STEAM-WATER RELATIVE PERMEABILITY - Stanford · PDF filesteam-water relative permeability a dissertation submitted to the department of petroleum engineering and the committee on graduate

STEAM-WATER RELATIVE PERMEABILITY

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PETROLEUM ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

BY

John Raymond Counsil

May 1979

@ Copyright 1979

John Raymond Counsil

ii

ACKNOWLEDGEMENT

I sincerely appreciate and want to recognize the contributions of

many friends, too numerous to be listed below.

Kathy, my wife, provided unceasing support both at home and in the

lab. Professor H. J. Ramey, Jr., served as my advisor and, with Pro-

fessors w. E. Brigham and M. B. Standing, provided many helpful sugges-

tions. The Reading Committee included Emeritus Professor F. G. Miller

and Professor H. Cinco-Ley. The Oral Examination Committee included

Professor S. S. Marsden, Jr. Dr. Paul Atkinson and Dr. Roland Horne

served as Geothermal Project Managers. Jon Grim, machinist, and Paul

Pettit, technician, helped develop the experimental apparatus. Dr.

H. K. Chen, C. H. Hsieh, Kai Lanz, and Joe Counsil provided the support

necessary to make the capacitance probe useful. Elizabeth Luntzel typed

the manuscript and the figures were drafted by Terri Ramey. Photographs

were taken by Abdurrahman Satman.

This research was carried out under Research Grants GI-34925 and

AER-03490-A03, provided by the National Science Foundation, and Research

Grant DOE-LBL-167-3500, provided by the Department of Energy (Energy

Research and Development Administration) through a subcontract with the

Lawrence Berkeley Laboratory.

iv

ABSTRACT

Steam-water relative permeability curves are required for mathematical

models of two-phase geothermal reservoirs. In this study, drainage steam-

water relative permeabilities were obtained from steady, two-phase, non-

isothermal bench scale flow experiments. Liquid water saturations were

measured along the length of the synthetic sandstone cores using a capaci-

tance probe.

In addition, nitrogen-water relative permeabilities were obtained for

a synthetic sandstone core. These isothermal, unsteady gas-drive experi-

ments were used to determine drainage relative permeabilities and calculated

water saturations. Experiments were conducted at several temperatures and

pressures. It was established that nitrogen-water relative permeability was

not strongly temperature dependent below 300'F at confining pressures less

than 300-500 psig.

A comparison of the steam-water and nitrogen-water relative permea-

bility curves indicated that at high water saturations, the external gas

drive (nitrogen-water) gas relative permeabilitieswerelarger than the

internal gas drive (steam-water) steam relative permeabilities.

V

TABLE OF CONTENTS

ACKNOWLEDGEMENTS . . . . , . . . . . . . . . . . . . . . . . . . . i v

ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . x i i i

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 . LITERATURE SURVEY . . . . . . . . . . . . . . . . . . . . . . 2

2 . 1 D e f i n i t i o n of Steam-Liquid Water Relative Permeabi l i ty . 2

2.2 Steam-Liquid Water Re l a t i ve Permeabi l i ty Curves Cur- r e n t l y Avai lable . . . . . . . . . . . . . . . . . . . 4

2 .2 .1 Experimental Re l a t i ve Permeabi l i ty Curves . . . . 4

2 .2 .2 F i e l d Data Re l a t i ve Permeabi l i ty Curves . . . . . 1 2

2.2.3 Other Re l a t i ve Permeabi l i ty Curves . . . . . . . . 1 4

2.3 Experimental Techniques Used t o Obtain Gas-Liquid Re l a t i ve Permeabi l i ty . . . . . . . . . . . . . . . . . 22

2.3.1 Methods of Experimentally Determining Re l a t i ve Permeabi l i ty . . . . . . . . . . . . . . . . . . 22

2.3.2 Methods of Ca l cu l a t i ng Re l a t i ve Permeabi l i ty f o r Gas o r Liquid Drive Displacement Data . . . . . 22

2.3.3 Fac tors Af f ec t i ng Gas Drive Re l a t i ve Permeabi l i ty Measurements . . . . . . . . . . . . . . . . 23

2.4 Heat Transfe r i n Nonisothermal Flow Experiments . . . . . 28

2.5 The E f f e c t of Frequency on t h e Capacitance of Water- Sa tura ted Porous Media . . . . . . . . . . . . . . . . 28

v i

3 . STATEMENT OF THE PROBLEM . . . . . . . . . . . . . . . . . . . . . 3 1

4 . EXPERIMENTAL APPARATUS . . . . . . . . . . . . . . . . . . . . . . 32

4 . 1 CoreHolde r . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 .2 Liquid Water S a t u r a t i o n Measurement . . . . . . . . . . . . . 35

4.3 A i r Bath . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 .4 Temperature Measurement . . . . . . . . . . . . . . . . . . . 4 1

4.5 Pres su re Measurement . . . . . . . . . . . . . . . . . . . . 4 1

4.6 Pump and Accumulator . . . . . . . . . . . . . . . . . . . . 42

4.7 Electric Furnace and Temperature C o n t r o l l e r . . . . . . . . . 42

4.8 Flowline Sight-Glass . . . . . . . . . . . . . . . . . . . . 43

4 .9 High Pressure Gas-Liquid Sepa ra to r . . . . . . . . . . . . . 43

4.10 PorousMedia . . . . . . . . . . . . . . . . . . . . . . . . 43

4.10.1 Unconsolidated Sand Packs . . . . . . . . . . . . . . 43

4.10.2 Consol idated Sandstone Cores . . . . . . . . . . . . 44

4.10.3 Syn the t i c Consolidated Sandstone Core Mold . . . . . 45

5 . EXPERIMENTAL PROCEDURE . . . . . . . . . . . . . . . . . . . . . . 48

5 . 1 Probe C a l i b r a t i o n . . . . . . . . . . . . . . . . . . . . . . 48

5 .2 Syn the t i c Sandstone Fab r i ca t ion Technique . . . . . . . . . . 49

5 .3 Steam-Water Relative Permeabi l i ty Experiments . . . . . . . . 50

5.4 Nitrogen-Water Relative Permeabi l i ty Experiments . . . . . . 5 1

6 . RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . 53

6 . 1 Capacitance Probe C a l i b r a t i o n . . . . . . . . . . . . . . . . 53

6 . 1 . 1 E f f e c t of Frequency on Probe Response . . . . . . . . 58

6.1.2 E f f e c t of Temperature on Probe Response . . . . . . . 62

6 .1 .3 E f f e c t of Gravi ty Segrega t ion on Probe Response . . . 65

6.1.4 Future Improvements . . . . . . . . . . . . . . . . . 65

v i i

6 . 2 Nonisothermal. Steady. Boi l ing Flow Experiments . . . . . . 66

6 .2 .1 Method of -C a l c u l a t i o n . . . . . . . . . . . . . . . 67

6.2.2 Axial Thermal Conduct iv i ty . . . . . . . . . . . . 69

6 .2 .3 Overa l l Heat Trans fe r C o e f f i c i e n t . . . . . . . . . 70

6.2.4 Water S a t u r a t i o n Measurement . . . . . . . . . . . 73

6.2.5 Run SW1 . . . . . . . . . . . . . . . . . . . . . . 73

6.2.6 KunSW2 . . . . . . . . . . . . . . . . . . . . . . 77

6.2.7 Run SW3 . . . . . . . . . . . . . . . . . . . . . . 85

6.2.8 Comparison of Three Steam-Water Runs . . . . . . . 94

6.2.9 . Gravi ty Segregat ion . . . . . . . . . . . . . . . . 97

6.2.10 C a p i l l a r y End E f f e c t s . . . . . . . . . . . . . . . 97

6.2.11 Confining Pressure Sleeve Gas Product ion . . . . . 97

6.2.12 Fu ture Improvements . . . . . . . . . . . . . . . . 99

6 .3 Isothermal . Unsteady. Ex te rna l Gas Drive Experiments . . . 99

6 .3 .1 C a l c u l a t i o n Procedure . . . . . . . . . . . . . . . 100

6.3.2 F l u i d P r o p e r t i e s and t h e E f f e c t s of Water Vapori- z a t i o n . . . . . . . . . . . . . . . . . . . . . 105

6.3.2.1 E f f e c t of Water Vapor izat ion on Gas Mix- t u r e Viscos i ty . . . . . . . . . . . . . 106

6.3.2.2 E f f e c t of Water Vapor izat ion on Water S a t u r a t i o n . . . . . . . . . . . . . . . 109

6.3.2.3 E f f e c t o f Water Vapor izat ion on Gas Mix- t u r e Volume . . . . . . . . . . . . . . 122

6.3.2.4 Nitrogen S o l u b i l i t y i n Liquid Water . . . 126

6.3.3 Gravi ty Segregat ion . . . . . . . . . . . . . . . . 1 2 9

6.3.4 C a p i l l a r y End E f f e c t s . . . . . . . . . . . . . . . 130

6.3.5 Klinkenberg S l i p . . . . . . . . . . . . . . . . . 130

6.3.6 E f f e c t of Temperature on Absolute Permeabi l i ty . . 132

v i i i

6.3.7 Addi t ional Considerations . . . . . . . . . . . . . . 132

6.3.8 Future Improvements . . . . . . . . . . . . . . . . . 133

6.4 Comparison of I n t e r n a l (Nonisothermal) and Exte rna l (Iso- thermal) Drive Experimental Resu l t s . . . . . . . . . . . . 134

7 . RESULTS AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . 138

7.1 Capacitance Probe C a l i b r a t i o n Conclusions . . . . . . . . . . 138

7.2 Steam-Water Flow Data Conclusions . . . . . . . . . . . . . . 138

7.3 Nitrogen-Water Flow Data Conclusions . . . . . . . . . . . . 139

7.4 Comparison of I n t e r n a l and Exte rna l Drive Flow Data . . . . . 140

8 . NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 1

9 . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

APPENDIX 1: EQUIPMENT MANUFACTURERS AND SUPPLIERS . . . . . . . . . . 154

APPENDIX 2: CAPACITANCE PROBE DETAILS . . . . . . . . . . . . . . . . 157

APPENDIX 3: TABULATED EXPERIMENTAL DATA AND CALCULATIONS . . . . . . . 163

A3.1 Capacitance Probe C a l i b r a t i o n . . . . . . . . . . . 163

A3.2 Nonisothermal. Steady. Steam-Water Flow . . . . . . 163

A3.3 Isothermal. Unsteady. Nitrogen-Water Flow . . . . . 163

ix

LIST OF FIGURES

2-1 WATER-STEAM RELATIVE PERMEABILITY FOR RUN NO. 4 , SYNTHETIC SANDSTONE (ARIHARA, 1 9 7 4 ) . . . . . . . . . . . . . 5

2-2 COMPARISON OF STEAM VAPOR AND LIQUID RELATIVE PERMEABILITIES FOR BOISE AND BEREA SANDSTONE (TRIMBLE AND MENZIES, 1 9 7 5 ) . . 8

2-3 RELATIVE PERMEABILITY TO STEAM AND WATER VS WATER SATURATION FOR A SYNTHETIC CONSOLIDATED SANDSTONE CORE (CHEN, 1 9 7 6 ) . . 10

2-4 Fw AGAINST Fs FROM WYCKOFF & BOTSET AND FROM GRANT ( 1 9 7 7 ) . . 1 3

2-5 STEAM-WATER RELATIVE PERMEABILITIES FROM WAIRAKEI WELL DATA (HORNE, 1 9 7 8 ) . . . . . . . . . . . . . . . . . . . . . . . . 15

2-6 RELATIVE PERMEABILITY VS FLOWING MASS FRACTION FROM WAIRAKEI WELL DATA (SHINOHARA, 1 9 7 8 ) . . . . . . . . . . . . . . . . . 16

2-7 A TYPICAL EXAMPLE OF RELATIVE PERMEABILITY CURVES REFERRING TO A LIQUID AND A GAS (AFTER WYCKOFF AND BOTSET, 1 9 3 6 ; SCHEIDEGGER, 1 9 7 4 ) . . . . . . . . . . . . . . . . . . . . . 1 8

4- 1 SCHEMATIC DIAGRAM OF NONISOTHERMAL, STEAM-WATER FLOW APPARATUS . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4-2 SCHEMATIC DIAGRAM OF ISOTHERMAL, NITROGEN-WATER FLOW APPARATUS . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4-3 CORE HOLDER FOR CONSOLIDATED SANDSTONE CORE . . . . . . . . . 36

4-4 DETAILS OF OUTLET FITTINGS (ARIHARA, 1 9 7 4 ) . . . . . . . . . 37

4-5 COMPARISON OF THE CALIBRATION OF THE PROBE IN DIFFERENT MEDIA AND AT DIFFERENT OPERATING TEMPERATURES (CHEN, 1976 ) . . . . 39

4-6 CORE HOLDER FOR UNCONSOLIDATED SAND PACK (CHEN, 1 9 7 6 ) . . . . 40

4-7 CORE MOLD APPARATUS USED TO FABRICATE SYNTHETIC CONSOLIDATED CORES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6

6-1 WATER SATURATION VS NORMALIZED PROBE SIGNAL IN AN UNCONSOLI- DATED SAND PACK (18-20 MESH) (BEFORE REPAIRING BAKER-TYPE ELECTRONICS) . . . . . . . . . . . . . . . . . . . . . . . . 56

X

6-2 WATER SATURATION VS NORMALIZED PROBE SIGNAL FOR SEVERAL SAND GRAIN SIZES IN UNCONSOLIDATED SAND PACKS . . . . . . . . . . 57

6-3 WATER SATURATION VS NORMALIZED PROBE CAPACITANCE FOR SEVERAL FREQUENCIES IN AN UNCONSOLIDATED SAND PACK (20-30 MESH. 310°F) . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6-4 WATER SATURATION VS NORMALIZED PROBE CAPACITANCE IN AN UN- CONSOLIDATED SAND PACK (20-30 MESH. 305.F. 7.5 MHz) . . . . . 60

6-5 OVERALL HEAT TRANSFER COEFFICIENT VS MASS INJECTION RATE . . 71 6-6 TEMPERATURE VS DISTANCE. RUN SW1 . . . . . . . . . . . . . . 74 6-7 NORMALIZED PROBE SIGNAL VS DISTANCE. RUN SW1 . . . . . . . . 75 6-8 WATER SATURATION VS DISTANCE. RUN SW1 . . . . . . . . . . . . 76 6-9 STEAM-WATER RELATIVE PERMEABILITY VS WATER SATURATION FOR

MEDIUM FLOW RATE. RUN SW1 . . . . . . . . . . . . . . . . . . 80 6-10 TEMPERATURE VS DISTANCE. RUN SW2 . . . . . . . . . . . . . . 81 6-11 NORMALIZED PROBE SIGNAL VS DISTANCEy RUN SW2 82 . . . . . . . . 6-12 WATER SATURATION VS DISTANCE. RUN SW2 . . . . . . . . . . . . 83 6-13 STEAM-WATER RELATIVE PERMEABILITY VS WATER SATURATION FOR LOW

FLOW RATE. RUN SW2 . . . . . . . . . . . . . . . . . . . . . 87

6-14 TEMPERATURE VS DISTANCEy RUN SW3 . . . . . . . . . . . . . . 88 6-15 NORMALIZED PROBE SIGNAL VS DISTANCE. RUN SW3 . . . . . . . . 89 6-16 WATER SATURATION VS DISTANCE. RUN SW3 . . . . . . . . . . . . 90

6-17 STEAM-WATER RELATIVE PERMEABILITY VS WATER SATURATION FOR HIGH FLOW RATE. RUN SW3 . . . . . . . . . . . . . . . . . . . 93

6-18 GAS-WATER DRAINAGE RELATIVE PERMEABILITY VS WATER SATURATION FOR SEVERAL TEMPERATURES . . . . . . . . . . . . . . . . . . 101

6-19 WATER VAPOR CONCENTRATION IN NITROGEN-WATER VAPOR MIXTURE (CALCULATED USING RAOULT'S LAW AND DALTON'S LAW) . . . . . . 107

6-20 SATURATION VS DISTANCE: SCHEMATIC FIGURE DURING DRY GAS IN- JECTION WITH WATER VAPORIZATION AT INLET . . . . . . . . . . 114

6-21 CORRECTION FACTOR REQUIRED TO ESTIMATE NITROGEN-WATER VAPOR MIXTURE VOLUME FROM NITROGEN VOLUME (CALCULATED USING RAOULT'S LAW AND DALTON'S LAW) . . . . . . . . . . . . . . . 123

xi

6-22 GAS-WATER RELATIVE PERMEABILITY VS WATER SATURATION FOR SEVERAL TEMPERATURES USING RESULTS NOT CORRECTED FOR GAS VOLUME EXPANSION DUE TO WATER VAPORIZATION . . . . . . . . . . 125

6-23 GAS-WATER DRAINAGE RELATIVE PERMEABILITY VS WATER SATURATION FOR SEVERAL TEMPERATURES USING: (1) THE VISCOSITY OF NITRO- GEN RATHER THAN A NITROGEN-WATER VAPOR MIXTURE, AND (2) RE- SULTS NOT CORRECTED FOR GAS VOLUME EXPANSION DUE TO WATER VAPORIZATION . . . . . . . . . . . . . . . . . . . . . . . . . 127

82-1 CAPACITANCE PROBE USED TO MEASURE WATER SATURATION (ARI- HARA, 1974) . . . . . . . . . . . . . . . . . . . . . . . . . .158

A2-2 CAPACITANCE PROBE CIRCUIT (ARIHARA, 1974) . . . . . . . . . . . 159 A2-3 DIGITAL TO ANALOG CONVERTER CIRCUIT USED TO RECORD FREQUENCY

(ARIHARA, 1974) . . . . . . . . . . . . . . . . . . . . . . . . 160

A2-4 SCHENATIC DIAGRAM FOR IN-SITU MEASUREMENT OF WATER SATURATION IN STEAM-WATER FLOW IN SYNTHETIC CONSOLIDATED SANDSTONE CORE (CHEN, 1976) . . . . . . . . . . . . . . . . . . . . . . . . . 161

A2-5 SCHEMATIC DIAGRAM OF THE PROBE CIRCUIT (CHEN, 1976) . . . . . . 162

xii

LIST OF TABLES

6-1 RUN SW1 CALCULATIONS FOR FLOWING GAS MASS FRACTION . . . . . 78 6-2 RUN SW1 CALCULATIONS FOR STEAM-WATER RELATIVE PERMEABILITY . 79



6-7 NITROGEN-WATER VAPOR MIXTURE VISCOSITY AT SEVERAL TEMPERA- TURES AND PRESSURES . . . . . . . . . . . . . . . . . . . . 110

6-8 AMOUNT OF WATER VAPORIZATION AT SEVERAL TEMPERATURES AND PRESSURES . . . . . . . . . . . . . . . . . . . . . . . . . 112

6-9 OUTLET END GAS SATURATION CORRECTED FOR WATER VAPORIZATION . 118 A3.1-1 CAPACITANCE PROBE CALIBRATION DATA: BAKER-TYPE ELECTRONICS

BEFORE REPAIR (350'F) . . . . . . . . . . . . . . . . . . . 165 A3.1-2 CAPACITANCE PROBE CALIBRATION DATA: BAKER-TYPE ELECTRONICS

BEFORE REPAIR (300'F) . . . . . . . . . . . . . . . . . . . 166 A3.1-3 CAPACITANCE PROBE CALIBRATION DATA: Q-METER (7.5 MHz) . . . 167 A3.1-4 ... (40 kHz) . . . . . . . . . . . . . . . . . . . . . . . . 168

A3.1-5 ... (750 kHz) . . . . . . . . . . . . . . . . . . . . . . . . 169

A3.1-6 ... (100 kHz) . . . . . . . . . . . . . . . . . . . . . . . . 170

A3.1-7 ... (180 kHz) . . . . . . . . . . . . . . . . . . . . . . . . 171

A3.1-8 ... (14MHz) . . . . . . . . . . . . . . . . . . . . . . . . 172

A3.1-9 ... (7.5 MHz) . . . . . . . . . . . . . . . . . . . . . . . . 173

x i i i

A3.1-10

A3.1-11

A3.1-12

A3.1-13

A3.2-1

A3.2-2

A3.2-3

A3.3-1

A3.3-2

A3.3-3

A3.3-4

A3.3-5

A3.3-6

A3.3-7

A3.3-8

A3.3-9

CAPACITANCE PROBE CALIBRATION DATA: BAKER-TYPE ELECTRONICS (80-170 MESH SAND. UNIT NO . 1) . . . . . . . . . . . . ... (20-30 MESH SAND. UNIT NO . 1) . . . . . . . . . . . ... (20-30 MESH SAND. UNIT NO . 2) . . . . . . . . . . . ... (20-30 MESH SAND. UNIT NO . 2) . . . . . . . . . . . NONISOTHERMAL. STEADY. STEAM-WATER FLOW DATA. RUN SW1

NONISOTHERMAL. STEADY. STEAM-WATER FLOW DATA. RUN SW2

NONISOTHERMAL. STEADY. STEAM-WATER FLOW DATA. RUN SW3

XSOTHERMAL GAS DRIVE DATA. RUN NW1 . . . . . . . . . . ISOTHERMAL. GAS-DRIVE CALCULATIONS FOR GRAPHICAL ANALYSIS. RUNNW1 . . . . . . . . . . . . . . . . . . . . . . . . . ISOTHERMAL. GAS-DRIVE RELATIVE-PERMEABILITY CALCULATIONS. RUNNW1 . . . . . . . . . . . . . . . . . . . . . . . . . ISOTHERMAL GAS-DRIVE DATA. RUN NW2 . . . . . . . . . . . . ISOTHERMAL GAS DRIVE CALCULATIONS FOR GRAPHICAL ANALYSIS. RUNNW2 . . . . . . . . . . . . . . . . . . . . . . . . . ISOTHERMAL GAS DRIVE RELATIVE PERMEABILITY CALCULATIONS. RUNNW2 . . . . . . . . . . . . . . . . . . . . . . . . . ISOTHERMAL GAS DRIVE DATA. RUN NW3 . . . . . . . . . . . . ISOTHERMAL GAS DRIVE CALCULATIONS FOR GRAPHICAL ANALYSIS. RUNNW3 . . . . . . . . . . . . . . . . . . . . . . . . . ISOTHERMAL GAS DRIVE RELATIVE PERMEABILITY CALCULATIONS. RUNNW3 . . . . . . . . . . . . . . . . . . . . . . . . .

174

175

176

177

178

180

182

184

185

187

188

189

191

192

193

195

x iv

ERRATA ( 11/81 ) Steam-Water R e l a t i v e Permeabi l i ty

John R . Counsil Petroleum Engineer ing PhD D i s s e r t a t i o n

S t a n f o r d University (May 1979)

Page

1

1

77

77

103

107

113

Change

1 ine 11 from "Toyoni" t o "Toronyi"

1 ine 16 from "curves have" t o "has"

l ine 17 from " e x t r a p o l a t i o n from" t o " i n t e r p o l a t i o n between''

1 ine 1 8 from " t o " t o "and"

Equation 6- 19 Rewri t e - sg = 1 - s w = - W n v,, c o r e

"b 'w, room

Figure 6-19 t i t l e : from ''law'' t o ' ' law)"

1 a s t 1 i ne rewrite :

X 2 ( L ) i s : J l s!l dx

- 'dry

115 E q u . 6-36 Rewrite r n

115 E q u . 6-37 rewrite denominator

r c

115

115

E q u . 6-39 from "S 92

t I-

119 E q u . 6-50 from "q" t o llqt'l

120 1 a s t 1 ine from " f w " t o " f w 2 "

121 E q u . 6-60 from "(6-60)" t o "(6-60a)" below Equ. 6-60 write

"krg = intercept (6-60b)

125 F i g . 6-22 from "Drainage ,I' t o "Drainage"

136

140

142

l i ne 13 from "lower" t o "higher"

1 ine 4 from "methods" t o "method"

l i ne 1 2 from " k = ef fec t ive permeability t o l iquid" t o "k = ef fec t ive permeability t o steam'' and "kW = ef fec t ive permeability t o 1 i q u i d water"

142 below I l k r l I' write

I' krs = steam re1 a t ive permeabi 1 i ty"

1. INTRODUCTION

Relative permeability curves are important because they are used to

obtain effective permeabilities, which are used in rate equations, such as

Darcy's law. Rate equations are used in mathematical models to calculate

the rate of mass and energy recovery from geothermal reservoirs. Thus

relative permeability curves are a key part of forecasting project energy

recovery and project economics.

The need for and use of relative permeability curves in petroleum

reservoir simulation computer models have been described by Crichlow (1977),

Staggs et al. (1971), and many others.

Steam-water relative permeability curves were used in reservoir simu-

lation or characterization by Donaldson (1967), Toyoni (1974), Atkinson

(1975), Faust and Mercer (1975) , Garg et al. (1975) , Martin (1975) , Mercer

and Faust (1975), Moench (1976) , Brownell et al. (1977), Herkelrath (1977) ,

and Thomas and Pierson (1978).

The important point is that although steam-water relative permeability

curves have never been studied in the laboratory in detail, estimated rela-

tive permeabilities have been used a great deal in reservoir simulation.

The purpose of this research is to establish a set of valid steam-

water relative permeability curves for a drainage process. Distinctive

characteristics of these curves should then be incorporated into the

relative permeability curves used for matching or forecasting geothermal

reservoir performance.

-1-

2, LITERATURE SURVEY

In order to characterize the nature of the problem and the state-of-

the-art, a brief discussion of the following subjects is required: (1)

definition of relative permeability, (2) relative permeability curves

currently used for drainage steam-water flow, (3) experimental techniques

traditionally used to obtain gas-liquid relative permeability, ( 4 ) heat

transfer considerations in nonisothermal bench scale experiments, and (5)

the effect of frequency on the capacitance of water-saturated porous

media ,

2.1 Definition of Steam-Liquid Water Relative Permeability

Amyx, Bass and Whiting (1960), Frick (1962), Craig (1971), Standing

(1975), and others have described relative permeability in the traditional

petroleum engineering sense for immiscible multiphase fluid flow. Rela-

tive permeability is the effective permeability normalized to a specific

base permeability, such as the absolute permeability, Effective permea-

bility is a measure of the conductivity of a rock to a fluid in the pres-

ence of at least one other fluid, and depends on:

a. pore size

b. pore size distribution

c. wettability

d . fluid saturation

e. fluid saturation history (drainage-decreasing wetting phase saturation, imbibition-increasing wetting phase saturation)

-2-

-3-

Effective permeabilities are used in flow rate equations, such as

Darcy's law for linear horizontal flow:

where :

w = weight rate of flow

A = cross-sectional area perpendicular to flow

v = fluid specific volume

1-1 = fluid viscosity

-

dp/dx = pressure gradient in flowing fluid

k = effective permeability

g = gas

R = liquid

The gas effective permeability is equal to k the gas relative rg ,

permeability, multiplied by the appropriate base permeability, which may

be the absolute permeability K. Likewise, kR = krR-K. The fluid used to

measure absolute permeability must be stated. Relative pemeabilities are

often presented in graphical form as functions of fluid saturation.

To understand steam-liquid water flow through porous media, the im-

portant physical processes must be understood. Kruger and Ramey (1973)

and Trimble and Menzie (1975) have discussed some of the important points.

