Investigation of Recovery Mechanisms in Fractured Reservoirs

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Investigation of RecoveryMechanisms in Fractured Reservoirs

by

Huiyun Lu

A dissertation submitted in fulfillment of the requirements for

the degree of Doctor of Philosophy of the University of London

and the Diploma of Imperial College

November 2007


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Abstract

My work focuses on developing a physical-motivated approach to modeling displacement

processes in fractured reservoirs. I use analytical expressions for the average recovery

as a function of time to find matrix/fracture transfer functions for different recovery

mechanisms such as gas gravity drainage and counter-current imbibition.

To account for heterogeneity in wettability, matrix permeability and fracture geometry

within a single grid block a multi-rate model is proposed, where the matrix is composed of

a series of separate domains in communication with different fracture sets with different

rate constants in the transfer function.

I show example results using a streamline-based dual porosity model for single and

multi-rate models and demonstrate that a multi-rate model predicts rapid initial recovery

followed by very low production rates at late time. The use of streamlines elegantly allows

the transfer between fracture and matrix to be accommodated as source terms in the one-

dimensional transport equations along streamlines that capture the flow in the fracture.

I studied the Clair oil field operated by BP using a fine-grid reservoir model. The

method is very efficient allowing million-cell models to be run on a standard PC. I also

applied this methodology to simulate recovery in a Chinese oil field to assess the efficiency

of different injection processes.

On the basis of the transfer functions study, the formulation for the matrix-fracture

transfer function in dual permeability and dual porosity reservoir simulation was rewrit-

ten. Based on one-dimensional analytical analysis in the literature, expressions for the

transfer rate accounting for both displacement and fluid expansion were found. The

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resultant transfer function is a sum of two terms: a saturation-dependent term repre-

senting displacement and a pressure-dependent term representing fluid expansion. The

expression reduces to the Barenblatt form for single-phase flow at late times, but more

accurately captures the pressure-dependence at early times. For displacement, both imbi-

bition and gravity drainage processes were considered. The transfer function is validated

through comparison with one and two-dimensional fine-grid simulations and compared

with predictions using the traditional Kazemi et al. formulation. This method captures

the dynamics of expansion and displacement accurately, giving better predictions than

current models, while being numerically more stable.

Publication as a results of this research is as follows:

Di Donato, G., Lu, H., Tavassoli, Z., Blunt, M. J. Multi-rate transfer dual porosity

modeling of gravity drainage and imbibition. SPEJ, Vol: 12, Pages: 77-88.

Lu H., Di Donato,G., Blunt Martin J. General Transfer Functions for Multiphase Flow.

SPE102542, proceedings of the SPE Annual Technical Conference and Exhibition held in

San Antonio, Texas, U.S.A., 24-27 September 2006.

Lu H., Blunt Martin J. General Fracture/Matrix Transfer Functions for Mixed-Wet

Systems. SPE107007, proceedings of the SPE Europec/EAGE Annual Conference and

Exhibition held in London, United Kingdom, 11-14 June 2007.

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Acknowledgements

I would first like to thank my supervisor Professor Martin J. Blunt for his constant

support, guidance and encouragement over the last three years. This project would not

have gone as smoothly as it did without his brilliant leadership and management skills.

I would also like to thank Ahmed Sami Abushaikha, Qatar Petroleum, and Olivier

Gosselin, Total, for making us aware of the interesting behaviour of mixed-wet systems

and inspiring us to do this work. I am grateful to the EPSRC, DTI and the sponsors of

the ITF project on Improved Simulation of Flow in Fractured and Faulted Reservoirs,

including - BP, especially Peter Cliford who helped to set up the Clair field model. I am

very grateful to Petro-China for providing me with data on the Liu7 oil field.

My great gratitude extends to Olivier Gosselin from TOTAL and Dr. Stephan Matthai

from Imperial College London for serving on my examination committee.

I would like to thank all the friends and colleagues for their great help and support. It

is impossible to put in words my gratitude for their help. It is hard to think how I would

have survived without their friendship.

