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SGP-TR-5 9
FLOW CHARACTERISTICS AND RELATIVE
PERMEABILITY FUNCTIONS FOR TWO
PHASE GEOTHERMAL RESERVOIRS FROM
A ONE DIMENSIONAL THERMODYNAMIC MODEL
Anthony J. Menzies
August 1982
A
.
Stanford Geothermal Program Interdisciplinary Research in Engineering and Earth Sciences
STAMFORD UNIVERSITY Stanford, California
SGP-TR-5 9
FLOW CHbJUCTERISTICS AND RELATIVE PERMEABILITY FUNCTIONS FOR TWO PHASZ GEOTHERMAL RESERVOIRS FROM A ONE DIMENSIONAL THERMODYNAMIC MODEL
BY
Anthony J. Menzies
August 1982
Financial support was provided through the Stanford Geothermal Program under Department of Energy Contract No. DE-AT03-80SF11459 and by the Department of Petroleum Engineering, Stanford University.
ABSTRACT
Theore t i ca l flow c h a r a c t e r i s t i c s f o r a f r a c t u r e d geothermal r e s e r v o i r
have been obtained by m d e l l i n g t h e system with a one dimensional
thermodynamic model. The model inc ludes t h e e f f e c t of heat t r a n s f e r from t h e
rock t o t h e f l u i d and i r r e v e r s i b l e processes , such as f r i c t i o n , by us ing an
e f f e c t i v e i s e n t r o p i c e f f i c i e n c y term. By approching the problem i n t h i s manner
i t has not been necessary to d e f i n e the f low geometry o r t o de f ine such
parameters as the two phase f r i c t i o n f a c t o r .
By comparing the t h e o r e t i c a l c h a r a c t e r i s t i c s generated by the model with
f i e l d d a t a it is p o s s i b l e t o estimate the flow area and an e f f e c t i v e f r a c t u r e
width f o r the two phase f low i n t o t h e wellbore from t h e rese rvo i r . It is a l s o
p o s s i b l e t o c a l c u l a t e under what cond i t ions choking w i l l occur i n t h e
r e s e r v o i r and hence, the maximum e x p l o i t a t i o n rate f o r the r e s e r v o i r / w e l l
system.
F i e l d examples are included t o i l l u s t r a t e how the flow area and e f f e c t i v e
f r a c t u r e width are ca lcu la ted . It w a s f u r t h e r found t h a t c e r t a i n
c h a r a c t e r i s t i c s of the f i e l d f low d a t a could be explained by the concept of
choked o r c r i t i c a l flow.
From t h e d a t a generated by the model i t w a s poss ib le t o de r ive a unique
set of r e l a t i v e permeabi l i ty curves , independent of the r e s e r v o i r temperature.
They were der ived as func t ions of t h e inp lace l i q u i d s a t u r a t i o n and could
t h e r e f o r e be used i n p resen t geothermal s imulators . They have been compared
wi th a number of o the r r e l a t i v e permeabi l i ty func t ions and i t is concluded
t h a t the r e l a t i v e permeabi l i ty func t ions developed here are probably more
c o n s i s t e n t f o r f r a c t u r e d geothermal r e s e r v o i r s .
ii
.
TABLE OF CONTENTS
--
ABSTRACT.. . . . . . a . . . . . . . . . . . . . . . . . . . ii TABLE OF CONTENTS. . . . . . . . . . . . . . . . . . . . iii LISTOFFIGURES. 0 . . v
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . v i i
1. INTRODUCTION. . rn 1
2. TWO PHASE GEOTHERMAL RESERVOIRS . . . . . . . . . . . . . . 3
2.1 Flow Characteristics . . . . . . . . . . . . . . . 3
2.2 R e l a t i v e Pe rmeab i l i t y Functions. . . . . . . . . . . 6
3. STREAMTUBE MODEL. . . . . . . . . . . . . . . . . . . . 9
3.1 S e l e c t i o n of Model . . . . . . . . . 9
3.2 Desc r ip t i on of Model . . . . . . . . . . . . 10
3.3 Mathematical Formulation . . . . . . . . . . . . 13
3.4 Ca lcu l a t i on of Relative P e r m e a b i l i t i e s . . . . 18
3.5 Computer Program (GEOFLOW) . . . . . . . . . . 20
23 . 4. FLOW CHARACTERISTICS. . . . . . . . . . . . 4.1 E f f e c t of Reservoi r P re s su re . . . . . . . . . . . . 23
4.2 ChokedFlow. . . . . . . . . 26
4.3 Flow Geometry. . . . . . . . . . . . . . . . . . . 33
5. COMPARISON OF FLOW CHAEUCTERISTICS WITH FIELD
AND EXPERIMENTAL DATA . . . . . . . . . . . . . . . 37
5.1 F i e l d D a t a . . . . . . . . . . . . . . . . . . . . . . 37
5.1.1 Well "Utah-State" 14-2, Roosevelt Hot
Spr ings , Utah, USA. . . . . . . . . 39
5.1.2 Well BR-21, Broadlands Geothermal F i e l d ,
W e w Z e a l a n d . . . . . . . . . . . . . . . . . . 42
5.1.3 Well KG-12, Kra f l a Geothermal F i e l d ,
I c e l a n d . . . . . . . . . . . . . . . . . . 45
5.1.4 Well 403, Tungonan Geothermal F i e l d ,
t h e P h i l i p p i n e s . . . . . . . . . . . . . . . . 47
5.2 Experimental Data. . . . . . . . . . . . . . . . . 51
iii
6 . RELATIVE PERMEABILITY FUNCTIONS . . . . . . . . . . . . . . 53
6.1 Effect of the Input Variables . . . . . . . . . . . . . 53
6.2 Comparison with Corey and X-type Relative Permeability Functions . . . . . . . . . . . . . . . . 55
6.3 Comparison with Field Derived Curves . . . . . . . . . 58
6.4 Comparison with Experimental Relative Permeability Curves . . . . . . . . . . . . . . . . . . 60
6.5 Comparison with Relative Permeability Curves for Vugular Cores . . . . . . . . . . . . . . . 60
7 . DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.1 Flow Characteristics . . . . . . . . . . . . . . . . . 64
7.2 Flow Geometry . . . . . . . . . . . . . . . . . . . . . 65
7 .3 Relative Permeability Curves . . . . . . . . . . . . . 66
8 . CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . 67
9 . RECOMMENDATIONS FOR FUTURE WORK . . . . . . . . . . . . . . 69 10 . NOMENCLATURE . e 7 1
11 . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . 73
APPENDIX A: Text of Paper by Wallis and Richter . . . . . . . . . 76
APPENDIX B: Listing of Program GEOFLOW with Typical Output 83
APPENDIX C: Output from GEOFLOW for Field Examples . . . . . . . 95
iv
LIST OF FIGURES
Figure
2.1
2.2
2.3
3.1
3.2
3.3
3.4
Description Page
Massflow Characteristics of Typical Geothermal Wells . . . 4
Enthalpy Characteristics of Typical Geothermal Wells . . . 4
Enthalpy, Massflow Crossplot for Two Phase -
Geothermal Well. . . . . . . . . . . . . . . . . . . 5
Enthalpy/Entropy Diagram Showing Formation
of Streamtubes (Wallis and Richter, 1978). . . . . . . . 11
Enthalpy/Entropy Diagram Showing Formation
of Streamtubes, Including Effect of Heat Transfer
(after Wallis and Richter, 1978) . . . . . . . . e . . . 12
Streamtube Model Flow Diagram. . . . . . . . . . . . 16
Massflux vs Pressure Drop - Comparison with Wallis and Richter . . . . . . . . . . . . . . . . 22
3.5
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Slip Ratio vs Pressure Drop - Comparison with Wallis and Richter . . . . . . . . . . . . . . . . . . . 22 Massflux vs Pressure Drop, T0=25O0C. . . . . . . . . . . . 24
Massflux vs Pressure Drop, To=300°C. . . . . . . . . . . . 24
Massflux vs Flowing Pressure, T0=3000C . . . . . . . . . 25
Massflux and Enthalpy vs Pressure Drop, T0=25o0C . 27
Massflux and Enthalpy vs Pressure Drop, T~=~~O'C . . . . . 28
Massflux and Enthalpy vs Pressure Drop, TO=2700C . . . . . 29 Massflux and Enthalpy vs Pressure Drop, T ~ = ~ ~ O ' C . . . . 30 Massflux and Enthalpy vs Pressure Drop, TO=29O0C . . . . . 31
Massflux and Enthalpy vs Pressure Drop, T0=30O0C . . . . . 32
V
4.10
4.11
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
6.1
6.2
6.3
6.4
Fracture/Borehole Orientation used for Calculation
of Effective Fracture Width. . . . . . . . . . . . . . . . 34 Massflux vs Temperature, for Estimation of Flow
Area when Flowing Surveys Unavailable. . . . . . . . . . . 35
Massflow vs Flowing Downhole Pressure,
Well "Utah-State" 14-2 . . . . e . . e . . . . . 41 Massflow vs Flowing Downhole Pressure, Well ER-21. e . e 44
Enthalpy vs Flowing Downhole Pressure, .Well BR-21. . . . 44
Massflow vs Flowing Downhole Pressure, Well KG-12. . . . 46
Enthalpy vs Flowing Downhole Pressure, Well KG-12. . . . 46 Massflow vs Flowing Downhole Pressure, Well 403. . . . . . 49
Enthalpy vs Flowing Downhole Pressure, Well 403. . . . 49 Enthalpy/Massflow Crossplot, Well 403. .. . . . . 50
Relative Permeability Curves for Steam and Water
as Generated by GEOFLOW. . . . . . . . . . . e . 54 Corey, X-type and GEOFLOW Relative Permeability Curves . 56
Flowing Enthalpy vs Water Relative Permeability ,To=25O0C
(after Bodvarsson, O'Sullivan and Tsang, 1980) . . . . e 57
Horne and Ramey(1978), Shinohara(l978) and
GEOFLOW Relative Permeability Curves . . . . . 59 6.5 Experimental Relative Permeability Curves for
Steam and Water (Counsil, 1979). . . . . . . . . . . . 61 6.6 Relative Permeability Curves f o r Vugular Dolomite Core
(Sigmund and McCafferty, 1979) . . . . . . . . . . . . . . 62
vi
LIST OF TABLES
Table
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
Descr ipt ion
F ie ld Data - Summary of W e l l and Reservoir Data. . 38
Measured Flow Data, Well "Utah-State" 14-2 . . . . 39
Calculated E f f ec t i ve Frac ture Width f o r -
Well "Utah-State'' 14.2 . . . . . . . . . . . . . . . . . 40
Calculated E f f ec t i ve Frac ture Width f o r Well BR-21 . . . 43
Calculated E f f ec t i ve Frac ture Width f o r Well KG-12 . . . 45
Measured Flow Data 'from Well 403 . . . . . . . . 47
Calculated E f f e c t i v e Frac ture Width f o r Well 403 . . . . . 48
P rope r t i e s of Cores used by Arihara(1974). . . . . . . e . 51
Ef f ec t i ve Flow Area from Experimental Data, Arihara( l974) . 52
.
v i i
...
1. INTRODUCTION
c
Study of the flow characteristics of geothermal wells has been largely
limited to either liquid dominated reservoirs where flashing occurs in the
wellbore, Nathenson(l974), Ryley(1980), or to vapor dominated systems,
Rumi(1972). In both these cases flow in the reservoir is single phase and
essentially isothermal. The reservoir behaviour can therefore be analyzed
using flow equations developed in the groundwater hydrology or petroleum
engineering literature. When two phase flow occurs in the reservoir the
situation is more complex and cannot be analyzed in the same manner. The
interactions between the two phases and the flow system become important in
describing the flow behaviour. These interactions are accounted for in
petroleum reservoir engineering by the use of the concept of relative
permeability .
-
Geothermal applications have an additional complication since the two
phase flow of oil and gas is essentially isothermal whereas a two phase flow
of steam and water is not. Temperature drops as high as 5OoC have been
measured in geothermal reservoirs. In spite of this, relative permeability
curves, particularly those developed by Corey(1954) for oil reservoirs, are
still used in simulation models of geothermal reservoirs.
Two phase compressible flow has been studied in detail, particularly in
nuclear reactor engineering, but very little of this research has been applied
to geothermal systems. Choked or critical flow has formed a central part of
this research effort and although it forms the basis of the James(1962) method
for measurement of output parameters in geothermal wells, the idea that choked
flow could occur in reservoir flow systems, thereby limiting the systems'
-1-
o u t p u t , has not been widely discussed.
The purpose of t h i s r esea rch was t o s tudy the flow c h a r a c t e r i s t i c s of two
phase geothermal r e s e r v o i r s , p a r t i c u l a r l y the p o t e n t i a l problem of choked o r
c r i t i c a l flow. As a r e s u l t of the s tudy it has a l s o been poss ib le t o genera te
1
r e l a t i v e permeabi l i ty func t ions t o account for the observed flow
c h a r a c t e r i s t i c s .
- 2-
2. TWO PHASE GEOTHERMAL RESERVOIRS
2 . 1 Flow Characteristics
The flow characteristics of geothermal reservoirs are inferred from the
measurement of enthalpy, total massflow and concentration of chemical
components as functions of the wellhead pressure. By plotting these -
characteristics under both transient and steady state conditions the general
processes occurring in the reservoir can be inferred. In this study only the
enthalpy and massf low changes are considered although the chemical changes are
also important.
Figures 2.1 and 2.2 show specific examples of measured output
characteristics from the Tungonan geothermal field, the Philippines and from
the Larderello field in Italy, Rumi(1972). They illustrate the major
.- differences in measured flow characteristics between single phase water,
single phase steam and two phase geothermal reservoirs. The important
characteristics of the two phase system are the almost constant massflow and
increasing enthalpy at low wellhead pressures.
Figure 2 .3 shows a crossplot of the enthalpy and massflow data for the
two phase well, showing how the enthalpy rises very quickly over a small
change in massflow. This is in disagreement with the observation of Sorey,
Grant and Bradford( 1980) : "In two phase wells these measurements usually show
enthalpy varying linearly with massflow (to a first approximation)". This
characteristic is difficult to explain and is a central aspect of this
research effort.
-3 -
100 L
80 n
\ m Y
a
2 40 v) cn u I
20
0
1500 n en Y \ T) Y
t 1400
CI
$ a 1 1 3 0 0 x c
WELL MB-3 (SINGLE PHASE)
WELL 403 (TWO PHASE)
GABBRO 1 (STEAM)
1.0 2.0 3.0 WELLHEAD PRESSURE, pwh (MPa.a)
WELL 403
FIGURE 2.1: MASSFLOW CHARACTERISTICS OF TYPICAL GEOTHEWMAL WELLS
FIGURE 2.2: ENTFIALFT CHARACTERISTICS OF TYPICAL GEOTHERMAL WELLS
1200 I I I I 1 I 0 1 .o 2.0 3.0
WELLHEAD PRESSURE, pwh (MPa.a)
...
-4-
1500
1400
1300
1200 0
0 WELL 403
10 20 MASSFLOW, W (kg/s)
30
FIGURE 2.3: ENTHALPY, MASSFLOW CROSSPLOT FOR TWO PHASE GEOTHERMAL WELL
- 5-
2.2 Rela t ive Permeabil i ty Functions
A petroleum engineer ing approach t o two phase flow i n r e s e r v o i r s i s the
use of r e l a t i v e permeabi l i ty func t ions . Their bas i c use i s t o account f o r t h e
i n t e r a c t i o n s between one f l u i d and t h e o the r and a l s o with the surroundings.
Corey( 1954) developed formulas t o relate the o i l and gas r e l a t i v e
pe rmeab i l i t i e s t o t h e inp lace l i q u i d s a t u r a t i o n based on numerous measurements
of t h e flow of o i l and gas through consol ida ted sedimentary cores.