Other facets of two-phase flow, such as liquid holdup, are brought out in

principle by Gould's (1974) discussion of steam-water flow in geothermal

-4-

wellbores, Miller's (1951) discussion of nonisothermal, two-phase boiling

propane flow through a sandpack, and in the Culham et al. (1969) study of

two-phase hydrocarbon flow.

Steam-liquid water flow may differ from oil-gas or oil-water flow

due to the thermodynamic and interfacial characteristics of water-rock

systems. Furthermore, interphase mass transfer occurs between steam and

liquid water resulting in both changing quality and saturation along the

flow path.

For immiscible fluids, it has been observed that only one fluid flows

through a given pore at a time. For steam-water flow, it is not clear

whether dry steam flows through some pores and liquid water through others,

or whether a locally homogeneous wet steam flows at differing liquid and

vapor velocities. It is not clear that fluid interference is similar for

steam-water flow and for immiscible gas-oil, or water-oil flow. The next

section presents the steam-water relative permeability curves currently

available.

2.2 Steam-Liquid Water Relative Permeability Curves Currently Available

In the following sections we will consider available experimental

relative permeability data, data extracted from field performance, and

some miscellaneous sources of information.

2.2.1 Experimental Relative Permeability Curves

Arihara (1974) developed the steam-water relative permeability curve

shown in Fig. 2-1 using equations developed by Miller (1951) for steady,

single-component, two-phase, nonisothermal adiabatic flow:

-5-

E 0

0 0

.- t

Y- I,

L

E W a,

W > t- -I W Q:

- a

FIG. 2-1: WATER-STEAM RELATIVE PERMEABILITY FOR RUN NO. 4 , SYNTHETIC SANDSTONE (ARIHARA, 1974)

-6-

where :

h = enthalpy of inlet fluid w = total mass flow rate (w +w )

f i g

Arihara measured the temperature profile through the boiling region

in his synthetic sandstone core to obtain the fluid properties, enthalpy,

and pressure at each point in the two-phase region. Thermal equilibrium

and no vapor pressure lowering effects were assumed. Although vapor pres-

sure lowering has been studied by Calhoun, Lewis, and Neuman (1949) and

Chicoine, Strobel, and Ramey (1977), no quantitative understanding exists

yet. It does appear that vapor pressure lowering may be significant in

consolidated sandstones at low water saturation. Work in this area is

continuing by Hsieh and Ramey (1978) and Moench and Herkelrath (1978).

With regard to local thermal equilibrium, Atkinson (1977) has dis-

cussed the assumption of uniform, local rock-fluid temperature, and Miller

(1951) and Culham et al. (1969) have discussed local phase equilibrium.

The relative permeabilities and their ratio areeasily calculated

using the above equations. Since Arihara was not able to measure saturation,

-7-

he used his calculated relative permeability ratio and Weinbrandt's (1972)

water-oil permeability ratio vs water saturation curve to obtain a water

saturation for each steam or water relative permeability. The obvious

drawback here is that the oil-water permeability ratio curve may not be

appropriate for the steam-water system.

Trimble and Menzies (1975) developed the curves shown in Fig. 2-2

for Boise and Berea sandstone cores. It was not stated, but Trimble and

Menzies assumed the liquid water and steam velocities were equal when they

determined their water saturations from calculated steam quality. It is

generally expected that gas flows at a higher velocity than liquid in two-

phase flow. Miller (1951) discussed this point in detail. To summarize,

the quality, or gas mass fraction, f, in an element A x at an instant in -

time is:

where S = liq W

pid

- 1

water saturation, fraction pore volume.

In contrast, the quality f = w /w of the two-phase fluid passing a g

point x in unit time is:

f = I -

+ (A)( 2)(:)

-8-

0.8

0.7

0.6

a 0.4

W > I- A W K

- a 0. I

0. z

(

I I I I

0 BOISE CONDITIONED 0 BEREA CONDITIONED

VAPOR PHASE PHASE

5 LIQUID

IO WATER

15 20 SATURATION

25 O/O

FIG. 2-2: COMPARISON OF STEAM VAPOR AND LIQUID RELATIVE PERMEABILITIES FOR BOISE AND BEREA SANDSTONE (TRIMBLE AND MENZIES, 1975)

-9-

Equations 2-7 and 2-8 indicate that f = r only when the gas microscopic velocity u is equal to the liquid velocity u Trimble and Menzies

incorrectly used f in place of r in Eqs. 2-4 and 2-5. g 2 '

Chen (1976) presented the drainage relative permeability curves

shown in Fig. 2-3 for a synthetic sandstone. Chen used the same equations

as Miller and Arihara, except that he obtained water saturation directly

using a capacitance probe. Chen extrapolated limited relative permeability

data using Corey-type equations (X =2): C

where : s * = 'w-'wi w l-swi

(2-9 1

(2-10)

(2-11)

Sw = volumetric liquid water saturation

Swi = irreducible water saturation

The critical gas saturation S was assumed to be zero.

Chen calculated the irreducible water saturation, Swiy for several

k (Sw) values and noted that Swi increased with increasing water satura-

tion and temperature. This led Chen to extrapolate his data as a function

of temperature.

gc

rR

Poston et al. (1970), Weinbrandt et al. (1975), Casse and Ramey

(1976), and Aruna (1976) have demonstrated either increased irreducible

water saturation or decreased permeability to water in sandstones with in-

creasing temperature. Sinnokrot et al. (1971) also concluded that sandstones

-10-

1.0- I I I I I I I I - SWi =0.77; T ~ 3 0 0 ° F

c 0 - 2 -S, i=O.83; Tz315"F .- t 0 3 - Swi=0 .87 ; T = 3 2 4 81 329 OF 0.8 -

Y- L

> Krs I--

- K r w } EXPERIMENTAL DATA

2 0.6 - m x, =2 a 2 DRAINAGE PROCESS

w 0.4 - a

W

nz

-

W > I-

A W [1z

- - < 0.2-

- I I 1 I I I I 0.

0 0.2 0.4 0.6 0.8 I .o WATER SATURATION, fraction

FIG. 2-3: RELATIVE PERMEABILITY TO STEAM AND WATER VS WATER SATURATION FOR A SYNTHETIC CONSOLIDATED SANDSTONE CORE (CHEN, 1976)

-11-

become more water wet at high temperatures. Arihara ( 1 9 7 4 ) and Casse and

Ramey ( 1976 ) concluded that significant (more than 5%) temperature effects

were observed at confining pressures greater than 450 psi. Chen did not

report his confining pressure, although it is known to be less than 400 psi,

The effect of slippage on steam-water relative permeability curves

has not been discussed before, to the author's knowledge. The effect

of gas slippage on gas-liquid relative permeability measurements has

been studied by Estes and Fulton ( 1 9 5 6 ) , Fulton ( 1951 ) , and Rose ( 1 9 4 8 ) . It was demonstrated experimentally that the effect of gas

slippage on the measured effective gas permeability decreased with an

increase in liquid saturation. Rose also showed that no slip correction

is required for the gas relative permeability if the same mean pressure

is used to determine the gas effective permeability and the gas absolute

permeability. Estes and Fulton showed that the slip correction:

~~ ~

k -k c = - x 100 k

g (2-12)

where :

k = gas permeability at mean pressure

k = liquid or slip corrected permeability g 9 Pm

R

was roughly constant at all oil saturations studied (0 < S < 0 . 7 0 ) . Fulton

also observed that k and bk (where b is the Klinkenberg constant) de-

creased with increasing oil saturation.

0

R R

For the case of steam-water relative permeabilities, slip could be

reduced by running experiments at very high pressures, and therefore very

high temperatures. The use of high temperatures poses a severe materials

selection problem for the experimental apparatus.

-12-

The next section presents relative permeability curves obtained from

geothermal field data.

2.2 .2 Field Data Relative Permeability Curves

Grant (1977) presented "permeability reduction factors" obtained

using production data from the fissured Wairakei Field, New Zealand.

Grant was not able to obtain liquid saturations, and presented his re-

sults as steam relative permeability versus water relative permeability,

as shown in the cross-hatched region in Fig. 2- 4.

Grant assumed that the driving pressure gradient and temperature

did not change with producing enthalpy change. He developed the following

two equations with two unknowns:

W (GI p" -=krp"+k - - W 0

(2- 13)

(2- 14)

where w is the 100% liquid water flow rate taken from a graph of log well-

bore discharge rate versus discharge enthalpy. Individual wellbore graphs

were shifted to obtain the best common match. Grant describes his method

as being crude and having a large uncertainty. However, he did feel that

steam-water flow in a fissured medium differed from that in a sandstone

porous medium. It appears that the two phases do not interfere with each

other in a fissured rock as they do in a sandstone-type porous rock. Grant

used the Wyckoff and Botset (Scheidegger, 1957) water-gas curves to draw

his comparison.

0

-13-

I .c

F W

0.5

0 I

I

I \ - \ \ \ \ \ \

0.5

FIG, 2- 4: F AGAINST Fs FROM WYCKOFF AND BOTSET, AND FROM GRANT (1977) W

-14-

Horne (1978) extended the work of Grant by (1) considering wellhead

pressure change with time for each well, and (2) using downhole tempera-

tures and fluid properties. Horne graphed his relative permeabilities

versus a "flowing" water saturation that did not consider the immobile

fluid in the reservoir. Horne expected a low immobile liquid saturation

in the Wairakei data because flow was through fissures. The resulting

curves are shown in Fig. 2-5.

Horne's curves are not relative permeability curves in the tradi-

tional sense. The flowing water saturations are actually flowing liquid

mass fractions, and not resident volumetric "water saturations." As dis-

cussed in Section 2.2.1 in regard to Trimble and Menzies' (1975) experi-

mental data, flowing mass fractions or flowing volumetric fractions are

usually not equal to water saturation because the gas and liquid microscopic

velocities are unequal and unknown.

Shinohara (1978) further refined the methods of Grant (1977) and

Horne (1978), and an example of his results is shown in Fig. 2-6. It is

felt that the work of Grant, Horne, and Shinohara is not yet directly use-

ful in the traditional sense because the relative permeabilities are not

graphed as a function of conventional water saturation, However ,

these results do constitute an interesting method of comparison of relative

permeability data. This work is continuing.

sW'

The next section presents other forms of relative permeability

curves, including the Corey-type equations.

2.2.3 Other Relative Permeability Curves

This section presents the Wyckoff-Botset curve and Corey-type

equations.

I .o

0.8

0.6

0.4

0.2

-15-

I I I I I I I i I

O

0 0

n

STEAM PERMEABILITY

6

I

0.2 0.4 0.6 0.8

WATER SATURATION

1.0

FIG. 2-5: STEAM-WATER RELATIVE PERMEABILITIES FROM W A I W E I WELL DATA (HORNE, 1978)

-16-

3 Y

a, > .- t

-4-

++

0.4 0.6 0.8 I

Flowing Water Mass Fraction

FIG. 2-6: RELATIVE PERMEABILITY VS FLOWING MASS FRACTION FROM WAIRAKEI WELL DATA (SHINOHARA, 1978)

-17-

Donaldson (1967) and, it appears, Martin (1975) used the Wyckoff

and Botset(Scheidegger, 1974) CO gas-water curve shown in Fig. 2-7 to

represent two-phase flow in geothermal reservoirs. 2

Toyone (1974) , Atkinson (1975) , Faust and Mercer (1975) , Garg et

al. (1975), Mercer and Faust (1975), Moench (1976) , Brownell et al. (1977) ,

Herkelrath (1977), and Thomas and Pierson (1978) used variations of Corey's

(1954, 1977) equations to characterize steam-water relative permeability.

Corey used a pore size distribution index of x = 2 in his two-phase drain-

age relative permeability equations. A value of X = 2 represents a wide

range of pore sizes.

C

C

Atkinson (1975) adapted the Corey equations for a drainage process

from Corey, Rathjens, Henderson, and Wyllie (1956) as:

k = K[l-(SR*) 2 ] [l-SR*] 2 (2-15) g

where :

kR = K[SR*] 4

s -s R Rr s * = R l-Sgc-str

(2-16)

(2-17)

'Rr

S = critical gas saturation, a linear function of temperature

K = absolute permeability, a linear function of temperature

= residual liquid saturation, a linear function of temperature

gc

Actually, the temperature-independent equations reduce to the Corey

equations only for the case of S = 0 and S = 0. The proper expressions

for the Corey equations are presented clearly by Corey (1954) and less

clearly by Corey et al. (1956):

Rr gc

-18-

W z a W Q

PERCENT LIQUID SATURATION

FIG. 2-7: A TYPICAL E W L E OF RELATIVE PERMEABILITY CURVES REFERRING TO A LIQUID (k ) AND A GAS (k ) (AFTER WYCKOFF AND BOTSET, 1936; SCHEIDEGEER, 1974)

-19-

when (l-S ) SR > S gc Rr

(2-18)

when 1 > SR > ’Rr (2-19)

The relative permeability curves are normalized to give R = 1 when g

SR - - Sir. R = 0 when S = R = 1 when S = 1, and R = 0 when S -

For the oil, water, and/or gas systems that Corey studied, it was also

understood that R = 0 when S > S > 0, RR = 0 when SRr > SR > 0, and that

g g sgc’ R R R R -

g gc g R = k = k /K only when SRr = 0 or K = k g rg g g@SR r.

Corey (1956), Johnson (1968), and Standing (1975) discuss techniques

of estimating S and S from gas-oil relative permeability data. Standing

(1975) also presented a general relationship for ._B :

Rr gc @‘fir K - = 1.08 - 1.11 SRr - 0.73(SRr) @’Rr 2

K

when 0.2 < S < 0.5 Rr (2-20)

Geothermal systems are somewhat different from oil/gas/water systems

in that residual (nonflowing) liquid water may vaporize and flow as steam

(Kruger and Ramey, 1974). Thus, R would increase over the unit R @ S

value as the liquid saturation decreased below the S value. Corey’s

equations also assume that the fluids do not interact with the rock. This

g g Rr

Rr

-20-

may not be a good assumption for hydrothermal systems with/without dis-

solved salts. Since no measured steam-water relative permeability curves

have been presented in the literature to date, it cannot be demonstrated

that Corey's equations do or do not properly characterize steam-water flow.

Garg, Pritchett, and Brownell (1975) treated modified Corey equa-

tions (Corey, 1956) in a way similar to Atkinson (1975). However, Garg

et al. also assumed R = 1 when SR < SRr and RR = 1 when S > (l-S ) . Although the above formulation may accurately describe some gas

g R cg

relative permeability curves, it would be expected that R would increase

beyond unity as the liquid vaporizes and interferes less with the total g

gas flow. Using the same logic, R should increase as the gas saturation R

decreases from S to zero. According to the proper Corey equations,

should be unity when S = 0, not when S = S

cg RR

g g gc' Faust and Mercer (1975) and Toyoni (1974) used modified Corey equa-

tions that were in agreement with Corey (1954) when SRr = S (= 0.05>, and gc

k @ Sir = K. rg

Moench (1976) studied steam transport in vapor-dominated systems

using :

2 2 k = (1-Si) ( l - S i ) rg

(2-21)

This expression agrees with Corey's equation when S - = 0. gc - 'Rr

Herkelrath (1977) treated steam-water flow in a hypothetical frac-

tured material using:

3 krR = (SR)

k = l-SR rg

(2-22)

(2-23)

-21-

This is the same as the Corey-type equations with X = 03 and C

'gc = 'Rr

represents an unconsolidated sand or a uniform pore size in a drainage

process. Herkelrath's study dealt with the "heat-pipe" effect in a vapor-

dominated system; that is, very high heat transport caused by convection

of condensible vapors.

= 0. From Standing (1975) and Frick (1962), a value of Xc = 03

Thomas and Pierson (1978) used:

s -s -s 4

rw = gc] Rr gc

(2-24)

tr = 0.3; S = 0.05 (2-25) gc

These equations reduce t o the proper Corey equations when S = 0 and

kg @ SRr = K. Notice that for SR = Str = 0.3 and S = 0.05, k = 1.15. gc

gc rg It is not clear why this formulation for relative permeability was used.

Relative permeability curves used in oil thermal recovery simulators

usually consider oil, water, and steam plus gas, and therefore are not

directly applicable to a geothermal system. For the case of zero oil satu-

ration, Crookston, Culham, and Chen (1977) use:

k = Sg 3 [2-Sg] rg

s -s 4

rw

(2-26)

(2-27)

(2-28)

-22-

These equations reduce to Corey's equations when S = S = 0 and

= K. The dominant process modeled here would be steam conden- wr gc

kg @ swc

sation, which is an imbibition process. Corey's equations are for

drainage processes.

The next section presents the experimental techniques used to obtain

gas-liquid relative permeability.

2 . 3 Experimental Techniques Used to Obtain Gas-Liquid Relative Permea-

bility

In the following sections we will briefly discuss experimental

methods of determining relative permeability. The gas-drive method will

be covered in detail.

2 . 3 . 1 Methods of Experimentally Determining Relative Permeability

Osoba, Richardson, Kerver, Hafford, and Blair (1951) described and

evaluated five methods of measuring relative permeability in the labora-

tory. The five methods were: (1) Penn State, (2) single core dynamic,

(3 ) gas drive, ( 4 ) stationary liquid, and (5) Hassler.

For determining gas-liquid relative permeabilities, the gas drive

technique is often used because it is rapid and reliable.

2 . 3 . 2 Methods of Calculating Relative Permeability from Gas or Liquid

Drive Displacement Data

Welge (1952) was the first to modify the Buckley-Leverett theory

and present the equations required to calculate (relative) permeability

ratios from linear displacement data. Johnson, Bossler, and Naumann (1959)

later extended this theory to allow the calculation of individual relative

permeabilities. The base permeability was the pre-drive effective

-23-

permeability at the initial wetting phase saturation. Jones and Roszelle

(1976) then presented a simplified graphical technique that yielded indi-

vidual relative permeabilities with the absolute (brine) permeability as

a base.

The next section presents some of the factors that influence gas-

drive relative permeability measurements.

2.3.3 Factors Affecting Gas-Drive Relative Permeability Measurements

Osoba et al. (1951) described several techniques of reducing capil-

lary end effects. One method was to use high rates of flow without exceed-

ing the darcy flow limitation. If end effects were not reduced, the

calculated wetting phase effective permeability was too low. Hysteresis

effects caused by drainage-imbibition were also discussed.

Geffen, Owens, Parrish, and Morse (1951) described reduction of end

effects by increasing the pressure gradient across the core. The expan-

sion of gas along the length of the core increases the gas saturation

while the capillary end effect increases the liquid saturation. This gas

expansion effect can be reduced by using pressure drops small compared to

a high average pressure.

Welge (1952) stated that gas (at nearly constant pressure) dis-

placing liquid may be considered an immiscible displacement if the concen-

tration of the gas in the liquid is constant. If the pressure is nearly

constant with respect to space and time, changes in gas density and solu-

bility are negligible.

With regard to scaling, Rapoport and Leas (1953) found that for an

oil-water system defined by its viscosity ratio, interfacial tension, and

contact angle, a critical scaling factor LVu (core length, injected fluid w

- 24-

velocity, injected fluid viscosity) can be evaluated. Linear floods per-

formed at scaling factors larger than the critical value yield identical

recovery VS injection curves and were independent of rate, length, and

capillary end effects. However, recovery did vary with viscosity ratio

changes. Values of the scaling coefficient required to reach stabilized

flow appeared to increase with permeability. For the materials tested, the

critical scaling coefficient was between 0.5 and 3.5 (cp-cm 2 )/min. In

addition, Rapoport demonstrated that most practical field operations were

operated under stabilized conditions.

It has been suggested that the relative permeability curve for a

solution gas drive case may differ from that for an external gas drive

due to the different saturation distributions that exist at a given average

fluid saturation. However, Stewart, Craig, and Morse (1953) found that

permeability ratios (Welge's method) determined for solution gas drive

and external gas drive were: (1) identical for sandstones, and (2) simi-

lar for limestones with intergranular sandstone-type porosity. Stewart

et al. also found that for 6-11 in length intergranular limestone cores of

2-300 md, a pressure differential of 20 psi was sufficient to reduce capil-

lary end effects.

Owens, Parrish, and Lamoreaux (1956), in a paper written before

Johnson et al. (1959), evaluated the gas drive method for determining

(relative) permeability ratios. They showed that gas drive and steady-

state gas-oil relative permeability tests were in good agreement on homo-

geneous samples. A wide variation resulted for non-uniform samples. It

was also concluded that the magnitude of pressure differential required

to minimize end effects varied inversely with the core permeability. The

length of the "stabilized zone" was inversely related to injection rate.

-25-

Therefore, to reduce the effects of the "stabilized zone" and "end effects , I 1

it was determined that ''the pressure differential should be of such magni-

tude that a volume of gas approximately equal to one-half the pore volume

of the test sampleshouldbe produced at the downstream pressure conditions

in a time interval of 60 seconds." Owens et al. (1956) also concluded

that, contrary to common belief, gas expansion did not influence gas drive

data using Welge's method. Thus, high static pressures were not required.

Corey and Rathjens (1956) studied the effect of stratification on

relative permeability measurements made on laboratory cores. When flow was

parallel to the bedding planes, the critical gas saturation was low (close

to zero) and S values often exceeded unity. S was the extrapolated end-

point saturation when k = 0. When flow was perpendicular to the strati-

fication, critical gas saturation was high, the oil relative permeability

curve was steep at high oil saturations, and S values were often less than

unity. Oil relative permeabilities were often less sensitive to slight

stratification than were gas relative permeabilities.

m m

rg

m

Kyte and Rapoport (1958) determined that stabilized water flooding

conditions for water-wet cores could be obtained by maintaining a total

pressure drop of 50 psi or greater, regardless of core length. They used

cores up to 32.80 cm in length.

Davidson (1969) studied the effect of temperature on the permeability

ratio of oil-water and gas-oil systems. Davidson concluded that nitrogen-

oil permeability ratios appeared to increase with temperature due to molecu-

lar slippage (Klinkenberg effect), and possibly due to changes in inter-

facial properties. Davidson also concluded that neither: (1) gas-in-oil

solubility changes with temperature and pressure, nor ( 2 ) changes from

Darcy to non-Darcy flow were important for his experiments. He did not

evaluate oil vaporization effects.

-26-

Jones and Roszelle ( 1 9 7 6 ) concluded that a graph of (Ap/q) vs

volume injected must be a straight line unless either: (1) the initial

saturation throughout the core was non-uniform, or ( 2 ) the core was not

homogeneous.

Richardson and Perkins ( 1957 ) observed gravity segregation as water

displaced oil in a lucite model that was 6 ft long, 6 in high, and 3 /8 in

thick. Waterfloods were made at 0.14 cc/sec and 0.31 cc/sec, and corres-

ponded to reservoir velocities of 0.047 and 0.10 ftlday. Gravity segre-

gation was observed at both flow rates, the water under-running the oil

more at the lower injection rate. However, rate affected the recovery

injection curve to a very minor degree. The important point was that at

these same rates, a gas-liquid system would show even more gravity segre-

gation due to the greater density difference.

Craig, Sanderlin, Moore, and Geffen ( 1957 ) used scaled reservoir

models in their laboratory study of gravity segregation in frontal drives.

In linear gas or water drives, gravity effects caused recovery at break-

through to be 20% of that otherwise expected. Although these results were

obtained with immiscible fluids, Craig et al. believed them to be appli-

cable to gas-drives in which capillary forces were "insignificant."

Using scaling equations similar to those of Rapoport ( 1 9 5 5 ) , a convenient

correlating term was obtained by multiplying the geometric dimension

scaling parameter,

by the ratio of viscous pressure gradient to the gravity gradient,

-27-

The result,

was characterized by q the injection rate divided by cross-sectional

area; po, the displaced fluid (oil) viscosity; x, system length; y,

system thickness; kx, horizontal specific permeability; Ap, fluid density

difference; and g, gravitational constant. Smaller density differences,

higher rates, and thinner and longer systems tended to reduce gravity

effects.

i'

One can appreciate the magnitude of gravity segregation by com-

paring the vertical and horizontal components of: (1) pressure or potential

gradient, (2) velocity gradient, or (3) time required for gas to flow some

characteristic distance.

Goode (1978) suggested dividing a characteristic length by fluid

velocity to obtain a characteristic time. For instance:

tV - - - t H 2

LH 'Pg

In this case, % can be the radius and LH the length of a horizontal, cylindrical core. The larger tV/tH, the less likely gravity -segregation

will influence the results.

(2-29)

The next section presents important aspects of the heat transfer

involved in nonisothermal flow through posous media.

-28-

2.4 Heat Transfer in Nonisothermal Flow Experiments

Miller and Seban (1951) determined that thermal conductivity in the

direction of flow was not important forthevaporizingpropane flowexperiments

of Miller (1951). Atkinson and Ramey (1977) concluded that axial thermal

conductivity was not important for the steady-state, nonisothermal liquid

flow experiments of Arihara (1974) and Arihara and Ramey (1976). Atkin-

son estimated thermal conductivity from the results of Adivarahan, Kunii,

and Smith (1962). Arihara (1974) determined the overall heat transfer co-

efficient for his experiments by injecting cold water into a hot core, hot

water into a cold core, and finally, by injecting condensing steam into a

cold core.

For the case of vaporizing liquid flow, Arihara evaluated axial

thermal conductivity and radial heat transfer through the coreholder.

Thermal conductivity was estimated using correlations prepared by Anand,

Somerton, and Gomaa (1972), and by Gomaa and Somerton (1974). Arihara

concluded that the heat flux due to conduction and radial convection was

minor compared to the convective heat flux of the flowing fluid. In

other words, he had nearly isenthalpic flow.

The next section discusses the effect of frequency on the capaci-

tance of water-saturated porous media. A s shown by Chen (1976), water

saturation can be measured using a capacitance probe.

2.5 The Effect of Frequency on the Capacitance of Water-Saturated Porous

Media

Dielectric phenomena have been described in detail by Smyth (1955).

Hill (1969) has discussed the effect of the applied field frequency on

polarizability. Hill also discussed the effect of frequency on interfacial

-29-

Polarization as it occurs in heterogeneous mixtures. Sillars (1937; Smyth,

1955) and van Beek (1967) have presented specific examples of frequency and

distribution effects in heterogeneous mixtures.

Keller and Licastro (1959) and Parkhomenko (1967) have presented a

large amount of data demonstrating the effect of frequency and water satu-

ration on core dielectric constant and resistivity.

A number of studies of porous media capacitance versus water con-

tent are discussed next. Anderson (1943) used an A.C. Fheatstone bridge

at a frequency of 1 kHz and found low capacitance sensitivity at high

water contents. Thomas (1966), using a bridge at 30 MHz, found high sensi-

tivity at high water saturations with a nearly linear water content versus

logarithm capacitance relationship. Laws and Sharpe (1969), using a bridge

at 30 MHz, found that their results varied with salt concentration and

cable length.

Meador and Cox (1975) recently described their successful efforts

in developing a dielectric constant logging device to estimate formation

brine saturation. Using a theoretical model, laboratory-scale experi-

ments, and field trials, they concluded that a two-frequency sonde was re-

quired to account for both the dielectric constant and the formation re-

sistivity. Meador and Cox found that 16 MRz and 30 MHz signals were

adequate for the oil-brine systems they studied. Meador and Cox modified

the Lichtnecker and Rother equation to obtain a general equation for the

dielectric constant of porous media-fluid mixtures. However, no frequency

dependence was built into this equation.

Chen (1976), using a 7.5 MHz frequency difference method, obtained

a nearly linear relation between capacitance probe signal and water content.