I would also like to thank my family for supporting me from the beginning of my study.

Bingjun and Lufu showed their patience and love to support and encourage me. I would

like to dedicate my thesis to them for their endless love!!!

Finally I would like to thank my father Huanxiang who came here twice to help me

during my transfer and the final stage of my PhD study, my sister in law Fengying who

came here to help me in spite of her busy work schedule and my brother, sister and all

the relatives who give me their endless encouragement.

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Contents

1 Introduction 1

1.1 Recovery mechanisms in fractured reservoirs . . . . . . . . . . . . . . . . 11.2 Dual porosity modelling and shape factors . . . . . . . . . . . . . . . . . 5

1.3 Transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Streamline-based simulation . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Streamline-based dual porosity simulation 16

2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Streamline-based dual porosity formulation . . . . . . . . . . . . . . . . . 17

2.3 Single and multi-rate transfer functions . . . . . . . . . . . . . . . . . . . 19

2.3.1 Capillary-controlled imbibition . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Gravity drainage . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.3 Extensions to the approach . . . . . . . . . . . . . . . . . . . . . 26

2.3.4 Multi-rate transfer function . . . . . . . . . . . . . . . . . . . . . 272.3.5 Multi-rate model . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.6 Coupling with the fracture fractional flow . . . . . . . . . . . . . 29

3 Oil field case studies 32

3.1 Synthetic fractured reservoir simulation . . . . . . . . . . . . . . . . . . . 32

3.1.1 Reservoir description and computing comparison . . . . . . . . . . 32

3.1.2 Transfer rates and different transfer functions comparison . . . . . 34

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3.1.3 Permeability-dependent transfer rates . . . . . . . . . . . . . . . . 35

3.1.4 Results for a single rate model . . . . . . . . . . . . . . . . . . . . 37

3.1.5 Results for multi-rate models . . . . . . . . . . . . . . . . . . . . 40

3.1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Liu7 oil field simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.1 Reservoir description . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.2 Geological characteristics . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.3 Fluid characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.4 Parameters for imbibition modeling . . . . . . . . . . . . . . . . . 48

3.2.5 Parameters for gravity drainage . . . . . . . . . . . . . . . . . . . 51

3.2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Clair field recovery investigation . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.1 Reservoir description . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.2 Reservoir simulation . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.3 Recovery comparison with different simulators . . . . . . . . . . . 67

3.3.4 Initial conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 General Transfer Functions for Multiphase Flow 72

4.1 Formulation for multiphase flow . . . . . . . . . . . . . . . . . . . . . . . 73

4.1.1 Traditional formulation . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1.2 Our formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.3 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . 81

4.1.4 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 General Fracture/Matrix Transfer Functions for Mixed-Wet Systems 95

5.1 Formulation for mixed-wet media in two dimensional models . . . . . . . 96

5.1.1 Horizontal and vertical displacement . . . . . . . . . . . . . . . . 96

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5.1.2 Transfer rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.1 Base case study for the general transfer function . . . . . . . . . . 99

5.2.2 One-dimensional and two-dimensional model tests with capillary

pressure and gravity . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2.4 Comparison with other models . . . . . . . . . . . . . . . . . . . . 112

5.2.5 Sensitivity studies for the capillary pressure . . . . . . . . . . . . 119

5.2.6 Sensitivity studies for gas gravity drainage . . . . . . . . . . . . . 128

5.2.7 Correct factor of the gas gravity drainage . . . . . . . . . . . . . . 135

6 Conclusions and future work 139

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.1.1 Development of a general transfer function . . . . . . . . . . . . . 139

6.1.2 Developed a multi-rate model for displacement processes in frac-

tured reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.1.3 Sensitivity studies of the transfer function . . . . . . . . . . . . . 140

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.2.1 Upscaling strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A Barenblatt et al.s analytical solution for counter-current imbibition 1