Tsang and Wang(1980) reviewed t h e cu r ren t p r a c t i c e i n s imula t ion of
geothermal recovery processes and observed t h a t a l l the two phase models use
t h e r e l a t i v e permeabi l i ty func t ions developed by Corey( 1954). This is i n s p i t 4
of t h e f a c t t h a t few geothermal r e s e r v o i r s are sedimentary and most are h igh ly
f r a c t u r e d volcanics . The r a t i o n a l f o r us ing t h e Corey equat ions i n models of
f r a c t u r e d geothermal r e s e r v o i r s is t h a t i f a l a r g e enough con t ro l volume i s
used the he te rogene i t i e s due t o the f r a c t u r e s w i l l average out. Howevers t h i s
is only true when the two phase condi t ions are widespread over the r e s e r v o i r
and i s l i k e l y t o be i n s e r i o u s e r r o r where l o c a l changes i n flow condi t ions
occur. I n add i t ion , another major problem is t h a t the r e l a t i v e permeabi l i ty
func t ions f o r steam and water are not w e l l eshabl ished.
I n terms of steam and water t h e Corey type r e l a t i v e permeabi l i ty
func t ions are (Sorey e t a1.(1980)):
krw = [S*I4
k = [ l - ( S* ) 2 ] [ ( 1 - S*)2] rs
where :
(2.2-1)
(2.2-2)
(2.2-3)
...
4
-6-
and : S,= r e s i d u a l water s a t u r a t i o n
Srs= r e s i d u a l steam s a t u r a t i o n
A major problem with t h e use of these func t ions is the determinat ion of
Srw and Srs.
Experimental work by Counsil(1979) has def ined r e l a t i v e permeabi l i ty
func t ions f o r steam and water f low, based on measurements i n consol idated
cores . These func t ions have not received widespread use i n geothermal
s imulat ion.
A f u r t h e r method of d e f i n i n g t h e r e l a t i v e permeabi l i ty func t ions is t o
use measured flow c h a r a c t e r i s t i c s . The b a s i c approach is descr ibed by Sorey e t
a1.(1980) and Horne and Ramey(1978) and Shinohara(l978) present r e l a t i v e
pe rmeab i l i ty curves c a l c u l a t e d from procedures based on t h i s approach. Using
product ion d a t a from w e l l s i n t h e Wairakei geothermal f i e l d i n New Zealand,
they were a b l e t o o b t a i n r e l a t i v e pe rmeab i l i ty curves as func t ions of t h e
f lowing water mass f r a c t i o n . The Corey r e l a t i v e permeabi l i ty curves are
func t ions of the inp lace l i q u i d s a t u r a t i o n (vol . b a s i s ) and t h e f i e l d de r ived
curves need t o be converted t o t h i s b a s i s before being used i n p resen t
geothermal s imulators . This i s u s u a l l y n o t poss ib le .
The problem involved i n conver t ing from flowing t o inp lace s a t u r a t i o n s is
discussed by Miller(1951). In h i s paper he s tates: " the weight f r a c t i o n of gas
i n t h e mixture ins tan taneous ly at x is q u i t e d i f f e r e n t from t h e weight
f r a c t i o n of gas i n the mixture pass ing x i n u n i t t i m e " . This arises because
t h e vapor has a h igher mobi l i ty and hence a higher v e l o c i t y than t h e l i q u i d .
The r a t i o between the vapor and l i q u i d v e l o c i t i e s i s c a l l e d the " s l i p r a t i o "
and i t must be known t o convert from flowing t o inp lace s a t u r a t i o n s . This i s
c l e a r l y not poss ib le i n a f i e l d s i t u a t i o n .
- 7-
This study the re fore set out t o i n v e s t i g a t e t h e o r i t i c a l l y the two phase
flow of steam and water mixtures i n f r a c t u r e s i n an a t t e m p t t o de r ive r e l a t i v e
permeabi l i ty curves more appropr ia te t o geothermal r e s e r v o i r a p p l i c a t i o n s .
The next s e c t i o n deals with the s e l e c t i o n and desc r ip t ion of t he
thermodynamic model used i n t h i s research.
A
-8-
.
3 . STREAMTUBE MODEL
3.1 Selec t ion of Model
I n an a t tempt t o b e t t e r understand the processes involved i n t h e two
phase f low of steam and water i n a f r a c t u r e d geothermal r e s e r v o i r , a model of
steam/water flow i n a confined condui t was sought.
There are a number of e x i s t i n g one dimentional models f o r the s tudy of
two phase vapor l i q u i d flow. They are normally c l a s s i f i e d as homogeneous, s l i p
o r separa ted flow models, depending on t h e assumptions made i n t h e i r
d e r i v a t i o n . The homogeneous models assume t h a t the vapor and l i q u i d phases
have the same v e l o c i t y , hence no meaningful r e l a t i v e permeabi l i ty f u n c t i o n s
can be derived.
The m d e l found t o be most appropr ia te t o t h i s research was t h e
"streamtube" model of Wallis and R i c h t e r ( 1978). This model overcomes t h e
d i f f i c u l t i e s inheren t i n t h e usua l s l i p flow theory by al lowing t h e v e l o c i t y
and thermodynamic state t o vary normal t o t h e f low d i r e c t i o n . It does t h i s by
cons ider ing the two phase flow f i e l d t o be d i s t r i b u t e d between a number of
d i s c r e t e s t reamtubes , hence the name streamtube model. The streamtube model
has been found t o p r e d i c t c r i t i c a l f low i n nozzles more a c c u r a t e l y than o t h e r
s l i p models.
The text of Wallis and R i c h t e r ' s paper is reproduced as Appendix A.
- 9-
3 . 2 Descr ipt ion of Model
The model uses a series of d i s c r e t e pressure s t e p s t o approximate the
continuous f l a sh ing . After each pressure s t e p a streamtube is c rea ted . In t h i s
newly c rea ted streamtube i n i t i a l l y only s a tu r a t ed steam flows. A t the same
time the steam i n streamtubes t h a t a l r eady e x i s t e d is assumed t o expand
i s e n t r o p i c a l l y . When t h i s occurs some of t he steam condenses; t h i s small
amount of l i q u i d is assumed t o have t h e same v e l o c i t y as the steam. Thus
w i th in each streamtube t h e homogeneous model is assumed t o apply. This basic
process is shown i n Figure 3 . 1 . There is assumed t o be no i n t e r a c t i o n between
t h e streamtubes and thus no t r a n s f e r of energy, mass or momentum. Each
streamtube has a d i f f e r e n t v e l o c i t y , hence a ve loc i t y p r o f i l e e x i s t s normal t o
t h e flow d i r ec t i on . The f i r s t vapor streamtube has the h ighes t v e l o c i t y and
t he l i q u i d streamtube has the lowest ve loc i t y . The e f f e c t i v e s l i p r a t i o i s
found from the r a t i o of the average vapor v e l o c i t y t o the l i q u i d ve loc i t y .
I n the o r i g i n a l form of the model, t he energy balance only considered t h e
changes i n enthalpy and v e l o c i t y while t he assumption of i s e n t r o p i c expansion
requ i red t h a t the o v e r a l l process be r eve r s ib l e . Thus the decrease i n en tha lpy
must be equal t o t he i nc r ea se i n ve loc i t y . This impl ies t h a t t he system i s
f r i c t i o n l e s s and o the r energy changes such as g r a v i t a t i o n a l e f f e c t s have n o t
been taken i n t o account. I n s t u d i e s on nozzles these assup t ions have been
found t o be va l id . However, f o r geothermal r e s e r v o i r app l i ca t i ons hea t
t r a n s f e r t o the f l u i d i s important and must be included i n the model
formulation.
-10-
' SATURATION LINE - STAGNATION +e
POINT
LlQU STREAM
- . CONSTANT PRESSURE LINES
FIGURE 3.1: ENTHALPP/ENTROPY DIAGRAM SHOWING FORMATION OF STREAMTUBES (WALLIS AND RICHTER, 1978)
-11-
I
s t STREAMTUBE
LI STRE
SATURATION LIN 1 s t STREAMTUBE
2nd STREAMTUBE S TA GNAT ION
3rd STREAMTWBE
2nd STREAMTUBE
3rd STREAMTWBE
CONSTANT PRESSURE LINES I 1
6
FIGURE 3.2: ENTHALPY/ENTROPY DIAGRAM SHOWING FORMATION OF STREAMTUBES, INCLUDING EFFECT OF HEAT TRANSFER (AFTER WALLIS AND RICHTER, 1978)
-12-
Using bas ic thermodynamic r e l a t i o n s h i p s , t h i s energy gain i s given by:
where :
= the e f f e c t i v e i s e n t r o p i c e f f i c i e n c y nS
(3.2-1)
The i s e n t r o p i c e f f i c i e n c y term is used s ince not a l l the heat. t r a n s f e r r e d
r e s u l t s i n an inc rease i n the f l u i d s i n t e r n a l energy. Some i s used t o
counterac t the i r r e v e r s i b l e processes not included i n the energy balance of
the bas ic model.
This heat t r a n s f e r s t e p changes the basic model as ind ica ted i n Figure
3 . 2 . It extends the bas ic model by al lowing t h e f l a s h i n g process t9 be
approximated by a two s t e p process r a t h e r than the basic s i n g l e s t e p p rocess t
3 . 3 Mathematical Formulation
The bas ic s t e p s involved i n t h e model are shown i n Figure 3 . 3 . This sho*s
the formation of the f i r s t two vapor s treamtubes and as the pressure cont inues
t o dec l ine f u r t h e r s treamtubes are formed and expand i n the same fashion. To
s impl i fy the computation, the model is normalized on the b a s i s of u n i t
massflow ie. Yi + yi = 1.
I f we consider the f i r s t i s e n t r o p i c expansion s t e p ( l a b e l l e d 1 i n Figure
3 . 3 ) w e have a l i q u i d massflow, Yo, wi th enthalpy, ho, entropy, so, and
v e l o c i t y , vo, ca lcu la ted from:
2 v 0 = 2(Po - Psat)/Pf (3 .3- 1)
- 1 3-
expanding to a liquid massflow, Y1, with properties hlt, sl', v1 and a vapor
massflow, yl, with properties hltt, sl" and vl. Note that the liquid and
vapor streamtubes are assumed to have the same velocity, when the .vapor
streamtube is first formed.
Applying the usual conservation equations,
mass :
yo = y1 + Y1 ( 3 . 3 - 2 )
2 2
( 3 . 3 - 3 ) 1 v 1 V 2
and energy:
vO Yo(ho' + 2/ = Yl(hl' + 2/ + y (h " +F) 1 1
and the assumption of isentropic expansion:
Y s ' = Y 1 s l f + ylsl P I 0 0
we get from ( 3 . 3 - 2 ) and ( 3 . 3 - 4 ) :
and from ( 3 . 3 - 2 ) and ( 3 . 3 - 3 ) :
2 2 + 2[h0' - hlt -Y\hlst yl - h l ' ) ]
0 v1 = vo
( 3 . 3 - 4 ) .. .
( 3 . 3 - 6 )
It is a simple matter to extend these equations to their more general
form. (see Appendix A for the more general derivation).
- 1 4 -
A s ind ica ted i n Figure 3 . 3 , the l i q u i d and vapor streamtubes now undergo
h e a t t r a n s f e r which i n c r e a s e s both the l i q u i d and vapor enthalpy and entropy.
The b a s i c equat ions are :
Ah = Q
(3 .3-7)
(3.3-8)
(3.3- 9) Q AS e- T1
The a c t u a l change w i l l depend on the e f f e c t i v e i s e n t r o p i c e f f i c i e n c y and
t h e s p e c i f i c heat of the f l u i d i n each p a r t i c u l a r streamtube. It is a l s o
assumed t h a t t h e f l u i d v e l o c i t i e s remain cons tap t dur ing t h e hea t t r a n s f e r
s t e p .
During t h e second i s e n t r o p i c expansion a new vapor streamtube is c rea ted .
A t t h e same t i m e the f i r s t vapor streamtube expands and some of the vapor may
condense r e s u l t i n g i n a steam mass f r a c t i o n o f :
if X1,J < 1:
h = x h " + ( 1 - ~ ~ , ~ ) h ~ ~ 192 1 , 2 2
i f x > 1: 192
- 1 5 -
(3 .3- 10)
( 3.3-1 1 )
( 3 3- 12)
P
o +
P
P P >
t a >
i
e P
* P
* e 0 '
a 1
9 9 a c c c o
-7- r -- w
> a
I
-16-
H CI
The homogeneous mixture is assumed to have a uniform velocity and this is
calculated from:
(3.3-13)
The two vapor streamtubes and the liquid streamtube then undergo heat
transfer, thereby increasing.their respective enthalpies and entropies before
the next isentropic expansion. This process is continued until the total
pressure drop is reached, at which stage there will be n vapor streamtubes
where n is the number of pressure steps.
Defining i as the streamtube number and n as the total number of pressuae
steps, the homogeneous density in the ith streamtube is:
1 "i,n (1 - xi n> x
Y i ,n + I 1
'n "n
and the total massflux can be calculated from:
G = [ + yi
P V i-1 i,n i,n
-17-
(3.3-14)
(3.3-15)
From the generated v e l o c i t y p r o f i l e average vapor and l i q u i d v e l o c i t i e s
can be c a l c u l a t e d , from which an e f f e c t i v e s l i p r a t i o can be found:
??
and :
- i=1 ' Yivi ,nx i ,n - V - ' JTiXi,n i=l
I n a d d i t i o n t h e f lowing enthalpy can be found from:
n
i=1 ] + Y h
n n
(3 .3-16)
( 3.3- 17)
( 3.3- 18)
.. (3.3-19)
3 . 4 Calcula t ion of Re la t ive P e r m e a b i l i t i e s
The l i q u i d s a t u r a t i o n a f t e r n p ressure drops can be ca lcu la ted from t h e
s l i p r a t i o and the l i q u i d mass f r a c t i o n ; a f t e r Miller(1951):
(3.4-11
- 1 8 -
where :
( 3 . 4- 2 )
The r a t i o of r e l a t i v e pe rmeab i l i t i e s can then be ca lcu la ted from:
This is equiva len t t o the more common formula derived i n Grant and
Sorey( 1 9 7 9 ) :
(3 .4- 3)
(3 .4- 4)
.-
.
The streamtube model assumes no i n t e r a c t i o n between the streamtubes which
imp l i e s :
k + k = 1 rw rs ( 3.4-5)
Where flow i s con t ro l l ed by a f r a c t u r e system the assumption t h a t t h e
phases do not i n t e r a c t is gene ra l ly thought t o be reasonable.
Using Equation (3.4- 1) the model c a l c u l a t e s the inplace l i q u i d s a t u r a t i o n
and from Equations (3.4- 3) and (3.4- 5) t h e corresponding values of the water
and steam r e l a t i v e pe rmeab i l i t i e s are ca l cu l a t ed .
- 1 9 -
3.5 Computer Program (GEOFLOW)
A computer program has been written to solve the streamtube model
equations and generate the relative permeability values. A listing of the
program with a typical output is included as Appendix B.
Input to the program involves the following variables:
PI: reservoir pressure (MPa.a)
PS : saturation pressure (MF'a.a)
DP : pressure step size (kPa)
N: number of pressure steps
EIE : effective isentropic efficiency, n S
At present it is assumed by the program that the reservoir pressure is
greater than or equal to the saturation pressure but it only requires minor
modifications for GEOFLOW to accept two phase initial conditions. ..
After each pressure/heat transfer step the program prints out the
following variables:
PRESS :
TEMP :
MASS FLUX:
SLIP RATIO:
ENTHALPY:
Yw:
SATW:
KS :
Kw:
pressure (MPa.a)
saturation temperature ( O C )
total mass flux (kg/m s )
effective slip ratio
total flowing enthalpy (kJ/kg)
liquid flowing mass fraction
liquid inplace volume fraction
steam relative permeability
water relative permeability
2
- 20-
To c a l c u l a t e t h e thermodynamic p r o p e r t i e s needed by the program,
subrout ines developed by Reynolds(l979) were used. In the c a l c u l a t i o n of t h e
hea t t r a n s f e r and r e l a t i v e p e r m e a b i l i t i e s values of the s p e c i f i c heat and
dynamic v i s c o s i t y were required. Curve f i t s were developed for these
p r o p e r t i e s , a t s a t u r a t i o n cond i t ions , based on t h e d a t a presented i n
Schmidt( 1969).