-30-

The next s e c t i o n d e s c r i b e s t h e primary o b j e c t i v e of t h i s s tudy ,

which i s t o e s t a b l i s h t h e b a s i c c h a r a c t e r of steam-water r e l a t i v e permea-

b i l i t y curves .

3. STATEMENT OF THE PROBLEM

The primary objective of this study is to establish the basic

character of drainage steam-water relative permeability curves. Any

difference existing between gas-liquid and single-component steam-water

relative permeability curves should be identified. It is therefore neces-

sary to perform experiments that will provide both boiling flow and gas

drive relative permeabilities as a function of liquid water saturation.

The next section describes the experimental apparatus used in this in-

vestigation.

- 31-

4 . EXPERIMENTAL APPARATUS

The experimental apparatus used in the steady, nonisothermal steam-

water flow experiments is shown in the schematic diagram in Fig. 4-1. The

core holder is located within the air bath. Deionized water is deaerated

in a flask by vigorous boiling. One pump and accumulator maintains the

confining pressure. The filtered, deaerated water is pumped through a

furnace for preheat before entry into the core inlet. An accumulator is

used to damp pump pulsations.

Effluent fluid from the core is passed through a heat exchanger,

filtered, and the backpressure maintained with a fine metering valve.

Timed weighing of the effluent water allows determination of total mass

produced, and thus the mass flow rate. Pressures are measured at the

inlet, outlet, and across the core with pressure transducers. Temperatures

are measured with thermocouples and recorded at the inlet, around the air-

bath, and along the length of the core. Liquid water saturation is mea-

sured along the core length with the capacitance probe and its associated

electronics.

Slight modifications were made in this apparatus for the unsteady,

nitrogen displacing water, isothermal experiments. A s shown in Fig. 4 - 2 ,

a gas injection line was added at the inlet, and a high-pressure gas-liquid

separator was added at the outlet.

Details of the major parts of the apparatus are discussed in the

remainder of this section.

-32-

e - 33-

W (3 3 a (3

W a 3 v) v) W

a a

W > -I

s

W

3 4

3 (3 W a

W

3 s

H

0 z 2 Fr 0

W H f i

.. rl

I e

W

3 v)

a -34-

-35-

4 . 1 Core Holder

The core holder used in the two-phase flow experiments is of the

Hassler-type, as shown in Fig. 4-3. The core holder is similar to that

used by Arihara ( 1 9 7 4 ) and Chen ( 1 9 7 6 ) , and is based on a design by S.

Jones of the Marathon Oil Co.

The core holder is made of 304 stainless steel for sufficient

strength and corrosion resistance at elevated pressures and temperatures.

The end caps are made of brass to reduce thread seizure problems. The

brass inlet plug has two ports for fluid inflow and for pressure measure-

ment. The brass outlet plug has three ports for fluid outflow, pressure

measurement, and for the capacitance probe guide. The heat exchanger

fitting shown in Fig. 4-4 was used to pass the thermowell through the out-

let flow port.

The silicone rubber sleeve allows the application of confining

pressure in the annular region to prevent fluid bypass at the sleeve-core

boundary. Water is the preferred confining fluid because nitrogen gas can

pass through the sleeve. Ethylene propylene, another type of elastomer,

was also used as a sleeve material.

With the compression ring in place and a vacuum on the sleeve annu-

lus, cores can be removed and inserted into the core holder assembly by

removing the outlet end bracket and removing the outlet plug.

4.2 Liquid Water Saturation Measurement

The Baker capacitance probe described by Arihara ( 1 9 7 4 ) and Chen

( 1 9 7 6 ) is used to determine the liquid water saturation profile along the

length of the synthetic consolidated sandstone core. Liquid water has

a dielectric constant of 56 at 212°F (100°C) and 28 at 464°F (24OoC) ,

M

(3 Z - a

a Q z- 0 u v) v) * 0 M

-36-

W

2 0 V

m e I ..

-37-

3/8" O.D. TUBULAR SEAL FOR SATURATION PROBE

OUTLET PLUG I- =.

GUIDE

THERMOCOUPLE WELL

HEAT TEE

OUTFLOW TUBING

FIG. 4-4: DETAILS OF OUTLET FITTINGS (ARTHARA, 1974)

-38-

whereas sandstone is 4-6 and dry steam or gas is about 1. This difference

in dielectric constant allows the probe to detect liquid water saturation

as the probe is moved through a probe guide cast into the center of the

synthetic sandstone. Details of the probe and probe electronics are

presented in Appendix 2.

The capacitance probe must be calibrated to provide a signal that

can be related to liquid water saturation. Chen (1976) demonstrated that

the same calibration resulted f o r synthetic consolidated sandstone short

cores at room temperature as for long unconsolidated sand packs at ele-

vated temperature, as shown in Fig. 4-5. Long consolidated cores at ele-

vated temperature were not used for calibration due to the difficulty of

obtaining uniform saturations along the core length.

Minor changes in the components of the probe and probe electronics

and subsequent tuning adjustments required that the probe be recalibrated.

Calibration was performed using long unconsolidated sand packs in the core

holder shown in Fig. 4-6. Results are given in Section 6.

A Hewlett-Packard Q-meter was used to obtain sandpack capacitance

as a function of frequency and liquid water saturation.

4.3 Air Bath

A NAF'CO air bath maintains the core holder in a constant, high-

temperature environment. A high-speed fan and louvres were added to main-

tain a uniform (+5OF:3'C) temperature distribution around the core holder.

The working volume is 41-114 in by 17-112 in by 24 in. The unit is

-

rated at 2,800 watts with a maximum operating temperature of 400°F (204OC).

-39-

I I I I I I I

-CALIBRATION IN A SMALL SAND P - FILLED WITH STEAM AND WATER

MIXTURES AT HIGH TEMPERATURES - O A V CALIBRATION IN SYNTHETIC

CONSOLIDATED SANDSTONE CORES AT ROOM

I I I I I I I I 1 0 0.2

NORMALIZED

0.4

PROBE

0.6

SIGNAL

0.8 ms - a

F I G . 4- 5: COMPARISON OF THE CALIBRATION OF THE PROBE I N DIFFERENT MF,DIq AND AT DIFFERENT OPERATING TE.WEBATURES (CHEN, 1976)

n

PI

i2 0 V

a I

..

-41-

4.4 Temperature Measurement

Type J, iron-constantan, sheathed thermocouples were used to measure

all flowline, air bath, and core temperatures. Axial temperatures in the

core were measured with a 0.040 in (1.02 mm) outside diameter thermocouple

sliding within a 0.072 in (1.83 mm) outside diameter, 0,009 in (0.23 mm)

wall thickness stainless-steel tube. The tube was closed (welded) at one

end and cast in the synthetic consolidated sandstone core. The rated ac-

curacy of the thermocouples was 5 3/8 - 3/4% or +2-4"F (1-2OC) at 350°F (177OC).

-

A 24-point recorder was used to record all temperatures. The re-

corder has a sensing time of 1.5 sec per point. The rated accuracy of the

recorder is +0.3% of span, or +1.8"F ( l . O ° C ) . - -

A single-point thermocouple-recorder calibration check was made

using boiling water. At this temperature, the additive rated error is

+4-6"F (2-3OC) at 350°F (177OC). The thermocouples indicated the boiling

temperature of water to be 212'F +2'F.

-

-

4.5 Pressure Measurement

Diaphragm-type variable magnetic reluctance pressure transducers

were used to measure pressure drop across the core and pressure at the core

inlet and outlet. Pressure displaces a stainless steel diaphragm and varies

the inductive loop between two "E" cores. A digital voltmeter is used with

a carrier demodulator transducer indicator to provide digital pressure read-

out. The unit is calibrated on a nitrogen gas dead weight tester. Trans-

ducer linearity is rated at 0.5% full scale for the best straight line.

Hysteresis is rated at 0.5% of pressure excursion. The transducer indicator

accuracy is rated at +1% meter full scale or 0.01% static change with -

-42-

suppression control. The 3-1/2 digit, digital voltmeter has a 0.1% of

reading + one digit accuracy. Calibration indicated an overall reproduci-

ble accuracy of +1% of full scale. -

Several bourdon tube pressure gages were also used to allow a visual

check of confining pressure and core inlet pressure.

4.6 Pump and Accumulator

The controlled volume pump is designed to displace up to 5.1 gallons

per hour (5.4 cc/sec) at 100 psig (6.8 atm). A plunger operates in oil

at a fixed stroke. The resulting pressure actuates a Teflon diaphragm,

which displaces the pumped fluid. Although a fixed volume of oil is dis-

placed by the plunger, pumping capacity is controlled by adjusting the

volume of oil that bypasses the diaphragm cavity. The rated repetitive

accuracy of the discharge volume is +1% of range, or 20.051 gallons per

hour (+0.054 cc/sec). Observed accuracy was about +5% setting in the

range of settings used. A 60-micron filter is located upstream of the

Pump

-

- -

A hydropneumatic accumulator immediately downstream of the pump

helps suppress the pump pulsations. Nitrogen gas, precharged to 150 psig

(10-2 atm), is separated from the pumped fluid by a rubber bladder. Nitro-

gen pressure can be adjusted to optimize pumping performance.

4.7 Electric Furnace and Temperature Controller

A 2,000"F (1,093OC) rated high-temperature tubular furnace is used

to heat the core inlet water. The stainless steel injection tubing makes

three passes through the 2 in (5.1 cm) diameter by 24 in (61 cm) length

furnace working volume. A percentage timer-type controller is used to

maintain the desired furnace temperature. The furnace exit fluid tempera-

ture for a given control setting is a function of mass flow rate.

-43-

4.8 Flowline Sight-Glass

For the isothermal gas-drive experiments, a glass tube Fisher-

Porter flowrator is used as a sight glass to identify the instant of gas

breakthrough. The produced fluids are passed through a heat exchanger

before passing through the sight glass.

4.9 High-pressure Gas-Liquid Separator

For the isothermal gas-drive experiments, the fluids are produced

into a stainless steel, high-pressure gas separator. A regulating valve

allows the liquid to be produced from the separator at a rate such that

the gas-liquid level remains constant. A sight glass is used to identify

the liquid level. Another regulating valve is continually adjusted to

maintain a constant gas backpressure on the separator and core outlet.

To establish separator pressure, nitrogen is injected into the separator

while water only is pumped through the core. A change in water level in

the low volume separator can be read to +1 mR water using the graduated

sight glass. The gas balance is easy to maintain when the separator

pressure and liquid level are kept constant.

-

4.10 Porous Media

Two types of porous media were used in this study. Unconsolidated

sand was used for the capacitance probe calibration and a synthetic con-

solidated sandstone was used for the two-phase flow experiments.

4.10.1 Unconsolidated Sand Packs

The unconsolidated sand packs used for the static probe calibration

runs were made with either 18-20 mesh sand or a uniform, 20-30 mesh silica

sand. Unconsolidated sand was used for calibration because thermal

-44-

equilibrium was reached faster with the sand pack than with a consolidated

core in a Hassler sleeve, as well as for other reasons.

4.10.2 Consolidated Sandstone Cores

Wygal (1963) Heath (1965) , Evers et al. (1967) , Arihara (1974),

and Chen (1976) have described techniques for making synthetic, consoli-

dated porous media. The cores used in the two-phase flow experiments in

this study were made using the techniques and materials suggested by Ari-

hara (1974) and Chen (1976).

The Fondu cement (Lone Star Lafarge Company) used in the consoli-

dated cores is primarily mono-calcium aluminate (CaO-AR 0 ) rather than

the tricalcium silicate (3 CaO'S 0 ) found in portland cement. On i 2

hydration, Fondu cement forms di-calcium aluminate and the relatively

inert alumina is liberated.

2 3

Fondu cement was developed for making a concrete that would not be

attacked by sea water or sulphate groundwater. It is also useful for

high-temperature environments. The resistance to pure water is high, but

the resistance to fluids with a pH of 4 or less is low.

Fondu cement sets slower than portland cement, but its 24-hr strength

is greater than the strength of portland cement after 28 days. There is

little volume contraction when the concrete is wet cured. One potential

problem is the conversion that may occur over long periods of time in the

presence of water at temperatures greater than 77°F. When Ca0*AR2O3-10 H20

converts to (CaO) A!?, 0 *6'H 0, there may be a reduction in strength, and

an increase in porosity and permeability.

3 2 3 2

-45-

Silica sand and gravel undergo a volume change and may cause

spalling if used in concrete subject to temperatures greater than 400'F.

Despite these limitations, Fondu cement appears to have good over-

all qualities for use in synthetic, consolidated sandstone geothermal

cores.

4.10.3 Synthetic Consolidated Sandstone Core Mold

The core mold shown in Fig. 4-7 is made of a 3 1 in length of 2 in

inside diameter polypropylene tubing cut lengthwise to form two halves.

The inside of each half-tube is covered with heavy duty duct tape to help

provide a clean, smooth surface for the core mold. To assemble the mold,

the two halves of polypropylene are then taped together along the cut

edges and clamped between two halves of 2 in inside diameter brass tubing

using stainless steel hose clamps. The clamped tubing is then assembled

in the bracket and endpieces shown in Fig. 4-7. The bottom, or outlet,

end of the core mold is an aluminum disk with an O-ring seal for the

clamped tubing and 3 ports. One port is for the capacitance probe guide,

which is a 9 mm outside diameter, 24- 1 /4 in (61.6 cm) long pyrex tube with

one end closed. The open end has a 3 / 8 in ( 9 . 5 mm) outside diameter, 6-1/2

in (16.5 cm) long stainless steel tubular seal. The stainless steel end

passes through the outlet end disk of the core mold. The second port is

for the thermowell (thermocouple probe guide), which is a 28-30 in (71- 76

cm) length of 0.072 in ( 1 .83 mm) outside diameter, 0.009 in ( 0 .23 mm)

wall thickness, 321 stainless steel tubing. The third port is for the

mold outlet valve.

Before assembly, the glass portion of the probe guide is inserted

into a zero-gage, thin-walled (0 .015 in - 0.38 mm) teflon tubing to reduce

the risk of glass breakage due to the glass, metal, and core thermal

-4 6-

Fig, 4-7: CORE MOLD APPARATUS [DEVELOPED BY ARIHARA (1974) AND CHEN (1975)l USED TO FABRICATE SYNTHETIC CONSOLIDATED CORES

- 4 7-

expansion. P r i o r t o i n s e r t i o n of t h e g l a s s i n t o t h e t e f l o n tub ing , t h e

t e f l o n tub ing i s f i r s t s t r e t c h e d s l i g h t l y w i th a rod whose t i p i s f l a r e d

t o a diameter s l i g h t l y l a r g e r than t h e diameter of t h e t e f l o n tub ing .

The g l a s s tub ing w i t h i n t e f l o n tub ing must b e placed i n an a i r b a t h a t

over 300'F t o ach ieve t h e shr inkage t h a t might occur under ope ra t ing

cond i t i ons .

Quartz g l a s s was a l s o used f o r t h e probe guide. However, i t was

found t h a t t h e q u a r t z g l a s s broke o r cracked almost as e a s i l y as t h e

pyrex g l a s s .

The next s e c t i o n p r e s e n t s t h e procedure used i n t h e c a l i b r a t i o n

and re la t ive permeabi l i ty experiments .

5. EXPERIMENTAL PROCEDURE

The experimental procedure is described in detail for: (1) cali-

bration of the capacitance probe at high temperature, (2) fabrication

of the synthetic consolidated sandstone cores used in the flow experi-

ments, ( 3 ) steam-water, steady, nonisothermal, two-phase relative permea-

bility experiments, and ( 4 ) nitrogen-water, unsteady, isothermal, gas-

drive relative permeability experiments.

5.1 Probe Calibration

Either 18-20, 20-30, or 80-170 mesh sand was poured into the inlet

end of the core holder shown in Fig. 4-6. The glass probe guide and

the thermowell were fixed in place at the core holder outlet end. The

core holder was lightly tapped on the side and the sand surface lightly

tapped from time to time, using the centralizer. The unconsolidated sand

pack was then installed in the air bath.

At the temperature of interest, the sand pack was evacuated over-

night. The next day, the core was saturated (under vacuum) with filtered,

deionized water. The water was deaerated by vigorous boiling.

In a manner similar to that reported by Chen (1976), the probe

signal was recorded at several locations along the length of the core.

The core was partially depleted, allowed to reach thermal equilibrium (1-

10 hours), and the probe signal again recorded at several locations. This

process was repeated until the core was dry. Average core water saturation

was known after each depletion by a mass balance. Probe response was

-4 8-

-49-

normalized a t each p o i n t us ing ((3 -(3)/(@ -0 ) and averaged f o r t h e

core .

S s w

This c a l i b r a t i o n technique was used wi th both t h e Baker- type

e l e c t r o n i c s a t 7.5 MHz, and wi th t h e Hewlett-Packard Q-meter from 40 kHz

t o 1 4 MHz.

The next s e c t i o n p r e s e n t s t h e techniques used i n t h e f a b r i c a t i o n

of t h e s y n t h e t i c sandstone cores .

5.2 Syn the t i c Sandstone F a b r i c a t i o n Technique

The fo l lowing procedure was used t o form co res w i th a permeabi l i ty

of 20-100 md and a p o r o s i t y of 30-40%.

1) S i f t about 2,400 g (80 w t %) of 80-170 mesh Ottawa s i l i ca sand.

Wash and dry.

2 ) Mix sand thoroughly wi th 12- 15 g (0 .5 w t %) water t o w e t sand

g r a i n s .

3 ) Gradual ly blend i n about 600 g ( 2 0 w t X) Fondu calcium alumi-

n a t e cement. Mix thoroughly.

4 ) Pour small q u a n t i t i e s of t h e mix ture i n t o t h e co re mold. V i -

b r a t e s p a r i n g l y and l i g h t l y . Tap mixture s u r f a c e w i t h c e n t r a l i z e r from

time t o time t o eliminate b r idg ing .

5 ) I n j e c t de ionized water from co re mold top wi th a head of 3-5 f t

of water.

6 ) Shut i n i n l e t and o u t l e t valves a f t e r water breakthrough i s

observed (2- 5 h r s ) .

7) Allow c o r e t o hydra t e i n mold f o r 17 t o 24 h r s .

8 ) Af te r 17- 24 h r s , c a r e f u l l y remove co re from mold.

-50-

The fo l lowing s t e p s should b e taken t o avoid co re cracking:

1) Do no t u se o l d cement t h a t has hard lumps. Using cement

which has absorbed mois ture from t h e a i r causes i nc reased s e t t i n g times

and decreased co re s t r e n g t h .

2) Core mold should be c l e a n b e f o r e packing. Hold should b e

l i n e d wi th new duct tape . The co re may s t i c k t o a d i r t y mold and make

s u c c e s s f u l removal d i f f i c u l t .

3 ) Wet cu re t h e co re a f t e r removal from mold. Wrap co re i n w e t

paper towels o r spray w i t h water and wrap i n p l a s t i c t o keep co re mois t .

The next s e c t i o n p r e s e n t s t h e procedure used i n t h e steam-water,

nonisothermal flow experiments .

5.3 Steam-Water Relative Permeabi l i ty Experiments

A t t h e temperature of i n t e r e s t , t h e co re was evacuated ove rn igh t .

Confining p r e s s u r e was maintained i n t h e 300-500 p s i g range. The next

day, t h e co re was s a t u r a t e d (under vacuum) wi th f i l t e r e d , de ionized

water. The water was deaera ted by v igorous b o i l i n g . Seve ra l pore vol-

umes of waterwerepumped through the core . The o u t l e t end r e g u l a t i n g

valve was then opened t o main ta in t h e d e s i r e d f l o w r a t e and p r e s s u r e drop

ac ros s t h e core . I n l e t end p r e s s u r e was high enough above t h e vapor

p r e s s u r e t o i n s u r e t h a t on ly l i q u i d water flowed i n t o t h e core . Ou t l e t

end p r e s s u r e was maintained below t h e vapor p re s su re .

Once t h e temperature p r o f i l e a long t h e co re was s t a b i l i z e d , i t was

recorded along wi th t h e probe s i g n a l p r o f i l e . I n a d d i t i o n , i n l e t and

o u t l e t p r e s s u r e s , p r e s s u r e drop, and a i r b a t h temperature were a l s o re-

corded. The cons t an t water flow rate was determined by timed mass

weighings. The co ld probe s i g n a l i n a i r was a l s o recorded t o i n s u r e t h a t

-51-

no zero s h i f t occurred. The probe s i g n a l p r o f i l e was obta ined f o r t h e

cases where S = 1.0 and S = 0 so t h a t thenormal ized probe s i g n a l ,

a* = (Q -@)/(?Ps-Qw) could b e c a l c u l a t e d . An i so the rma l , compressed water

f low test was a l s o run t o o b t a i n t h e a b s o l u t e permeabi l i ty t o water.

W W

S

The nex t s e c t i o n p r e s e n t s t h e procedure used i n t h e unsteady, i so-

thermal , gas- drive experiments .

5.4 Nitrogen-Water Relative Pe rmeab i l i t y Experiments

The co re was s a t u r a t e d as descr ibed i n Sec t ion 5 . 3 . Water was

pumped through t h e co re , through t h e s h o r t h e a t exchanger, and i n t o t h e

high p r e s s u r e gas- l iqu id s e p a r a t o r . Water w a s d ra ined from t h e s e p a r a t o r

u s ing a r e g u l a t i n g va lve such t h a t t h e l i q u i d level i n t h e s e p a r a t o r re-

mained cons t an t . The backpressure on t h e c o r e w a s maintained above t h e

vapor p r e s s u r e by i n j e c t i n g n i t r o g e n i n t o t h e s e p a r a t o r . Water flow rate,

p r e s s u r e drop, core , a i r ba th and room tempera tures , and conf in ing pres-

s u r e were recorded t o determine t h e a b s o l u t e permeabi l i ty t o water.

To i n i t i a t e t h e gas- drive, t h e water i n j e c t i o n l i n e was c losed a t

t h e co re i n l e t as t h e gas i n j e c t i o n l i n e was opened. A t t h e same time,

a l abo ra to ry c lock was s t a r t e d . The water r e g u l a t i n g valve was ad jus t ed

t o main ta in a cons t an t l i q u i d level i n t h e s e p a r a t o r . T ime , cumulative

water product ion i n t o a graduated c y l i n d e r , i n l e t and o u t l e t p re s su re ,

and p re s su re drop were recorded. When gas breakthrough was f i r s t no t i ced

i n t h e o u t l e t f l o w l i n e s i g h t g l a s s , t h e o u t l e t gas r e g u l a t i n g va lve was

ad jus t ed t o main ta in a n e a r l y cons t an t p r e s s u r e drop a c r o s s t h e co re .

I n l e t p r e s s u r e was r egu la t ed from a c y l i n d e r , and was t h e r e f o r e almost

cons t an t . Gas product ion i n t o a w e t test meter w a s a l s o recorded as a

f u n c t i o n of time. Atmospheric p r e s s u r e was a l s o recorded, a l though i ts

-52-

v a r i a t i o n was no t important a t t h e h igh p re s su re s used i n t h e s e experi-

ment s . The next s e c t i o n p r e s e n t s t h e r e a u l t s of t h e c a l i b r a t i o n and

r e l a t i v e pe rmeab i l i t y experiments ,

6 . RESULTS AND DISCUSSION

The results of this study are presented in the following three

sections. The first section discusses the calibration of the capacitance

probe. The effect of frequency on the calibration curve is also dis-

cussed. The second section presents steam-water relative permeability

curves generated from nonisothermal, steady, steam-water flow experi-

ments. Water saturation is measured with the capacitance probe. Finally,

the third section presents gas-water relative permeability curves ob-

tained from isothermal, unsteady, nitrogen-displacing-water flow experi-

ments. The effect of temperature on relative permeability is also dis-

cussed.

6.1 Capacitance Probe Calibration

The capacitance probe calibration curves used later in this study

were developed by obtaining the 100% water probe signal, @ at a number

of locations along the length of a sand pack completely saturated with

liquid water. A s the core was stepwise depleted, the probe signal, @, was

recorded at each location. After the final depletion, the signal for

100% steam, Q was recorded at each location in the steam-saturated sand

pack. For each location, the normalized probe signal:

W’

S Y

@ 4 S @* = - %-@w

-5 3-

-54-

was calculated. The @* values along the length of the core were then arithmetically averaged at each average water saturation, S t o obtain

a probe signal - water saturation correlation. W’

The problem of returning to precisely the same location to measure

water Saturation after each depletion was a minor problem because satura-

tion gradients across the sand pack were not large after thermal equili-

brium had been attained.

Water saturations for the calibration runs were approximated within

1% error as the mass fraction. Chen (1976) obtained the following expres-

sion for volumetric water saturation:

-w - - I 17

where : - v = specific volume of water W - v = specific volume of steam

V = pore volume of sand pack

S

P m = m + mw = mass of steam and water in sand pack

S

Since the pore volume equals the product of the 100% liquid saturated water

mass, m and the water specific volume, the saturation equation can be

written:

i’

= - m ’w m i

V W -- _ -

v -v s w

-55-

With less than 1% error, S = m/m. is a good approximation for the W 1

saturated temperature and pressure ranges investigated in this study.

The capacitance probe was first calibrated for water saturation

response to verify Chen's calibration curve (Fig. 4- 5 ) . However, using

an 18-20 mesh sand and four twelve-volt batteries for the power supply,

a much different calibration curve resulted, as shown in Fig. 6-1.

A great deal of time and effort was spent trying to isolate the

cause of this shift in the calibration curve. Several months had passed

since the equipment was last used. Finally, several electronic components

were replaced, the 7.5 MHz operating frequency was verified, and the built-in

regulated power supply repaired.

Figure 6-2 shows four calibration runs using either 80-170 or 20-30

mesh sand at 310°F. This calibration curve is in good agreement with the

one developed by Chen. The recently repaired Baker-type electronics pack-

age was used in all four runs. The data expressed as triangles were ob-

tained using a different power supply and frequency difference-to-analog

converter than that used to obtain the data expressed as a circle or a

square. Since the regulated power supplies appeared to be adequate, the

four 12 volt lead-acid storage batteries used by Chen were no longer used.

Limited data taken with an 80-170 mesh sand yielded more scatter in

the calibration curve than did runs with 20-30 mesh sand. This may have

been caused by fines migration or non-uniform packing. Fine-grained sands

tended to plug the outlet filters and cause long depletion run times.

During the course of the calibration experiments, a number of the

probes were broken and rebuilt. The repair generally required replace-

ment of the silver-plated glass tubing. It is therefore suspected that

slight differences in probe construction had a minor effect on the

-56-

6 @ s - @ NORMALIZEQ PROBE SIGNAL a @ = 4 S 4 W

FIG. 6-1: WATER SATURATION VS NORMALIZED PROBE SIGNAL IN AN UNCONSOLIDATED SAND PACK (18-20 MESH)(BEFORE REPAIRING ELECTRONICS)

-57-

0 80-170 MESH, 305°F - c 0 20-30 MESH , 302 OF

L V 20-30 MESH , 305°F

0 .- Z 0.8 - A 20-30 MESH , 303°F a Y-

L - 3

v,

z 0 I- Q

- 0.6 - - - - -

- a W

- -

- -.I

0 0.4 0.6 0.8 I .o

FIG. 6-2: WATER SATURATION VS NORMALIZED PROBE SIGNAL FOR SEVERAL SAND GRAIN SIZES IN UNCONSOLIDATED SAND PACKS

-58-

c a l i b r a t i o n response of each probe. Never the less , t h e c a l i b r a t i o n r e s u l t s

shown i n Fig. 6-2 show small s c a t t e r and are considered s a t i s f a c t o r y .

