fluid

Injection , Tum ., AIME

11959)

2]6,

312·314.

i. Muskat, M.: Physical Principles

of

Oil Produrtion.

lst Ed .

McGraw Hill Book Co., N. Y. (1949) 194. 215.

R

Owens. \Y.

D. and Suter.

V.

E.: Steam

Stimulation-Nf'wf'st

Form of Secondary Pf'troleulll Rf'co\'f'ry ,

Oil Gas

JOUf.

(April 26, 1965) 82.

9.

Payne,

R. \V. and Zambrano, G.: Cyt:iie Steam Injedion

Helps

Raise Venezuf'la Productio n . Oil tIC Gas Jllur. (May

24. 19(5)

78.

10. Ramey. H. J .•

Jr.:

Wellborf'

Heat

Transmission , Jour. Pel.

Tech.

(April,

19(2)

427-435.

II.

Squier,

D.

P.,

Smith. D. D. and Dougherty, E.

L.:

Calclllatf'd

Telllperatn re Behavior of Hot· \Vatpr InjPetion Wells ,

Jour.

Pet. Tech.

(April, 1962) 436-440.

PPENDIX

DERIVATION

OF EQUATIONS

NEGLIGIBLE ENERGY

REMOVAL

WITH PRODUCED FLUIDS

The heat transfer model for conduction cooling

of

the

heated oil sands following termination of steam Injection

consists

of

approximating the sand-shale sequence by a

stack

of

variable thickness. equal radii sand cylinders.

These cylinders initially at constant temperature

T,

COII_

duct

heat as a function

of

time to variable thickness inter

bedded shales and to semi-infinite overburden and under

burden having thermal properties identical to the oil sands.

The

geometric approximation

of

the sand-shale sequence

is

illustrated in Fig. 12.

When conduction radially and vertically are the only

mechanisms of heat transfer. the temperature

of

any point

within the originally heated cylinder can be expressed as

the product:

v = V v,

(A-l)

where v, and are unit solutions of component conduc

tion problems in the rand

z

directions, respectively. Sim

ilarly, an integrated average temperature for the heated

regions may be computed:

v V v

A-2)

DECEMBER 1966

The

average unit solution forv, is obtained by solving

the one-dimensional heat conduction problem in the radial

direction:

~ ~ , - - ( r ~

= ~ ,

r?r (Jr i l l

A-3)

with boundary and initial conditions:

v

=

1

t = t,

0<

r

r

v,

= 0 t

=

t,

r>

r

A-4)

v,. =

0

t

t,

r __

Ct

I f

the thermal properties

do

not vary with

r,

the tempera

ture solution for r < r . is'

if

4 f e-b' fi,(y)J,,(ry/r,,)dy

v, = ' y' [J, (y) Y

n

(y) -

in

(y) Y, (y)]'

(A-5)

o

where b

=

aU

-

t,)/r,, .

Since J

1

(y) Y (y)

- J

(y) Y

1

(y) =2... Eq. A-5 reduces to:

....y

v,,

f)

f e-I>'II J,(y)J,,(ry/r,.)dy

o

The average

of

I',

from 0 to

r

is given by:

,

I

I

I

I

I

OVER URDEN

,

2

t -

UNDER URDEN

FIG.

12-GEOMETRIf . ApPROXIMATION

OF

MULTIPLE

SAND,SHALE

SEQUENCE.

A-6)

1621



r

y

e- h'''''},(y)dy

(A-7)

gives

the

following solution

of

the integral

I I I

Eq.

1

00

-

V

r

- -b - Sf.

/ :=0

(A-8)

s, is defined as:

(

l),

- IT

1 (1.5

+ k)T'( 1 + k)

(A-9)

S, =

yr. k

1 (2 +

k)

1 (3 +

k)

Eq. A-9:

1

S, ,=

4

(A-lO)

it follows that:

[

I) (k+1.5)

]

S t l

= -

IT 2 + k) (3 + k) .1',.

(A-II)

Eq. A-8. together with Eqs. A-lO and A-II. form

V,.

A

plot of

v \ s b' is given in Fig. 4.

The average unit solution for

v,

is obtained by solving

heat conduction problem in the verti

direction:

il'v,

ill',

(t ilz' = T t

(A-12l

boundary

conditions

\ ,

=

0,

t

>

t,

as

z

:x.

The

in-

condition is

-

(

)

_

0 all

z

outside the regions

of

thickness h

v. t z - j -

. I all

z within

the regions of thickness II,.