Before us ing t h e program t o s tudy t h e system of i n t e r e s t , it w a s t e s t e d
a g a i n s t an example presented i n Wallis and Richter(1978). Using a p ressure
s t e p of 100 kPa, i n i t i a l temperature of 25OoC with no heat t r a n s f e r , GEOFLOW
w a s run and the r e s u l t s compared i n Figures 3 . 4 and 3.5. The s l i g h t
d i f f e r e n c e s are believed t o be due t o the f a c t t h a t d i f f e r e n t thermodynamic
c o r r e l a t i o n s were probably used by Wallis and Richter .
The next s e c t i o n p resen t s the d a t a generated by GEOFLOW f o r a range of
i n p u t condi t ions .
- 21-
3 x 1 0 4
I I WALLIS AND RICHTER (1978)
2 . 5 ~ 1 0 ~ E
x
1 x 1 0 4 0 0.4 0.8 1.2 1.6
PRESSURE DROP, Ap (MPa)
FIGURE 3 , 4 : MASSFLUX VS PRESSURE DROP - COMPARISON WITH WALLIS AND RICHTER -.
8 I
h
3 f) \
Y
a .... J a3
0 I I I 0 ' 0.4 0.0 1.2 1.6
PRESSURE DROP, A p (MPa)
FIGURE 3.5: SLIP RATIO Vs PRESSURE DROP - COMPARISON WITH WALLIS AND RICHTER
-22-
4. FLOW CHARACTERISTICS
Using the computer program, GEOFLOW, theoritical curves of massflux and
enthalpy as functions of the production zone flowing pressure can be
generated. The initial reservoir conditions and the effective isentropic
efficiency were used as input variables. A constant pressure step of 50 kPa
was used for this study.
4.1 Effect of Reservoir Pressure
The effect of reservoir pressure is shown, for reservoir temperatures of
25OoC and 3OO0C, in Figures 4.1 and 4.2. The graphs indicate that within the
two phase region the reservoir pressure has negligible effect on the
calculated massflux. If the reservoir pressure is greater than the saturation
pressure the data can be extrapolated into the single phase region by assuming
that the massflux is zero when the well flowing pressure is equal to the
reservoir pressure. In this case the effect of the reservoir pressure is
important, as shown in Fig~re~4.3.
If the reservoir pressure is greater than the saturation pressure it has
no effect on the increase in flowing enthalpy as heat transfer only occurs
after flashing has started.
These results imply that meaningful comparisons between field and model
data can be made even when the reservoir pressure is not accurately known,
provided it is greater than the saturation pressure. This is important as the
reservoir temperature is normally known more accurately than the initial
reservoir pressure.
- 2 3-
3x104
n 0 > 2 . S X l O ' % D Y Y
Q .) 2x104
X 3 A . LL Q)
1 . 6 ~ 1 0 ~ I
1x104
4x104
3x10'
2x104
1x104
T,=250°C =3.970 MPa.a
'88t
po = 3.978 MPa.a -- po = 7.978 MPa.a
- - / 0.4 0.8 1.2 1.6 c
PRESSURE DROP, A p = pSat-pwf (MPa) 0
FIGURE 4.1: MASSFLUX VS PRESSURE DROP, T0=25o0C
To = 300 O C 0.985
-
- po= 8.593 MPa.a -- po = 12.593 MPa.a -
I I I I 0.6 1 .o 1.6 2.0 2.6
PRESSURE DROP, A p = paat -pWf (MPa)
FIGURE 4.2: MASSFLUX VS PRESSURE DROP, T0=30O0C
-24-
.-
,
4 x 1 0 4
3 x 1 0 4
2 x 1 0 4
1 x 1 0 4
0 4 6 8 10 12 1
FLOWING PRESSURE, p (h4Pa.a) wf
FIGURE 4 . 3 : MASSFLUX VS FLOWING PRESSURE, T0=3000C I
-25-
4.2 Choked Flow
The d a t a generated by GEOFLOW have been p l o t t e d (Figures 4.4 t o 4.9) as
graphs of massflux and enthalpy vs t h e pressure d rop based on the s a t u r a t i o n
p ressure . The choked flow condi t ion is ind ica ted by the dashed h o r i z o n t a l
l i n e s .
The d a t a from GEOFLOW showed t h e massflux inc reas ing as the p ressure drop
inc reased , u n t i l a maximum value w a s reached. After t h a t po in t the c a l c u l a t e d
massflux s t a r t e d t o decl ine . This is a c h a r a c t e r i s t i c of the thermodynamic
models but the phenomena is n o t observed i n p r a c t i c e , Moody(1965). The choked
massflux is taken t o be t h e maximum pred ic ted value; the value where
swap = 0.
The graphs (Figures 4.4 t o 4.9) i n d i c a t e t h a t as the e f f e c t i v e i s e n t r o p i c
e f f i c i e n c y decreases the choked massflux decreases and choking occurs a t a
lower value of pressure drop. This is i n agreement wi th the statement from
Reynolds and Perkins( 1977): "for a given f lowra te the re is a maximum h e a t
i n p u t f o r which the prescibed flow can be passed by the duct . Compressible
f lows t h e r e f o r e e x h i b i t choking due t o heat ing" . The water s a t u r a t i o n a t which
t h i s occurred was found t o be 0.6 - 0.7.
The output from GEOFLOW is i n terms of the massflux and t o conver t t h i s
t o a massflow f o r comparison wi th f i e l d d a t a , the flow area i s required.
Conversely if the model i s being used t o s tudy f i e l d d a t a i t i s p o s s i b l e t o
c a l c u l a t e t h e flow area from t h e r a t i o of the measured massflow t o t h e
c a l c u l a t e d massflux. This procedure is descr ibed i n the next sec t ion .
-26-
3x104
2.5~ lo4
2x104
1 . 5 ~ 1 0 ~
T o = 25OoC
Psat =3.078 MPa.a
'S
18 /--- 0.8
I/ I--- I I / / I
1oool I I I r 0 1 .o 2.0 3.0 Ap (MPa) I
I I I I I I 0.2 0.4 0.6 0.8 1 .o 1.2 1
PRESSURE DROP, A p = pSat -pWf (MPa)'
1 x 1 0 '
FIGURE 4 . 4 : MASSFLUX AND ENTHALPY VS PRESSURE DROP, To=25O0C
-27-
I I
3 x 1 0 4
2 . 5 ~ 1 0 ~
2x104
1 . 5 ~ 1 0 ~
2 0 0 0
n m Y \ 2 1500 Y c c
' 8 0 To= 260°C
Psat =4 .694 MPa.a .905
0 . 2 0 . 5 0 .8
0.92
0.96
0.995 1000' I I
0 1 .o 2.0 3.0 Ap(MPa)
1x104
PRESSURE DROP, Ap=PSat -Pwf (MPa)
FIGURE 4.5: MASSFLUX AND ENTHALPY VS PRESSURE DROP, T0~260'C
-28-
L
4 x 1 0 4
3 x 1 0 4
2x104
1x104
To= 27OoC
S a t = 5 . 5 0 6 MPa.a
- 0.995 7
7 X Y
C
1500
0.2 0 .5 -
0 1 .o 2.0
A P (MPa) 3.0
I I I I I 0
PRESSURE DROP, A p = p - (MPa) 8at 'Wf
FIGURE 4 . 6 : MASSFLUX AND ENTHALPY VS PRESSURE DROP, TO=2700C
-29-
I 1
0
4x104
I I I I 0 0.4 0.8 1.2 1.6 2.0 ' 2.4
3x10'
2x104
1 x 1 0 4
To = 28OoC
b a t d . 4 2 0 MPa.a
- 0 .
- 0.2
n 0 x \ 7 Y Y
c. t
5
2000
1500
0. $95
I 's 0.2 0.5
0 1 .o 2.0 A p,MPa
3.0
PRESSURE DROP, A p = psat -pwr (MPa)
c
FIGURE 4.7: MASSFLUX AND ENTHALPY VS PRESSURE DROP, T0=2800C
-30-
I 1 1
4 x 1 0 4
3 x 1 0 4
2x104
1 x 1 0 4
T o = 29OoC
Psat =7.446 MPa.a
0 3 9 5
2 0 0 0 n
x 7 Y
\
Y - 1500 c
0
1, 0.2 0.5
1 .o 2.0 3 A p,MPa
0.4 0.8 1.2 1.6 2.0 2.4
PRESSURE DROP, A p = pSat -pWf (MPa)
FiGURX 4.8: MASSFLUX AND ENTHALPY VS PRESSURE DROP, T0=2900C '
-31-
4 x 1 0 4
3 x 1 0 4
2x104
1x104
To=3000C 8.593 MPa.a Psat'
2000 0 a x \ 7 Y Y
* 1500 c
0
0.995
I S 0.2 0.5
0.8
0.92 0.96 0.995
1 .o 2.0 A p,MPa
3.0
- 0 I I I I I I
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8
PRESSURE DROP, A p = p - (MPa) sat 'vvf
FIGURE 4 . 9 : MASSFLUX AND ENTHALPY VS PRESSURE DROP, T0=30o0C
-32-
.
4.3 Flow Geometry
One use of the graphs i n Figures 4.4 t o 4.9 i s t o compare f i e l d and m d e l
data t o ob ta in information on the flow geometry as the f l u i d e n t e r s the w e l l .
Knowing the r e s e r v o i r temperature, flowing pressure a t the production zone and
the measured enthalpy it i s poss ib le t o c a l c u l a t e the flow area. By assuming a
f r a c t u r e o r i e n t a t i o n and borehole geometry it is f u r t h e r poss ib le t o estimate
an e f f e c t i v e f r a c t u r e width. In t h i s s tudy a s i n g l e f r a c t u r e perpendicular t o
t h e borehole is assumed, as shown i n Figure 4.10.
The f i r s t s t e p i n e s t ima t ing the flow area is t o select the graph
app l i cab le t o the f i e l d data . Using the enthalpy/pressure drop p l o t t h e
appropr ia te value of the e f f e c t i v e i s e n t r o p i c e f f i c i e n c y is es t imated and the
corresponding massflux i s found from the massflux/pressure drop graph, us ing
t h e e f f e c t i v e i s e n t r o p i c e f f i c i e n c y as a parameter. The flow area is then
c a l c u l a t e d from t h e r a t i o of the measured massflow t o the ca lcu la ted massfluk.
Note t h a t the pressure drop is defined with respec t t o the s a t u r a t i o n p ressure
and not t h e r e s e r v o i r pressure .
Using the f r ac tu re /boreho le geometry shown i n Figure 4.10, an e f f e c t i v e
f r a c t u r e width can be es t imated. The e f f e c t of f r a c t u r e o r i e n t a t i o n i s
included i n t h e e f f e c t i v e i s e n t r o p i c e f f i c i e n c y , hence a hor izon ta l f r a c t u r e
o r i e n t a t i o n should be used i n the es t ima t ion of the e f f e c t i v e f r a c t u r e widthh
I f the flowing pressure and/or t h e flowing enthalpy are unknown, an
approximate e s t ima t ion of t h e flow area can be found using the massflux/
temperature graph of Figure 4.11.
Knowing the flow area i t i s poss ib le t o convert the ca lcu la ted massflux
values t o massflows and compare the ca lcu la ted massflow and enthalpy
r e l a t i o n s h i p s with the f i e l d da ta .
-33-
f W
FIGURE 4.10: FRACTURE/BOREHOLE ORIENTATION USED FOR CALCULATION OF EFFECTIVE FRACTURE WIDTH
- 34-
.
-.
L
4 x 1 0 4
3 x 1 0 4
2 x 1 0 4
1 x 1 0 4
0 - 250 270 290 310
TEMPERATURE, T0(OC)
FIGU~E 4.11: MASSFLUX vs TEMPERATURE, FOR ESTIMATION OF FLOW AREA WHEN FLOWING SURVEYS UNAVAILABLE
-35-
The est imated values of e f f e c t i v e f r a c t u r e width can be compared with
e a r l i e r approximate methods, provided da t a is ava i l ab l e under condi t ions of
both s i n g l e and two phase flow. These methods were proposed by James(1975) and
Bodvarsson(l981) and they relate the pressure drop and massflow t o the
e f f e c t i v e f r a c t u r e width, under condi t ions of s ing le phase incompressible
flow:
James( 1975) :
1.85 0.15 3 pw
Wf = 106Ap d0.85pw
a f t e r Bodvarsson(l981):
2 2
r 2r (2d 2 6 10 Ap pW)wf 3 - ( 5 )w, - - W f d = O
2
where f = f r i c t i o n f a c t o r
(4.3-1)
(4.3-2)
Both de r iva t ions are based on a ho r i zon ta l f i s s u r e of constant th ickness
but James assumes t h a t a l l t he k i n e t i c energy is converted t o s t a t i c p re s su re ,
hence the k i n e t i c energy term is dropped from the equation.
The next s ec t ion cons iders d a t a from fou r geothermal w e l l s and compares
the ca l cu l a t ed flow c h a r a c t e r i s t i c s t o the c h a r a c t e r i s t i c s measuted i n the
f i e l d . E f f ec t ive f r a c t u r e widths have a l s o been ca l cu l a t ed t o see i f
" reasonable" values could be obtained.
- 3 6 -
5. COMPARISON OF FLOW CALCUJATIONS WITH FIELD AND EXPERIMENTAL DATA
.
. ..
GEOFLOW was used to investigate the flow characteristics and flow
geometry of four geothermal wells from fields with widely differing reservoir
conditions. The flow data from Arihara(l974) for two phase steadwater flow an
consolidated cores has also been studied to find the effective flow area of
the core and to compare this with the flow areas obtained from the field data.
The output from GEOFLOW for the four field examples is reproduced as
Appendix C.
5.1 Field Data
A summary of the well
is presented in Table 5.1.
GEOFLOW and where possible
characteristics.
and reservoir conditions for the four wells studi d
The flow characteristics have been calculated usir/g
compared with the measured massflow and enthalpy
7
To obtain the flow characteristics from GEOFLOW, a value for the flowin
pressure opposite the production zone was required and also the correspondin$
enthalpy and massflow measurements. The lowest pressure available was ~
generally used as this corresponded to the highest value of massflow. GEOFLOY
I
was run, using a trial and error technique, until the value of effective
isentropic efficiency gave the required value of enthalpy at the measured
flowing pressure. The flow area was determined from the ratio of measured
massflow to the corresponding calculated massf lux and the effective fracture
width estimated as described in Section 4.2.
Using the calculated flow area, the massflux values were converted to
massflows and plotted as a function of the flowing downhole pressure. The
- 3 7 -
n 0 m m
n z n
n
W m d m 2 N
W
* e: H i kl
h 6
U m
U '4
m Q) N
e
n
? 6
B W
d In U 03
m \o
U '4 N 4
m - m m
0 4 1
3 %-I
d 0 \o N
I In In N
0 ro N 0 N
m In m N 6 H
n a
W N N 0
u N 0
u N N
0 J El
n 0 0 h h
W E kl z s
0 m U
e
E z p1 P
W
E Q
3 0 h
3 a E
0 \o m
0 0 N N
m P
n
2 cl
- Y
3 E H E h;
si m 0 .5
I u u
.
-38-
c a l c u l a t e d e n t h a l p i e s were p l o t t e d i n a similar fashion. Both graphs and the
c r o s s p l o t of enthalpy and massflow could then be compared with the measured
f i e l d da ta .
5.1.1 Well "Utah-State" 14-2, Roosevelt Hot Springs, Utah, USA
Two f low tests have been repor ted on t h i s w e l l ; t h e f i r s t i n May 1978 aad
t h e second i n May 1979. Flowing pressure surveys were conducted a t a number O f
massflows, but problems with t h e flow measuring equipment precluded t h e
measurement of the t o t a l f l u i d enthalpy. The f lowra te measurements are s a i d t o
have an accuracy of f15%, Butz and P loos te r ( l979) . The measured flow d a t a i s
presented i n Table 5.2.
TABLE 5.2: MEASURED FLOW DATA, WELL "UTAH-STATE" 14-2
DATE
May 1978
FLOWING PRESSURE, pwf
(MPa.a)
4.79
(5.99)
6.08
(6.72)
2.59
3.52
(4.22)
6.41
6.90
( ): est imated p ressure
May 1979
MASSFLOW, W
(kg/s 1
57.2
45.0
46.5
32.1
73.1
55.8
63.6
40.9
35.8
-39-
I n t h e c a l c u l a t i o n s us ing GEOFLOW, a value of rl = 0.995 w a s assumed. S
This r e s u l t e d i n the expansion process being v i r t u a l l y i s e n t h a l p i c .