6 .1.1 E f fec t of Frequency on Probe Response

A s tudy of t h e e f f e c t of frequency on probe response was a l s o made.

The d i e l e c t r i c l i t e r a t u r e suggested t h a t heterogeneous materials show a

s t r o n g frequency dependence. The r e s u l t s of t h i s s tudy are shown i n Fig.

6-3. A more d e t a i l e d run a t 7.5 MHz is presented i n Fig. 6-4. Normalized

capac i tance is : (1) n e a r l y independent of frequency i n t h e 750 kHz - 1 4 MHz

range, and (2) s t r o n g l y dependent on frequency i n t h e 40-180 kHz range.

This in format ion i s u s e f u l i n t h a t t h e h ighe r f requencies can g ive t h e

b e s t water s a t u r a t i o n r e s o l u t i o n over a wide range of water s a t u r a t i o n .

The lower f r equenc ie s would be most u s e f u l f o r d e t e c t i n g small water

s a t u r a t i o n changes a t very low water s a t u r a t i o n s .

The frequency-dependence s tudy was c a r r i e d out w i th a Hewlett-

Packard Q-meter. Each resonant frequency r equ i r ed a d i f f e r e n t c o i l . The

response of t h e Q-meter: c o i l assembly was a f f e c t e d by room temperature

v a r i a t i o n and r equ i r ed a c o r r e c t i o n of roughly -0.040 pf/OF. .The 14 MHz

readings were too uns t ab l e t o al low temperature c o r r e c t i o n . A temperature

c o r r e c t i o n of 0.04 pf/OF would s h i f t t h e 1 4 MHz curve t o t h e r i g h t f o r

a c l o s e r agreement w i th t h e 7.5 MHz and 750 kHz curves . The 47 mH c o i l

used a t 40 kHz d i d no t demonstrate a well def ined temperature s e n s i t i v i t y .

The Q-meter provided an ou tpu t i n terms of a capac i tance (p icofarad)

d i f f e r e n c e a t t h e resonant frequency. This was i n c o n t r a s t t o t he Baker-

type e l e c t r o n i c s , which provided a d i g i t a l ou tput t h a t was r e l a t e d t o t h e

resonant frequency d i f f e r e n c e between t h e tun ing (o r sens ing) c i r c u i t and

a r e f e rence frequency. The Baker- type e l e c t r o n i c s package w a s chosen f o r

gene ra l use because i ts ope ra t ion r equ i r ed much less e f f o r t than t h e Q-meter.

-59-

3 l- tn a

a

3

W

$

0.4

n n ! I I I I I I I I I

0 14 MHZ 0 7.5 MHZ V 750 KHZ

180 KHZ 0 100 KHZ A 40 KHz

/ /

/ 1 I --

0 0.2 0.4 0.6 0.8 1.0 .

t c s - c NORMALIZED PROBE CAPACITANCE a C = c s -c,

FIG. 6-3: WATER SATURATION VS NORMALIZED PROBE CAPACITANCE FOR SEVERAL FREQUENCIES IN AN UNCONSOLIDATED SAND PACK (310°F, 20-30 MESH SAND)

-60-

- 0.0 -

- 0.6 -

- -.

0.4 - -.

- .I

- -.

0 0.2 0.4 0.6 0.8 1.0 n yr

NORMALIZED PROBE CAPACITANCE , C* = Ls - L

c s - cw

FIG. 6-4: WATER SATURATION VS NORMALIZED PROBE CAPACITANCE IN AN UNCONSOLI- DATED SAND PACK (20-30 MESH, 305"F, 7.5 MHz)

-61-

It is interesting that the normalized capacitance (C*) vs water

saturation (S ) and the normalized frequency (f *) vs water saturation

(or O* vs S ) calibration curves are similar at 7.5 MHz.

Since :

W r

W

c = (2JP C where :

C = capacitance in oscillating circuit

L = inductance in oscillating circuit

f = resonant frequency of circuit

C

r

Then :

(f r+f rs) rw

Cfrw+frs) fr

L

c * = f * r 2

where : c -c

c* = - S

c -c s w

f -f

r f -f rs r

rs rw f * =

Subscripts: s = C or f measured in 100% steam-saturated core

w = C or f measured in 100% water-saturated core r

r

The group (fr+frs)frw /[(frw+frs)fr 3 is almost unity because the L L

resonant frequency for the steam-saturated core varies by only about 1%

from the resonant frequency for the water-saturated core. Thus, f * C*

is a good approximation at 7-10 MHz. These statements have been verified

r

-62-

by conver t ing t h e change i n capac i tance d a t a , AC, t o capac i tance , C y and

then t o frequency, f . I d e n t i c a l va lues are obta ined f o r @* and C* t o

two decimal p l aces .

r

The next s e c t i o n p r e s e n t s some of t h e temperature e f f e c t s t h a t

should be considered when us ing t h e s a t u r a t i o n probe.

6.1.2 E f f e c t of Temperature on Probe Response

A s t h e probe is p u l l e d o u t of t h e co re ho lde r , more of t h e probe

is exposed t o t h e room temperature. This causes a gradual cool ing of t h e

probe wi th time. For a p a r t i a l l y e x t r a c t e d probe, probe output can de-

c r e a s e by up t o 0.6 mv given a coo l ing time of 2-3 minutes. This tends

t o i n c r e a s e t h e apparent steam s a t u r a t i o n wi th time. To o b t a i n r e p e a t a b l e

r e s u l t s : (1) t h e probe should be f u l l y i n s e r t e d i n t o t h e co re , ( 2 ) pu l l ed

ou t and t h e reading taken immediately, and (3) t h e probe should then be

l e f t f u l l y i n s e r t e d i n t o t h e co re f o r 2-3 min be fo re t h e next reading .

A long time i s r equ i r ed t o o b t a i n a p r o f i l e of many p o i n t s a long t h e

l e n g t h of t h e co re u s ing t h i s procedure.

S ince t h e maximum v a r i a t i o n i n probe response i s 15-30 mv, a

v a r i a t i o n of 0.6 mv caused by probe temperature change r e p r e s e n t s a 2-4%

e r r o r . To decrease t h e time requ i r ed f o r a complete traverse, an alter-

n a t e procedure w a s a c t u a l l y used. The same time schedule was used f o r

t h e probe traverse a t each s t a g e i n t h e co re d e p l e t i o n process . The use

of t h e time schedule allowed t h e probe temperature change t o b e similar

f o r each traverse, thus reducing t h e e r r o r below t h e above-mentioned

va lue of 2-4%.

-63-

As noted by Chen (1976), the calibration curves for the capaci-

tance probe were found to be nearly independent of temperature.

- Because the dielectric constant of water decreases with increasing >

: temperature, the capacitance also decreases and the resonant frequency - - (f = [~ITJZ-~]'~) increases. The normalized signal, or frequency (fs-f)/

(fs-f,), discussed by Chen would therefore be expected to decrease with

increasing temperature because f > f. S

However, a second temperature effect also exists. At higher tem-

peratures, the brass rod within the probe expands more with temperature

increase than does the silver-plated glass tubing. For a parallel plate

capacitor, capacitance is indirectly proportional to the distance between

the plates. As the gap in the probe capacitor increases with increasing

temperature, capacitance is expected to decrease, resonant frequency (or

probe signal) increase, and normalized frequency (or normalized probe sig-

nal) decrease. Apparently, the parallel plate capacitor model is inadequate

because Chen observed a decrease in probe signal with increasing tempera-

ture when the probe was immersed in air. Similar tests recently completed

have verified this result. For instance, a probe that was partially

removed from a hot, steam-saturated core for 3 minutes, provided a signal

that was 0.6 mv larger than the signal from the probe when fully inserted

into the core. As Chen concluded, the effect of temperature on the pr0b.e

geometry compensates, in some as-yet-undefined way, for the effect of

temperature on the dielectric constant of water.

Although the normalized probe signal appears to be independent of

temperature, the use of the calibration curve requires some care. The

main use of the probe is in nonisothermal, two-phase flow experiments.

-64-

Because water saturation determinations are required through a region of

changing temperature, the 100% water-saturated (@ ) and steam-saturated

(@ ) probe signals should be determined as a function of temperature so

that @* can be calculated properly for the probe signal at each temperature.

W

S

Using QW and @ values determined at the higher temperature of the S

injected, compressed water will cause an unpredictable error in the calcu-

lated normalized probe signal. @ will tend to increase with increasing

temperature due to the lowering of the dielectric constant of water, and

@ will tend to decrease with increasing temperature due to the thermal

expansion character of the probe. These effects should be measured, not

estimated. It is expected that 0 will only decrease with increasing

temperature due to probe expansion.

W

W

S

For the moment, assume that @ has a negligible change for some W

temperature range and that @ increases by 1.0 mv with decreasing tempera-

ture. The assumption of a @ value at the high inlet temperature (it is

convenient to deplete the core at the end of the experiment to get as

at the air bath temperature) will cause QS to be less than actual. This

will cause the normalized probe signal, @*, to be less than actual because @ - @ will be percentage-wise reduced more than @ - 0 . The net effect

is to cause the apparent steam saturation to be larger than it actually

is. Recall that this example is for the case of constant QW.

S

S

S S W

Additional experiments indicated an average @ of -86 mv at 298"F, W

and a @ of -82 mv at 84°F for a particular zero and span setting. This

surprising result indicates that the probe signal decreases with increasing

temperature rather than increasing as forecast. It is not understood why

this occurs. At about 33OoF, Os = -69 mv.

W

-65-

The next section describes the effect that steam-water vertical

separation may have on probe response.

6.1.3 Effect of Gravity Segregation on Probe Response

The effect of gravity segregation on the steam-water calibration

curves is not known. If significant effects do occur, the following

errors would probably result: (1) if the steam-water "interface" is far

enough above the core centerline that the probe does not detect steam,

then the normalized probe signal, @*, will be larger than it would be for a homogeneous steam-water saturated medium at large water saturation values,

and (2) if the steam-water "interface" is far enough below the core center-

line that the probe does not detect liquid water, then the normalized

probe signal, @*, will be smaller than it would be for a homogeneous steam- water saturated medium at low S values. A proper analysis of the fluid

distribution effects must include the effect of changing dielectric inter-

faces on the probe field; it is beyond the scope of this study.

W

The final comments concerning this study of the capacitance probe

present recommendations for further study.

6.1.4 Future Improvements

This section presents guidelines for further study of the capaci-

tance probe. In general, the Baker-type capacitance probe appears to

work well. However, at times, stability is poor, resulting in a low con-

fidence level in the water saturation data. The electronic circuit de-

sign is fairly simple, and perhaps can be modified to improve stability.

Using the calibration curves, probe response correctly represents

water saturation to - +5-10% of pore volume. The lack of reproducibility is

-66-

highest in the 70-100% water saturation range. An analytic or numerical

study of the effect of core holder metal parts, fluid distributions, and

other dielectric interfaces on the probe field and probe response may

lead to a better understanding of the shape of the calibration curve.

More effort should also be directed toward understanding the effect of

temperature on probe response. Calibration curves should be developed

for brine-steam mixtures.

This concludes the section on the capacitance probe. We now turn

to the nonisothermal, steady, steam-water flow experiments in which the

saturation probe was used.

6.2 Nonisothermal, Steady, Boiling Flow Experiments

This section presents data for three steady, nonisothermal, boiling

flow experiments. Three different synthetic sandstone cores were used.

The experiments were performed in a manner similar to that of Arihara (1974)

and Chen (1976), as discussed in Sections 3, 4, and 5.

Compressed water was injected through the consolidated core. The

outlet end pressure was maintained at a level such that boiling occurred

within the core. The measured temperatures, pressures, probe signals, and

flow rate data are presented in Appendix A3.2.

The following presents the method of interpretation, as well as the

determination of axial thermal conductivity, overall heat transfer coeffi-

cient, and water saturation. Then the results of the three runs are dis-

cussed. The three runs presented in this section are representative of

the many runs made in this investigation.

-67-

6.2.1 Method of Calculation

Arihara (1974) and Chen (1976) chose to assume that their flow data

was not influenced by axial heat conduction or by radial heat transfer from

the environment to the core. The assumption of an adiabatic process re-

sulted in the isenthalpic, steady flow equations for drainage process

relative permeability:

fw (UT) kg/K =

AK [- 31 (l-f)w(uT)

kR/K = x

AK [- %]

where :

k = effective permeability, md

K = absolute permeability, md

f = flowing mass fraction of gas

1-1 = viscosity

h = enthalpy, Btu/lb

v = specific volume, ft /lb

w = w + w = total weight rate of flow, lb/hr

p = pressure, psi

x = linear, horizontal distance, in

A = core cross-sectional area, ft

- 3

g R

2

(6-10)

-68-

Subscripts: R = liquid (liquid water)

g = gas (water vapor)

Rg = liquid to vapor

The adiabatic flow assumption is inadequate for the low flow rates used

in this study. Isenthalpic flow does not allow the development of a

large vapor saturation. The required steady, non-adiabatic (or non-

isenthalpic) flow equations follow:

(2) forced + (2) axial + (2) = 0 (6-11) radial

convection conduction heat transfer

- d dx [ (wh)& + (whIg] + dx d [ - XA (E)] + PU(Tm-T) = 0 (6-12)

Integrating along the length of the core from zero to x, assuming

compressed water of enthalpy h is injected at the inlet end at weight

rate w, and solving for the flowing gas mass fraction f :

L

where :

X = axial thermal conductivity, Btu/(hr-ft-OF)

U = overall heat transfer coefficient, Btu/(hr-ft -OF)

T = axial core temperature, O F

2

Too = air bath temperature, OF

P = core perimeter, ft

A = core cross-sectional area, ft 2

qH = heat flow rate , Btu/hr

(6- 13)

-69-

The absolute permeability, K, is the 100% water-saturated core

permeability determined from a separate flow test. Viscosity, specific

volume, enthalpy, and pressure gradient can be obtained in the two-phase

region from the measured temperature profile. It is assumed that there

is no vapor pressure lowering and no capillary pressure effect.

The elimination of a major assumption, adiabatic flow, has created

the need for several lesser assumptions to describe the axial and radial

heat transfer. The axial heat transfer is discussed in the next section.

6.2.2 Axial Thermal Conductivity

Heat conduction along the length of the core can be calculated

using Fourier's law. An axial thermal conductivity of X = 1 Btu/(hr-ft-OF)

was used in this study. No data is available for the thermal conductivity

of flowing two-phase systems. Adivarahan,Kunii, and Smith (1962) pre-

sented data for the countercurrent flow of single-phase fluid and heat.

They concluded that thermal conductivity was rate dependent. However,

analysis of the Adivarahan (1961) data by the present author indicated

that a radial heat gain (or loss) term was neglected. Inclusion of this

heat transfer term can eliminate most of the supposed rate dependency.

Use of the Anand, Somerton, and Gomaa (1972) correlations indi-

cated X = 0.3 Btu/(hr-ft-'F) for a dry rock, and X = 0.9 Btu/(hr-ft-'F)

for a water-saturated rock with (p = 0.35, B = 30 md, and T = 250°F.

Because (1) many rocks demonstrate a thermal conductivity larger

than 0.9 Btu/(hr-ft-OF) and (2) the correlations used were based on rocks

of lower porosity and higher permeability than the synthetic cores used

in the present study, it was decided that A = 1 Btu/(hr-ft-OF) was a

reasonable value to use. The thermal conductivity was assumed constant,

although it may vary with water saturation.

-70-

The following section discusses the selection of the core holder

overall heat transfer coefficient.

6.2.3 Overall Heat Transfer Coefficient

The radial heat gain to the horizontal, cylindrical core can be

characterized using Fourier's law and an overall heat transfer coeffi-

cient. The overall heat transfer coefficient, U, for the core holder was

discussed in detail by Arihara ( 1 9 7 4 ) . Overall heat transfer coefficients

determined by Arihara for a core holder similar to the one used in this

study is shown in Fig. 6-5. For the single-phase flow of hot water in a

cold core, the overall heat transfer coefficient was lower, and a stronger

function of flow rate than for the case of cold water injection into a hot

core. The differences between the two cases is a result of the increased

film coefficient caused by the use of a fan in the high temperature air

bath runs. No fan was used in the room temperature air bath runs. At

the low flow rates of 0.1 - 0.2 lb/hr, the hot air bath curve extrapolates

down to roughly U = 2 Btu/ (hr-ft -OF). 2

The overall heat transfer coefficients obtained early in this study

are also shown in Fig. 6-5. The heat transfer coefficients are larger

than those obtained by Arihara because of the different core holder - hot

air film coefficient. This film coefficient was different because the

present fan: (1) is located in a different position than for Arihara's

runs, (2) has a more powerful motor, and (3) can be controlled with ad-

justable louvers. At low flow rates ( 0 .2 lb/hr) , a value of U = 3 Btu/

(hr-ft -OF) is expected using the same slope as Arihara's cold water, 2

hot air bath curve.

-71-

v,

c - r

i W z 0 I- v,

z n

z 0

a v)

Q

0

-72-

Heat transfer coefficients in this study were determined from an

energy balance using the temperature profile and miss flow rate:

-w(h-hll) + AX U = L (6- 14)

P J ( Tm-T) dx 0

The axial thermal conduction term is often negligible for these single

phase experiments because the inlet and outlet temperature gradients are

similar. Heat transfer coefficient variation along the length of the core

can be detected with Eq. 6-14 by using the equation for the core holder

segment of interest.

The value of the overall coefficient was assumed constant, although

the brass end plugs and end caps may have caused variations near the core

holder ends. Also, air flow around the core holder was probably not uni-

form.

Since the heat transfer coefficients obtained in the single-phase

tests may be larger than actual coefficients in a two-phase fluid due to

different core-silicone rubber sleeve film coefficients, a value of U =

2 Btu/(hr-ft -OF) was used in this study. Louvers, which direct air flow 2

across the core holder, were also arranged to provide a low air flow, thus

lowering the core holder film coefficient.

In addition to the overall heat transfer coefficient data discussed

earlier, Arihara ( 1 9 7 4 ) also obtained values for condensing steam injection

in a cold air bath. Arihara found that for steam injection rates of 0.10

to 0.15 lb/hr, the overall heat transfer coefficient varied from 1.4 to

1.6 Btu/ (hr-ft -OF) . 2

-73-

6.2.4 Water Saturation Measurement

The large scatter in the water saturation measurements represents

one of the major problems encountered in this study. One of the undesir-

able characteristics of the capacitance probe noted in the calibration

study was the large data scatter present at high water saturations.

Since: (1) the liquid water dielectric constant is much greater than the

sand or water vapor dielectric constant, and ( 2 ) the sand and water vapor

dielectric constants are much closer in magnitude than the sand and

liquid water constants, this saturation data scatter is much greater at

high liquid water saturations than at very low liquid water saturations.

It was found to be easier to smooth the normalized probe signal

data before obtaining the water saturation profile. Due to the shape

of the probe signal - liquid water saturation curve, Fig. 6- 2, more data

scatter is induced by graphing the water saturation obtained from each

normalized probe signal data point.

We turn now to consideration of the results obtained from the

nonisothermal boiling runs.

6.2 .5 Run SW1

This section presents the results of the medium flow rate, boiling

flow run. Compressed water was injected into the inlet end of the 26 md

core at a rate of 0.212 lb/hr. Confining pressure was applied to a sili-

cone rubber sleeve. The temperature, normalized probe signal, and water

saturation profiles are presented in Figs. 6- 6, 6-7, and 6- 8.

In this experiment, there was some uncertainty in the inlet pres-

sure measurement. It is possible that a two-phase mixture was injected

rather than single-phase compressed water.

-74-

.

LL 0

W

3

.. a

s a W n z W I-

W

0 0

-I

X

a

a a -

1 I

DISTANCE FROM INLET END, IN.

FIG. 6-6: TEMPERATURE VS DISTANCE, RUN SW1

-75-

8 3

' I . 0 C

e M

0.9 6

0.92

0.8 8

0.84

\ I '. 0 0 I I I I 1

0' 0

- 0 0 0

ooo 000 0 0 0 0 0

0 0 0 0 0 0 0

- -; 0 0 "_I

n Z w

DISTANCE FROM INLET END IN.

FIG. 6-7: NORMALIZED PROBE SIGNAL VS DISTANCE, RUN SW1

-76-

I .c

0.9

0.0

I I I I 1

DISTANCE- FROM INLET END, IN.

FIG. 6-8: WATER SATURATION VS DISTANCE, RUN SW1

-77-

The calculation of flowing gas mass fraction in Table 6-1 demon-

strates that radial heat gain near the outlet end of the core dominated

the formation of vapor. The steam-water calculations are presented in

Table 6-2, and the relative permeability curves are shown in Fig. 6- 9.

Both the steam and liquid water curves are steep and cover only a narrow

range of water saturation.

6.2.6 Run SW2

The results for a low injection rate run are presented in this

section. Compressed water was injected into the inlet end of the 34 md

core at a rate of 0.106 lb/hr. Confining pressure was applied to an

ethylene propylene sleeve. The temperature, normalized probe signal, and

water saturation profiles are presented in Figs. 6-10, 6-11, and 6-12.

A crack in the core, perpendicular to flow, had no apparent effect

on steam-water flow. The crack was located 7.5 inches from the inlet end

of the core. The temperature profile demonstrates that flow rate was so

low that a dry region of higher temperature developed near the outlet end

of the core. Pressures were estimated in this region by extrapolation

from the two-phase region to the measured outlet pressure. The ambient

air bath temperature exceeded the inlet fluid temperature, thus contri-

buting to vaporization.

The calculated flowing gas mass fraction in Table 6-3 demonstrates

that radial heat gain from the environment dominated the formation of

vapor. The values of overall heat transfer coefficient and axial thermal

conductivity discussed earlier gives reasonable results and helps support

the use of X = 1 Btu/(hr-ft-'F) and U = 2 Btu/(hr-ft -OF). 2

m + .!I u

I

P rl

M\ w 3 s o m

P rl

d\

c 2 m

al

E-r a 0

X .dl G

0

0

0

0

m

0 co m

co rl b N

0

0

N 0 m

0

m b U 0 0 0

0

m I 0 rl

m X Q rl 0

I

m I 0 rl

rl X

Q rl

m I 0 rl

0 X rl

rl

N

0 m m

co 0

N b

m 0 0

0 I

0 U

m

rl 0 m

N

N

N N

0 0

0

m I 0 rl

rl X

N b

m I 0 rl X e rl 03

m I 0 rl X rl m N

0

0

m rl

b

m Q N

m m rl

0 N

b

0 0 m

U

0

U b

0 0

0

m I 0 rl

N X Q

rl

m I 0 rl X N U

rl

m I 0 rl X 0 m U

m

rl rl

m

b

b Q N

0 0 m

Q

N rl

a3 m N

Q

-7 8-

0 h

0 rl

d

m I 0 rl

m X co m

m I 0 rl

N X 0

N

m I 0 rl X co 0

m

m U

m rl

m

Q m N

m M)

rl

0

co rl

U m N

0 rl

c o r n c o d U b 0 0 0 0

m m I 1 0 0 r l r l

2 : o m m e

m m I I 0 0 r l r l

3 3 m m m m

m m I 0 rl X

N

N rl

m N N m

N

N m N

b h

m

0

0 m

m co N

U rl

m m m 0

0

m I 0 rl X b

m Q

m I 0 rl

m x hl

U

m I 0 rl

co x m N

N

03 hl m

rl

U U N

3-4

m rl

U

co m

m b N

co rl

I 0 rl X U

m b

rl

m m m

m

m N

rl

a3 rl

N

m m

co Q N

N 0

0 rl X rl

m rl

rl

N U m

b

N m N

\c

U cu

m

r l N 0

II II II

X P P I

O N r l O m . b . o w 0

II II II II

-79-

4 -x -x -x M \

L.4 3

nl

0 0

rl

0 0

rl

0

U h

h rl 0

0

h co N

a

0 co rl

0

0 U rl 0

0

0

-x rl h Cr)

0

0

h m 0

h U co 0

m 4

0 rl

0

a U h rl 0

0

h h m a

rl co rl

0

0 U 0 rl

0

m U h

O 0 0

0

0 U U

0

N

03 03

0

N U m 0

U U m 0

0

m U

rl h

0

0

a a U

a

N co rl

0

0 U rl 0

0

N

N N

0 0

0

m 0

a 0

U

a 03

0

U a m 0

co m U 0

0

U m h rl 0

0

N

a m

a

co m rl

0

m m rl 0

0

0

U h

0 0

0

0 m

rl

ro

m 03

0

m 03 N

0

rl m co 0

0

U rl

h rl 0

0

U U 03

a

co m rl

0

cn m rl 0

0

a co m 0 0

0

0 m rl

a3

m 03

0

m U N

0

N h

rl

0

cn m h rl 0

0

N U 0

b

h 03 rl

0

m m 0 rl

0

0 h rl 0

0

m m rl

0 rl

rl m 0

m N rl

0

rl m e

0

h m h

0 rl

0

U m m h

m a3 rl

0

co m 0 rl

0

a N h

0

0

m b

rl

N rl

0 m 0

m 0 N

0

a U m 0

m N h

0 rl

0

h a N

co

a m rl

0

a m 0 rl

0

m U a3

0

0

co h

rl

U rl

cn h

0

m co rl

0

m rl

U

0

a N h

0 rl

0

m h h

co

m m rl

0

m m rl 0

d

m rl h 0

0

m 0

N

a rl

co h

0

m a rl

0

m a rl

0

N N h rl 0

0

0 N m m

m 0 N

0

U m rl 0

0.

m m m 0

0

m N

N

03 rl

h h

0

a N rl

0

U 0 h

0

a h rl

rl 0

0

co m 0 rl

0

N rl

0

m m 0 rl

0

N h

rl

0

U m N

0 N

h h

0

0

m 0

0

m m 03

0

m 0 h rl 0

0

U h

N rl

c\I m N

0

0 m 0 rl

0.

0 h rl

0

Q m m

N N

/> M

u @J G

5 ld b M

M G .rl

N 0 0 rl

u 0

-80-

WATER SATURATION, S, fraction

FIG* 6-9: STEAM-WATER RELATIVE PERMEABILITY VS WATER SATURATION FOR MEDIUM FLOW RATE, RUN SW1

-a 1-

LL 0 L

W

a a W a E W I-

W


FIG. 6-10: TEMPERATURE PROFILE VS DISTANCE, RUN SW2

-82-

e I

v) e

I .o 3

e I

v)

e 0.8

0.6

0.4

0.2

0


FIG. 6-11: NORMALIZED PROBE SIGNAL VS DISTANCE, RUN SW2

-83-

c 0.8 0 .- c 0 0 L YI

0.6 v) c

z 0 -

1.0 q

I - I

1 - I

S 0.4 - a

!z 5 0.2-

3

co

I- 4 0 0 I I I , , I #

4 8 12 16 20 24


FIG. 6-12: WATER SATURATION VS DISTANCE, RUN SW2

-84-

m + N

I

rl

II W

U U N 0

0 0

m

m o U b r l V l O N r n r l r l N

0 0 r l r l N

0 0 0 0 0

I - c v m N N m

0 0

m m I I 0 0 r l r A

u o N m it3

m m I I 0 0 r l r l x x m a u c o a r -

m c n

m m c o m c o m m u m 0 0 0

x n a w I w a

$ 0

W X

m I 0 rl

e X

t-n I 0 rl

U X

U N

0

0

m 03 0 m

co rl

N h

c\l In

N

N 0 m

N

m m m m

N

3 P4

0

0

0

m W 0 m

co rl

N h

0

0

N 0 m

0

r l r l d

m m m I l l 0 0 0 r l r l r l

c o m u X X X * m a u m m

m m m I 0 I+

c\l X

m rl

m I -0 4

rl X

N m

I-

N U m

b

N N N

0'. I- U

\D

m

U m N

N rl

m t m m I

0 rl X 0 rl

rl

c.3

m m 0

00

0 I- N

0 b

U

0 ? m

rl 0 m

U

0 rl X m I-

\o

0

m rl m

a m N

0 rl

0 rl

0'. m rl

\o m N

co

I 0 rl

N X

rl rl

0: m m rl

"I rl a N

"I m rl

? U N

N m cv

0 rl

IO rl X rl

0 N

b

rl N m

m 5 m N

b

N N

03

03 N

U

N co

U rl

. . I 0 rl

N X

U m

? rl m m

0

U 0

N

rl

m m

co N m

rl b N

co rl

IO

Q: rl X

a U

m m m m

b

b N N

m

U m

9. m Q

m m cv

0 F*l

m a I ..