B Numerical implementation 5

C Downscaling permeability data 10

D Calculation of the average saturation in gravity/capillary equilibrium 11

E Input deck of the Base Case (mixed-wet) for Eclipse model 13

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List of Figures

1.1 Schematic of different recovery processes in fractured reservoirs . . . . . . 2

1.2 Capillary imbibition from the fracture to matrix . . . . . . . . . . . . . . 3

1.3 Displacement of oil by gas gravity drainage forces . . . . . . . . . . . . . 4

1.4 Idealized fractured reservoir model represented by Warren and Root(1963) 6

1.5 Streamline-based dual porosity model . . . . . . . . . . . . . . . . . . . . 12

1.6 The principle of streamline-based simulation . . . . . . . . . . . . . . . . 13

2.1 Oil production rate comparisons of grid-based and streamline-based model 22

2.2 Run times of different grid numbers . . . . . . . . . . . . . . . . . . . . . 23

2.3 A single grid block in a field-scale simulation . . . . . . . . . . . . . . . . 27

2.4 Intersection fracture sets in a limestone and shale outcrop . . . . . . . . 28

2.5 Schematic of the conceptual model used to develop our dual porosity model 30

3.1 The porosity distribution of North Sea reservoir model . . . . . . . . . . 33

3.2 Comparison of oil rate using a fixedav . . . . . . . . . . . . . . . . . . 37

3.3 Fracture and matrix saturation . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Recovery curve for different transfer rate . . . . . . . . . . . . . . . . . . 39

3.5 Oil production rate as a function of time for different model . . . . . . . 41

3.6 Comparisons of oil production for different model . . . . . . . . . . . . . 42

3.7 Fracture permeability distribution field . . . . . . . . . . . . . . . . . . . 44

3.8 Capillary pressure curve of Liu7 . . . . . . . . . . . . . . . . . . . . . . . 45

3.9 Oil and water relative permeability curves in the matrix . . . . . . . . . 46

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3.10 Sections of Liu7s x direction permeability . . . . . . . . . . . . . . . . . 48

3.11 Water saturation of horizontal slices in the reservoir . . . . . . . . . . . . 53

3.12 Oil rate prediction of the field by different model . . . . . . . . . . . . . 54

3.13 Oil rate prediction for imbibition model . . . . . . . . . . . . . . . . . . . 55

3.14 Oil rate of the field by different wettability . . . . . . . . . . . . . . . . . 56

3.15 Field recovery of imbibition vs. injected pore volumes . . . . . . . . . . . 57

3.16 Vertical slices of gas saturation . . . . . . . . . . . . . . . . . . . . . . . 58

3.17 Field recovery for gravity drainage . . . . . . . . . . . . . . . . . . . . . 59

3.18 Clair field development location . . . . . . . . . . . . . . . . . . . . . . . 61

3.19 Schematic of N-S cross-section of Clair field . . . . . . . . . . . . . . . . 62

3.20 Clair field relative permeability curves in the matrix. . . . . . . . . . . . 63

3.21 Clair field capillary pressure curves in the matrix. . . . . . . . . . . . . . 63

3.22 Clair field permeability distribution . . . . . . . . . . . . . . . . . . . . . 65

3.23 Well placement for water flooding . . . . . . . . . . . . . . . . . . . . . . 66

3.24 Oil production comparison for different model . . . . . . . . . . . . . . . 68

3.25 Oil production prediction with different grid refinement . . . . . . . . . . 69

3.26 Oil production for dual porosity model with different grid number . . . . 70

4.1 Predicted and simulated oil transfer rate for Case 1 . . . . . . . . . . . . 87

4.2 Predicated and simulated oil transfer rate for Case 2 . . . . . . . . . . . 88

4.3 Predicted and simulated oil transfer rate for Case 3 . . . . . . . . . . . . 89

4.4 Predicted and simulated matrix water saturation for case 3 . . . . . . . . 90

4.5 Predicted and simulated average oil pressure for Case 3 . . . . . . . . . . 91

4.6 Predicted and simulated matrix water saturation for Case 3 . . . . . . . 92

4.7 Predicted and simulated matrix gas saturation for Case 4 . . . . . . . . . 93

5.1 Capillary pressure curve for the mixed-wet case . . . . . . . . . . . . . . 100

5.2 Base case for a block length of 1m . . . . . . . . . . . . . . . . . . . . . 102

5.3 Base case for a block lenght 10m . . . . . . . . . . . . . . . . . . . . . . 102

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5.4 The three cases considered in this section . . . . . . . . . . . . . . . . . . 103