There were two f lowra tes a t which f l a s h i n g occurred i n the r e s e r v o i r and . t h i s da ta was used with the output of GEOFLOW t o c a l c u l a t e the flow area and
e f f e c t i v e f r a c t u r e width. I
Single phase flow d a t a was a l s o a v a i l a b l e i n t h i s w e l l (pwf > psat) and
e f f e c t i v e f r a c t u r e widths were c a l c u l a t e d us ing t h e formulas of James(1975)
and Bodvarsson(1981). A f r i c t i o n f a c t o r of 1.0 was used i n Bodvarsson's
formula. This i s t h e l i m i t i n g value suggested by Smith and Ponder( 1982) f o r
self propped f r a c t u r e s .
The r e s u l t s of t h e c a l c u l a t i o n s f o r e f f e c t i v e f r a c t u r e width are shown in
Table 5.3:
TABLE 5.3: CALCULATED EFFECTIVE FRACTURE WIDTH FOR WELL "UTAH-STATE" 14-2
W G A Pwf
( MPa . a ) (kg/s) (kg/m2s) (m2>
2.59 73.1 27695.76 0.00264
4.22 63.6 22560.94 0.00282
6.08 46.5 0.00505
0.00311
6.90 35.8 0.00470
0.00282
c a l c u l a t e d from GEOFLOW 1
2 c a l c u l a t e d from James(1975)
3 c a l c u l a t e d from Bodvarsson( 1981)
wf
(mm)
3.8l
4.1'
7.32
4.53
6.82
4. i3
-40-
1 0 0
..
80
60
40
20
0
o 1978 FLOW TEST 0 1 9 7 9 FLOW TEST - MODEL ( 9 ,SO. 9 9 5 )
2 4 6 8
FLOWING PRESSURE, p I (MPa.a) W t
I
FIGURE 5.1: MASSFLOW VS FLOWING WWNHOLE PRESSURE, WELL “UTAH-STATE”
’4 ~
14-2
-4 1-
Using the average flow area from GEOFLOW the massflow/flowing pressure
curve was calculated and is compared with the field data in Figure 5.1. The
data has been extrapolated into the single phase region by assuming that the
massflow is zero when the flowing pressure is equal to the reservoir pressure
(when pwf = 9.845 MPa.a).
GEOFLOW predicted that choking would occur when the flowing pressure was
less than 3.44 MPa.a, suggesting that the maximum flowrate available from
"Utah-State" 14-2 would be approximately 75 kg/s.
5.1.2 Well BR-21, Broadlands Geothermal Field, New Zealand
This well has been tested a number of times since it was completed in
June 1970. The latest series of tests were conducted in March/April 1982 as
part of a study on high enthalpy wells, Grant(1982).
Enthalpy and pressure data were available at a single flowrate and this
was used in GEOFLOW to obtain the effective isentropic efficiency and hence,
the flow characteristics. The reservoir pressure is equal to the saturation
pressure, suggesting that the fluid is either saturated water or a two phase
steam/water mixture. GEOFLOW assumes that the fluid is saturated water. If the
inplace fluid is in fact a steam/water mixture, the inplace enthalpy w i l l be
greater than the saturation enthalpy assumed by GEOFLOW, resulting in a higher
value for the effective isentropic efficiency. The effective isentropic
efficiency was found to be 0.58, substantially lower than the value for the
other field examples, suggesting that two phase conditions do in fact exist in
the reservoir. This would also mean that the calculated flow area and
effective fracture width would be maximum values as the calculated massflux
values will be lower than the true values.
-42-
The calculation of the effective fracture width is summarized in Table
5.4. A s single phase flow does not occur in the reservoir the calculation
methods of James(1975) and Bodvarsson( 1981) cannot be used. I
TABLE 5.4: CALCULATED EFFECTIVE FRACTURE WIDTH FOR WELL BR-21
3.51 21.7 16857.47 0.00129 2.0
.I
4
The calculated flow characteristics for massflow and enthalpy, as
functions of the flowing pressure are shown in Figures 5.2 and 5.3. No
reliable measured flow characteristics are available at lower massflows as tbe
well did not stabilise during the flow test, Grant(1982).
Choking was predicted to occur at a flowing pressure of 4.2 MPa.a but 1
I
this is probably a high estimate because of the initial conditions used in the
calculation by GEOFLOW.
-43-
30 n 0
1 I I I 1
c i 10 0
Q) Q) < L
FIGURE 5.2: MASSFLOW VS FLOWING DOWNHOLE PRESSURE, WELL BR-21
1800
A 0, x \ 3 1600 c: Y
2 1400
2 w 1200
\ \ \ \
I I I 0 1 2 3 4
FLOWING PRESSURE, pWf (h4Pa.a)
FIGURE 5.3: ENTHALPY VS FLOWING DOWNHOLE PRESSURE, WELL BR-21
-44-
5.1.3 Well KG-12, Krafla Geothermal Field, Iceland
The Krafla field is a liquid dominated field which produces saturated and
superheated steam in a number of wells. The measured massflows are low, with
KG-12 producing 6 . 7 kg/s but no decrease in massflow is seen as the wells are
back pressured, Stef ansson and Steingrimsson( 1980).
A flowing pressure survey was available from KG-12 and the corresponding
enthalpy was estimated to be 3000 kJ/kg. Using this data, GEOFLOW was found to
fit with an effective isentropic efficiency of 0.95.
The calculation of flow area and effective fracture width is summarised
in Table 5.5:
TABLE 5.5: CALCULATED EFFECTIVE FRACTURE WIDTH FOR WELL KG-12
wf W G A Pwf
(MPa. a) ( W s ) (kg/m2s (m2> (mm>
2.10 6.7 32580.45 0.00021 0.3
Based on the calculated flow area, the flow characteristics were
calculated and are shown in Figures 5.4 and 5.5.
The massf low/f lowing pressure curve indicates that choking occurs when
the flowing pressure is less than 9.6 MPa.a, resulting in a constant massflob
which is independent of the flowing pressure. This is consistent with the
observed well characteristics.
-45-
I 1 1
l - MODEL
I I I
0 4 a 12 16 2 FLOWING PRESSURE, p (MPa.a)
wf F I G U R E 5 .4 : MASSFLOW VS FLOWING DOWNHOLE PRESSURE, WELL KG-12
F I G U R E
3000
2500
2000
1 so0 I I I I 4 8 12 16 i
. FLOWING PRESSURE, P (MPa.a) wf
5.5: ENTHALPY VS FLOWING DOWNHOLE PRESSURE, WELL KG-12
-46-
5.1.4 Well 403, Tungonan Geothermal Field, the Philippines
Flow characteristics for this well are available from a flow test and
from flowing pressure and temperature surveys conducted between August 1980
and Feburary 1981.
The data from these tests is summarized in Table 5.6:
TABLE 5.6: MEASURED FLOW DATA FROM WELL 403
WELLHEAD E'LOWING MASSFLOW ,W
PRESSURE,PWh PRESSURE ,p&
( MPa . a) (MPa . a ) (kg/s)
0.95 30.2
1.26 3.73* 28.8
1.80 26.6
2.46 7.20 22.8
2.58 11.33 9.0
* estimated from flowing temperature survey
ENTHALPY, ht
(kJ/kg)
1440
1400
1370
1330
1270
The saturation water enthalpy at 295OC is 1317 kJ/kg; greater than that
measured at the lowest massflow. This suggests that, although the production
zone at 2000-2200 m is the predominant zone, other lower enthalpy zones do
feed into the well under high wellhead pressure. Unfortunately this is a
common problem when trying to analyze geothermal well behavioiir. This data was
used t o calculate the effective fracture width using James(1975) and
Bodvarsson( 1981) but is not included in the graphs of flow characteristics.
The calculation of fracture width is summarized in Table 5.7. To obtain
the data from GEOFLOW, an effective isentropic efficiency of 0.987 was used.
-47-
TABLE 5.7 : CALCULATED EFFECTIVE FRACTURE WIDTH FOR WELL 403
W G A Pwf
( MPa . a ) (kg/s 1 (kg/m2s) (m2>
3.73 28.8 34088.3 1 0.00084
7.20 22.8 29063.45 0.00078
11.33 9.0 0.00420
0.00240
calculated from GEOFLOW 1
2 calculated from James(1975)
calculated from Bodvarsson( 1981) 3
"f
(=I
1.21
1.11
2 6.1
3.53
The discrepancy between the GEOFLOW and the James/Bodvarsson results, is
probably due to error in the assumed reservoir pressure, This would not affect
the GEOFLOW calculations but does influence the results from James and
Bodvarsson.
Using the average flow area from GEOFLOW, the flow characteristics were
calculated and plotted in Figures 5.6 and 5.7. A crossplot of the enthalpy and
massflow data was also prepared and is compared with the field data in Figure
5.8.
Choking was calculated to occur when the well flowing pressure is less
than 6.15.MPa.a; indicating that the total system massflow would be limited to
approximately 28 kg/s.
-48-
30
- n 0
P Y
\
Y
1500
n
2 1450 \ 7 c
c Y c,
+- 1400 a 4 < I c E 1350
1300
FLOWING PRESSURE, pWf (MPa.a)
FIGURE 5.7: ENTHALPY VS FLOWING WWNHOLE PRESSURE, WELL 403
-49-
1 1
1450
1400
135C
0 WELL 403 - MODEL 7,=0.987
0
0
1300 I I I i I I I 0 10 20 30
MASSFLOW, W ( k g h )
FIGURE 5.8: ENTHALPY/MASSFLOW CROSSPLOT, WELL 403
-50-
5.2 Experimental Data
Experimental work on the flow of f l a s h i n g steamlwater mixtures i n porous
media was one of the aspec t s of Ar ihara ' s ( 1974) resea rch on non- isothermal
f low through consol idated sandstone cores. Seven runs were made; f i v e with a
s y n t h e t i c core and two with a Berea sandstone core. A summary of the core
p r o p e r t i e s is presented i n Table 5.8:
TABLE 5.8: PROPERTIES OF CORES USED BY ARIHARA(1974)
CORE
SYNTHETIC BEREA
Permeabi l i ty , k (md) 100 400
Poros i ty , #I ( X ) 35.9 22.0
Diameter, dc (mm) 50 50
Length, 1 (mm) 597 597
I n a l l cases, except f o r run 3 , hot p ressur ized water was int roduced i n t o
t h e core and allowed t o f l a s h wi th in the core. I n run 3 it appears t h a t some
f l a s h i n g may have occurred before t h e water w a s i n j e c t e d i n t o t h e core.
GEOFLOW w a s used t o analyze t h e d a t a i n o rder t o c a l c u l a t e the e f f e c t i v e I
f low area. An e f f e c t i v e i s e n t r o p i c e f f i c i e n c y of 0.992 was assumed which 1
~
approximated an i s e n t h a l p i c process. The d a t a i s presented i n Table 5.9.
m? The average flow a r e a f o r the s y n t h e t i c core was found t o be 5 x
and f o r t h e Berea core , 2.1 x 10
lower than the flow a reas ca lcu la ted f o r t h e f i e l d examples, suggesting t h a t
the exper imental s e t u p w a s n o t an adequate r e p r e s e n t a t i o n of flow i n a
geothermal system.
m2. These values are orders of magnitude
-51-
n U h
2
i s
Q:
A ;4
VI
ha
VI N
0 0 N d
.-p
U
El
5 M N
W
h m d
VI N d
a 2
d m d
d
N
OD U
Q:
z r(
U 0
x W ro
m
4
U U
0
2 d
In
2
4
? d
N
m m 0 0 In 4
a
U
51
El x
d
0
0 ?
r(
9 r(
W
g
m h
d
m
d
? *o
N m a
d
U I 2 X
m VI m
W U
0
VI VI
d
m a c(
U b
d
U
El 0 4
m N N
a 0 d U
m
d d
U U
51 2
? Y X X 0 h
N VI
a
a ‘4
2 0 0
U I El
2 x
m
In
3 4
In m d
m 2
z 2 . . 0 -
V I d
d d
r o m . .
I- VI ? d
‘0. d
VI ro
VI ? d
b
.-
I
-52-
6. RELATIVE PERMEABILITY FUNCTIONS
The relative permeabilities of steam and water were generated by GEOFLOk
at each pressure step to account for the calculated values of flowing
enthalpy. The data was calculated for a range of input conditions.
6.1 Effect of the Input Variables
It was found that the calculated relative permeability functions were
virtually insensitive to reservoir temperature and effective isentropic
efficiency. This may be due to the changing kinematic viscosity ratio(v /v )
as the flashing occurs. To illustrate how insensitive the relative
permeability functions are to the input variables, values of the steam and
water relative permeabilities at 25OoC and 3OO0C for TI = 0.92 and 0.5 are
plotted in Figure 6.1.
s w
S
The data suggests that it is possible to define a unique set of relative
permeability curves. Using a power law curve fit on the water relative
permeability, the following functions were derived:
0.6 - Sw > 0.4, krw - 'w
0.7 k = Sw rw 0.4 > Sw > 0.2 ,
- 0.77 krw - sw sw < 0.2,
and : k = l - k rs rw
-53-
(6.1-1)
(6.1-2)
(6.1-3)
(6.1-4)
1.0
0.8
0.6
0.4
0.2
0.2 0.4 0.6 0.8 S (VOL. BASIS)
W
I 0 0
FIGURE 6 . 1 : RELATIVE PERMEABILITY CURVES FOR STEAM AND WATER AS GENERATED BY GEOnOW
3
-54-
6.2 Comparison with Corey and X-type Relative Permeability Functions
Bodvarsson,O'Sullivan and Tsang( 1981) studied the sensitivity of
geothermal recovery processes to relative permeability parameters. Their study
considered the Corey and X-type relative permeability functions and included a
study of the effect of the residual water and steam saturations.
For the comparison with the relative permeability curves generated by
GEOFLOW, only the basic Corey and X-type curves were used. These are shown in
Figure 6.2. For the Corey curves a residual water saturation of 0.3 and
residual steam saturation of 0.05 had been assumed.
As mntioned in Section 2.2, the relative permeability functions can be
estimated from output characteristics, in particular the flowing enthalpy. In
the same way the flowing enthalpy can be calculated knowing the relative
permeability functions and the f hid properties:
k rw + hs F) k
ht = Qhw W S
where :
rs k rw t w S
k 1 +v - 2 : -
U U
(6.2-1)
( 6.2- 2)
The relationship between the relat-ve permeabilities and the flowing
enthalpy was studied by Bodvarsson et a1.(1980), for the basic Corey and X-
type curves. They presented their results as a function of the water relative
permeability for the specific example of a 25OoC reservoir. This is reproduceid
in Figure 6.3 along with the corresponding data from GEOFLOW. Bodvarsson et
a1.(1980) considered the Corey and X-type curves to "represent the likely
extremes of what the real relative permeability functions may be" and "it is
- 5 5 -
1 .o
0.8
0.6
0.4
0.2
0 0.2 0.4 0.6 0.8 1 .o WATER SATURATION, S, (VOL. BASIS)
0
FIGURE 6.2: COREY, X-TYE'E AND GEOFLOW RELATIVE PERMEABILITY CTfRVES
Y
- 56-
2800
A
2 2400
Y
2000
1600
1200
T= 2 5 0 OC
I I I I 0
800 0.2 0.4 0.6 0.8 1 .a
RELATIVE PERMEABILITY TO WATER, k,,
FIGURE 6.3: FLOWING ENTHALPY VS WATER RELATIVE PERMEA.BILITY, T0=250*C (AFTER BODVARSSON, O'SULLIVAN AND TSANG, 1980)
-57-
probable that krw/ht values determined from field data will fall within this
zone" (the envelope enclosed by the Corey and X-type curves in Figure 6.3). It
can be seen that the data from GEOFLOW does in fact fall within this envelope.
6 . 3 Comparison with Field Derived Curves
Using production data from the Wairakei geothermal field in New Zealand,
Horne and Ramey(1978) and Shinohara(l978), using slightly different
procedures, derived the relative permeability functions. The main assumption
used in their derivations was that flashing did not occur in the reservoir or
to reflect wellbore. This implies that the wellhead conditions were assumed
the corresponding reservoir conditions.