I II II II II

-85-

The steam-water relative permeability calculations of Table 6-4

are graphed in Fig. 6-13. These curves show a wide range of steam-

water relative permeabilities over a wide range of water saturations.

The critical gas saturation appears to be nearly zero. We now turn

to a third boiling flow experiment using a third core.

6.2.7 Run SW3

The results of a high injection rate run are presented in this

section. Compressed water was injected into the inlet end of the 36 md

core at a rate of 0.244 lb/hr. Confining pressure was applied to a

silicone rubber sleeve. The temperature, normalized probe signal, and

water saturation profiles are presented in Figs. 6- 14, 6-15, and 6-16.

The declining temperature along the first five or six inches of

the core may be caused by the core temperature exceeding the ambient air

bath temperature. The normalized probe signal curve was smoothed, con-

sidering that evaporative cooling does not start until 6 inches from the

inlet end.

The flowing gas mass fraction calculations in Table 6-5 show that

radial heat gain dominated the data to a lesser extent than in the lower

injection rate runs, SW1 and SW2.

The calculated steam-water relative permeabilities are listed in

Table 6-6. The relative permeability curve in Fig. 6-17 covers a very

narrow steam relative permeability range. The apparent critical gas

saturation is much higher for Run SW3 than for Run SW2.

The core used in this run was also used in the isothermal gas drive

experiments discussed in Section 6.3. A comparison of the three steam-

water runs is presented in the next section.

-86-

. . d O d d d d d d d d d d O * -

. . . . . . . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

u r n 0

0 ~ 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . .

-87-

WATER SATURATION S, fraction

FIG. 6-13: STEAM-WATER RELATIVE PERMEABILITY VS WATER SATURATION FOR LOW FLOW RATE, RUN SW2

-88-


FIG. 6-14: TEMPERATURE VS DISTANCE, RUN SW3

0.9 2

0.8 8

0.8 4

0.80

0.

-89-

0

0 W

0 I


FIG, 6-15: NORMALIZED PROBE SIGNAL VS DISTANCE, RUN SW3

-90-


FIG, 6-16: WATER SATURATION VS DISTANCE, RUN SW3

-91- m + N

I rl

II w

x a E-r I

m H

n

8 W

N

rl 'IC Po

s c

0

0

0

0

u3

0 b co

co rl N m

0

0

0 m m

0

m N

0 m 0

0 I

m I 0 rl X N m U

I

m IO rl

co X U rl

m I 0 rl

m X rl

rl

m rl b co

00

0 N m

0

rl I

d * rl

U m m

N

rl m 0 m 0

0 I

m I 0 rl X m m 00

I

m I 0 rl X m m rl

m I 0 rl X m m m

\o

d b co

\o

u3 rl m

r- rl

I

0 m rl

m U m

U

N m N 0 0

0 I

m I 0 rl

rl X co co I

m I 0 rl X U co rl

m I 0 rl X m m co

m u3 b 03

m U rl m

00

rl I

0 co rl

d m m

\o

m U 0 rl

0

0

m I 0 rl

N X m b I

m I 0 rl

m X co rl

m I 0 4 b X

0 rl

m b b co

U

N rl m

v)

rl I

u3

co rl

U rl

m

CQ

m rl

0 rl

0

m I 0 rl X N m N I

m I 0 rl X 0 m rl

m I 0 rl X b

u3 rl

rl

N

co

rl

b 0 m

u3

C I

03

03 rl

u3 m m

0 rl

m 0 N 0

0

m I 0 rl X co co m

m I 0 rl

0 X 0

N

m I 0 rl

0 X

m rl

b

m 00 co

0

VI 0 m

co 0

co m rl

U m m

N rl

r- N

0 m

0

m I 0 rl X

Lo N rl

m I 0 rl X U N

N

m I 0 rl

m X

N N

rl

u3 03 co

m rl 0 m

u3

N

9 U N

rl m m

U rl

W 0 m 0

0

m I 0 rl X m m N

cr) I 0 rl

m X rl

m

m I 0 rl

m X

co N

rl

0 o\ co

u3

u3 m N

m m

03

m rl

u3 N m

u3 rl

m rl b

0, 0

m I 0 rl

N X

N U

m I 0 rl X u3 m U

m I 0 rl X m

m m

0

U m 03

m

m rl

N

03

co

u3

m U

rl N m

co rl

0 LA

rl

0

m I 0 rl

m X

U u3

m I 0 rl X \o m F

m I 0 rl X -j.

U r-

m m 0 m

0

m r- N

UY

m rl

3 U b

0 m m

0 N

* co rl

0

m I 0 rl

b X

b m

m I 0 4 X N

\o rl

m I 0 rl X m u3 \o

b

u3 rl Dl

m

W 0

N

m 0 N

m a u3 rl

m rl

N

N N

II II II II

E-r 8 3 2-4 s

-92-

0 0 3 U J m U m N N 0 0 0 0 0 3 h h h h ~ h h

r l r l r l r l o o o o o o o o

c * * M 1 0

a3 r; 0 0 0 0 0 0 0 0

- X I 03 r n 0 h r l h h h U m 0 3 U J h m r l 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

O N V l n W h E 2

03 03 h

m co h

rl 03 h

0 rl

0

03 h h 4 9 0

N h h

0 rl

0

t- U m b rl

9 0

h m h

0 rl

0

UJ h

rl 0

0

rl 0

rl 0

0 0

a al V

U 0

h

m

\o 0

03 m m

m h

03 9 V

m rl m N

U

U 03 m m U

V

U m 03

*

m V 0

UJ m m m

.rl a (d k M

t-

m w 3 h

al k 1 rn VI al k a

H H

h m rl

0

0 UJ rl

rl W rl

N \D rl

0

\o m rl

0

03 UJ rl

0

UJ h rl

0

m 03 rl

0 0 0

d m rl 0

0

Q\ U d 0

0

03 U rl 0

0

H (A

m U rl 0

0

h U rl 0

0

UJ U rl 0

0

m U rl 0

0

03 m rl 0 0

H 0 0 rl

2 0

m N

rl m (? 0 0

N

(\I m 0 0

m U rl 0 0

0

0 U m

rl rl 0

0

m 0 w 0

h N m 0

0

UJ 0 m 0

m rl h 0

0

.. m 0 0

m 0 rl

0

03 U rl 0 0

I 0 0 0 0

I 0

I m (A 5

m II

UJ I

w

..

s E-l

' M

a3 c c -x x .dl

C O N U V J 0 3 O N U U J 0 3 O N r l r l r l r l r l h l h l

-93-

WATER SATURATION , S, fraction

FIG. 6-17: STEAM-WATER RELATIVE PERMEABILITY VS WATER SATURATION FOR HIGH FLOW RATE, RUN SW3

-94-

6.2.8 Comparison of Three Steam-Water Runs

The most noticeable difference between the three sets of steam-

water relative permeability curves is the location of the steam curve.

Because the core properties of the three different synthetic cores are

expected to be similar, the differences may be a function of flow rate,

as described later in this section. The critical gas saturation appears

to increase as the flow rate increases.

Using the present apparatus, flow rates can only be varied over

a narrow range. Low flow rates result in a wider saturation range, but

radial heat transfer and thermal end effects dominate at these lower

flow rates. In addition, low flow rates require very low permeability

cores in order to maintain sufficient backpressure to keep the inlet fluid

compressed. Outlet pressure must remain low to encourage vaporization.

Very low volume pumps are also required. It is not practical to use gas

displacing water pumps, because the inlet fluid should be degassed prior

to injection. Low flow rates may also contribute to gravity segregation

effects.

Higher flow rates decrease the saturation range experienced and,

as stated earlier, there is less resolution in the water saturation values

at high water saturations. An apparent probe signal gradient for the

case of 100% liquid water, QW, exists and may be caused by: (1) fines

migrating to close to the outlet end during initial flushing of the core,

(2) gas, produced by a reaction between the elastomer sleeve and water,

that tends to migrate downstream, or (3 ) electronic or electrostatic

reasons not apparent at this time. The presence of produced gas is prob-

ably not the explanation, because this effect does not vary with the

pressure level of the compressed water. Many experimental runs were made

-95-

in this study; those shown here produced only minor amounts of non-

condensable gas that are believed to be caused by a water elastomer re-

action.

One possible explanation for the apparent rate effect is that at

low flow rates, radial heat transfer dominates, causing a higher gas

saturation near the outer edges of the core. Because the probe measures

saturations near the core axis, the lower the flow rate, the greater the

error in measured saturation. The measurement of a higher-than-actual

average water saturation would cause the low flow rate curves t o shift

to the right, and thus give the impression of a low critical gas satura-

tion. One problem with this explanation is that the water relative

permeability curves then become inconsistent for the three runs. This

would require explanation.

The formation of an annular ring of flowing water vapor and liquid

water around a plug of mainly flowing liquid should also influence the

relative permeability curves. The actual vapor and liquid effective

permeabilities would be higher in each of their respective regions than

they would be assuming homogeneous flow. To correct the low flow rate

data, lowering the water relative permeability curve as flow rate de-

creased would contribute to both: (1) bringing the water relative permea-

bility curves into closer agreement after making a qualitative correction

shift to lower water saturation as flow rate decreased, and (2) forming

a relative permeability curve of more reasonable shape compared to tradi-

tional gas-liquid relative permeability curves.

At this point, it becomes obvious that gas-water relative permea-

bility curves are needed for these synthetic consolidated cores. Steam-

water and gas-water relative permeability curves could then be compared.

-96-

The next section summarizes a study of the effect of temperature and

pressure level on nitrogen-water drainage relative permeabilities at low

confining pressures.

The effect of temperature on the relative permeability curves is

expected to be minor because the confining pressure was maintained at a

low value of 300-500 psig.

The lowering of the steam relative permeability curve for Run SW3

is also suggestive of turbulence. However, Reynolds number calculations

indicate that vapor turbulence is probably not a major factor for Run SW3

(0 .244 lb/hr, f = 0 . 1 4 8 ) :

N =- - R All

qdp - 0.029

where d = characteristic pore diameter, 0.018 cm

or :

where S = 0.28 g

The above calculations are qualitative because Reynolds number

criteria are based on single-phase flow. Since the calculated Reynolds

numbers are low (less than unity), it is believed that turbulence is not

important (Amyx, Bass, and Whiting, 1960).

The next section evaluates the importance of gravity segregation

in the boiling flow experiments.

-9 7-

6.2.9 Gravity Segregation

It is believed that gravity segregation does not dominate the data

in these experiments. Using a form of Eq. 2-29 presented in the litera-

ture survey, characteristic time ratio values, tV/tH, of about three for

Runs SW1 and SW2 and six for the high rate Run SW3 indicate that vapor

tends to flow horizontally. An example calculation for Run SW3 follows:

L AP (0.0833 ft)(lOl psi) tV V

tH _ - - - -

1 1 = 5.7 (1.925 ft) -

3 .01737 - l b ft 7.3536 - lb

The next section discusses the possible influence of capillary end

effects.

6.2.10 Capillary End Effects

Outlet end effects, caused by capillary retention of liquid water

at the outlet face of the core, were not measured, and are not expected

to influence this data significantly. Runs SW1, SW2, and SW3 had flowing

pressure drops of 52 psi, 53 psi, and 101 psi, respectively. The water

relative permeability curves do not increase in magnitude with increasing

pressure drop.

The next section discusses the problem of confining pressure

sleeve gas production.

6.2.11 Confining Pressure Sleeve Gas Production

Past studies have experienced trouble with: (1) gases emanating

from the elastomer confining sleeves, and (2) confining gases passing

through the elastomer sleeves. Chen (1976) solved the second problem by

-98-

using water as the confining fluid rather than nitrogen. Viton and sili-

cone rubber tend to evolve gas when in the presence of steam and water.

Materials selection charts indicated that ethylene propylene could with-

stand steam and hot water. Tests indicated that ethylene propylene sleeves

also produced gas. Ethylene propylene discolored the produced water and

produced a strong odor. Silicone rubber sleeves also discolored the water

and had a slightly less offensive odor. Both silicone rubber and ethylene

propylene maintained their integrity at temperatures above 350°F. Ethylene

propylene smoked badly when placed in 250°F air. Silicone rubber was much

more stable in hot air and had only a slight odor. A s a result, silicone

rubber was used in most of the experiments in this study. The fact that

produced water was obviously affected by the sleeve material led to ef-

forts to separate the elastomer from the core. Both kitchen quality

aluminum foil and a soldered, thin copper foil were used to isolate the

core from the elastomer. Both attempts failed. Permeability measurements

of a Berea core with and without the foil showed the core-foil permeability

was higher than the core permeability and decreased with increasing con-

fining pressure at room temperature. This indicated fluid bypass along

"wrinkles" formed at the core-foil interface. At this stage of the inves-

tigation, the elastomers did not have an observed, significant effect on

the core. Subsequent investigation suggested that the silicone rubber

sleeve causes the cores to lose their natural water wettability.

The next section suggests improvements for future boiling flow

experiments.

-99-

6.2.12 Future Improvements

Future improvements to this type of experiment should include

adiabatic sections along the core length in a manner similar to that of

Miller (1951). This improvement may be difficult to implement with a

Hassler-type core holder. Provision for temperature control along the

core length by a set of separately controlled heat tapes will help re-

duce any problems caused by radial heat transfer from the environment.

The drawback of adiabatic flow is the limited water saturation range that

will be encountered.

Another possibility is to inject two-phase fluids of known enthalpy

and quality. Otherwise, experiments similar to those conducted in this

study should use a pump designed for very low rates, lower permeability

cores should be fabricated, and greater control exercised over the air

bath temperature distribution. A study of the thermal conductivity and

overall heat transfer coefficients for multiphase flow in porous media

should be made. Other methods of measuring water saturation profiles

should also be investigated.

This concludes the section on the nonisothermal, boiling flow

experiments. The next section presents the results of the isothermal,

gas-drive experiments.

6.3 Isothermal, Unsteady, External Gas-Drive Experiments

This section presents the results for three unsteady, isothermal,

nitrogen gas-displacing-water experiments. The experiments were performed

at a variety of temperatures and mean pressures, using a high pressure

gas-water separator at the outlet

The synthetic sandstone core used

the 36 md core used in Run SW3 of

end, as described

in the isothermal

the nonisothermal

in Sections 4 and 5.

flow experiments was

boiling experiments.

-100-

Gas and water relative permeabilities can be obtained from the

cumulative gas and water production data recorded as a function of time.

Inlet and outlet pressures, as well as pressure drop and temperature,

were also recorded. The data and calculations required to develop gas-

water relative permeabilities are presented in Appendix A3.3.

The gas-water relative permeability curves presented in Fig. 6-18

demonstrate that at the temperatures (70-300°F) and low confining pres-

sures (300-500 psig) used in this study, gas and water relative permea-

bilities were not strong functions of temperature.

The calculated gas-water relative permeability data at 7S°F, 19S°F,

and 294°F almost fall on a single set of relative permeability curves.

Much of the scatter in the gas curve is due to: (1) experimental error,

(2) graphical analysis error, and ( 3 ) approximations used to describe

the effects of water vaporization in the core. The three runs presented

in this section are representative of the many runs made at different tem-

perature and pressure levels.

T h e EolloxainS section sunmarizes the calculations requi red to

generate the relative permeability values.

6.3.1 Calculation Procedure

The calculation procedure suggested by Jones and Roszelle (1976)

can be modified in the following way t o determine gas-liquid relative

permeabilities at various temperatures and pressures. Cumulative separa-

tor gas production is corrected from room conditions to average core con-

ditions by suming the incremental production from time t to time t : j -1 j

(6-15)

-101-

78OF, Pavg 423 psia , A p =71 psi

198"F, Pavg =I79 psia , A p = 6 5 psi

. O kg

WATER SATURATION , S , fraction

FIG. 6-18: GAS-WATER DRAINAGE RELATIVE PERMEABILITY VS WATER SATURATION FOR SEVERAL TEMPERATURES

-102-

where :

Pbar core AGsep(Tcore’Pavg) = AGsep(Troom’Pbar I - - p

avg Troom

T (6-16)

and G = separator gas production after breakthrough seP

core T = core absolute temperature

T = room absolute temperature room

’bar = barometric absolute pressure

- ’avg

core room

- (pin+pout)/2 = average absolute pressure across core

Z = z = 1.0

The cumulative gas injected into the core is:

(6-17)

where :

Ginj = cumulative gas injected into core

W = cumulative water produced from core P

- V w, core = water specific volume at core temperature - V w, room

’H2 0

= water specific volume at room temperature

= mole fraction water vapor in nitrogen-water vapor gas mixture

The term l/(l-yH o) is the correction factor that considers the 2

gas expansion due to water evaporation.

The gas injection rate, q , is calculated for each time increment

as :

-103-

(G. 1nJ .> . -(Ginj> q =

j -1 t -t j j-1

(6-18)

The cumulative gas injection in units of pore volume is Q.=G 1 injfvp’

where V is the pore volume of the core sample. Average gas saturation, P

- - w v s = 1 - - p core - 1 --- g sW v -

P Vroom

The average relative reciprocal mobility term:

(6-19)

(6-20)

can also be calculated where Ap is the pressure drop across the core and

(qvfAp>b is the single-phase data that is representative of the core

absolute permeability t o water. The term 1-l is called the average rela-

tive reciprocal mobility because it is related to the relative permea-

bilities and viscosities of the two phases:

(6-21)

where :

k = water effective permeability

k = gas effective permeability

The next step is to graph 5 vs Qi on coordinate graph paper.

W

g

g The outlet face gas saturation, S is then obtained as a, function of

cumulative gas injection, Qi, by recording the zero Qi intercept of the

tangent at several values of Q The initial s vs 9 data points should

82

i‘ g i

-104-

be linear to the time of gas breakthrough, This line should extrapolate

back to S = S = 0 . 82 gi -

Finally, graph x-' vs Qi on coordinate graph paper, The outlet

face mobility term, X2 , is obtained as a function of Q by recording the

zero Q intercept of the tangent at several values of Qi. The initial

X-' vs Qi data points should be linear to the time of gas breakthrough.

This line should extrapolate back to the value of the water viscosity, 1-1,.

-1 i

- i

The relative permeabilities can then be calculated:

where :

A 2

f = 1 - 82 fw2

(6-22a)

(6-22b)

(6-23a)

(6-23b)

fw2 = fractional flow of water

f = fractional flow of gas 82

The above equations assume no capillary effects (p = pg - pi = 0 ) . C

A computer program was used to process the raw lab data and calcu-

late relative permeability. The data was smoothed using cubic spline

subroutines from the International Mathematical and Statistical Library.

-105-

Optimum knot locations for the cubic splines were automatically deter-

mined by minimizing the least squares error. The computed slopes, inter-

cepts, relative permeabilities, and saturations showed good agreement

with values obtained by hand graphical analysis.

A review of the equations required to calculate gas and water

relative permeabilities indicates that the porosity (or pore volume)

estimate does not influence the calculated values of k /K or k /K.

However, the outlet face saturations are dependent upon the porosity of

the core. For the core used in this study, porosity was determined by

first evacuating the core and then saturating it at a pressure of 300

psig. The difference in the weight of the dry core and the saturated

core was then related to core porosity (34%).

g W

We turn now to a discussion of important fluid properties and the

impact of water vaporization on the experimental results.

6.3.2 Fluid Properties and the Effects of Water Vaporization

In this section, gas density and fluid viscosity relations are dis-

cussed. The importance of liquid water vaporization is also discussed in

detail.

Using methods described in Reid, Prausnitz, and Sherwood (1977)

and Katz et al. (1959), it was found that nitrogen density varies with

temperature and pressure in the same manner as an ideal gas for the range

of conditions encountered in this study (0-200 psig, 70-300°F).

Because the injected nitrogen is not saturated with water at core

temperature and pressure, three water vaporization effects must be con-

sidered: (1) gas mixture viscosity reduction as the mole fraction of

water vapor increases, (2) core drying, or water saturation reduction

-106-

caused by dry nitrogen injection, and (3) gas volume expansion as the

mole fraction of water vapor increases. The potential importance of each

factor is discussed in the following three sections.

6.3.2.1 Effect of Water Vaporization on Gas Mixture Viscosity

Dry nitrogen is diluted by water vapor when the nitrogen gas

comes in contact with liquid water at elevated temperatures. This is

caused by the high vapor pressure of water.

The mole fraction of water vapor in the nitrogen gas can be esti-

mated from Dalton's and Raoult's laws:

(6-24)

where :

'H2 0 = mole fraction water in vapor phase

X = mole fraction water in liquid phase (Henry's Law indicates XH z 1,O) 2

pP

pVP

= partial pressure of water in vapor phase

= vapor pressure of liquid water (from Steam Tables)

pT = total pressure of vapor phase mixture

The effect of vapor phase pressure and temperature on water vapori-

zation is presented in Fig. 6-19.

The viscosities of water, steam, and nitrogen are required to

calculate gas-mixture and water relative permeability, Liquid water vis-

cosity was obtained from the Electrical Research Association 1967 Steam

-107-

0 0

0 /

.II

-108-

Tables. Steam viscosity was estimated using an equation presented by

Farouq Ali (1970):

= 0.17~10 -4 T1. 116 "steam (6-25)

where:

1-1 = viscosity, cp

T = temperature, OK = O C + 273.1

Nitrogen gas viscosity was estimated using the Sutherland equa-

tion obtained from the International Critical Tables (V-2):

(6-26)

where :

1-1 = viscosity, cp

T = temperature, O K

Using methods described by Reid, Prausnitz, and Sherwood (1977),

it was found that the effect of pressure on the viscosity of nitrogen was

not important for the range of conditions encountered in this study. The

same conclusion is reasonable for the steam and the nitrogen-water vapor

mixture viscosity. Although Farouq Ali (1970) presented equations charac-

terizing the pressure dependence of steam viscosity, these relationships

were not used because there is significant disagreement in the literature

as to the magnitude and direction of viscosity change caused by pressure

level. In any event, the low pressure levels used in this study would

not affect gas mixture viscosity more than a few percent.

-109-

Table 6-7 indicates the potential importance of water vapor on

vapor phase viscosity. The Herning and Zipperer (Reid et al., 1977; Ali,

1970) mixture law is used here (although more complicated techniques are

often recommended for mixtures that include polar gases):

(6-27)

where :

M = molecular weight

y = mole fraction

1.1 = viscosity

6.3.2.2 Effect of Water Vaporization on Water Saturation

Water saturation at the core inlet may be less than the water

saturation immediately downstream due to the vaporization of water by the

dry nitrogen gas. The amount of core drying can be estimated in terms of

grams of water produced per liter of nitrogen produced at room tempera-

ture and atmospheric pressure:

where :

R = gas law constant, 0.08207 (R atm)/(moll K O )

T = temperature, OK

p = pressure, atm

'H2 0 = mole fraction water in vapor phase

(6-28)

fn

E H v3 0 u m

H 2 .. b I

\o

f/ Lo 5

3 1 0

H

U b rl 0

0

b U N 0

0

\L) b d 0

0

b \o m 0 0

0

b

U rl

0 b

\o

rl b

0

0

hl 03 rl 0 0

0

\o b 4 0

0

b \o m 0 0

0

0 0 hl

E

In 4 0

0

U co b

0

0 b

N 0

0

U hl d 0

0

b

U d

0 0 N

F? 0 N 0

0

\L) b m 0

0

b 0 N 0

0

U N rl 0

0

0 0 hl

0 0 N

I

I

m N hl 0

0

m U rl 0

0

b

U rl

0 0 m

-110-

m N 0

0

0

Lf-l F? cr)

0

m N N 0

0

e m

0 rl

0

0 0 N

0 0 0

-111-

AS shown in Table 6-8, core drying is only significant at high

temperatures and low pressures. Elevated pressure tends to reduce t:he

magnitude of core drying. For the case where the core is at 294°F and

184 psia, 36.9 grams of water may be removed from the core due to va.pori-

zation, not gas-water immiscible displacement, for every 100 liters of

nitrogen produced.

For a steady flow process, the evaporated water collected in the

heat exchanger should be subtracted from the liquid water flow rates that

are based on separator liquid effluent. Because the process under con-

sideration is unsteady, the solution should be obtained using the fr,ac-

tional flow equation, the frontal advance equation, and a Welge-type

equation that relates average water saturation to outflow end saturation.

Afterwards, a correction should be made in the mobility terms.

If it is assumed that all water vaporization occurs at the inlet

end of the core, then it can also be expected that a dry region (S = 1.0)

may extend a distance x into the core. The distance x will in-

crease with increased cumulative nitrogen injected.

g

dry dry

The fractional flow equation:

1 f = g

1 +x- lJ kw l-Iw kg

and the frontal advance equation:

will apply for that part of the core downstream of the dri,ed zonec

(6-29)

(6-30)

m w

Fr 0 E-l 2 P s co .. \o I

n

a

PC k

o m m $4 u a

k PC

v) co N 0 0

0

\o m ch

\ 0

\D co 0 m 0

0

I-

U rl

co r-

m hl 4

v)

U r-

0

co I-

0

U Q\

0

0

co Q\ m

0 \ 0

\o N

9 0

r- U rl

m I-

m I- 4

rl

rl rl

co m 4

-112-

-113-

I n t e g r a t i n g t h e f r o n t a l advance equat ion from x t o L , and d ry

from time zero t o t h e time when cumulative i n j e c t i o n is Q : i

where :

and t h e c o r r e c t i o n has been made f o r gas volume expansion caused by

evapora t ion .

Welge's equat ion:

- s = s g g2 + Q i f w 2

(6-31)

(6-32)

(6-33)

(6-34)

i s no t app ropr i a t e , and another equa t ion must be der ived t o re la te average

and producing end s a t u r a t i o n s dur ing t h e gas d r i v e . The new equat ion can

be obta ined i n a way similar t o t h a t descr ibed i n Craig (1971) .