5.5 Grid blocks for one and two-dimensional models . . . . . . . . . . . . . . 105

5.6 The oil recovery for Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.7 The oil recovery for Case 1 of 10 m . . . . . . . . . . . . . . . . . . . . . 108

5.8 The oil recovery for Case 2 of 1m . . . . . . . . . . . . . . . . . . . . . . 109

5.9 The oil recovery for Case 2 of 10 m . . . . . . . . . . . . . . . . . . . . . 110

5.10 The oil recovery for Case 3 of 1 m . . . . . . . . . . . . . . . . . . . . . . 111

5.11 The oil recovery for Case 3 of 10m . . . . . . . . . . . . . . . . . . . . . 112

5.12 Case1: Oil recovery comparison by Sonier function . . . . . . . . . . . . . 113

5.13 Case1:Oil recovery comparison using Quandalle and Sabathier function . 114

5.14 Case2:Oil recovery comparison by Sonier function . . . . . . . . . . . . . 115

5.15 Case2:Oil recovery comparison of Case2:Oil recovery comparison using

Quandalle and Sabathier function . . . . . . . . . . . . . . . . . . . . . . 116

5.16 Case3:Oil recovery comparison by Sonier function . . . . . . . . . . . . . 117

5.17 Case3:Oil recovery comparison by Quandalle and Sabathier function . . . 118

5.18 Case3:Oil recovery comparison by Quandalle and Sabathier function . . . 119

5.19 The oil recovery for different capillary pressure curves . . . . . . . . . . . 120

5.20 The oil recovery for Case 1 of different capillary curves . . . . . . . . . . 120


5.22 The oil recovery for Case 2 of 1 m . . . . . . . . . . . . . . . . . . . . . . 121


5.24 Case 3 for a block height of 1 mwith different capillary pressures. . . . . 122

5.25 Different capillary pressure curves for the base case . . . . . . . . . . . . 123

5.26 The different capillary pressure for Case 3 of 10 m . . . . . . . . . . . . . 124

5.27 The different capillary pressure for Case 3 of 10 m . . . . . . . . . . . . . 125

5.28 Matrix relative permeabilities with different exponents . . . . . . . . . . 126

5.29 The curves of different relative permeabilities exponents for 10 mblock . 126

5.30 The perm curves used for the mixed-wet system . . . . . . . . . . . . . . 127

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5.31 Oil recovery curve with permeabilities curves in Fig. (5.28) . . . . . . . . 128

5.32 The oil viscosity is 103 Pa.s . . . . . . . . . . . . . . . . . . . . . . . . 129

5.33 The oil viscosity is 102 Pa.s. . . . . . . . . . . . . . . . . . . . . . . . 130

5.34 The oil viscosity is 0.1 Pa.s . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.35 The oil viscosity is 1 Pa.s . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.36 The matrix block is 10mhigh . . . . . . . . . . . . . . . . . . . . . . . . 132





5.41 Gravity drainage with an oil viscosity of 103 Pa.s . . . . . . . . . . . . 136



5.44 Gravity drainage with an oil viscosity of 0.1 Pa.s . . . . . . . . . . . . . 137

5.45 Gravity drainage with an oil viscosity of 0.1 Pa.s . . . . . . . . . . . . . 138

6.1 Simulations using a discrete fracture model at water breakthrough after

136 days . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2 Relative permeability curves determined from waterflood simulation with

the model above . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

D.1 The capillary curves used for the mixed-wet system . . . . . . . . . . . . 11