The relative permeability curves were presented as function of the
flowing water mass fraction and in this form they are unsuitable for use in
geothermal simulators. Unfortunately it is impossible to convert the data to
the inplace water saturation (volume basis) without knowing the slip ratio or
the immobile water saturation. ^I
The relative permeability curves from GEOFLOW are available on a flowing <
water mass fraction basis and can be compared with the curves from Horne and
Ramey(1978) and Shinohara(l978) on this basis, as in Figure 6.4. A reservoir
temperature of 25OoC and effective isentropic efficiency of 0.92 were assumed
for the comparison.
The consistency between the shapes of the curves, particularly at high
water mass fractions indicates that the assumptions used in GEOFLOW give
results in agreement with measured field data from a fractured geothermal
reservoir.
-58-
_ -
8
1 .o
0.8 > I- 4 - - ai w 0.6 z a W Q
>” 0.4 I- < w
I
a 0.2
*\. A \ \
- HORNE AND RAMEY (1978) \ - - AFTER SHINOHARA (1978) - - a - T=250°C. n -=0 .92
\ ‘ .
0 0 0.2 0.4 0.6 0.8
Yw (MASS BASIS)
FIGURE 6 . 4 : HORNE AND RAMEY(1978), SHINOHARA(1978) AND GEOFLOW RELATIVE PERMEABILITY ClTRVES
-59-
6.4 Comparison with Experimental Relative Permeability Curves
An experimental study of steam/water relative permeability was undertaken
by Counsil( 1979), using synthetic cores with an average permeability of 32 md.
The water saturation within the core was measured using a capacitance probe
but due to the low flowrates and radial heat transfer effects, it is believed
that a saturation profile existed normal to the flow direction. The probe
measured the saturation near the axis of the core, which may have been higher
than the average saturation of the cross section.
Counsil(1979) presented three examples of flow data and the derived
relative permeability curves. One of these curves is reproduced as Figure 6.5.
The other two examples have the same functional form but cover lower ranges of
water saturation. The graph in Figure 6.5 shows that the residual water
saturation is high, approximately 50%, while the residual steam saturation is
not well defined, although it is assumed to be zero in this case, The shape of
the curves is similar to the Corey(1954) relative permeability curves for
consolidated porous media.
6.5 Comparison with Relative Permeability Curves for Vugular Cores
There has been some work reported in the literature on the effect of
stratification, Corey and Rathjens(l956), and heterogeneities such as vugs,
Ehrlich( 1971) and Sigmund and McCafferty( 1979), on relative permeability
curves. An example from Sigmund and McCafferty( 1979) is reproduced in Figure
6.6 for water displacing oil in a core from a dolomite reservoir. The core
contained a compact crystalline matrix and vugs of various sizes. Curves for
the other examples in Sigmund and McCafferty(1979), were similar to the Corey-
type curves, suggesting that they were in fact homogeneous or had well
distributed heterogeneities.
-60-
..
1.0
0.8
>.
2 0.6 m a W z U
-
0.4
A
0
I
WATER SATURATION, S,
FIGURE 6.5: EXPERIMENTAL RELATIVE PERMEABILITY CURVES FOR STEAM AND WATER (COUNSIL, 1979)
-6 1-
1.0
0.8 ..
0.6
0.4
0.2
C
PHASE
0.2 0.4 0.6 0.8 1.0
NORMALIZED WETTING PHASE SATURATION
FIGURE 6.6: RELATIVE PERMEABILITY CURVES FOR WGULAR DOLOMITE CORE (SIG?KlND AND I@CAFFERTY, 1979)
..
?
-62-
The shape of the r e l a t i v e permeabi l i ty curves i n Figure 6.6 a re similar
i n shape t o the GEOFLOW r e l a t i v e permeabi l i ty curves, suggesting t h a t a
vugular system where the he t e rogene i t i e s a r e not w e l l d i s t r i b u t e d has similar
flow p r o p e r t i e s t o the system modelled i n GEOFLOW.
The next s ec t ion d iscusses the r e s u l t s obtained from using the GEOFLOW
program t o study the two phase flow of steam and water under simulated
geothermal r e s e r v o i r condi t ions.
..
.
- 6 3 -
7 . DISCUSSION
7.1 Flow Characteristics
One of the aims of this research was to investigate why the flowing
enthalpy increased as a non-linear function of the massflow. It appears that
this may be explained by the concept of choked flow. In the field examples all
the wells exhibited choked flow characteristics at low wellhead pressures but
only in well 403 from the Tungonan geothermal field, the Philippines, was both
enthalpy and massflow data available. Taking into account the errors involved
in the measurement of the enthalpy and massflow and the possibility that more
than one zone could be contributing to the total flow, the agreement between
GEOFLOW and the field data supports the contention that choked flow may cause
this phenomena.
Choking appears to occur when the inplace water saturation is about 0.6-
0.7, but it is not immediately apparent where this occurs in relation to the
wellbore. It is generally found in simulation studies of radial systems, for
example Jonsson(l978),that most of the pressure drop occurs close to the well.
This may suggest that choking occurs near the wellbore and furthermore since
the Krafla wells can produce saturated or superheated steam it suggests that
the choking occurs in the reservoir and not as the fluid enters the wellbore.
This is important as it is generally assumed that choking occurs at an abrupt
change in geometry, such as at the outlet of a pipe discharging to the
atmosphere.
The value of effective isentropic efficiency used to fit the field data
was generally found to be greater than 0.9. This suggests either that limited
heat is being "mined" from the rock or that most of the heat i s lost in
- 6 4 -
..
i r r e v e r s i b l e processes , such as f r i c t i o n . It is poss ib le t h a t a s teady state
s i t u a t i o n develops where the f l a s h i n g f r o n t i s v i r t u a l l y s t a t i o n a r y . Under
t h i s condi t ion the heat contained i n the rock where the f l a s h i n g process i s
occurr ing w i l l be r a p i d l y deple ted and the rock temperature g rad ien t w i l l
approximate the f l u i d temperature g rad ien t . When t h i s s i t u a t i o n develops the
heat t r a n s f e r w i l l be c l o s e t o zero and is r e f l e c t e d by a high e f f e c t i v e
i s e n t r o p i c e f f i c i e n c y .
It appears t h a t the da ta from GEOFLOW can be success fu l ly ex t rapo la ted
i n t o the s i n g l e phase region t o g ive an i n d i c a t i o n of the expected f low
c h a r a c t e r i s t i c s . This is important i n w e l l s where both two phase and s i n g l e
phase flow condi t ions can e x i s t .
7.2 Flow Geometry
An important reason f o r us ing t h e f i e l d d a t a i n t h i s research w a s t o see
i f GEOFLOW could p r e d i c t reasonable values f o r t h e flow area and e f f e c t i v e
f r a c t u r e width. The r e s u l t s ranged from 0.3 - 4.1 mm which do appear t o be
wi th in the expected order of magnitude. The c a l c u l a t i o n method of
Bodvarsson(1981) f o r s i n g l e phase incompressible flow w a s found t o g ive
comparable f r a c t u r e widths when a f r i c t i o n f a c t o r of 1.0 w a s used. James(l975)
formula give c o n s i s t e n t l y high va lues , sugges t ing t h a t James' assumption t h a t
the k i n e t i c energy term w a s n e g l i g i b l e may not be va l id .
The flow areas of 5 x 10 -' m2 and 2.1 x 10 -8 m2 ca lcu la ted from t h e
r e s u l t s of Ar ihara( l974) sugges ts t h a t h i s experiments may not r e f l e c t t h e
s i t u a t i o n i n a geothermal r e s e r v o i r , p a r t i c u l a r l y i n the area c lose t o the
w e l l , where the f l a s h i n g is l i k e l y t o occur. This i s probably due t o the low
permeab i l i t i e s (100-400 md) of the consolidated cores used i n Ar iha ra ' s s tudy.
-65-
7 . 3 Relative Permeability Curves
It was mentioned in Section 2 . 2 that the rational for using porous medium
type relative permeability functions to model flow in fractured reservoirs,
was that heterogeneities should average out if a large enough control volume
could be assumed. However, in a geothermal system it appears that flashing,
and hence two phase flow, occurs close to the wellbore and only over a
relatively short distance. This implies that the use of Corey relative
permeability curves to describe the flow in a fractured geothermal reservoir
will probably give misleading results.
The relative permeability functions measured in vugular cores show
similar properties to the relative permeability curves from GEOFLOW further
suggesting that the functional form of the relative permeability curves for
fractured systems is very different from the basic Corey-type curves.
Experimental data on steam/water relative permeabilities has been
restricted to low permeability consolidated cores and the resulting curves
are, not unexpectedly, found to resemble the Corey curves.
The relative permeability curves generated by GEOFLOW are at the other
extreme; an open fracture with no steam/water interaction. They do, however,
appear to give results that may be closer to reality than either the Corey o$
X-type curves. They also agree reasonably closely with the field derived
curves of Horne and Ramey( 1978) and Shinohara( 1978). Therefore it is
considered that the GEOFLOW curves represent the most appropriate functional
form for steam/water relative permeabilities for fractured geothermal system$.
- 6 6 -
8. CONCLUSIONS
From this study it can be concluded that:
1. Choked flow may occur within a two phase geothermal reservoir,thereby
limiting the ultimate exploitation rate.
2. The choked flow condition occurs when the liquid saturation falls
below 0.6-0.7.
3. The concept of choked flow may explain observed flow characteristic6
such as the enthalpy rise and constant massflow at low wellhead
pressures in two phase geothermal systems.
4 . The streamtube model can be used to estimate values for flow area abd
effective fracture width.
5 . The mining of heat from the rock by the flowing fluid does not appesr
to be a very efficient method of energy recovery from geothermal
systems.
6 . Relative permeability curves for consolidated sandstone may give
misleading information when applied to fractured geothermal
reservoirs.
- 6 7-
7 . Using r e l a t i v e permeabi l i ty curves of the fol lowing form
n k r w = Sw
krs = 1 - k, where n = 0.6 - 0.8 .
may better s imula te energy recovery processes i n f r a c t u r e d geothermal
r e s e r v o i r s than the r e l a t i v e permeabi l i ty funct ions p resen t ly used i n
geothermal r e s e r v o i r s imulat ion.
-
8. The r e l a t i v e permeabi l i ty curves from GEOFLOW are not temperature
dependent and the re fo re r ep resen t a s i n g l e set of curves app l i cab le
t o any geothermal system.
- 6 8-
9. RECOMMENDATIONS FOR FUTURE WORK
-.
5
At present GEOFLOW assumes that the reservoir initially contains either
saturated or compressed water. The field examples indicate that it would be Bn
advantage to modify GEOFLOW to accept two phase initial reservoir conditions.
This could be accomplished by either using the initial water mass fraction or
the inplace fluid enthalpy as additional input parameters.
It would be difficult to modify GEOFLOW beyond considering two phase
initial conditions. If further terms were incorporated in the energy balance
it would require some definition of the system geometry and GEOFLOW would lose
the advantage of being a completely general thermodynamic model. However the
effective isentropic efficency should be analyzed to see what extra
information it can provide about the system. F o r example, in the case of BR-21
the low value of effective isentropic efficiency suggested that the reservoir
was naturally two phase.
A common problem in the analysis of geothermal well behaviowr, is the
existence of multiple production zones. It would therefore be useful to derive
a multiple zone model based on GEOFLOW.
An attempt was made to use the derived relative permeability curves in
the geothermal simulator, GEONZ, described in Horne, Ogbe, Temeng and Ramey
Jnr.(1980). Due to technical problems no useful results were obtained. It is
recommended that this work should be continued and the results compared with
simulations using Corey and X-type relative permeability curves. The
simulations should be based on transient massflow and enthalpy measurements
from field data.
Experimental studies on the relative permeability of steam and water need
-69-
to be continued. However, the experiments should be modified to reflect the
likely flow conditions in a fractured geothermal reservoir. Therefore the
synthetic cores should be constructed so that they adequately represent the
heterogeneities within the reservoir. The size of the experimental apparatus
and the required massflow through the system should also be considered,
particularly where heat transfer effects are likely to be - important.
1 -
c
,
-70-
10. NOMENCLATURE
A
cP
dC
d
f
G
h
hi
ht
krw
kr s
k
1
P
pi Q S
si S*
'rw 'rs sW T
Tf
vi
wf w X
yi Y
Yi
flow area
specific heat at constant pressure wellbore diameter core diameter friction factor total massflux en t ha1 py
enthalpy after ith pressure step total mixture flowing enthalpy permeability
water relative permeability steam relative permeability core length pressure
pressure after ith pressure step heat transferred
entropy
entropy after ith pressure step normalized liquid saturation
residual water saturation residual steam saturation water saturation temperature
fluid temperature
velocity after ith pressure step effective fracture width total massf low
steam mass fraction steam mass fraction in ith streamtube water mass fraction
water mass fraction after ith pressure step
2 m
kJ/kg°C m m
mm
MPa.a MPa .a
kJ/kg kJ/kg°C
kJ/kg°C
OC O C
m/ s mm
- 7 1 -
A
nS P
E
P V
V t
4
difference effective isentropic efficiency density
slip ratio dynamic viscosity kinematic viscosity total mixture kinematic viscosity porosity
SUPERSCRIPTS
t water steam ( 1
* . property after heat transfer step - average value
SUBSCRIPTS
i rn n
0 S
sat W
wf wh
ith streamtube after nth pressure step nth pressure step initial condition steam property at saturation conditions water well flowing (downhole) wellhead property
kg/m3
Pa.s m2/s m2/s
-72-
11. REFERENCES
c
Arihara, N.: "A Study of Non-Isothermal Single and Two-Phase Flow Consolidated Sandstones", Ph.D. Thesis, Petroleum Engineering Stanford University, Stanford, California (November 1974).
Bodvarsson, G.: "Interstitial Fluid Pressure Signal Propagation along Fracture Ladders", Proc., Seventh Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California, 139-141, (15-17 December 1981).
Bodvarsson, G.S., O'Sullivan, M.J. and Tsang, C.F.: "The Sensitivity of Geothermal Reservoir Behaviour to Relative Permeability Parameters", 1
Proc. , Sixth Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California, 224-237, (16-19 December 1980). I Butz, J. and Plooster, M. : "Subsurface Investigations at the Roosevelt ~
KGRA, Utah", Report No. DOE/ET/28389-1, performed under DOE Contract Noi. AS08-77ET28389 by Denver Research Institute, University of Denver, Denver, Colorado, (October 1979).
Corey, A.T. : "The Interrelation Between Gas and Oil Relative Permeabilities", Producers Monthly, 38-41, (November 1954).
Corey, A.T. and Rathjens, C.H. : "Effect of Stratification on Relative Permeability", Trans., AIME, - 207, 358-360, (1956).
Counsil, J.R. : "Steam-Water Relative Permeability", Ph.D. Thesis, Petroleum Engineering Department, Stanford University, Stanford, California (May 1979)
Ehrlich, R.: "Relative Permeability Characteristics of Vugular Cores - Their Measurement and Significance", paper SPE 3553 presented at the 461th Annual Fall Meeting of the Society of Petroleum Engineers, New Orleans,, La. (October 1971).
t Grant, M.A. and Sorey, M.L. : "The Compressibility and Hydraulic Diffusivity of a Water-Steam Flow", Water Resoures Research, 15, 3, 684 686 (June 1979).
-
Grant, M.A., Applied Mathematics Division, New Zealand Department of Scientific and Industrial Research, personal communication (1982).
Horne, R.N. and Ramey Jnr., H.J.: "Steadwater Relative Permeabilities from Production Data", Trans., Geothermal Resources Council Annual Meeting, - 2, 291-293 (July 1978).
~
,
-73-
Horne, R.N., Ogbe, D.O., Temeng, K. and Ramey Jnr., H.J.: "Geothermal Reservoir Engineering Computer Code Comparison and Validation Using the GEONZ Simulator Program", Final Report, performed under DOE Contract No. DE-AC03-80SF11450 by the Petroleum Engineering Department, Stanford University, Stanford, California, (November 1980).