For t h e s a tu ra t ion- d i s t ance r e l a t i o n s h i p shown i n F ig . 6-20,

t h e average d i s p l a c i n g f l u i d s a t u r a t i o n f o r t h e reg ion from x (x ) t o

L is:

1 dry

(6-35)

-114-

-115-

From Eq. 6-31, f ' is p r o p o r t i o n a l t o (L-X ) : g d r y

- 4 ' s g d f '

s = g f '

I n t e g r a t i n g by p a r t s :

S f ' S f ' 1' f 'dS g2 g2- g l g1- g g

The q u a n t i t y f ' is unknown a t x = x . Assume f' +. 0 and 81 1 d r y g l

f = 1. The r e s u l t is: g l

- (f 2-1) s = s - g 82 f ' 82

Using Eq.6-31 f o r t h e outf low end of t h e core:

The above express ion i s t h e average gas s a t u r a t i o n f o r t h e reg ion f-rom

x t o L. The average gas s a t u r a t i o n f o r t h e e n t i r e c o r e is: d r y

(6-36)

(6-37)

(6-38)

(6-39)

X

= (1) dry + [ Sg2 - [+] fw2Qi] [*] (6-41) d r y

= d r y + X s [ f Q L 82 w2 i (6-42)

where :

-116-

N w,core 2 X = t

dry

X = A@R

= cumulative volume of n i t rogen i n j e c t e d a t room cond i t i ons

- V w, core

P A = c ros s- sec t iona l area of co re

@ = co re p o r o s i t y

= s p e c i f i c volume of l i q u i d water a t co re cond i t i ons

V = co re pore volume

(6- 43)

( 6- 44 )

The usua l graph of 2 v s Qi on coord ina te graph paper w i l l provide g

t angen t s o f :

s l o p e = f w2 (6- 45)

X

zero Q~ i n t e r c e p t = dry + s L 8 2 (6- 46)

The o u t l e t end gas s a t u r a t i o n , S and water f r a c t i o n a l f low, fw2, 8 2 ,

can be c a l c u l a t e d because L i s known and x can be computed: dry

L-x (6- 47)

- S - i n t e r c e p t

Qi fw2 - - (6- 4 8 )

-117-

The above results indicate that: (1) if no vaporization occurs,

X = 0 and the results reduce to the usual displacement equations, (2) dry

- using the intercept value, the correct fw2 and f values wiii be calcu- - 82 t hted whether or not vaporization occurs (x > 0), and (3) relative - dry -

permeability will be graphed as a function of incorrect saturations if

the intercept value is assumed to be the outlet face saturation, S 82 '

when vaporization occurs (x > 0). dry

Table 6-9 shows that only small gas saturation corrections are

required at large injection volumes. In general, the required correc-

tion is small except for very large cumulative injection volumes at 'high

temperatures and low pressures. The experiments in this study were termi-

nated before the vaporization of liquid from the core affected the re-

sults appreciably.

It is interesting that inlet end water saturation reduction due

to vaporization requires a correction in the outlet end water saturation.

The assumption that all water evaporation occurs at the core in-

let is not strictly correct because, due to the flowing pressure gradient,

some vaporization occurs along the entire length of the core. The region

of unit gas saturation, x in length, could perhaps be more appro-

priately considered a region of (1-S .) gas saturation of length .. - . ~~~ .. ~ ~ dry . . ~ . . .- - ~ . - ~

w1 . . ~ - . - .-.. . ~- - ~- - ~.

X dry [&] , where Swi is the irreducible water saturation. But a similar __ _ _ ~ ~ ~. wi - ~ ~ _ _ _

analysis would again indicate small corrections to the outflow liquid

saturation.

Now, all that remains is to correct the relative rec-iprocal - mobility terms, X-1 and Xi1, to consider the inlet region of -lowered water

saturation. This decreased water saturation should cause an increased

z I? i; i3 H

5 0 ..

-118-

N N d - m o m m u d m u d w

0 0 0 0 0 0 0 ? ? ? m m m * . . . .

rn co m 0

cn m m 0

m 0 m 0

N m N

m d d

w

-119-

flow rate and decreased pressure drop compared to the case of no ev,apora-

tion. The decreased pressure drop (Ap) will lower the reciprocal mo- -

bility terms (X-' and X ) and increase the calculated values of relative -1 2

permeabilities over values obtained with no vaporization.

The methods of Johnson et al. (1959) and Jones and Roszelle

(1976) can be combined to derive an approximate expression characterizing

the relationship between relative reciprocal mobility (X ), cumulative

injection (Q.), and relative permeability to water (k ).

-1

1 rw

At a given time in the displacement process, the total pressure

drop across the core may be considered by a succession of steady states

in space:

(6- 49)

0

Using Darcy's law for the water phase over an incremental distance, and

treating the total flow rate as a function of time (not space):

x - x = Lf'Q dry g i

(6-50)

(6-51)

-120-

Thus at a given time:

dx = LQ,dfl 1 g

and at x = L:

(6- 52)

(6- 53)

Incorporating these results into Eq. 6-50 and differentiating both sides

yields :

(6- 54)

= [ L-tdry] d [ &-] ( 6- 55 ) f

k rw

It was again assumed that f' = 0 at x = x g dry'

Using the definition of the drivative of a quotient:

(6- 56)

Thus, a graph of (Ap/q ) vs Qi provides tangents whose zero Qi t

intercept is:

-121-

The equation can be presented in terms of X (Jones and Roszelle) and

Qi because :

-1

Thus :

(6-57)

I (6-58)

- In this case, a graph of X vs Q. provides tangents whose zero Q inter-

cept is:

-1 1 i

intercept = - k rw

The water relative permeability is:

k = fw21-lw [ (L-rdry)] rw intercept

(6-59)

(6-60)

where x is obtained from E q . 6-43 and fw2 is obtained from E q . 6-48. dry The preceding Table 6-9 shows that near the end of Run NW3, x dry

is only 2.05 inches compared to the core length L of 23.1 inches. Thus,

the correction to the gas and water relative permeabilities was small for

all runs. Except for the last data point in Run NW3, the corrections for

water saturation reduction due to vaporization are believed to be negli-

gible. Therefore, the gas-drive relative permeabilities presented in

this study were not corrected for water saturation reduction caused by

vaporization. If larger volumes of nitrogen had been injected at ele-

vated temperature, a correction would have been required.

-122-

The next section presents a method of correcting for the gas

mixture volume increase resulting from water vaporization. This is the

third and final water vaporization effect discussed in this report.

6.3.2.3 Effect of Water Vaporization on Gas Nixture Volume

This section demonstrates that the gas injection volume cor-

rected for water vaporization must equal the volume of flowing nitrogen

and water vapor. Because only the noncondensible nitrogen can be mea-

sured

clude

where

vapor

tures

after the separator, the nitrogen volume must be increased to in-

the water vapor:

r -n

vol(H20+N2) = vol N2 11-iH20j

(6-61)

'H20 = mole fraction H 0 in vapor phase. 2 This equation includes the assumptions of instantaneous liquid-

phase equilibrium, and that nitrogen, water vapor, and their mix-

behave like ideal gases. Figure 6-21 presents the correction factor

as a function of temperature and pressure.

If one calculated the gas volumes, relative reciprocal mobili-

ties, and gas-water relative permeabilities neglecting the impact of

volume expansion, and labeled them Qi',

approximate corrections might be made by the relations:

1-1 I 9 krg I , and krw', obvious

(6-62)

(6-63)

-123-

0 0

- 0

LL

0 10 M

0

- I /

I ( A ) ) ( 'N +O 'H) ' 1 0 A

-124-

(6- 64)

k - rw - krwl

Equation 6-17 shows this method of correction is improper.

(6- 65)

Furthermore,

there is enough scatter in the X vs Q. graph at low Q. values that the

relation X-' 1-1, at Qi = 0 must be used to draw the proper straight line

through the early data before breakthrough. This may also influence the

-1 1 - 1

decision on where to position the curve just after breakthrough.

Without the volume expansion correction, X , or -1

is incorrectly set to 1-1 at Qi = 0. It is not sufficient to set: W

( 6- 66 )

L L

at Qi = 0. The only way to correct the analysis is to regraph X vs Qi

using the correct rates and volumes in X and Qi. The final correction

may affect both gas and water relative permeability at low gas saturation.

-1 - -1

The calculated gas-water relative permeability data without

correction for gas expansion due t o water vaporization is shown in Fig.

6- 22. The uncorrected gas data show an apparent decrease in gas re:lative

permeability with increasing temperature. The water relative permeability

appears to be temperature independent.

-125-

I .o

x \

Y ’ 0.8 .. x \

Y 9,

- 0.6 > I- J - - m a 0.4

U W a.

2 0.2 a

W

I-

J W U

0 0 0.2 0.4 0.6 0.8 1.0

WATER SATURATION , S, ,fraction

FIG. 6-22: GAS-WATER DRAINAGE, RELATIVE PERMEABILITY VS WATER SATURATION FOR SEVERAL TEMPERATURES USING RESULTS NOT CORRECTED FOR GAS VOLUME EXPANSION DUE TO WATER VAPORIZATION

-126-

The Fig. 6-22 gas relative permeabilities at 294'F would be

increased by 12% if the viscosity of nitrogen (0.0228 cp) was used :in

Eq. 6-22b rather than the viscosity of a nitrogen-water vapor mixture

(0.0204 cp). Figure 6-23 presents the calculated gas-water relative

permeability data for the case where: (1) the gas viscosity used in

Eq. 6-22b is that of nitrogen (not a nitrogen-water vapor mixture), and

(2) there is no correction for gas expansion due to water vaporizat:ion.

A s mentioned earlier, no data in this report is corrected for water

saturation reduction caused by water vaporization. Comparison with the

corrected relative permeability curves in Fig. 6-18 shows that the correct

gas relative permeability curves at various temperatures are close together

at low water saturations. Thus, there appears to be no strong gas-water

relative permeability temperature dependence in this synthetic core.

A related fluid behavior topic is the solubility of nitrogen in

water. The next section demonstrates that the solubility changes far

nitrogen-water systems are not significant for these experiments.

6.3.2.4 Nitrogen Solubility in Liquid Water

This section demonstrates why no allowance was made for changes

in the nitrogen solubility of liquid water as a function of temperature

and pressure. Before displacement begins, the core is completely satu-

rated with partially degassed liquid water. The resulting equilibrium

between injected gas and water may have a minor effect on water viscosity

or water volume factor.

Henry's law for the solute nitrogen in the solvent water is:

pP N2 H = x k (6-67)

-127-

I- - I I

e

- W >

0.2 -

n

I 0 P

0.2 0.4 0.6 0.8

WATER SATURATION S, ,fraction

FIG. 6-23: GAS-WATER DRAINAGE RELATIVE PERMEABILITY VS WATER SATURATION USING: (1) THE VISCOSITY OF NITROGEN RATHER THAN A NITROGEN- WATER VAPOR MIXTURE, AND (2) RESULTS NOT CORRECTED FOR GAS VOLUME EXPANSION DUE TO WATER VAPORIZATION

-128-

where :

p = partial pressure of nitrogen in vapor phase

x = mole fraction of nitrogen in liquid phase PN2

N2

k = Henry's law constant H

For the case of maximum loss of nitrogen vapor phase to solution

in the liquid water phase, assume zero nitrogen in the initial core water

and nitrogen saturation at 70°F, 150 psia:

pvP *363 ~ 2.4x10-3 mole H20(v)

yH20 PT =-=-u

150 mole(H20fN2)(v)

L

n

pPN2 (150 psia/l4.7 psia/atm) mole N2 x = - =

N2 kH 8 . 5 ~ 1 0 4 atm/mol frac = 20x10-4 mole (N2+H2@

Therefore, at 70°F (294.2'K) and 150 psia (10.2 atm):

(6-68)

where :

R = 0.08207 ( R atm)/(mol K O )

-129-

Even this maximum nitrogen loss effect is too small to include

in the gas balance equations.

This concludes the discussion of important vapor-liquid equili-

bria effects. The next section presents an analysis of the importance

of gravity segregation in the gas-drive experiments.

6.3.3 Gravity Segregation

The vertical separation of the vapor and liquid phases must: be

evaluated in two-phase flow experiments. The characteristic time ratio

discussed earlier in this section and in the literature survey may be

used to conclude that gravity segregation was probably not a major i.n-

fluence in these experiments:

where : - v = yN* [e] + yH20 [ $1


R = gas constant

T = absolute temperature

p = absolute pressure.

(6-69)

(6-70)

(6-71)

Experiments NW1, NW2, and NW3 had t /tH values between 3 and 4 , V

which means that vertical flow was much less important than horizontal

flow.

-130-

The following section presents a brief evaluation of the capil-

lary end effects.

6.3.4 Capillary End Effects

The repeatable results obtained at various pressures, pressure

drops, and temperatures suggested that capillary end effects did not:

strongly influence the experimental data. Additional gas-drive runs for

various pressure levels and pressure drops at a set temperature also sug-

gested that end effects did not markedly influence the relative permea-

bility curves shown in Fig. 6-18. With end effects, the calculated value

for water relative permeability would be lower than for a run without end

effects. The pressure drop across the core for the 78"F, 198"F, and 294°F

runs shown in Fig. 6.3.1 was 71 psi, 65 psi, and 52 psi, respectively.

The slightly lower water relative permeability values at higher tempera-

tures (and lower pressure differences) when water saturation is in t:he

80-90% range may be caused in part by end effects.

Another laboratory phenomenon, Klinkenberg slip, is investigated

in the next section.

6.3.5 Klinkenberg Slip

Klinkenberg slip, discussed in Section 2, was not considered in

the analysis of the gas flow data. Figure 7 of Arihara (1974) suggests

that the expected correction for slip at the high (8-12 atm) pressures

used in this study would be fairly small. Ideally, the gas relative

permeability curve should be lowered slightly to correct the minor slip

effect. Amyx, Bass, and Whiting (1960) show that:

kuncorrected = + - b

kcorrected Pm (6-72)

-131-

where :

b = Klinkenberg slip factor

'm = mean pressure

If b = 0.2 atm and p = 10 atm, then: m

b 1 +- = uncorrected Pm kcorrected

k = 1.02

The above calculations are representative of the treatment of

absolute permeability data. Slip effects for relative permeability data

have been discussed in the literature survey and in the steam-water re-

sults sections. Rose (1948) showed that gas relative permeability can

include slip by dividing the effective gas permeability by the absolute

gas permeability determined at the same mean pressure. Correcting the

water absolute permeability in this experiment by a factor of 1.02 is a

negligible correction.

Davidson (1969) observed that the gas-oil permeability ratio,

kg/ko, increased with increased temperature. Davidson believed this

temperature dependence was caused by gas molecular slippage (Klinkenberg

effect). Using either nitrogen or steam and Chevron USP Grade No. 15

white oil, Davidson did not consider the effect of gas volume expansion

or gas-mixture viscosity change caused by the vaporization of oil.

Reasonable values can be estimated for oil vapor pressure and

oil vapor viscosity. Assuming that vaporized oil does not condense while

passing through the heat exchanger, the produced gas volumes measured may

approximately represent the total gas (nitrogen + oil vapor) volume. The

observed temperature dependence might be caused by a large

-132-

mole-fraction of low viscosity oil vapor in the gas stream at elevated

temperatures. Additional study is required to support this explanation.

The next section summarizes a brief study of the effect of tem-

perature on the absolute permeability to water of the consolidated syn-

thetic sandstone core.

6.3.6 Effect of Temperature on Absolute Permeability

Although the results are not shown here, the water absolute

permeability of the synthetic sandstone did not depend upon temperature

in the range 70-300°F and at the low confining pressures used in th:is

study. This result is in agreement with the results of Arihara (19'74)

and Cas& (1976) in that minor permeability reductions with increased

temperature have been observed at low confining pressures (300-500 psig).

Additional experimental considerations and problems are presented

in the next section.

6.3.7 Additional Considerations

Problems not considered in the preceding are core homogeneity,

elastomer gas generation at high temperatures, and the possibility of

turbulent flow.

Previous studies by Arihara (1974) and Chen (1976) with the arti-

ficial consolidated sandstone cores showed reasonable homogeneity w:ith

the same fabrication methods used in this study. However, sleeve giSS

generation was a major problem in Chen's study. Sleeve gas was not ob-

served in the isothermal flow experiment because of the presence of

nitrogen in the flowstream. Prior to nitrogen breakthrough, no noneonden-

sible vapor bubbles were observed flowing through the outlet sight glass.

-133-

This lack of apparent gas generation in total liquid flow was expected

because sleeve gas is primarily generated when the sleeve is in the :pres-

ence of steam.

At the end of the experimental runs, it was determined that the

choice of core sleeve elastomer may affect core wettability. By observ-

ing water droplets placed on dry cores, it was concluded that unused

synthetic cores, unused Berea cores, and synthetic cores used at elevated

temperatures in ethylene propylene sleeves were more water wet than syn-

thetic cores used at elevated temperatures in silicone rubber sleeves.

Finally, it is believed that gas phase turbulence was not impor-

tant because, as shown in Fig. 6-18, there was no gas relative permeability

lowering with increased pressure drop.

6 . 3 . 8 Future Improvements

Future experiments should be run to determine the effect of con-

fining pressure on the temperature sensitivity of relative permeability.

These experiments should be conducted with both synthetic and natural

consolidated sandstones. Capillary pressure data should also be obtained

for each sandstone.

Several improvements in the apparatus are also planned. The use

of automatic regulating valves, an inlet flowrator, and the injection of

nitrogen gas already saturated with water at core temperature and in:Let

pressure will help reduce the labor required to obtain and analyze the

data from these experiments. A more durable, high pressure liquid-gas

separator should also be designed and fabricated. It is also highly de-

sirable to use elastomer confining sleeves or other materials that do not

degas at high temperatures.

-134-

Finally, a study should be made to determine the extent of core

anisotropy caused by: (1) steam-water hydrothermal alteration, ( 2 ) fines

migration, and (3) fabrication technique.

We turn now to a comparison of the nonisothermal, steam-water

relative permeability curves discussed in Section 6 . 2 , and the isothermal,

gas-water relative permeability curves just presented in Section 6 . 3 .

6 . 4 Comparison of Internal (Nonisothermal) and External (Isothermal

Drive Experimental Results

A comparison of the isothermal and nonisothermal relative permea-

bility curves indicates that: (1) the high flow rate steam (nonisothermal,

internal drive) relative permeability is less than the gas (isothermal,

external gas drive) relative permeability for high water saturations, (2)

the steam relative permeability appears to be greater than the gas (iso-

thermal) relative permeability for the lowest water saturations examined,

and ( 3 ) the water relative permeability for the nonisothermal, internal

drive is less than the water relative permeability for the isothermal

external gas drive.

One possible explanation for the difference between the steam and

gas relative permeability curves is core anisotropy, as explained by Corey

and Rathjens ( 1 9 5 6 ) in Chapter 2 . 3 . A high permeability channel may exist

along the length of the core axis at the probe guide - core boundary. The

gas drive data would be influenced more by this possible anisotropy than

the boiling flow (nonisothermal) data. The gas-drive data tends to indi-

cate a low critical gas saturation ( 4 % ) , which is in agreement with the

Corey-Rathjens model. Unfortunately, the probe guide is required for the

-135-

boiling flow experiments unless a gamma ray or other external saturation

measuring device is available.

A second possible explanation for the difference between steam

and gas relative permeability curves is a difference in vapor saturation

distribution.

In the case of the external gas drive, nitrogen would be expected

to enter and flow through the largest pores first. A s flow progresses,

and the nitrogen and water vapor mixture saturation at a given cross-

section increases, more of the gas enters smaller pores. As gas enters

smaller pores, water would be displaced into the larger pores to flow

through the core. Thus, some of the larger pores may become partially

blocked with liquid water at high gas saturations.

In the case of the internal boiling drive, as the pressure is

lowered, water vaporizes both in small and large pores. Vapor (steam)

in the large pores, which dominates volumetric flow, represents only a

part of the total vapor saturation. Although vapor may flow in smaller

pores, there will be a great resistance to flow. This distribution effect

could explain why the steam relative permeability is so low at low vapor

saturations. However, as vaporization continues, the larger pores clomi-

nate flow and channel the vaporized water. The small, or high flow re-

sistance, pores already have some vapor saturation (depending upon

vapor pressure lowering) and evaporation may allow the transport of

water molecules without significant blockage of vapor flow by liquid

water. This last comment may explain the high steam relative permea-

bilities at high vapor saturations.

It is believed that the above explanation is a viable possl-

bility despite the report of no difference in external and internal gas

-136-

drive relative permeabilities by Stewart et al. (1953) . In many of the

petroleum industry internal gas drive studies, only one of several com-

ponents vaporize. Much of the original fluid remains liquid. In this

boiling flow study, a single component fluid, water, was used. Thus,

liquid-vapor interference in porous media may be different for a single

component fluid than for a multi-component fluid.

One other difference between the isothermal and nonisothermal

experiments lies in the method of determining water saturation. Water

saturation in the isothermal external gas-drive experiments was obtained

by material balance calculations. Water saturation for the nonisothermal

boiling flow experiments was obtained with the capacitance probe. As

discussed earlier in Section 6.2, when radial heat transfer dominates in

the vaporization process, measured water saturations may be lower than

actual average values due to the distribution of water vapor in the core

cross-section normal to the axis. At high water saturations in the three

nonisothermal experiments, the more radial heat gain dominated the results,

the higher the steam relative permeability.

Thus, the high steam relative permeabilities should probably be

graphed at higher gas saturations or lower water saturations. This would

bring the results of the low flow rate runs, SW1 and SW2, into closer

agreement with the high flow rate run, SW3. Correcting the nonisothermal

results for these complex heat transfer effects would also help support

the explanation for the difference between the isothermal and nonisothermal

experimental results. Of course, a nonhomogeneous radial variation in

steam saturation is not considered in the interpretation of data. The

point is that the less the radial heat transfer dominates the steam

saturation, the greater the difference between nonisothermal and isothermal

experimental results.

-137-

This concludes the discussion of the results of this experimental

study. The next section presents the conclusions drawn from the study of

the capacitance saturation probe, steam-water relative permeability, and

gas-drive relative permeability at elevated temperatures.

7. RESULTS AND CONCLUSIONS

The conclusions of this study concern two major studies: calibration

and evaluation of the capacitance probe for liquid detection, and rela-

tive permeability determination. Thus the results of these studies will

be discussed under the topics: (1) capacitance probe calibration data,

(2) analysis of the steam-water flow data, (3) analysis of the nitrogen-

water flow data, and ( 4 ) comparison of the internal drive steam-water

flow data and the external gas-drive nitrogen-water flow data. The

conclusions drawn from the relative permeability experiments apply only

to the synthetic cores used in this study. However, it is believed that

similar results will be obtained for natural sandstone cores.

7.1 Capacitance Probe Calibration

The capacitance probe, using the 7.5 MHz, Baker-type electronics

developed by Chen (1976), is useful for measuring water saturations when

a wide range of saturations is encountered (0 < S < 1). W

Using a Q-meter in the kHz range, the capacitance probe is useful

for measuring water saturation when only low saturations are encountered.

The apparent accuracy of water saturation measurements using t'he

capacitance probe calibration curves is about - +lo% pore volume.

7.2 Steam-Water Flow Data Conclusions

Axial conduction and radial heat transfer are important to the vapori-

zation process at low flow rates for the apparatus used in this study.

-138-

-139-

Radial heat transfer may cause the measured vapor saturation at the

core axis to be less than the actual cross-sectional average vapor satura-

tion for low flow rates.

Steam-water, drainage, boiling flow relative permeability curves

appear to be rate dependent for the apparatus used in this study. The

rate dependence may be caused by complex heat and mass transfer phenomena.

This rate effect is related to the radial heat gain which tends to domi-

nate vaporization at low flow rates.

7.3 Nitrogen-Water Flow Data Conclusions

External gas drive, nitrogen-water, drainage relative permeabilities

are not a strong function of temperature for water saturations ranging

from 0.4 to 0.9 fraction of pore volume.

External gas-drive relative permeability calculations must inchde

the elevated temperature effects of liquid vaporization on gas-mixture

volume and viscosity. A new displacement theory which includes vapori-

zation was developed in this study. Liquid vaporization may have been a

factor in the gas-oil permeability ratio temperature dependence reported

by Davidson (1969).

External gas-drive relative permeability calculations may requi.re

correction for liquid saturation reduction due to vaporization at high

temperatures, low pressures, and at large gas injection volumes.

External gas-drive relative permeability experiments can be run using

a high-pressure gas liquid separator. It was found to be easier to regu-

late the flow of two single-phase fluids than one two-phase fluid mixture.

-140-

7.4 Comparison of Internal and External Drive Flow Data

Gas relative permeability values at high water saturations for

boiling nonisothermal flow are lower than gas relative permeabilities

for the external gas-drive methods for isothermal flow. This result: may

be caused by the different positions occupied by gas within the porous

media for the two processes.

The water relative permeability curves are higher at high water

saturations for the external gas drives than for the boiling drive.