The solution of the conduction equation subject to the

and boundary conditions

is

v =

C/J

I (

2\/-:rl'it

J

-C f

l ',(t z')

exp

[ -

(z

-

z')'/4l'it)dz'

A- l3)

the integration with the initial condition

I [ z .

h,

- z z -

B,

V

Z

= 2

e r l - = +

erf-=+ erl------===-+ erl

2ya t 2y'at 2v'at

B, + ii, - Z + ert z - B, +

-I

B., + ii. - z + ]

_ el

.....•

2\ /at

2v'at

2v al

(A-14)

B, = B,_, + h

j

_,

+ i

j

_,

B

j

= 0

1622

h

j

=

h

j

+

y

i, = - ~ I,

-

y (if i, < 0,

the two sands of

thickness

Iz

j

and h

+, plus

the shale

of thickness I, are

treated

as

a single sand. The y is then re

calculated to account

for

one less sand).

h, = thickness

of

sand j

I

=

thickness

of

shale

j

y = [M. (X, h'q + hI, - hl,V(-:rr,,'(pC), L -

T,)

N J - h (y

is

a hypothetical thickness

which. when

added

to the individual

sand

thicknesses,

accounts

for all the energy in

jected including that lost to

the

shale dur

ing the injection phase.)

The

average

temperature V, can

then be found by in

tegrating over all in.dividual sands:

Bl+hl

N.,

f

, =

;=1 /

N,

_

F,dz

II, .

;=1

(A-l5)

B,

Substituting Eq. A-l4 into Eq. A-15

and

integrating gives:

IN . N. [

W.

: c

W,

er t ,

+ w, erl ,-

T m= l n = l

VW

yw

_ I'I

F , =

2

m = 1

U

l

, W,

~

W·.

-- W, erl .

-

W, ert-;=-+ [exp(

- ,-Iw)

+

yw

V

\I

exp(

- W,'lw)

-exp(

-

W,'lw) -exp - W/lw) ] (A-16)

where U', = B", + iim - B

W, =

B

+ h - Bm

I·V =

B,.,

- B

W; = B + h

-

Bm - ji ,

If

= 4a(

-

t,)

.

For

a single sand reservoir. \ , is plotted vs 4 Y t - l ') lh,'

in Fig. 4.

CASES \\HERE ENERGY REMOYAI.

WITH PRODUCED

FLUIDS

MUST

HE TAKEN INTO ACCOUNT

I f significant energy

is

removed with the produced flu

ids,

some account should

be

made for

this energy removal.

Consider an energy balance taken on the sum of the thick

nesses

of the originally heated regions Z and radius rio:

t

Z r . r , , ( p C ) l ~

= Z-:rr,,' (pC), V

-

f

H,dx - H..,

(A-17)

ti

where H,

= the energy conducted to the shale and oil-sand

outside

the

originally

heated

region, Btu.

Eq. A-17 states

that

the energy contained within the

originally

heated

region

at

any time is

equal to that con

tained immediately

prior to production less

that

produced

with

the

fluids

removed from

the

formation and

less

that

conducted to shale and oil sand outside of the originally

heated region. A rigorous evaluation of

H,

has not been

possible.

However,

a satisfactory

approximation

for

H,

is

as follows.

(A-IS)

where

t

1 12 J ldx

V= V -

t,

Z-:rr,,' (pC) ,

(A-19)

Where conduction

is the sole

mechanism of temperature

JOt:R: I AL OF PETROLEIJM TECH:, \ ,OLOGY



decay, and the energy removed with the produced fluids is

negligible:

and Eqs. A-I7, A-IS and A-20 reduce to Eq. A-2.17 may

be considered an effective driving force for conduction

which takes into account the effect of temperature reduc

tion due to the removal of energy by the produced fluids.

The definition of 8 from Eq. 5 combined with Eqs. A-

17 through A-19 gives Eq. 4. By Eqs. A-19 and 5:

i = V I -

8

A-20)

Combining Eq. A-20 with Eqs.

A-I7

and

A-IS.

and divid

ing by Z. .r,,'(pCl, gives v

=

V - 2 V8 - V l - 8 I -

v,vol

which simplifies to E4.

4.

Of

several approximations for H, which were tried, the

one used here gave best agreement with field data. Note

that for the limiting case where conduction IS negligible

v,v,

7

1 ,

this approximation gives:

T,,, = T . T, -

T,)[l

- 28]

A-22)

From the definition of 8 this may be seen as giving the

correct value for T

t lvg

in this limiting case.

spe 1578 pa boberg lantz

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