James, R.: "Steam-Water Critical Flow Through Pipes", Proc., Institution of Mechanical Engineers, 176, 26, 741-748 (1962).
James, R. : "Drawdown Test Results Differentiate Between Crack Flow and Porous Bed Permeability", 2nd United Nations Symposium on the Development and Utilization of Geothermal Resources, San Francisco, 3, 1693-1696 (1975).
- J
-
Jonsson, V.: "Simulation of the Krafla Geothermal Field", &port No. LBL- 7076, UC-66a, performed under DOE Contract No. W-7405-ENG-48 by Lawrence Berkeley Laboratory, University of California, Berkeley, California (August 1978).
Miller, F.G. : "Steady Flow of Two-Phase Single-Component Fluids through Porous Media", Trans., AIME, 192, 205-216 (1951). - Moody, F.J.: "Maximum Flow-Rate of a Single-Component Two-Phase Mixture", Trans., ASME, Journal of Heat Transfer, 134-142 (February 1965).
Nathenson, M.: "Flashing Flow in Hot Water Geothermal Wells", Journal qf Research, U.S. Geological Survey, 2, 6, 743-751 (November-December 1974) -
.. PNOC.: unpublished data (1981)
Reynolds, W.C.: Thermodynamic Properties in S.I., Department of Mechanical Engineering, Stanford University, Stanford, California (1979).
.I
Reynolds, W. C. and Perkins, H. C. : Engineering Thermodynamics ( 2nd Edition), McGraw-Hill Book Company, New York (19/ / ) . Rumi, 0.: "Some Considerations on the Flowrate/Pressure Curve of the '
Steam Wells of Larderello", Geothermics, 1, 1, 13-23 (1972).
Ryley, D.J.: "The Mass Discharge of a Geofluid from a Geothermal Reservoir-Well System with Flashing Flow in the Wellbore", Geothermics , - 9, 221-235 (1980).
-
Schmidt, E.: Properties of Steam and Water in S.I. Units, Springer-Verlmg (New York) Inc., R Oldenburg, Munchen (1969).
Shinohara, K. : "Calculation and Use of SteamIWater Relative Permeabilities in Geothermal Reservoirs", M.S. Report, Petroleum Engineering Department, Stanford University, Stanford, California (June 1978).
-74-
Sigmund, P.M. and McCafferty, F.G.: "An Improved Unsteady-State Procedure for Determining the Relative Permeability Characteristics of Heterogeneous Porous Media", Journal of the Society of Petroleum Engineers, - 19, 1, 15-25 (February 1979).
Smith, M.C. and Ponder, G.M. (Editors): "Hot Dry Rock Geothermal Energy Development Program, Annual Report, Fiscal Year 1981", Report No. LA- 9287-HDR, UC-66a, performed under DOE Contract No. W-7405-ENG-36 by the Los Alamos National Laboratory, Los Alamos, New Mexico (April 1982).
Sorey, M.L., Grant, M.A. and Bradford, E.: "Nonlinear Effects in Two Phase Flow to Wells in Geothermal Reservoirs", Water Resources Researchl, - 16, 4, 767-777 (August 1980).
Stefansson, V. and Steingrimsson, B.: "Production Characteristics of Wells Tapping Two Phase Reservoirs at Krafla and Namafjall", Proc., Sixkh Workshop on Geothermal Reservoir Engineering, Stanford University, Stanford, California, 49-59 (16-19 December 1980).
Tsang, C.F.. and Wang, J.S.Y.: "State-of-the-Art of Models for Geothermal Recovery Processes", Journal of Energy Resources Technology, - 103, 12, 291-295 (December 1981).
Wallis, G.B. and Richter. H.J.: "An Isentropic Streamtube Model for ~
Flashing Two Phase VaporlLiquid Flow", Journal of Heat Transfer, - 100, 1~1, 595-600 (November 1978). I
L
-75-
APPENDIX A
TEXT OF PAPER
BY
WALLIS AND RICHTER
-76-
6. B. Wallis H. J. Richter
RvtrMdCngk.rkp --. ~.norr. n n. 03735
An Isentropic Streamtube Model fbr Flashing Two-Phase Vapor-Liquid Flow
Introduction
'fleshing" can occul when liquid flows into a region where the l d pressure is below the saturation pressure corresponding to the liquid temperature. An a result of the depressurization, vapor is formed. If the drop in pressure is large a two-phase flow with considerable vapor content is created. In some applications, such aa a postulated break in the coolant circuit of a pressurized water reactor or in a boiler feedwater system, the downstream pressure can be only a small fraction of the upstream saturation pressure and the discharge rate k limited by choked flow at or near the smallest moas section of the M e . Flashing OCCUR in several etages. If the incoming liquid is subeooled,
the initial stage is the nucleation of the fmt vapor, usually in the form of bubbles. These bubbles grow rapidly and tend to agglomerate, forming continuous regions of vapor that are accelerated more rapidly than the denser liquid. If the void fraction becomes large enough, a vapor core, probably containing some liquid droplets, is likely to de- velop, while the liquid may be displaced to the wall. The development of these successive flow patterns depends on many phenomena in- cluding the initial “nucleation centers” present in the fluid, three dimensional inertial effects that may cause phase separation, trace impurities that inhibit agglomeration, fluid properties that determine rates of interphase heat, ma88 and momentum transfer and so on. Since analysis of these effects in difficult, it is convenient to have available a few self-consistent analyses of certain “limiting casea” that may approximately describe the overall characteristics and may form the basis for more elaborate studies.
This paper presents a new model for the flashing flow of a two-pbase liquid-vapor mixture under the innuence of steep pressure gradients. A method for predicting choked or “eritid” flow is developed. The theory describes an idealized situation in which there are no irre- versible processes. The description is thermodynamically and me- cbanidy consistent and requires no additional assumptions beyond ltraightforward ones of reversible equilibrium flow without mixing, heat transfer or friction across streamlines.
It is not claimed that thii model gives a realistic picture of the de- hile of the flow. However, it provides a useful “ideal caae” for com-
- Contributed by the Heat Transfer Division for publication in the JOURNAL
W HEAT TRANSFER Mrnusaipt received by the Heat T d e r DivLion JUIW 26.1978.
occur. It also appear8 to predict cr i t id flbw r a t a at least as $== ell as parison with practical situations in which several irreversible p
previous theories and avoids some of the barlier conceptual d ficul- ties. t
~ Previous Work
sional, have previously been taken to this critical flow probl
ties and temperatures. 2 Slip Flow. The vapor and liq
velocities. The ratio between these ways, often without taking account
locities are unequal. The first two approaches have been followed about as f d as is
feasible by numerous previous workers [la]. The homog+eous equilibrium model in self-consistent and compatible with as- sumption of reversibilitr, its disadvantage is inaccuracy since t fails to account for differences in behavior between the phases. e slip
of equaI velocity is reIaxed. UsuaUy this appears as a form& for calculating the velocity ratio (U~/U/); for example, Fauske 111 ted it to ( p ~ / p ~ ) ~ / * while Zivi [4] or Moody (21 chose (pr /pr ) l I3 . A y as- sumption about relative motion tends to conflict with the no T on of reversibility (which is often assumed at the same time) since, when phase change occurs, the transferred mass is required to be suddenly accelerated from the liquid velocity to the vapor velocity, presmably by irreverisble friction or mixing. The one-dimensional approach is forced to compromise somewhere and it is apparently impossible to conserve energy, momentum and entropy without introducinq con- cepta such a8 “effective interface velocity” or apparent intedfacid forces that may appear artificial 151.
and may eventually provide more accurate and realistic predic ions.
flow model requires some additional assumption. since the con T L t
v The separated flow model is the subject of much current r
However, at present, proven methods terms,” including both reversible and exkt.
Journal of Heat Transfer NOVEMBER 1978, VOL 100 / 595
The Present Theory The model which we will describe gets around the difiicultiea with
the usual slip flow theory by allowing velocity and thermodynamic state to vary normal to the main flow direction.
The vapor flow is assumed to develop into different streamtubea that are independent of each other. These s t reamtuh form at the liquid-vapor interface (Fig. I). There is no friction, mixing nor heat transfer acroae streamlines, nor is there any impulsive velocity change upon evaporation (or condensation). Flow in each vapor streamtube is isentropic, yet each streamtube is different because it originates from a different point on the liquid-vapor interface (and hence at a different saturation temperatye when pressye changea are present in the flow field). The liquid k assumed to have a uniform velodtymd temperature a single stnamtube and to be in equiliirium with the vapor which contacta it. The preaawe b assumed to be uniform across the cross section normal to the main flow direction. It ia elso aesumed that the flow b sufticiently onedimenaional for the neglect of velocity components perpendicular to the main flow direction.
Saturated Inlet Stagnation Conditionr Assuming saturated liquid at the entrance into a nozzle, the pressure drop by a certain small amount isp will cause the fmt flashing. creating a vapor-liquid mixture. The assumption is now that the fmt vapor formed due to the pressure drop A p wil l flow in a streamtube (which we have arbi- traily located at the centerline of the nozzle). A hxther decrease by another Ap will flash more liquid and form a second streamtube in which initially saturated steam flows, decreasing the amount of liquid assumed to flow along the wall (or indeed anywhere in the nozzle as long as it forms a continuous stream; for example the liquid could flow as a jet down the center of the nozzle, surrounded by the vapor).
The vapor in the center streamtube created in the preceding pressure drop step will expand isentropically as a result of this f'urther pressure drop by Ap. The initially saturated steam will condense partially but the liquid fraction is very small. Therefore this small amount of liquid, probably droplets, will be assumed to have the same velocity as the steam in thii streamtube.
Each discrete drop in pressure will create one new streamtube in which initially saturated steam flows. At the same time the homoge- neous mixtures in each existing streamtube expand k n t r o p i d y M
indicated in the enthalpy-entropy diagram (Fig. 2). If the step Ap b taken very small a continuous expansion and flow field is created. For computation purposes a finite step size is chosen, sufficiently small for it to have negligible effect on the overall result. (With decreaeing step size certain calculation instabilities were observed depending upon the accuracy of the steam tables used in this computer prognrm. This led to some oscillations in the results. However, the predictions of the choked flow condition and the corresponding velocity profile were insensitive to these variations fox Ap d e r than 1 bar. as shown in Fig. 7).
Let us normalize on the basis of unit mass flow rate. Denote the fraction of the total mass flow rate in the ith vapor streamtube, created in the ith Ap step. by yI and the corresponding n o d i liquid flow rate after the ith tlaeh by Y,. Then the ith fleshing "stage" consista of isentropic conversion of a liquid flow rate Yl-t with ve- locity v I - ~ enthalpy hl-i , and entropy al-<, to a liquid rate Y,, with properties uI. hl', and a,', and a vapor flow rate yl, with propertiea ui, h,', and .sl* (see Fig. 3). Mass is conserved if:
h
hi"
& hi'
CJmassflux G, - critical mass flux h' = enthalpy of saturated water h" = enthalpy of saturated steam p - pressure pur = saturation pressure a' = entropy of saturated water a' = entropy of saturated steam
T = temperature v = velocity z = quality IO = initial quality Y = normalized liquid maan flow rate (di-
menaionlees) yi traction of tobl maas flow rate in ith .
streamtube (vapor + droplet& yo = initial moisture content yo = 1 - IO
W = mass flow rate p' = density of saturated water p* * density of saturated steam f = slip ratio
sukeripta 0 = stagnation value i, n = numbers of s t e p
596 / VOL 100, NOVEMBER 1978 Transactions of the A*
urd the velocity
U, [ 2 ( h - h,n) + Uiq'fi (11)
Since the homogeneous density in the ith streamtube is
1 P1.n = (12)
(1 - I i , n ) +% Pn' Cn
The total maes flow per unit overall c r o e e - d o n men ie ob the reciprocal of the sum of the area of all streamtubes, normalized flow as
...
Combining (1) and (2) we may solve for yi:
(4)
Since the thermodynamic properties are known from the prewue dePe, (4) and (5) c ~ l l be used to calculateyi aad ~i in lruccessinw dflaehing. Yi follow from (1).
modynamic identity, An interesting interpretation of (5) is poesible if we use the ther-
hi" - hi' T&i" -ai')
Substituting (4) in (5) and using (6) yields
If Ap is small thin equivalent to
(7)
(8) ap u A ~ = Ah' - T&'= - P'
which is just what would be expected if Bernoulli's equation had been applied to the liquid (a reasonable approach since there is no force besides the presswe that acta on the liquid stream and no reaction from the fleshing vapor since it suffers no finite change in veloci- ty).
Once the vapor is created it expands isentropically with si. the lpecitic entropy of the ith streamtube. equal to si". the vapor specific entropy at the originating pressure (Fig. 4). The initial conditions, the pressure at which the streamtube is created and the flow rate yi are known, therefore the quality, enthalpy, velocity, density and flow men of the streamtube can be calculated as a function of downstream PreCleUre.
For the ith streamtube, created in the ith Ap step. the quality at the nth Ap step downstream is
ai - an'
sn" - en' 1i.n -
The enthalpy is then
h
The criterion for critical flow ir dG
dP - = 0
Le.. the m m flow per unit area ie a maximum. Since the fluid in each streamtube has a different velocity,
veloped in the nozzle.
can be applied starting with a finite velocity equal to 12(p0 t p 3 / ~111'~ a t the onset of flashing.
Tbo-Phane Inlet Conditions. A s imi i approach can be adopted
and liquid velocities at the entrance.
velocities at the n o d e inlet. The calculation procedure is illuetraw on an
diagram in Fig. 5. For the fust pressure drop by a ce Ap it is assumed that the phases have qd velocities.
lowed. I
An Example This calculation procedure will be illustrated by means of an er-
ample. The initial state is chceen as saturated water with zero locity and an entrance preseure of p o = 3.98 W a , corresponding TO = 250"C. The pressure drop step size Ap h 0.1 ma. Fig. 6 sh predicted maes flux versus the pressure drdp. I t can be Been that a maximum is reached at about a pressurt drop of 1.05 MPa. Fig. 7 show the corresponding velocity profile at thii "critical flo " mn- dition for a cylindrical duct and a total flow rate of W * 1 kg 8; two
ma. different predictions are shown for Ap = 0.1 MPa and Ap i:
l i t amamlube
rd streamtube
Journal of Heat Transfer NOVEMBER 1978, VOL 100 f 597
We ab0 calculated average phase velocities at each step, using the defmitiom
n
n. Yisi,m
and deduced an effective slip ratio,
u
(17) U t 8
The nsult b compared with two previous theoria in Fq. 8.
Prediction of Critical MMS Flux Calculations were pursued f i r saturated water expanding from
various stagnation pressures. In Fig. 9 the critical m a s flux G, b plotted versus the stagnation pressure po at the entram to then& The present theory ia compand with the homogeneow theory and two classical slip flow theories. The results obtained from thin theory are between the extremes of homogenous flow and the maximum flux for a slip ratio of the cube root of the demity ratio.
Comparison with Data Fig. 10 show comparison with expcrimenta using saturated wabr
598 I VOL 100, NOVEMBER 1978
w u )
a 5i
-1 4
20s t P W
I 1
20
..
entering a n o d e (Schrock, Starkmnn, et nl. [6]).1 The predictions of the streamtube model seem to give better agreement with the widely d-tered data than the curve plotted in reference 161.
Comparison of the atreamtube model with other experimental re- UrHS from the m e authors [6,7] for a different shaped nazle for raturated BB well 88 eubcooled water entering the nozzle shows good meement (Fig. 11).
Earlier data of Starkman. Schrock, et al. [a] for steam-water mix- turea of different qualities at the n o d e entrance are compared with the etreamtube model in Fig. 12. The agreement is very good for low P-- In the paper by Dei& et al. 191 experiments in nozzles were de-
STAGNATION PRESSURE MPo
Re 11 Carrpvhonkhwnlhhthooryndoxp.rkrwnbby xhrodr, ot u[I(](auturt.d and rUk0d.d w r t r rt hM Into th. noak
Journal of Heat Transfer
acribed for different moisture contents, yo = 1 - I& at the inlet (Fig. 13). The agreement with the preeent thcory k good for low qualities and the data appear to lie between our predictions and the calcula- tions based on the homogeneous equilibrium model.