8. NOMENCLATURE

A

b

C

cS

C

C*

D

f

W

f

f

f

f' g f r

f rs

g

gl

g2

frw

f * r

fw2 - f

FS

FW

g

ENGLISH

= cross-sectional area perpendicular to flow

= Klinkenberg slip factor

= capacitance in oscillating circuit

= C measured in 100% steam-saturated core

= C measured in 100% water-saturated core

= normalized capacitance, (Cs-C)/ (Cs-Cw)

= core diameter

= flowing gas mass fraction (gas mass fraction of gas-liquid mixture passing a point in unit time)

= flowing gas volume fraction

= flowing gas volume fraction at point 1 in core

= flowing gas volume fraction at core outlet

= df /dS g g

= resonant frequency in oscillating circuit

= f in 100% steam-saturated core

= f in 100% water-saturated core

r

r

= normalized frequency,

= flowing water volume fraction

= gas mass fraction in a given section at a given time

= steam permeability reduction factor (relative permeability) as used by Grant (1977)

= water permeability reduction factor (relative permeability) as used by Grant (1977)

= acceleration due to gravity

-141-

-142-

= cumulative gas injection into core

= cumulative gas production from core as measured downstream of separator

= fluid enthalpy

= gas enthalpy

= enthalpy of injected fluid

= liquid enthalpy

= difference in enthalpy between gas and liquid

= effective permeability to liquid

= effective permeability to gas

= Henry's Law constant

= effective permeability to liquid

= gas relative permeability

= gas relative permeability calculated neglecting gas valume expansion due to water vaporization

= liquid relative permeability

= water relative permeability

= water relative permeability calculated neglecting gas volume expansion due to water vaporization

= effective permeability in the x direction

= effective permeability in the y direction

(k/p)H = horizontal mobility

(k/p)V = vertical mobility

K = absolute permeability

L = core length

LC

53 =V

= inductance in oscillating circuit

= horizontal characteristic length

= vertical characteristic length

m = mass of fluid in core

-143-

m = i n i t i a l mass of f l u i d i n c o r e

m = mass of steam i n core

i

S

m = mass of water i n core W


NR = Reynolds Number

P = p r e s s u r e

Pavg

pb ar

= a r i t h m e t i c average of in le t and o u t l e t c o r e a b s o l u t e p r e s s u r e s

= a b s o l u t e ba romet r i c p r e s s u r e

= c a p i l l a r y p ressure , p -p

= gas p r e s s u r e

= i n l e t p r e s s u r e

= l i q u i d p r e s s u r e

= a b s o l u t e mean p r e s s u r e i n gas phase

= p a r t i a l p r e s s u r e of component

PC

pg

P i n

PI1

Pm

pP

g R

pvP = vapor p r e s s u r e

PT

P = per ime te r around core , 21'r*radius of c y l i n d r i c a l co re

= t o t a l p r e s s u r e

q = gas flow rate

qH

qi

q t

= h e a t flow ra te

= i n j e c t i o n rate d iv ided by c r o s s- s e c t i o n a l area

= t o t a l volumetr ic flow rate

(qp/Ap) = product o f flow rate, v i s c o s i t y , and r e c i p r o c a l p r e s s u r e drop f o r s ingle- phase f low of a "base" f l u i d through a c o r e

Qi = cumulative gas i n j e c t i o n i n t o co re expressed i n pore volumes

R = u n i v e r s a l gas c o n s t a n t

R = e f f e c t i v e gas pe rmeab i l i ty normalized t o e f f e c t i v e gas permea- ?I b i l i t y a t t h e r e s i d u a l l i q u i d s a t u r a t i o n

-144-

= e f f e c t i v e l i q u i d permeabi l i ty normalized t o t h e a b s o l u t e l i q u i d pe rmeab i l i t y

= gas volumetr ic s a t u r a t i o n , f r a c t i o n pore volume

= cr i t i ca l gas s a t u r a t i o n

= i n i t i a l gas s a t u r a t i o n

= gas s a t u r a t i o n a t o u t l e t end of co re - S = average gas s a t u r a t i o n i n co re

sa

Rr

%* sm

sW

s w i

= l i q u i d volumet r ic s a t u r a t i o n , f r a c t i o n pore volume

= r e s i d u a l l i q u i d s a t u r a t i o n

= normalized l i q u i d s a t u r a t i o n , (Sa-Sar) / (l-Sgc-Sar)

= ex t r apo la t ed endpoint s a t u r a t i o n when k = 0

= o i l s a t u r a t i o n r g

= water volumet r ic s a t u r a t i o n , f r a c t i o n pore volume

= i r r e d u c i b l e water s a t u r a t i o n

= average water s a t u r a t i o n i n co re

sW* = normalized water s a t u r a t i o n , (S -S .) / (1-Sgc-Swi)

= time

w w 1

t

= c h a r a c t e r i s t i c time t o travel c h a r a c t e r i s t i c h o r i z o n t a l l eng th tH

= c h a r a c t e r i s t i c time t o travel c h a r a c t e r i s t i c vertical l eng th

T = temperature

= temperature of a i r b a t h environment surrounding coreholder

= gas microscopic v e l o c i t y U g

"H = h o r i z o n t a l c h a r a c t e r i s t i c v e l o c i t y

= l i q u i d microscopic v e l o c i t y

= v e r t i c a l c h a r a c t e r i s t i c v e l o c i t y Yl u = o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t for c y l i n d r i c a l co re

= gas s p e c i f i c volume

-145-

- VR = l i q u i d s p e c i f i c volume

= steam s p e c i f i c volume

= water s p e c i f i c volume

= cumulative volume of N i n j e c t e d a t co re cond i t i ons

V = pore volume

- V S - V

W

vN2 2

P W = t o t a l weight rate of flow

W = gas weight rate of flow g

R W = l i q u i d weight rate of flow

W = 100% l i q u i d water flow rate taken from a graph of l o g wel lbore 0 discharge rate vs d ischarge en tha lpy (Grant, 1977)

W = cumulative water produced from co re through s e p a r a t o r

X = h o r i z o n t a l d i s t a n c e P

X = h y p o t h e t i c a l d i s t a n c e from t h e i n l e t t h a t a dry reg ion , caused dry by water vapor i za t ion , extends into the core

%20

xN2 = mole f r a c t i o n H 0 i n l i q u i d phase 2

= mole f r a c t i o n N2 i n l i q u i d phase

= h o r i z o n t a l d i s t a n c e t o co re o u t l e t x2

Y = vert ical d i s t a n c e

'H,O = mole f r a c t i o n H 0 i n vapor phase 2

L

Yd2 = mole f r a c t i o n N2 i n vapor phase

Z = real gas compres s ib i l i t y f a c t o r

GREEK

A = d i f f e r e n c e

x = thermal conduc t iv i ty

x = pore s i z e d i s t r i b u t i o n index used i n Corey-type equat ions

= relative r e c i p r o c a l mob i l i t y a t co re o u t l e t

C

-146-

1-I- = average relative r e c i p r o c a l mob i l i t y i n core

% = gas v i s c o s i t y

= water vapor v i s c o s i t y

= l i q u i d v i s c o s i t y

= n i t r o g e n v i s c o s i t y

= o i l v i s c o s i t y

= steam v i s c o s i t y

= water v i s c o s i t y

= f l u i d d e n s i t y

= p o r o s i t y , in te rconnec ted pore volume f r a c t i o n of bu lk volume

= s a t u r a t i o n probe s i g n a l

= @ measured i n 100% steam- saturated co re

= Q measured i n 100% water- sa tura ted co re

= normalized probe s i g n a l , (as-@) / ('PS-Qw)

ARRREVIATIONS

ICH = k i l o h e r t z , 10 c y c l e s p e r second 3

= megahertz, 10 cyc l e s p e r second 6

-12

z

mz

P f = p ico fa rad , 10 f a r ad

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Hsieh, C.H., and Ramey, H . J . , Jr.: "An In spec t ion of Experimental Data on Vapor P re s su re Lowering i n Porous Media,'' Proc., 1978 Geothermal Resources Council Annual Meeting, Hilo, HA, July 25-27, 1978.

Johnson, C.E., Jr.: "A Two-Point Graphical Determinat ion of t he Constants S and Sm i n t h e Corey Equation f o r Gas-Oil Relative Permeabi l i ty &€io , : J. Pe t . Tech. (Oct . 1968), 1111-1113,

Johnson, E.F., Boss le r , D.P., and Naumann, V.O.: "Calculat ion of Relative Permeabi l i ty from Displacement Experiments," Trans., AIME (1959), -9 216- 370-372.

Jones, S.C., and Rosze l le , W.O.: "Graphical Techniques f o r Determining Relative Permeabi l i ty from Displacement Experiments," SPE Paper No. 6045, presented a t t h e 51st Annual F a l l Meeting, SPE o f AIME, Denver, CO, Oct. 3-6, 1976.

-151-

Katz, D.L., Corne l l , D., Kobayashi, R., Poettmann, F.H., Vary, J .A. , Elenbaas, J . R . , and Weinaug, C.F.: Handbook of Natura l Gas Engi- neer ing, New York: McGraw-Hill Book Company, 1959.

Klinkenberg, L . J . : "The Permeabi l i ty of Porous Media t o Liquids and Gases, 'I

API Drill. and Prod. Prac. (1941), 200-213.

Laws, V., and Sharpe, R.W.: "The Response o f a n H.F. Moisture Probe t o

Kruger, P., and Ramey, H. J . , Jr.: "Stimulation and Reservoir Engineering o f Geothermal Resources," SGP-TR-1, Stanford Univers i ty , June 1974.

Kyte, J.R., and Rapoport, L.A.: "Linear Waterflood Behavior and End Ef- fects i n Water-Wet Porous Media," Trans ., AIME (1958), 213, 423-426.

Lone S t a r Lafarge Company: " Proper t ies and Uses of Fondu Calcium Alumi- n a t e Cement" and "Refractory Appl ica t ions of Fondu Calcium Alumi- n a t e Cement," 'P .O. Box 448, 977 Norfolk Square, Norfolk, Vi rg in ia , 23501.

Martin, J , C . : "Analysis of I n t e r n a l Steam Drive i n Geothermal Reservoirs ,I1 J. Pet . Tech. (Dec. 1975), 1493-1499.

Meador, R.A., and Cox, P.T.: "Dielectric Constant Logging, A S a l i n i t y Independent Est imat ion of Formation Water Volume," SPE Paper No. 5504, p resen ted a t the 50th Annual F a l l Meeting, SPE of AIME, Dallas, Texas, Sept . 28-Oct. 1, 1975.

Mercer, J.W., and Faust , C.R.: "Simulation of Water and Vapor Dominated Hydrothermal Reservoirs ," SPE Paper No. 5520, p resen ted a t t h e 50th Annual F a l l Meeting, SPEof "E, Dallas, Texas, Sept . 28-Oct. 1, 1975.

Mickley, H.S., Sherwood, T.K., and Reed, C.E.: Applied Mathematics i n Chemical Engineering, 2nd Edi t ion , New York: McGraw-Hill Book Co., 1957, 54.

Miller, F.G., and Seban, R.A.: "Technical Note 319: The Conduction of Heat I n c i d e n t t o t h e Flow of Vaporizing F l u i d s i n Porous Media," J. Pe t . Tech. (Dec. 1955), 45-47.

Miller, F.G.: "Steady Flow of Two-Phase S ing le Component F lu ids Through Porous Media," Trans., AIME (1951), 192, 205-216.

-152-

Moench, A.F., and Herke l ra th , W.N.: "The E f f e c t of Vapor-Pressure Lower- i n g upon P res su re Drawdown and Buildup i n Geothermal Steam Wells, '' Proc. , 1978 Geothermal Resources Council Annual Meeting, Hi lo , HA, J u l y 25-27, 1978.

Moench, A.F.: "Simulation of Steam Transpor t i n Vapor-Dominated Geothermal Reservoi rs , " Open F i l e Report: 76-607, USGS, Menlo Park, CA, 1976.

Osoba, J.S., Richardson, J . G . , Kerver, J .K. , Hafford, J .A. , and Blair, P.M.: "Laboratory Measurements of Relative Permeabi l i ty ," Trans., AIM2 (1951), 192, 47-56. -

Owens, W.W., P a r r i s h , D.R., Lamoreaux, W.E.: "An Evalua t ion of a Gas Drive Method f o r Determining Relative Permeabi l i ty Rela t ionships ," -- Trans 9 AIME (1956), 207, 275-280.

Parkhomenko, E.: Dielectric P r o p e r t i e s o f Rocks, e d i t e d and t r a n s l a t e d from t h e Russian by G.V. Keller, New York: Plenum Pres s , 1967.

Poston, S .W., Ysrael, S. , Hossain, A.K.J.S., Montgomery, E.F., 111, and Ramey, H . J . , Jr.: "The Ef fec t of Temperature on I r r e d u c i b l e Water S a t u r a t i o n and Relative Permeabi l i ty of Unconsolidated Sands," SOC. Pe t . Eng. J. (June 1970), 171-180.

Rapoport, L.A., and Leas, W . J . : " Proper t ies of Linear Waterf loods," Trans., AIME (1953), 198, 139-148.

Rapoport, L.A.: "Scal ing Laws f o r Use i n Design and Operat ion of Water- O i l Flow Models, '' Trans., AIME (19551, *, 143-150 0

Reid, R.C., P r a u s n i t z , J .M. , and Sherwood, T.K.: The P r o p e r t i e s of Gases and Liquids, 3rd Ed i t i on , New York: McGraw-Hill Book Co., 1977.

Richardson, J .G. , and Perk ins , F.M., Jr.: "A Laboratory I n v e s t i g a t i o n of t h e E f f e c t of Rate on Recovery of O i l by Water Flooding," Trans AIME (1957), 210, 114-121.

-_a 3

-

Scheidegger, A.: The Physics of Flow Through Porous Media, Third Edi t ion , Canada: Un ive r s i t y o f Toronto P res s , 1974.

Shinohara, K.: " Calcu la t ion and Use of Steam/Water Relative Permeabi l i- ties i n Geothermal Reservoi rs , " M.S. Report, S tanford Univers i ty , June 1978.

Sinnokrot , A.A., Ramey, H . J . , Jr., and Marsden, S.S., Jr.: "Effect of Temperature Level upon Cap i l l a ry P re s su re Curves," SOC. Pe t . Eng. J. (Mar. 1971), 13-22.

- -

-153-

Smyth, C.P.: Dielectric Behavior and S t r u c t u r e , New York: McGraw-Hill Book Co., Inc . , 1955.

Staggs, H.J . , and Herbeck, E.F.: "Reservoir S imula t ion Models--An Engi- nee r ing Overview," J. Pe t . Tech. (Dec. 1971), 1428-1436.

Standing, M.B . : "Notes on Relative Permeabi l i ty Rela t ionships ," Lecture Notes, Petroleum Engineering Department, S t an fo rd Univers i ty , CA, 1975.

S tewar t , C.R., Craig, F.F., Jr., and Morse, R.A.: "Determination of Limestone Performance C h a r a c t e r i s t i c s by Model Flow Tests," Trans., AIME (1953), 198, 93-102.

Thomas, A.M.: "In S i t u Measurement o f Moisture i n S o i l and S imi l a r Sub- s t a n c e s by 'Fr inge ' Capacitance," J. Sc i . Instrum. (1966), s, 21-27.

Thomas, L.K., and Pierson , R.G.: "Three Dimensional Geothermal Reservoi r Simulat ion, SOC. P e t . Eng. J. (Apr . 19781, 151-161.

Toronyi, R.M. : "Two-Phase, Two-Dimensional S imula t ion of a Geothermal Reservoi r and t h e Wellbore System," Ph.D. D i s s e r t a t i o n , Pennsyl- van ia S t a t e Un ive r s i t y , Nov. 1974.

Trimble, A.E., and Menzie, D .E. : "Steam Mobi l i ty i n Porous Media, SPE Paper No. 5571, presented a t t h e 50th Annual F a l l Meeting, SPE of AIME, Dallas, Texas, Sept . 28-Oct. 1, 1975.

van Beek, L.K.H. : "Dielectric Behaviour of Heterogeneous Systems," Prog- ress i n Dielectrics, 7, J .B . Birks (Ed.), Cleveland, Ohio: Rubber Co. P re s s , 1967.

-

Weinbrandt, R.M., Ramey, H.J. , Jr., and Cass6, F.J.: "The Effect Of

Temperature on Relative Permeabi l i ty o f Consol idated Rocks," SOC. Pe t . Eng. J . (Oct. 19751, 376.

Welge, H.J. : "A S impl i f i ed Method f o r Computing O i l Recovery by Gas o r Water Drive," Trans., AIME (1952), 195 91-98.

-9

APPENDIX 1: EQUIPMENT MANUFACTURERS AND SUPPLIERS

The equipment manufacturers and/or s u p p l i e r s are l i s t e d below. Much of t h e equipment used i n t h i s s tudy was modified by o r designed and cons t ruc ted wi th t h e h e l p of P e t e r Gordon, Jon G r i m , and Paul P e t i t . The Hassler- type coreholder was const ructed from a des ign furnished by S.C. Jones, of t h e Marathon O i l Company.

1. Air Bath - NAPCO, Model 430 Van Waters & Rogers Redwood Ci ty , CA (369-5561)

2. Tubular Furnace - Model 1027 Varian Pa lo Al to , CA (493-4000)

3. Pump - Model Rl2lA Milton Roy San Mateo, CA (341-8796)

4 . Accumulator - Greero la to r Model 20-15TMR-S-l/2WS Hydraulic Controls Inc . Emeryville, CA (658-8300)

5. Temperature Recorder - Model Speedomax W Multi- Point Recorder Leeds & Northrup Co. San Mateo, CA (349-6656)

6 . Pressure Recorder - Model EU-20W Heathki t E l e c t r o n i c Redwood Ci ty , CA (365-8155)

7. Flowrator F i scher & P o r t e r Co. Walnut Creek, CA (933-8880)

8. Temperature C o n t r o l l e r - Model 61329-054 Van Water & Rogers Redwood Ci ty , CA (369-5561)

9. P ressure Transducer - Model -15, Celesco I n d u s t r i e s GADO Instrument S a l e s Mountain View, CA (961-2222)

10. Pressure I n d i c a t o r - Model CD25, Celesco I n d u s t r i e s GADO Instrument Sales Mountain View, CA (961-2222)

-154-

-155-

11. Pressure Gage - AMETEK, Model P1536, Hel icoid , Model KMonel 460 Jensen Instrument Co. So. San Francisco, CA (589-9720)

12. P ressure Regulator, Model 2-580 Matheson Gas Products Newark, CA (793-2559)

13. Sheathed Thermocouple, Thermocouple Wire Claud S . Gordon Co. San Carlos, CA (591-7070)

14. Valve, F i t t i n g , F i l t e r - Swagelok, NUPRO, WHITEY Van Dyke Valve & F i t t i n g Conax Instrument Laboratory Sunnyvale, CA (734-3145) Palo Al to , CA (328-1040)

15. Pipe , Tubing Tubesales San Francisco, CA (922-2240)

16. 0-Ring McDowell & Co. Hayward, CA (785-7744)

17. Viton Core Sleeve - Viton A Tubing West American Rubber Co . Orange, CA (714- 532-3355)

18. Core - Berea Sandstone Core The Cleveland Q u a r r i e s Co. Amherst, Ohio (216- 986-4501)

19. D i g i t a l Multimeter - Fluke 8000A Fluke Western Technical Center Santa Clara, CA (244-1505)

20. S t a i n l e s s S t e e l - Pyrex Tubular S e a l s Larson E l e c t r o n i c Glass Redwood Ci ty , CA (369-6734)

21. Fondu Calcium Aluminate Cement San Francisco Materials Co. San Francisco, CA (282-0133)

22. S i l i c o n e Rubber Core Sleeve Alasco Rubber and P l a s t i c Corp. Burlingame, CA (697-1420)

23. Ottowa S i l i c a Sand Smith I n d u s t r i a l Supply Co. San Francisco, CA (822-3600)

24. Teflon Tubing - Zero Gage, Thin-Walled C a d i l l a c P l a s t i c and Chemical Co. So. San Francisco, CA (589-1833)

-156-

25. Q-Meter - Model 4342-A Continental Rentals Santa Clara, CA (735-8300)

26. 20-30 Grade Silica Sand - Unisil - 50 lb net Unisil Corp . Gopher State Silica Div. Le Soeur, Minn. 56058

27. Wet Test Meter Precision Scientific Co. VWR Scientific San Francisco, CA (469-0100)

APPENDIX 2: CAPACITANCE PROBE DETAILS

F igu res A2-1 through A2-5 p re sen t d e t a i l s of capac i t ance probe equipment similar t o t h a t used i n t h i s s tudy . The f i g u r e s were taken from Ar ihara (1974) and Chen (1976), and a d d i t i o n a l d e t a i l s are found i n t h e i r r e p o r t s . The o r i g i n a l des ign of t he probe and the d e t e c t i o n c i r c u i t s were fu rn i shed by D r . Paul Baker through t h e cou r t e sy of t h e Chevron O i l F i e l d Research Company, La Habra, C a l i f o r n i a .

-157-

-158- CIC

I- O

u w

h v) w l

3" u

W \ lz t- n 4 \ E

u 0 2 W

I

Q Z = a

0 - l ' 0

0 H n w m 3

4

N I

.. 4

Frr

-159-

t >

(Y -I +

.

3 0

+ P I

r I I I I I I I

H u W

.. e4

I CJ c

-160-

I I

W L T c r 3 W t--I

I I

-161-

1 I I 1 1 TI

-162- 0 I-

H U

W 0 F9

Fr 0

U H

3 W X U m

APPENDIX 3: TABULATED EXPERIMENTAL DATA AND CALCULATIONS

This appendix con ta ins t h e experimental d a t a and c a l c u l a t i o n s

d iscussed i n Sec t ion 6. Appendix A3.1 p r e s e n t s t h e capac i tance probe

c a l i b r a t i o n d a t a . Appendix A3.2 p r e s e n t s t h e d a t a f o r t h e nonisothermal ,

s t eady , steam-water flow experiments. Appendix A3.3 con ta ins t h e d a t a

and c a l c u l a t i o n s f o r t h e i so thermal , unsteady, ni t rogen- water flow exper i-

ments.

A3 .1 Capacitance Probe C a l i b r a t i o n

This appendix con ta ins t h e experimental d a t a and c a l c u l a t i o n s d i s-

cussed i n Sec t ion 6.1. Tables A3.1-1 and A3.1-2 p r e s e n t d a t a ob ta ined

us ing t h e Baker- type e l e c t r o n i c s be fo re i t was r epa i r ed . Tables A3.1-3

through A3.1-9 p r e s e n t d a t a ob ta ined a t d i f f e r e n t f requencies us ing t h e

Q-meter. Tables A3.1-10 through A3.1-13 p r e s e n t t h e f i n a l c a l i b r a t i o n

d a t a ob ta ined w i t h two d i f f e r e n t Baker-type e l e c t r o n i c s packages.

A3.2 Nonisothermal, Steady, Steam-Water Flow

This appendix con ta ins t h e experimental d a t a d i scussed i n Sec t ion

6.2. Tables A3.2-1 through A3.2-3 p r e s e n t t h e co re temperatures and

normalized probe s i g n a l p r o f i l e s f o r Runs SW1, SW2, and SW3.

A3.3 I so thermal , Unsteady, Nitrogen-Water Flow

This appendix con ta ins t h e experimental d a t a and c a l c u l a t i o n s d i s-

cussed i n Sec t ion 6.3. Tables A3.3-1 through A3.3-3 p re sen t t h e d a t a

f o r Run NW1. Tables A3.3-4 through A3.3-6 p re sen t t h e d a t a f o r Run NW2.

Tables A3.3-7 through A3.3-9 p re sen t t h e d a t a f o r Run NW3. The i so thermal

-163-

-164-

gas d r i v e d a t a are conta ined i n Tables A3.3-1, A3.3-4, and A3.3-7. The

c a l c u l a t e d va lues used i n t h e g raph ica l a n a l y s i s are presented i n Tables

A3.3-2, A3.3-5, and A3.3-8. The r e s u l t s of t h e g raph ica l a n a l y s i s and

t h e r e l a t i v e pe rmeab i l i t y c a l c u l a t i o n s are conta ined i n Tables A3.3-3,

A3.3-6, and A3.3-9. The r e s u l t s presented i n Appendix A3.3 are sum-

marized g r a p h i c a l l y i n Fig. 6-18.

-165-

e v)

e

e

e

e

e

e

n P

W

e

e

e

e

e

e

e 3

E 0

mcnuF!omo~ocncnmNa3?u3cnd.ma3ud . . . . . . . . . . . . . . . . . . . .

N ? N - m u 3 u m N m d O u 3 O m . N m u 3 . ~ h m u 3 . . . . . . . . . . . . . . . . . .

-166-

? m b

b

m co

“1 U co

rl

m co

Qz m m

0: m m

N

h m

0

rn cn

0

Q 0 rl

U

N 0 rl

rl

m 0 rl

N

b

m b

rl

m co

rl

rl m

m m cn

CD

U cn

03

m m

0

b cn

CT)

cn cn

N

0 0 rl

? N 0 rl

m rl 0 rl

co

m m h

U

co m

* N m

U m

m m m

m

cn

m b m

rl

0 0 rl

9 U 0 rl

co e 0 rl

“1 N 0 rl

CD rl

\D

m h

U

m 03

y1 U m

Qz U m

m m m

b

CD m

m

m h

0

0 0 rl

‘9 m 0 rl

e m 0 rl

* N 0 rl

0 N

-167-

TABLE A3.1-3: CAPACITANCE PROBE CALIBRATION DATA: Q-METER (7.5 MHz);

11/22-23/77 310"F, 7.5 MHz, C,=170 pf 20-30 MESH SAND

G r a m H20 i n Core 726.5 610.6

Room Temp. OF 75 .O 68.0

Dis tance from I n l e t End, i n

3

5

7

9

11

13

15

17

19

21

23

AC W

-2.61

-2.22

-2.25

-2.32

-2.31

-2.39

-2.50

-2.82

-2.58

-2.78

-1.78

mc

-2.64

-2.19

-2.00

-2.30

-2.51

-2.43

-2.61

-2.78

-2.78

-2.81

-1.74

.L

414.6

72.5

AC, pf

AC

-1.56

-1.08

-.98

-1.32

-1.46

-1.68

-1.76

-1.95

-1.92

-1.72

- .91

207.3

74.2

AC

-0.22

+O .03

0 .oo

-0.32

-0.28

-0.39

-0.25

-0.19

-0.20

-0.08

-0.26

0

71.0

AC 5

+0.79

+O. 86

+O .89

+O. 79

+O. 79

+0.78

+O .81

+O .83

+O. 81

+0.91

+O. 56

Note: tempera ture c o r r e c t i o n = -0.04 pf/'F

11124-25177

Gram H20 in Core

Distance from Inlet End, in

2

4

6

8

10

1 2

14

16

18

20

22

724 .1

AC W

-3.52

- 2.21

-2.24

-2.49

-2.74

-2.48

-2.89

-2.79

-3.18

-3.29

-3.42

310"F, C1 = 194 pf

509.9 343.3

LC, Pf

AC AC

-3.41 -3.19

- 2.41 -2.13

-2.52 - 2.21

-2.52 -2.32

-2.90 -2.94

-2.60 -2.52

-2.88 - 2.81

-3.06 - 2.81

-3.39 -3.04

-3 44 -3 10

-3.31 -3 00

20-30 MESH SAND

124.3 0

AC

-3.19

-2.06

-2.19

- 2.41

-2.69

-2.45

-2.72

-2.71

-3.00

-3.05

AcS

+2.13

+2.19

+2.16

+2.15

+2.05

+2.13

+2.11

+2.06

t 2 I21

+2.20

-3.05 +2 3 1

Note: The 47 mH coil used at 40 kHz did not demonstrate a well-defined temperature sensitivity.