Even cornparisone with tube data 88 described by Moody show rather good agreement (Fig. 14). Since inertia effeda tend to daminate near critical flow the details of the uprtream flow in the pipe can probably be neglected aa long M the pipe ie not too long. Tbe name f m also shows Moody's theory which uses a. slip ratio qual to (Pi lpl) l l3. In order to obtain these predictions, which are baeed on quality at the point of critical flow, wt varied the "effective idet rtagnation quality" at ea& preseure until choking WM predicted at the desired exit quality.
NOVEMBER 1978, VOL 100 / 599
Conclusions This present model for prediction of choked or critical flow h mora
consistent in its assumptiolrcl than many other modeh and predicts observed critical flow rates competitively. It doea not represent the details of choking reahtically but it can be considered an a certain ideal limit, comparable to the hntropic predictiom of the charac- teristics of compression or expansion machines, which do not give the complete picture either but are very helpful for providing standarda for comparison with actual performance and aa starting points for the development of more elaborate theorien.
Acknowledgment The authorcl gratefully acknowledge the support for thin work fmm
the Electric Power Research Inatitutr, (EPRI), Contract (-44% 2).
References 1 Faurk, H. K. "Critial Two-Pbnw, Steam-Water Flow," ANL-8633.
1963. pp. 7989. 2 Moody, F. J.,"M.ximum Flow Fhte of a Sile Component, h o - P k
3 Levy, S.. "predictioo of W P h m Critical Fbr Rata," ASME JOIJFW~
4 Zivi. S. hi., "l%inution of Study S t a b B h . m Void Fraction by Maao. of thePrinciple ofMinimumEntropyPmduction,"ASME J ~ U R " . OF HEAT TRANS^ 1964, pp. 247-252
5 W&, G. B.. One D I ~ O I U L O M ~ Ttuo-phore Flow. McCraw-Hill, N w York, 1969. pp. 73-80.
6 Schrock V. E, E S. Stukrmn and R A. Brown, "Fl~hing Flow of h- t i d y Subcooled Water in Convergent-Diveqmnt Noel- " ASME p p e r 76- HT-12.
7 Schrock, V. E. E S. Starkmm md R A B r m , ''Fhhw Flow of hi- W y S u b l e d Watu in Convergent-Divergent Noel-" ASME J~URNAL OF HEAT TRANSFER. VOL 99. May, 1977. pp. 283-268.
8 Starlrrmn, E S.. V. E Schrock, K. F. N e m n . md D. J. M.nerly, "Ex- puuion of a Very Low Qwlity h O - P b Fluid Through I Conwrpnt-Di- v e w n t Nuzzle,"JocvMl of h i e Engineeriryl, June, 1964. pp. 247-
9 hi&, M. E, V. 5. Danilin, G. V. Tsiklaurj aad V. K. Sbanin, 3Invati- gation of h e Flow of W& Steam in Axi-Symmetric Lavd Nomica over I Wide Rmga of Moutum coOt.nt," High Temperature 7. V d 2,1969. pp. 2% rx).
Mixture." ASME JOURNAL OF HEAT TRANSPER. 1965, pp. 134442
OF HEAT TRANS- 1965. pp. 53-58.
600 / VOL 100, NOVEMBER 1978 Transactions of the ASYE
8
APPENDIX B
LISTING OF PROGRAM
GEOFLOW
W I T H TYPICAL OUTPUT
- 8 3 -
C C 'C C C C C C C C C C
C C C C C C C C C C
//GEOFLOW J O B // EXEC WATFIV //SYSIN DD * $1.1 AT F I V
STREAM TUBE MODEL TO CALCULATE STEAM/WATER MASS FLOW-RATES ASSUMING ISENTROPIC EXPANSION FOLLOWED BY HEAT TRANSFER AT CONSTANT PRESSURE BASED ON PAPER BY WALLIS, G.B. AND RICHTER, H.J. (1978)
STEAWWATER THERMODYNAMIC PROPERTIES CALCULATED USING SUBROUTINES DEVELOPED BY PROF. W.C.REYNOLDS, MECHANICAL ENGINEERING DEPT.,STANFORD UNIVERSITY
AUTHOR: A.J.MENZIES
IMPLICIT REAL*8 (A-H9O-Z) DIMENSION V I ( 2 0 0 ) ~ Y W ( 2 0 0 ) ~ Y S ~ 2 O O ~ , P ( 2 0 0 ~ ~ C P W ( ~ O O ~ ~ C P S ~ 2 0 0 ~
INPUT VARIABLES
PI - INITIAL PRESSURE P S - SATURATION PRESSURE DP - SIZE OF PRESSURE STEP N - NO. OF PRESSURE STEPS
EIE - ISENTROPIC EFFICIENCY
50 FORMAT(20A4) WRITE(6,60) (TI(J),J=1,20)
READ(5t100) PI,PS,DP,N,EIE FORMAT(ZF7.3, F5.1 , I31 F5.3) WRITE(6,150)
6 0 F O R M A T ( / / / / / / / , 1 9 X , Z O A 4 t / I / )
100
150 FORMAT(19X,'INIT. PRESS.'>lOX,'SATN. PRESS,',lOX#'DELTA P.', SlOX,'ISEN. EFFICIENCY') WRITE(6tl60) PI,PS,DP,EIE
160 FORMAT(ZOX,F7.3,' MPa.at,9X,F7.3,' MPa.a1,7X,F5.1,' kPavs $18X, F5.3 , / / I 1 WRITE(6,200)
200 F O R M A T ( l X , ' P R E S S . ' , l O X , ' T E M P . ' ~ l l X , ' M A S S FLUX'rlZX, C'SLIP RAT10',7X,'ENTHALPY'~9Xp'YW'~9X~'SATWt,llX, C'KS',llX,'KW')
C PI=PI*lD03 PS=PS*lDO3 FHT=1 .-EIE PA=PS 1=1 CALL STEAM(PA,I)
IF(PI.EQ.PS) VI(l)=O. IF(PI.GT.PS) VI(l)=DSQRT(2.*VW(I)*(PI-PS))
YW( 1 1 = 1 .
M=N+ 1 P( 11=PS DO 10 I=2,M P(I)=P(I-l)-DP PA=P(I)
C
C
C I
CALL STEAM(PAp1) C
PB=P(I)/lOOO. PBZ=P B SPB PB3=PB?*PB IF(PB.LT.2.) GO TO 5 IF(PB.LT.7.) GO TO 6
CPW(I)=3.2028+0.5352*PB-O'.O483*PB2+2.4!22D-3*PB3 CPS(I)=-3.0874+2.2944*PB-0.2316*PB2+0~Ol*PB~ GO TO 7
5 CPW(1)=4.2072+0.2236*PB-O.O2319*PB2 CPS(1)=2.0098+0.6689*PB-O.O8314*PB2
C
GO TO 7 6
7
C
8
9
1 1 C
C
C
C
25 22
C
C 30
26
20 C
C
C
C
C
C 40
IF(TD.LT.220.) GO TO 8 IF(TD.LT.290.) GO TO 9
VISW=87.5233+0.3404*TD-l.lO7ZD-3*(TD**Z.) V I S S = 1 0 8 . 0 2 6 3 - 0 . 6 6 1 9 R T D t 1 . 2 2 6 2 D - 3 ~ ~ T D ? + * Z . ~ GO TO 1 1 VISW=629.5949-5.20 18*TD+O. 0 1823*(TD**2. )-2.3066D-5*(TD**3.1 VISS=8.2206+0.03988*TD-1.363636D-5*(TD**2.) GO TO 1 1
QTS=FHT*CPS(I)*(TA(l)-TA(I)) DELSS=QTS/TA(I) QTW=FHT*CPW(I)*(TA(l)-TA(I)) DELSW=QTW/TA(I)
DO 70 J=ZYI IF(1.EQ.J) GO TO 30
GO TO 26
CONTINUE
GW=YW(I)*VW(I)/VI(I) GS=O. vs1=0. vs2=0. vw1=0. vw2=0. HSl=O. HWl=O. DSl=O.
DO 4 0 J=2yI
CONTINUE
V F = I . - ( l . / ( V F F + l . ) ) PWPS=(l./SR)*(VF/(l.-VF~)*(VISW/VISS) P S = l . / ( P W P S + l . ) Pw= 1 .- PS I F ( H A V E . L E . H S ( I 1 ) GO TO 4 1 SR=- 1 . p s = 1 . p w = o .
4 1 P E = P A / 1 0 0 0 . T B T A ( I 1 - 2 7 3 . 1'5 D 0
C
C
C
C
C C SUBROUTINE FOR C A L C U L A T I O N OF C STEAWWATER PROPERTIES C
10 CONTINUE
RETURN END
SUBROUTINE S T E A M ( P A , I )
I M P L I C I T REAL*8 ( A - H t O - 2 ) D I M E N S I O N V W S ( 2 0 0 ) COMMON
$ / A / V W ~ 2 0 0 ~ ~ V S ~ 2 0 0 ~ ~ S W ~ 2 O D ~ ~ S W S ~ Z D O ~ ~ S S ~ Z O O ~ $ /B / HW(200~~HWS(200),HS(ZOO~~TA(ZOO~
C COMMON /GRIT/ R,TC,VC,PC EXTERNAL PHZO,SHZO,DHZO
/ R = 4 6 1 . 5 1 T C Z 6 4 7 . 2 8 6 V C = 1 . / 3 1 7 . 0 P C = 2 2 . 0 8 9 D 6 T = 5 5 0 V = . 0 7 P = P A * l D 0 3 CALL SAT(T,P,DPDT,Z,SHZO) CALL PROP(T,P,V,U,H,S,Z,PHZO) CALL D H 2 0 ( T P D F ) V W ( I ) = l . / D F V S ( 1 1 =v v W s ( I ) = v s ( I ) - v w ( I ) H S ( I ) = H / 1 0 0 0 . HWS(I)=T*VWS(I)*DPDT/lOOO. HW ( I ) =HS ( I 1 -HWS( I )
RETURN END
C******************************~*************************** C
C C C C C C C C C C C C C
THE FOLLOWING ROUTINES ARE GENERAL ROUTINES G I V E N I N T P S I SUBROUTINE PROP(T,P,V,U,H,S,NOP,PH20)
R O U T I N E FOR THERMODYNAMIC P R O P E R T I E S E V A L U A T I O N
NOP DETERMINES THE TWO I N P U T PROPERTIES. T R I A L VALUES FOR T AND V MUST ALWAYS B E PROVIDED. I F N O P = l , ENTER W I T H T,V I F NOP=2, ENTER W I T H T,P, AND T R I A L V I F NOP=3, ENTER W I T H P ,V , AND T R I A L T I F NOP=4, ENTER W I T H ' V P H , AND T R I A L T I F NOP=5, ENTER W I T H T,H, AND T R I A L V I F NOP=6, ENTER W I T H S,V, AND T R I A L T I F NOP=7, ENTER W I T H SpT , AND T R I A L V
' I F NOP=8, ENTER W I T H S,P, AND T R I A L T,V
C IF NOP=9, ENTER WITH H,P, AND TRIAL T,V C IF NOP=lO,ENTER WITH S,H, AND TRIAL T,V C C C C C C C
THE INTERNAL PARAMETERS ERP,ERH, AND ERS CONTROL THE ACCURACY OF P, H, AND S ITERATIONS.
THE USER MUST FILL COMMON BLOCK CRIT WITH THE GAS. CONSTANT R AND THE CRITICAL T,V,P.
C PHZO(T,P,V,U,H,S) IS THE USER'S SUBSTANCE-SPECIFIC C ROUTINE THAT CALCULATES P,U,H,S FOR INPUT T,V. C C AL L QUANTITIES ARE DOUBLE PRECISION. C
IMPLICIT REAL*8 (A-H,O-Z) COMMON /GRIT/ R,TC,VC,PC DATA ERP,ERH,ERS/3*0.0001DO/
C INITIALIZATIONS DT=O. DO KBR=O DVBF=l.ODO VMIN=O . DO VMAX=l.OD30 PMIN=l.OD30 PMAXZO .DO DVSl=Z.ODO*VC DVS2=0.7DO*VC KTR= 1
1 RT=R*T C LOOP POINT
C TEST FOR CONVERGENCE CALL PHZO(T,PX,V,UX,HX,SX)
GO TO ~ 1 0 ~ 2 0 , 2 0 ~ 4 0 ~ 4 0 ~ 6 0 , 6 0 ~ 8 0 ~ 9 0 ~ 1 0 0 ~ , NOP 10 GO T O 700 20 IF (DABS(P-PX).LT.(ERP*P)) GO TO 7 0 0
GO TO 104 40 IF (DABS(H-HX).LT.(ERH*RT)) GO TO 700
GO TO 1 0 4 6 0 IF (DABS(S-SX).LT.(ERS*R)) GO TO 700
GO TO 1 0 4 80 IF ((DABS(S-SX).LT.(ERS*R)).AND.(DABS(P-PX).LT.(ERP*P))) GO TO 700
GO TO 104 90 IF ((DABS(H-HX).LT.(ERH*RT)).AND.(DABS(P-PX).LT.(ERP*P)))
1 0 0 IF ((DABS(S-SX).LT.(ERS*R)).AND.(DABS(H-HX).LT.(ERH*RT)))
104 IF (KTR.GT.20) GO TO 850
1 GO TO 700 GO TO 1 0 4
1 GO TO 700 GO TO 104
C CALCULATE THE NECESSARY PARTIAL DERIVATIVES
C PERTURB T 110 DT=O.OOlDO*T
T 1 =T+DT v1=v CALL PH20(Tl,Pl~Vl,Ul,Hl,Sl) GO TO ~ 8 8 0 ~ 8 8 0 ~ 1 4 0 ~ 1 4 0 ~ 8 8 0 ~ 1 4 0 ~ 8 8 0 ~ 1 2 0 ~ 1 2 0 ~ 1 2 0 ~ ~ NOP
IF (PX.LT.O.DO) GO TO 300 GO T O ~ 8 8 0 ~ 1 2 0 ~ 1 1 0 ~ 1 1 0 ~ 1 2 0 ~ 1 1 0 , 1 2 0 ~ 1 1 0 ~ 1 1 0 ~ 1 1 0 ~ ~ NOP
C ' PERTURB V 120 DV=O.OOlDO*V
V2=V+DV IF (V.LE.VC) DV=-DV
T 2 = T
IF (DPDV.GT.O;DO) GO TO 300 C THE POINT IS GOOD - UPDATE LIMITS
IF ( ( P X . G T . P ) . A N D . ( V . G T . V M I N ) ) VMINZV IF ((PX.LT.P>.AND.(V.LT.VNAX)) VMAXZV
IF (V.EQ.VMAX) PMAXZPX IF (VMIN.GE.VMAX) GO TO 8 4 0 IF ((VFlIN.GT.O.DO).AND.(VMAX.LT.l.OD30)) KBR=l DVBF=l.ODO IF (DPDV.EQ.O.DO) GO TO 226 DV=(P-PX)/DPDV DT=O. DO GO TO 4 0 0
IF (V.EQ.VMIN) PMIN=PX
C DPDV=O AT A GOOD POINT - TREAT BY BRACKETING
226 DVBFZ0.5DO
230 DPDT=(Pl-PX)/DT GO TO 300
DT=(P-PX)/DPDT DV=O .DO GO TO 4 0 0
DT=(H-HX)/DHDT 240 DHDT=(Hl-HX)/DT
DV=O. DO GO TO 4 0 0
DV=(H-HX)/DHDV DT=O. DO GO TO 4 0 0
DT=(S-SX)/DSDT DV=O. DO GO TO 400
DV= ( S-SX I /DSDV DT=O. DO GO TO 400
250 DHDV=(HZ-HX)/DV
260 DSDT=(Sl-SX)/DT
270 DSDV=(SZ-SX)/DV .-
280 DSDT=(Sl-SX)/DT ' DSDV=(SZ-SX)/DV DPDT=CPl-PX)/DT DPDV=(P2-PX)/DV DET=DSDT*DPDV-DPDT*DSDV DT=((S-SX)*DPDV-(P-PX)*DSDV)/DET DV=(DSDT%(P-PX)-DPDT*(S-SX))/DET GO TO 4 0 0
290 DHDT=(Hl-HX)/DT DHDV=(HZ-HX)/DV DPDT=(Pl-PX)/DT DPDV=(PZ-fX)/DV DET=DHDT*DPDV-DPDTSDHDV DT=((H-HX)*DPDV-(P-PX)*DHDV)/DET DV=(DHDT*(P-PX)-DPDT*(H-HX)) /DET GO TO 4 0 0
296 DHDT=(HI-HX)/DT DHDV=(HZ-HX)/DV DSDT=(SI-SX)/DT DSDV=(SZ-SX)/DV
D T = ( ( H - H X ) * D S D V - ( S - S X ) * D H D V ) / D E T DV=(DHDT*(S-SX)-DSDT*(H-HX))/DET GO TO 4 0 0
- DET=DHDT*DSDV-DSDT*DHDV
C SPECIAL TREATMENT FOR NOP=2p DESIGNED TO AVOID BAD ROOTS 300 IF (KBR.EQ.0) GO TO 320
C CALCULATE SLOPE FROM BRACKETING VALUES DPDV=(PMAX-PMIN)/(VMAX-VNIN) VZVMAX
DV=DVBF*(P-PX)/DPDV DT=O. DO DVBF=0.5DO*DVBF GO TO 4 0 0
PX=PMAX
C NOT YET BRACKETED - ALTER V TO SEEK GOOD POINT 320 IF (V.LE.VC) DV=-O.O5DO*V
IF (V.GT.VC) DV=O.ZDO*V IF (VMIN.GT.O.DO) DV=O.ZDO+V IF (VMAX.LT.l.OD30) DV=-0.05DO*V GO TO 4 0 0
C REGULATE THE MAXIMUM CHANGE 4 0 0 DVM=O.PDO*V
IF (V.LT.DVS1) DVM=0.5DO*DVM IF (V.LT.DVS2) DVM=0.5DO+DVM DTM=O.lDO+T IF (NOP.NE.2) GO TO 440
C SPECIAL PRECAUTIONS FOR N O P = 2 IF (KBR.EQ.0) GO TO 4 4 0 VT=V+DV IF ((VT.GE.VMIN).AND.(VT.LE.VNAX)) GO TO 4 4 0
DV=VMIN+(P-PMIN)*(VMAX-VMIN~/(PVAX-PNIN) - V 4 4 0 DVA=DABS(DV)
DTA=DABS(DT) IF (DVA.GT.DVM) DV=DV*DVM/DVA IF (DTA.GT.DTM) DT=DT*DTM/DTA T=T+DT V=V+DV
C BRA C K ET I N G L I Pl I TAT I 0 N
KTR=KTR+ 1 G O T O 1
C NORMAL RETURN 7 0 0 GO TO ( 7 1 0 , 7 2 0 ~ 7 2 0 ~ 7 4 0 ~ 7 4 0 t 7 6 0 ~ 7 6 0 ~ 7 8 0 ~ 7 9 0 ~ 7 9 6 ~ ~ HOP 7 1 0 P = P X
u=ux H=HX s=sx RETURN
7 2 0 U=UX H=HX s=sx RETURN
7 4 0 P = P X u=ux s=sx RETURN
' 7 6 0 P = P X u=ux H=HX RETURN
7 8 0 H=HX u=ux RETURN
7 9 0 S=SX u=ux RETURN
7 9 6 P=PX u=ux RETURN
C ERROR WRITES 8 4 0 W R I T E ( 6 , 8 4 2 ) T tP,V,VMIN,VMAX 8 4 2 FORMAT ( 'OPROP ERROR -. T,P,V,VMIN,VMAX= ' t 5 D 1 5 . 5 )
RETURN 8 8 0 W R I T E ( 6 , 8 8 2 ) 882 FORMAT ( 'OPROGRAM ERROR I N PROP ' )
RETURN 8 5 0 W R I T E ( 6 , 8 5 2 ) NOP,T,P,V,H,S,PXtHX,SX 852 FORMAT ( 'OPROP NOT CONVERGENT FOR NOP = ' , 1 3 /
1 1H ~7X,'T',14X,'P',14X,'V',l4X~'H',l4X,'S'~l4X~'PX'~l3X, 2 ' H X ' , 1 3 X , ' S X ' I l H 9 8 E 1 5 . 5 )
RETURN END
C**********************************************************
C C SATURATION PRESSURE- TEMPERATURE R O U T I N E C C C C C THE I N T E R N A L PARAMETER ERR CONTROLS THE I T E R A T I O N ACCURACY. C
SUBROUTINE SAT(T,PpDPDT,NOPpSHZO)
FOR N O P = l , CALCULATES PSATCT) AND D P I D T ON SAT. L I N E . FOR NOP=2, CALCULATES TSATCP) AND D P I D T ; A T R I A L T I S NEEDED.
c C c
THE USER MUST F I L L COMMON BLOCK C R I T W I T H THE GAS CONSTANT R AND THE C R I T I C A L T p V p P .
c. C SH20(T ,P ,DPDT) I S THE USER 'S S U B S T A N C E- S P E C I F I C R O U T I N E C THAT CALCULATES PPDPDT FOR I N P U T T . C C A L L Q U A N T I T I E S ARE DOUBLE P R E C I S I O N . C
I M P L I C I T R E A L s 8 (A-H,O-Z) COMMON I C R I T I R,TC,VCpPC GO TO (1*2)$ NOP
C S P E C I F I E D T 1 I F ( T . G T . T C ) GO TO 7 0 -
CALL SHZO(T,P,DPDT) RETURN
C S P E C I F I E D P - START W I T H THE T R I A L T 2 IF (P .GT .PC) GO TO 7 4
KTR=O E R R = l . OD-6*P
1 0 I F ( T . G T . T C ) T=TC-O.OOlDO CALL SHZO(T,PX,DPDT) DP=P-PX I F ( D A B S ( D P ) . L T . E R R ) GO TO 20 I F ( K T R . G T . 2 0 ) GO TO 8 0 DT=DP/ DP DT DTAZDABSCDT) DTM=O. lDO*T
IF (DTA.GT.DTM) DT=DT*DTM/DTA T=T+DT KTR=KTR+ 1 G O TO 10
20 RETURN
70 WRITE (6,921 T C ERROR WRITES
RETURN 7 4 WRITE ( 6 , 9 4 1 P
RETURN
RETURN 80 LJRITE (6,901 T,PtDPDT,PX
90 FORMAT ('OSAT NOT CONVERGENT FOR TPP,DPDT,PX=',4D15.5) 92 FORMAT ('OSAT CALLED FOR T=',F6.1,' >TC; GARBAGE RETURN') 94 FORMAT ('OSAT CALLED FOR P=',lPD12.4,' >PC; GARBAGE RETURN')
END C********* THERMODYNAMIC PROPERTIES O F H20, NH3, AND CO2 *** C c - DEVELOPED BY W.C. REYNOLDS, STANFORD UNIVERSITY C PROGRAMS USED FOR "THERMODYNAMIC PROPERTIES I N SI" C C**********THERMODYNAMIC PROPERTIES PACKAGE FOR H20
SUBROUTINE PHZO(T,P,V,U,H,S) IMPLICIT REAL*8 (A-HpO-2) DATA R/461.51DO/ RO=l.ODO/V CALL GH?O(T,CV,UG,SG) CALL QH20(T,RO,TAU,Q,DQDTAU,DQDRO) CO=ROSR*T P=CO*(l.ODO+RO*Q+RO*RO*DQDRO) TDQDT=TAU*DQDTAU U I C 0 *T DQDT +U G S=RO*R*(TDQDT-Q) - R*DLOG(RO)+SG H=U+ P *V RETURN E N D SUBROUTINE QHZO(T,RHO,TAU,Q,DQDT,DQDR)
C CALCULATES Q,DQ/DRHO,DQ/DTAU FOR INPUT TK AND RHO - FULL SI IMPLICIT REAL*8 (A-HsO-2) DIMENSION A( 10,7),JM(10)
DATA TAUP/2.5DO/ DATA R/461.51DO/
DATA JM/4*7 4*2 , 2*7/
DATA T O ~ T A U C , R H O A 1 ~ R H O A J , E ~ A / 1 . D 3 ~ 1 . 5 4 4 9 1 2 D O ~ 6 3 4 1 2.94929370D-02~-1.32139170D-04~ 2.74646320D-07, 2 3.42184310D-13~-2.44500420D-16~ 1.55185350D-19, 3 ~ 4 . 1 0 3 0 8 4 8 0 D ~ 0 1 ~ ~ 4 . 1 6 0 5 8 6 0 0 D ~ 0 4 ~ ~ 5 . 1 9 S 5 8 6 0 0 D ~ 0 3 ~ 4 ~ 3 . 3 3 0 1 9 0 2 0 D ~ 0 8 ~ ~ 1 . 6 2 5 4 6 2 2 0 D ~ 1 1 ~ ~ 1 . 7 7 3 1 0 7 4 0 D ~ 1 3 ~
6 6.83353540D-03,-2.61497510D-05~ 6.53263960D-08, 7 0.00000000D-01, 0.00000000D-01, 0.00000000D-01, 8 ~ 1 . 3 7 4 6 6 1 8 0 D ~ 0 1 ~ ~ 7 . 3 3 9 6 8 4 8 0 D ~ 0 4 ~ ~ 1 . 5 6 4 1 0 4 0 0 D ~ 0 4 ~ 9-9.27342890D-09, 4.31258400D-12, 0.00000000D-01, X 0.00000000D-01, 0.00000000D-01, 6.78749830D-03, 1-6.39724050D-03, 2.64092820D-05,-4.77403740D-O8~ 2 0.00000000D-01, 0.00000000D-01, 0.00000000D-01, 3 1.36873170D-01, 6.45818800D-04~-3.96614010D-03~ 4-2.91424700D-08, 2.95687960D-11, 0.00000000D-01, 5 0.00000000D-01, 0.00000000D-01, 7.98479700D-02, 6-6.90485540D-041 2.74074160D-06~-5.1028070OD-O9~ 7 0.00000000D-01, 0.00000000D-01, 0.00000000D-01,
5 1.37461530D-19, 1.55978360D-22, 3.37311800D-01,
S 1.30412530D-02, 7.15313530D-05/ TAU=TO/T S Q = O . DO SQR=O .DO SQT=O .DO EXAZEXRHO EX=O. DO IF (EXA.LT.70.0DO) EX=DEXP(-EXA)
B = O .DO DB=O. DO IF (J.EQ.1) RHOAZRHOAI IF (J.GT.1) RHOAZRHOAJ Cl=l .OD0 C2ZRHO-RHOA IF (DABS(RHO-RHOA).LT.(l.OD-OS*RHOA)) CZ=O.DO
IF (J.GT.JM(1)) GO TO 10
DO 40 J=1,7
DO 10 I=1,8
B=B+A(I,J)*Cl
.DOil.D3,4.8D-3, -3.60938280D-10, 5.97284870D-24, 7.77791820D-06, 1.27487420D-16,
-2.09888660D-04, -2.61819780D-11, 0.00000000D-01,
-7.25461080D-07, 0.00000000D-01, 1.04017170D-05, 5.63231300D-11, 0.00000000D-01, 1.54530610D-05, 0.00000000D-01, 3.99175700D-04, 3.96360850D-12, 0.00000000D-01,
IF (I.EQ.1) GO TO 4 DIN 1 =I- 1
-.
t
.
DB=DB+A(I,J)*C3*DIMl 4 C3=C1
ct =c 1 *c2 10 CONTINUE
C1=1 TODO CZZRHO DO 14 I=9,10 IF (J.GT.JM(1)) GO TO 14
B=B+C5 DB=DB-E*CS IF (I.EQ.9) GO TO 12 DB=DB+EX*A(I,J)*C3
ct =c l*C2
C5=EXsA(I,J)*Cl
12 C3=C1
DTF=O .DO GO TO 30
C J =2 22 IF (J.GT.2) GO TO 24
TNTCZTAU-TAUC TFZTMTC DTF= 1 . OD0 TNTP=TAU-TAUP IF (DABS(TNTP).LT.(l.OD-S*TAUP)) TMTP=O.DO C 7 T fl T C *T fl T P C8zTNTP C9=TNTC GO TO 30
C J 52 24 TF=C7
DTF=C8+(J-Z)*C9 C7=C7 *TMTP C8=C8*TMTP C9=C9*TMTP
30 SQ=SQ+TF*B SQR=SQR+TF*DB
4 0 SQT=SQT+DTF*B Q=SQ DQ DR=SQ R DQDTZSQT RETURN END SUBROUTINE GHZO(TX,CVtUG,SG) IMPLICIT REAL*8 (A-Hp0-Z) DIMENSION B(61 DATA R,B/46 1.51DO ,4.6D4,10 1
DATA T0/273.16DO/
.249D0,8.3893D-1,-2. 19989D-4, 1 2.46619D-7,-9.7047D-11/ DATA UO,SO/-0.23750207D7,-0 66965776D41
DATA L/O/ IF (L.EQ.0) GO TO 40
1 T=TX 2 DLT=DLOG(T)
T2=TST T3=T2*T T 4 =T 3 *T T5=T4*T T202=0.5DO*T2 T303=T3/3.ODO T404=0.25DOsT4 T505=0.2DO*T5 UG=+B(l)*DLT SG=-D(l)/T+B IF (L.EQ.0) UGZUG-UGO
+B(2)*T+B(3)*T202+B(4)*T303+B(5)*T404+B(6)*T505 (Z)*DLT+B(3)*T+B(4)*T202+B(5)*T303+B(6)*T40+ GO TO 4 2
SGZSG-SGO CV=B(l)/T+B(2)+B(3)*T+B(4)*T2+B(5)*T3+B(6)*T4 RETURN
4 0 T=TO R=R GO TO 2
42 L=t UGO =UG+U 0 SGO =SG+S 0
GO TO 1 END
IMPLICIT REAL*8 (A-H,O-Z) DIMENSION F(8) DIMENSION FA(8) DATA TOK/l.OD3/
SUBROUTINE SHZO(T,P,DPDT)
DATA TPK,TCK,PC,F~338.15DO~647.286DO,Z2.088D6~0.7419242OD3~ 1 -0.?9721000D2,0.1155~360D2,-0.8685635D0,-0.10940980D0, 2 0.43999300D0~-0.2520658OD0~0.521868~D-l~ TK=T S1=0. DO S2=0. DO Cl=l .OD0 CZ=O.OlDO*(TK-TPK) IF (DABS(C2).LT.(l.OD-IO*TPK)) CZ=O.DO C3=1. DO DO 4 I=1,8 S 1 =S 1 +F( I 1 *C 1 IF (I.EQ.1) GO TO 4 SZ=S2+F(I)*C3*(1-1) c3=c3*c2
4 c1=c1*c2 TAUX=TOK*l.OD-O5/TK
. TMTCZTK-TCK \
Z=TAUX*TMTC*Sl DZ=-Z/TK+TAUX*Sl+TAUX*TMTC*S2*0.0lDO EX=DEXP(Z) P=PC*EX DPDT=P*DZ RETURN END SUBROUTINE DH20(T,RF) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION G(8) DATA R H 0 C , G ~ 3 1 7 ~ 0 D 0 ~ 0 . 3 6 7 1 1 2 5 7 D 1 ~ ~ 0 ~ 2 8 5 1 2 3 9 6 D 2 ~ 0 ~ 2 2 2 6 5 2 4 0 D 3 ~
1 ~0.88243852D3~0.20002765D4~~0.26122557D4~0~1829767~D4~ 2 -0.53350520D3/ DATA TCK/647.286DO/ IF (T.EQ.TCK) GO TO 30 OT=l.OD0/3.0DO X=(l.DO-T/TCK)**OT IF (X.LT.1.OD-6) X=O.DO co =x SUM=l. OD0 DO 20 I=1,8 SUM=SUM+G(I)*CO
RHOF=RHOC*SUM GO TO 4 0
30 RHOFZRHOC GO TO 4 0
4 0 RFZRHOF RETURN END
2 0 co=co*x
C END THERMODYNAMIC PROPERTIES PACKAGE FOR H20 SDATA INITIAL RESERVOIR TEMPERATURE = 270 C 005.506005.506050.01000.800
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APPENDIX C
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