-169-

TABLE A3.1-5: CAPACITANCE PROBE CALIBRATION DATA: Q-METER (750 kHz)

11126-27177 310°F, C1 = 182 pf

G r a m s H20 i n C o r e 723.7 534.7 347.4

R o o m T e m p . OF 76.7 77.5 73.3

AC, Pf

D i s t a n c e f r o m In le t End, i n AC

W AC AC

2 -4.41 -3.29 -2.21

4 -3.41 -2.59 -1.55

6 -3.61 -2.81 -1.61

8 -3.65 -3.30 -1.81

10 -3.81 -3.22 -2.06

12 -3.65 -2.93 -2.00

1 4 -3.88 -3.05 -2.20

16 -3.75 -3.28 -2.33

18 -4.15 -3.59 -2.34

20 -4.21 -3.54 -2.42

22 -4.37 -3.19 -1.82

N o t e : t e m p e r a t u r e correction = -0.04 pf/OF

20-30 MESH SAND

140.5

71.0

AC

-0.80

-0.81

-0.91

-0.79

-0.81

-0.82

-0.65

-0.60

-0.58

-0.55

-0.72

0

68 .O

AcS

4-0.19

+O. 24

H . 2 2

+O .29

+O. 28

H . 2 9

H . 2 2

H.30

M.33

+O .39

+O .48

-170-


12/1/77 310°F, C, = 100 pf

Note: temperature correction = -0.03 pf/OF

Grams H20 in Core

Room Temp. OF


2

4

6

8

10

1 2

14

16

18

20

22

727.1

68.8

-4.38

-3.05

-3.61

-3.42

-3.71

-3.30

-3.69

-3.61

-4.03

-4.09

-4.21

I

576 .O

74 .O

AC

-4.11

-2.92

-3.16

-3.25

-3.55

-3.09

-3.50

-3.40

-3.89

-4.02

-3.99

405.1

77.7

AC, Pf

AC

-3.60

-2.72

-2.75

-2.79

-3.06

-2.72

-3.04

-3.02

-3.48

-3.62

-3.56

20-30 MESH SAND

145.3

72.8

AC

-3.10

-2.15

-2.31

-2.37

-2.64

-2.52

-2.55

-2.42

-2.55

-2.71

-2.75

0

71.8

+l. 29

+l. 30

+l. 31

+l. 30

+l. 31

+l. 31

+1.41

+l. 40

+l. 39

+1.41

+l. 4 1

-171-


G r a m s H20 i n C o r e

Room Temp. OF

D i s t a n c e f r o m In l e t E n d . i n

2

4

6

8

10

1 2

14

16

18

20

22

12/5/77 305"F, C1 = 55 pf

N o t e : t e m p e r a t u r e correction = 0-.05 pf/'F

727.1

76.9

-3.38

-3.19

-3.28

-3.09

-3.39

-2.86

-2.97

-2.92

-3.26

-3.48

-3.49

557.8

78.6

AC

-3.46

-2.94

-3.20

-2.69

-3.00

-2.62

-2.70

-2.76

-3.03

-3.22

-3.25

367.7

81.1

AC, P f

AC

-2.69

-2.13

-2.27

-2.11

-2.41

-2.24

-2.37

-2.39

-2.53

-2.74

-2.59

20-30 MESH SAND

237.4

78.3

AC

-1.52

-1.12

-1.32

-1.31

-1.60

-1.61

-1.62

-1.70

-1.81

-1.93

-1.78

0

75.9

AC 5

+l. 78

+1.81

+l. 82

+l. 82

+l. 86

+l. 86

+1.91

+1.95

+1.97

+1.95

+l. 98

-172-

TABLE A3.1-8: CAPACITANCE PROBE CALIBRATION DATA: Q-METER (14 MHz)

12/6/77

G r a m s H 0 i in C o r e 2

Room Temp. O F

D i s t a n c e f r o m I n l e t End, i n

2

4

6

8

10

1 2

1 4

16

18

20

22

725.5

74.1

AC W

-4.29

-3.71

-3.86

-3.39

-3.46

-3.31

-3.32

-3.28

-3.33

-3.41

-3.79

310"F, C1 = 35 p f

559.9 390.0

76.8 80.2

AC, Pf

AC AC

-2.97 -0.81

-2.02 +O .03

-2.53 0 .oo -2.21 -0.26

-2.29 -0.53

-2.05 -0.32

-2.30 -0.43

-2.26 -0.50

-2.41 -0.61

-2.75 -0.75

-2.51 +O .04

20-30 MESH SAND

185.5

81.6

AC

+2.21

+2.59

+2.52

+2.38

+2.19

+2.41

+2.25

+2.30

+2.41

+2.32

+2.79

0

77.5

AC 5

+4.08

+4.18

+4.02

+4.03

+4.11

+4.10

+4.18

+4.28

+4.29

+4.32

+4.38

N o t e : unstable response prevented t e m p e r a t u r e ca l ib ra t ion

-173-

0 m 0

I

N

0

* m Q

* m N d

d

h m d

0

0 Q N

h rl In

m m m h

m m Q m

b

m N h

rcl a

V h

a

m c o m c o b m m c o m o m m e * m m * * * m m m ooooooooooo . . . . . . . . . . .

+ + + + + + + + + + + m m m o c o m Q Q N Q a J d 0 0 d d 0 0 0 0 0 d

ooooooooooo I I l l l l l l l + +

. . . . . . . . . . .

m m O w d c o * N * m m m N m m e N N N d d 0 ooooooooooo I I I I I I I l l l l

. . . . . . . . . . .

Q * * m m m m m m m N N N d O a 3 N N N O r n N

ooooooooooo 1 1 1 1 1 1 1 1 1 1 1

. . . . . . . . . . .

r > ~ . . . . . . . . . . . 61 0 0 0 0 0 0 0 0 0 ~ 0

1 1 1 1 1 1 1 1 1 1 1

O d Q d O m O N m N N m N m m m O m m o d 0

I I I I I I I I I I I 4 . . . . . . . . . . . a N N N N N N N N N N N Fr 0 \ w

N d c o N N c o * N N m m a a 3 * h r n * O r n * N d N e N N N N N N N N N N N

0

I I I I I I I I I I I 0 I

. . . . . . . . . . .

. . . . . . . . . . . c o h h a h m b m m N Q m m N m Q m N O m N m

N N N N N N N N N N N I I I I I I I I I I I

c o m d m o m m d c n o N o m a 3 ~ c o Q Q h b m o ~ N N N N N N N N N ~ I I I I I I I I I I I

. . . . . . . . . . .

-174-

TABLE A3.1-10: CAPACITANCE PROBE CALIBRATION DATA: BAKER-TYPE ELECTRONICS (80-170 MESH, UNIT NO. 1)

219-10178

Gram H20 in Core


4

6

8

10

12

14

16

18

20

849.2

Q, w

85.3

85.8

84.9

84.2

85.2

86 .1

86 .1

85.9

85.7

305°F

692.2

Q

83.2

84.6

84 .O

81.4

85.2

85.6

85.3

85.5

83.3

488.2 367.3

Q, (-mV>

Q,

69.4

71 .4

70.2

70.3

74.6

74.7

73 .1

74.2

70 .5

Q,

67.2

69.2

67.8

67.8

73.9

73 .3

71.7

73 .3

68.8

80-170 MESH SAND

227.2

Q,

0

Q 5

63.6

65 .3

64 .3

66.5

71 .3

70.8

68.7

69.8

65.8

57 .1

57 .1

57.0

56.6

56.4

56.3

56.2

56 .1

56 .1

-175-

n rl 0

0 z

8 d ..

z 0 H

3

rl rl

I rl

..

2

0 m 0 I

N

Frr

N 0 m

0

co m co

h

U \D rl

w

.t I

\D

m N m

. . . . . 91 r - l \ D h m O c o N U m c o r l N u m m O r l * o ~ h u r l o r l o o m r l o o o o o o o o o o m m m m m h h h h h \ D h h ~ h ~ h h h h h ~ \ D \ D \ D \ D \ D

. . . , . . . . . . . . . . . . . .

h

m rl \D

. . . . . . . . . . . . . . . . . . . . . ~ N c o \ D M c o N ~ m h U m c o O h \ D \ D h N \ D u r l

h h h h h h h h h h h h h h h h h h h h h h ~ m N N N N N r l r l r l r l r l - I N r l r l r l 0 0 0 0 0

h

m 0 h

. . . . . . . . . . . . . . . . . . . . . . e3/

o m N m ~ u h r l r l c o ~ o m \ D o ~ o m m m N N m m o m m m m m m N N m N N N r l N ~ ~ r l r l r l h h h h ~ h h h h h h h h ~ ~ h h ~ ~ h h h

k .q w

-176-

n N

c

0 z f3 H z P

L

m cn k2

Y

2

X 6Y

53 0 m 0 I

N

0 cr)

m

rl

N r l m N c o m O O N O m r l N m m U Y r n r l U Y h r l ~ r l r l

rl e rlrlNrlN~N~0orl~d~ornrncocornrnrne u) hhhhhhhhhhhhhhhUYUYUYUYUYUYUYh

. . . . . . . . . . . . . . . . . . . . . . .

N

a3 N UY

a3

0 0 h

. . . . . ,LI . . . . . . . . . . . . . . . . . r n U Y r l r n o o m r n a 3 h m r l h o U Y N N r n h h U Y r l U Y r l N N N m ~ N r l r l r l r l r l r l N d d r l 0 0 0 o o r l h h h h ~ h h h h h h h h ~ h h h h h h h h h

-177-

0

rl

.3 h

X rn 9 0 m 0

I N W

w L

..

2 0 U

H

N

\D \D r(

m h m N

U \D U VI

\D

h N \D

h

,31 ~ ~ o ~ ~ ~ ~ o o o o o o o o o o o o o o o m . . . . . . . . . . . . . . . . . . . . . . .

h h h h h h h h h h h h ~ h h h h h h h h ~ \ D

-178-

TABLE A3.2-1: NONISOTHERMAL, STEADY, STEAM-WATER FLOW DATA, RUN SW1

RUN SW1 (5/26/78)

Absolute Permeability to Water, K = 26 md

Inlet Pressure = ? (calibration lost)

Outlet Pressure = 34 psig Confining Pressure = 470 psig

Mass Flow Rate = 1.60 gm/min

Inlet Temperature = 302°F

Airbath Temperature = 302°F

Core Length = 23 1 / 4 in

Core Cross-Sectional Area = 3.14 in 2

X Distance from Inlet End, in

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6 .5 7 7.5 8 8.5 9 9.5

10 10.5 11 11.5 1 2 12 .5 13 13.5 14 14.5 15

T core OF

302

301

300

299

298

297

296

295

294

292

291

286 284 283 282 279

@ ( -mv )

87.4 88.4 88.6 88.5 88.8 88.7 88.5 88.1 87.6 87.4 87.4 87.6 87.4 87.7 87.8 87,9 87.8 87.9 87.8 87.9 87.8 87.7 87.5 87.4 86.5 86.6 87.5 87,6 87.4

@S @ 302°F (-mV>

75.2 69.6 69.6 69.5 69.7 70.0 69.9 70.1 70.2 70.1 70.3 70.0 70.0 70.0 69.9 69.6 69.6 69.8 69.4 69.4 69.2 69.1 69,O 69 .O 68.8 68.7 68.9 68.9 68.6

@ W

@ 294°F ( -mv>

88.8 89.1 89.0 89.0 89.0 89.2 88.7 88.6 88.4 88.2 88.1 88.6 88.5 88.9 89.0 88.9 89.3 89.3 89.3 89.2 89.4 89.1 89.0 89.1 88.4 88 ,4 88.9 89.1 88.5

@ -@

@ -0 5 @* = ~

s w

0.95 0.96 0.98 0.97 0.99 0.97 0.99 0.97 0.96 0.96 0.96 0.95 0.94 0.94 0.94 0.95 0.92 0.93 0.92 0.93 0.92 0.93 0 ,93 0.92 0.90 0.91 0 ,93 0 ,93 0.94

Canttnued

-179-

TABLE A3.2-1, CONTINUED


15.5 16 16 .5 17 17.5 18 18.5 1 9 19.5 20 20.5 2 1 21.5 22 22.5

T core OF

279 279 278 278 276 275 273 272 270 268 265 263 259 255

cp ( -mV)

86.9 87.3 87.0 86.4 86.3 86.4 86.5 86.6 86.2 86.3 86.3 86 .1 85.9 85.9 81.4

cp S

@ 302°F (-mV)

68.4 68.5 68.4 68.3 68.4 68.3 68.3 68.3 68.3 68.4 68.4 68.2 68.1 68.0 70.5

cp W

@ 294°F (-mW

88.5 88.5 88.2 87.7 88.0 88.1 88.0 87.7 88.4 88.4 88.4 88.0 87.8 87.9 82.7

cps-cp cp* = - cp -@ s w

0.92 0.94 0.94 0.93 0.91 0.91 0.92 0.94 0.89 0.90 0.90 0.90 0.90 0.90 0.89

-180-


RUN SW2 (6/5/78) Absolute Permeability to Water, K = 34 md Inlet Pressure = 57.0 psig Outlet Pressure = 4.0 psig Confining Pressure = 352 psi Mass Flow Rate = 0.80 gm/min Inlet Temperature = 302°F Airbath Temperature = 315°F Core Length = 23-3/16 in

X Distance From Inlet End, in

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15

T core O F

302 302 302 301 301 301 301 300 299 299 298 298 29 7 296 296 295 294 293 292 291 289 288 287 286 286 285 284 283 280

(3

(-mV>

89.3 90.5 91.8 93.3 94.9 95.6 95.5 95.5 95.6 95.4 95.2 95.1 95.3 94.8 94.7 94.6 94.6 94.8 95.1 95.0 94.9 94.9 93.5 93.6 93.5 94.5 93.6 93.4 92.7

Core Cross-Sectional Area = 3.14 in 2

@ S

@ 316°F ( -mv>

77.2 77.2 77.2 77.3 77.6 77.7 77.5 77.5 77.5 77.1 77 .O 76.9 76.8 76.4 76.3 76.5 76.6 77 .O 77.1 76.8 76.6 76.4 76.4 76.3 76.2 76.0 76.0 75.9 76 .O

(3 W

@ 303°F (-mV>

94.0 95.6 96.7 97.6 97 .8 97.9 97.8 97.5 97.2 97.0 96.9 96.8 96.9 96.8 96.7 96.8 96.7 96.6 96.5 96.4 96.4 96.3 95.8 94.9 94.7 94.6 95.0 94.4 95.1

0.72 0.72 0.75 0.79 0.86 0.89 0.89 0.90 0.92 0.92 0.91 0.91 0.92 0.90 0.90 0.89 0.90 0.91 0.93 0.93 0.92 0.93 0.88 0.93 0.94 0.99 0.93 0.95 0.87

Continued

-181-


Tcore I n l e t End, i n O F

15.5 1 6 16.5 1 7 17.5 18 18.5 19 19.5 20 20.5 2 1 21.5 22 22.5 23

279 278 277 275 273 271 268 266 263 259 25 7 254 25 8 269 280 289

Q (-mV>

92.9 92.7 92.8 90.9 88.9 88.6 88.7 89.4 88.7 87.6 86.3 83.8 79.7 76.5 76.6

Q @ 316°F (-mV>

76.4 76 .O 76.0 75.7 75.7 75.6 75.7 76.0 75.8 75.8 75.6 75.4 75.4 74.9 76.6

0 W

@ 303°F (-mV>

94.4 94.2 92.7 91.1 90.7 91.4 92.8 92.7 93.5 92.2 92.8 92.8 93.0 92.3 84.4

Os-@ @* = -

@ -0 s w

0.92 0.92 1 .01 0.99 0.88 0.82 0.76 0.80 0.73 0.71 0.62 0.48 0.24 0.09 0.00

-182-


RUN SW3 (7/1/78, 6:lO pm) Absolute Permeability to Water, K = 35.8 md

Inlet Pressure = 137.0 psig Outlet Pressure = 36.0 psig Confining Pressure = 372 p s i g

Mass Flow Rate = 1.85 gm/min

Inlet Temperature = 350'F Airbath Temperature = 344°F Core Length = 23.1 in Core Cross-Sectional Area = 3.14 in 2

@ X

@ S W @ -0

0 -@ Distance from Tcore 0 @ 346'F @ 348'F @* = - Inlet End, in O F ( -mV 1 (-mV> ( -mV 1

5

P C

s w

1 13/16 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15

349 349

34 7

345

344

343

342

341

338

336

335

334

333 332 331 330 32 8

86.0 85.0 86.6 88.7 87.8 88.9 87.9 87.8 87.8 88.6 88.4 87.2 86.6 87.0 86.6 86.6 86.8 87 .O 86.8 85.8 86.2 85.8 84.8 85.5 86.3 85.8 86.0 85.7

71.6 69.6 69.9 70.1. 70.2. 70.4 70.7 70.6 70.3 70.1. 69.9 69.6 69.4 69.3 69.2 69.3 69.6 69.7 69.6 69.6 69.4 69.3 69.3 69.2 69.2 69 .0 69 .0 69.0

86.8 88.2 88.8 89.1 88.8 89.1 88.9 89.0 89.0 89.0 89 .O 89.0 89 .O 88.6 88.7 88.8 88.9 89.0 89.0 89.0 90.2 89.8 89.5 88.1 87.2 87.7 88.0 87.8

0.95 0.83 0.88 0.98 0.95 0.99 0.95 0.93 0.94 0.98 0.97 0.91 0.88 0.92 0.89 0.89 0.89 0.90 0.89 0.84 0.81 0.80 0.77 0.86 0.95 0.90 0.89 0.89

Continued

-183-



15.5 1 6 16.5 1 7 17.5 18 18.5 19 19.5 20 20.5 2 1 21.5 22 22.5 23 23.5

T core OF

32 7 326 326 325 32 3 3 2 1 319 316 312 309 306 301 296 2 9 1 286 2 7 1 239

cp

(-mV>

85.3 84.9 85 .O 84.9 84.6 84.3 84.3 83.7 83.4 83.1 83.5 83.9 83.6 83.7 83.6 81.4 79.3

0 S

@ 346°F (-mv>

69.1 69.0 68.9 68.8 68.7 68.6 68.6 68.5 68.5 68.7 68.8 68.7 68.3 68.3 68.2 67.6

cp W

@ 348°F ( -mv>

87.7 87 .1 87.1 87.2 87.2 87.2 87.4 87.0 86.6 86.6 86.7 86.7 86.4 86.5 86.7 83.9

Os-@ cp* = - cp -cp s w

_.

0.87 0.88 0.88 0.88 0.86 0.84 0.84 0.82 0.82 0.80 0.82 0.84 0.85 0.85 0.83 0.85

-184-

TABLE A3.3-1: ISOTHERMAL GAS-DRIVE DATA, RUN NW1

RUN NW1 (11/18/78)

V = 495 mR

L = 23.1 i n P

D = 2 i n $ = 0.34 K = 36 md s = o gi

A t sec

92 160 2 20 350 435 495 570 690 785 920

1,165 1,415 1 ,730 2,120 2,615 3 , 080 4,520 5,630 6,610 7,800 8,545 9 , 400

10 , 500

13 , 830 15 , 640 16 , 450 18 , 220 20,000

12,110

W

@ T P room

mR

6 9

20 43 54 64 72 78 83 90 99

109 115 124 133 1 4 1 15 7 16 6 172 178 181 184 188 194 198 203 205 209 212

T = 78°F

T = 78°F

’bar

core

room

= 14.82 p s i a

G seP

@ Troom’Pbar R

0 0 0 0.02 0.13 0.22 0.33 0.64 0.87 1.25 2.08 2.99 4.26 6.10 8.62

11.21 20.28 28.25 35.83 45.61 52.00 59.50 69.65 84.95

102.01 120.73 129.60 147.35 168.00

’in P s i g

147.6 147.6 147.6 145.1 145.1 145.1 145.1 144.6 144.6 144.6 144.6 144.6 144.6 144.6 144.6 144.6 144.6 144.6 144.6 144.6 144.6 144.6 145.6 145.6 145.6 145.6 145.6 145.6 145.6

AP ps i

70.9 70.9 71.8 72.6 72.6 72.6 72.6 71.8 71.1 71.1 71.1 71.1 7 1 . 1 71.1 71.1 71 .1 71.1 71.1 71.1 71.1 71.1 71.1 71.1 71.1 71.1 71 .1 71 .1 71.1 71.1

-185-

TABLE A3.3-2: ISOTHERMAL GAS DRIVE CALCULATIONS FOR GRAPHICAL ANALYSIS, RUN NW1

RUN NW1 (11/18/78)

Tcore = 78°F; p = 123 psia avg

Qi pv 0.0154

0.0231

0.0513

0.116

0.178

0.232

0.286

0.398

0.482

0.616

0.895

1.20

1.61

2.20

3.00

3.81

6.65

9.13

pv 0.0154

0.0231

0.0513

0.110

0.138

0.164

0.185

0.200

0.213

0.231

0.254

0.280

0.295

0.318

0.341

0.362

0.403

0.426

Time Step Average

x -

Qi pv

0.019

0.037

0.084

0.147

0.205

0.259

0.342

0.440

0.549

0.756

1.05

1.40

1.90

2.60

3.40

5.23

7.89

10.3

cp

1.36

0.327

0.310

0.215

0.177

0.216

0.170

0.176

0.154

0.135

0.126

0.119

0.102

0.0954

0.0875

0.0782

0.0690

0.0642

Continued

-186-

TABLE A3.3-2: CONTINUED

Time Step Average

Qi x-1 PV cp

- Time Step Average

Qi x-1 PV cp

- Qi

pv 11.5 0 .441

13.0 0.0605 14.5 0.456

15.5 0 .0581 16.5 0.464

0.472 17.6 0.0568

18.8 20.4 0.0542

21.9 0.482 24.3 0.0529

26.6 0.497 29.2 0.0507

31.8 0.508 34.7 0.0486

37.6 0 .521 38.9 0.0459

40.3 0.526 43.0 0.0502

45.7 0.536 48.9 0.0434

52.0 0.544

-187-

0 0 0 0 0 . . . . .

m a m o a a ln(Vhr\lcn O d r l N N

0 0 0 0 0 . . . . .

m o N h m c o o h m o h a N r l r l O O

0 0 0 0 0

r ; lN 4 x . . . . .

3 u 3 E H

H I4 m . . . . . I 0 0 0 0 0

a u a a u u a

N N 3

W I M W

m u 4 . u c o m u c o m m h m m m m 0 0 0 0 0 . . . . .

0 0 0 0

II II II II

. . w p?; 2 n

.A I

n ?

:--I . . . . .

0 0 0 0 0

'W ii w E VI 0

W

cd

Mq d c n m h o m c o c o m m r l h l m u l n 0 0 0 0 0

lm . . . . . .. m I

m m -4 w I4 F9

2

hi00

II II II II

E d

D = 2 i n 4 = 0.34 K = 36 md s = o gi

A t sec

64 120 225 300 385 485 5 70 825 950

1,170 1,385 1 710 1,910 2,360 2 885 3,435 4 750 6,695 7,590 8 400 9 740

10 , 960 1 2 910 15,170 1 7 140 19 860 21,990 24 430 26 330

-188-

TABLE A3.3-4: ISOTHERMAL GAS DRIVE DATA, RUN NW2

RUN NW2 (11/22/78)

v = 495 mR T = 198°F

L = 23.1 i n P co re

Troom = 73°F

’bar = 14.71 p s i a

W

@ T P room

mR

9 19 42 57 71 96

103 124 130 138 145 153 157 16 5 1 7 2 179 191 204 209 213 219 223 228 235 240 246 250 255 258

G SeP

@ Troom’Pbar R

0 0 0 0 0.06 0.14 0.31 0.96 1.40 2.21 3.10 4.63 5.19 8.03

11 .14 14.60 23.80 38.95 46.38 53.29 65.42 76.80 95.78

119.00 139.90 170.25 195.00 224.20 247.60

199.1 199.1 199.1 199.1 199.1 199.1 198.1 198.1 198.1 198.1 198.1 198.1 198.1 198.1 198.1 198.1 196.6 196.6 196.6 196.6 196.6 196.6 196.6 196.6 196.6 196.6 196.6 196.6 195.1

AP p s i

66.1 66.0 66.0 63.5 66.5 65.1 65.1 65.1 65.1 65.1 65.1 65 .l 65.1 65.1 65.1 65.1 65.1 65.1 65.1 65.1 65 .1 65.1 65.1 65.1 65.1 65.1 65.1 65.1 64.0

-189-


RUN NW2 (11/22/78)

Tcore

'avg

= 198°F = 179 psia

Time Step Average -

Qi pv 0.0239

0.0504

0.112

0.151

0.205

0.293

0.358

0.593

0.730

0.974

1.24

1.68

1.85

2.65

3.52

4.50

7.07

pv 0.0239

0.0504

0.112

0.151

0.188

0.255

0.273

0.329

0.345

0.366

0.385

0.406

0.417

0.438

0.456

0.475

0.507

Qi p v -

0.0372

0.0809

0.131

0.178

0.249

0.326

0.476

0.662

0.852

1.11

1.46

1.76

2.25

3.09

4.01

5.78

9.20

cp

0.302

0.246

0.264

0.223

0.161

0.184

0.153

0.128

0.127

0.115

0.104

0.171

0.0790

0.0846

0.0798

0.0719

0.0646 Continued

-190-


Ti.me S t e p Average - 1-1

Qi p v - cp

Qi PV

- S g pv 0 .541 11 .3

12.4 0.0607 13.4 0.555

14.4 0.0591 15.3

18.7

0.565 17.0

20.3

0.0558

0.0542 0.581

21.9 0.592 24.5 0.0520

27.2 0.605 30.4 0.0492

33.6 0.624 36.6 0.0477

39.5 0.637 43.7 0.0454

47.9 0.653 51.4 0.0436

54.8 0.664 58.9 0 , 0 4 2 3

62.9 ,0677 66.2 0.0406

69.5 0.685

-191-

rl I Nl

w51 0 0 0 0 0 0

W

a b N N a m m m 0 0 4 - m m O r l N N N m 0 0 0 0 0 0

n

m co rl

0

N a b

0

co N m

0

a3 a c\I

0

m b

co 0

m 0 rl

0

b m m 0

* co m 0

* a rl 0

0

* m 4-

0

0 m m 0

N 0 rl 0

0

m 0 m 0

a m m 0

b m m 0 0

0

a co m 0

m m m 0

m * rl 0 0

0

a u a U

o m m N O o m 0 0

I I II

. .

w E 0 m a *

N

N b m

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0

a 4-

0

H .. 0 0 0 a I m

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-192-

TABLE A3.3-7: ISOTHERMAL GAS DRIVE DATA, RUN NW3

RUN NW3 (11/29/78)

v = 495 mR

L = 23.1 i n

D = 2 i n @ = 0.34 K = 36 md s = o

P

g i

A t sec

115 160 2 20 290 340 545 660 7 70 880 980

1,150 1,480 1,620 1,960 2,450 2,720 3,260 3,570 3,920 4,340 4,930 6,470 7,810 9,150

11,360 12,680 15,540 18,528 20,110

T = 294°F

T = 75°F c o r e

room

Pvar = 14.89 p s i g

W

@ T P

room mR

23 30 43 57 69

104 114 128 133 137 1 4 3 15 3 155 16 3 170 174 182 186 189 193 198 211 219 225 236 243 255 268 274

G SeP

@ Troom’Pbar R

0 0 0 0 0 0.20 0.33 0.40 0.57 0.74 1.10 1.86 2.20 3.08 4.46 5.30 6.99 8.04 9.25

10.78 13.00 19.24 25.10 31.35 42.75 49.92 66.85 86.25 97.35

P i n p s i g

196.6 196.6 196.6 196.6 196.6 196.6 196.6 195.6 195.6 195.6 195.6 195.6 195.6 195.6 195.6 195.6 195.6 195.6 195.6 195.6 195.6 195.6 197.1 197.1 197.1 197.1 197.1 197.1 197.1

AP ps i

52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4 52.4

-193-


RUN NW3 (11/29/78)

T = 294°F = 184 psia

core 'avg

Time Step Average

Qi pv 0.0639

0.0833

0.119

0.158

0.192

0.376

0.460

0.529

0.618

0.703

0.877

1.24

1.39

1.80

2.42

2.80

- S

8 PV

0.0639

0.0833

0.119

0.158

0.192

0.289

0.317

0.355

0.369

0.380

0.397

0.425

0.430

0.453

0.472

0.483

Qi pv

0.0736

0.101

0.139

0.175

0.284

0.418

0.494

0.573

0.660

0.790

1.06

1.31

1.59

2.11

2.61

3.18

- A-1

CP

0.263

0.189

0.204

0.170

0.126

0.155

0.180

0.141

0.133

0.111

0.104

0.103

0.0948

0.0893

0.0810

0.0805

Continued

-194-


Time S t e p Average

Qi pv 3.56

4.03

4.57

5.25

6.23

8.99

11.5

14.2

19.2

22.3

29.7

38.1

42.9

S g pv 0.505

0.517

0.525

0.536

0.550

0.586

0.608

0.625

0.655

0.675

0.708

0.744

0.761

Qi pv cp

3.79

4.30

4.91

5.74

7.61

10.3

12.9

16.7

20.7

26.0

33.9

40.5

0.0748

0.0739

0.0701

0.0680

0.0632

0.0596

0.0564

0.0506

0.0481

0.0442

0.0403

0.0373

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STEAM-WATER RELATIVE PERMEABILITY - Stanford · PDF filesteam-water relative permeability a dissertation submitted to the department of petroleum engineering and the committee on graduate