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List of Tables

3.1 The parameters used in dual porosity simulation . . . . . . . . . . . . . . 34

3.2 Parameters used in the simulations . . . . . . . . . . . . . . . . . . . . . 51

3.3 Parameters used in the simulations . . . . . . . . . . . . . . . . . . . . . 62

3.4 Parameters used to calculate the transfer rate . . . . . . . . . . . . . . . 64

3.5 Run time of different simulators for 6000 days (minutes) . . . . . . . . . 71

4.1 Parameters used for case 1 and 2 (compressible fluids/no capillary pressure) 85

4.2 Parameters used for Cases 3 and 4 (capillary or gravity-driven flow with

compressible fluids) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.1 Parameters used for the mixed-wet case . . . . . . . . . . . . . . . . . . . 100

5.2 End-point saturations computed for the mixed-wet cases . . . . . . . . . 104

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Nomenclature

a inverse time constant

A area,L2,m2

b oil relative permeability exponent, dimensionless

B Vermeulen correction factor, dimensionless

c component concentration (fractional density),ML3,kgm3

D diffusion coefficient, L2T1, m2s1

f fractional flow, dimensionless

F weighting function, dimensionless

g acceleration due to gravity,Lt2, ms2

G gravity fractional flow, dimensionless

h height,L, m

H capillary rise,L, m

J entry dimensionless capillary pressure

K permeability, L

2

, m

2

or Dkr relative permeability, dimensionless

L, l length, m

Lc effective length, L, m

M mass per unit volume,ML3, kgm3

P pressure, ML1T2, P a

Pc capillary pressure,ML1T2, P a

qt total velocity, LT1, ms1

Q injection rate,L3T1, m3s1

r gravity/capillary ratio, dimensionless

R recovery, dimensionless

S saturation, dimensionless

S maximum saturation of invading fluid, dimensionless

t time, T, s

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t time step,T, s

T transfer function,T1, s1

V volume,L3, m3

Vt total velocity, LT1, ms1

rate in gravity transfer function,T1, s1

rate in imbibition transfer function,T1, s1

compressibility,ML1LT2, P a1

small convergence parameter, dimensionless

thermal diffusivity,m2/s

temperature, centigrade

porosity, dimensionless

mobility,M1LT, Pa1.s1

viscosity, M L1T1, Pa.s

density, ML3, kg.m3

coefficient,dimensionless

interfacial tension, L1t2, Nm1

transfer rate,ML3T1, kgm3s1

shape factor,L2, m2

time of flight,t, s

potential, ML1T2, P a

Subscripts

d diffusion

D dimensionless

e expansion

f flowing or fracture

g gas

i initial

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j grid block label

k fracture set label

m matrix or stagnant

N number of fracture sets

o oil

p, q phase label

r residual

s displacement

t total

th heat

w water

ultimate (at infinite time)

superscripts

n time level

H horizontal

V vertical

end point

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Chapter 1

Introduction

1.1 Recovery mechanisms in fractured reservoirs

In fractured reservoirs there are four principal recovery processes, illustrated schemati-

cally in Fig. (1.1): fluid expansion, capillary imbibition, diffusion and gravity-controlled

displacement. We will describe each of these processes in turn.

Initially the reservoir is at high pressure with oil in both fracture and matrix. During

primary recovery, the pressure will drop. Since the fractures are well connected, the

pressure will drop rapidly in them, while the lower permeability matrix will remain at

high pressure. This leads to a pressure difference between the matrix rock and the

fractures: slowly there will be flow of oil from matrix to fracture as the fluids expand.

When we drop below the bubble point, gas will evolve from solution and the expanding

gas will lead to further recovery from the matrix. This process is effective, but once thegas is connected in the system, principally only gas will be produced, leaving significant

quantities of oil in the matrix.

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Oil Oil

Oil Oil

Fracture

Matrix

Gas

(a) (b)

(c) (d)

Figure 1.1: Schematic of different recovery processes in fractured reservoirs: (a) fluid

expansion; (b) imbibition; (c) gravity-controlled displacement; and (d) diffusion. Weconsider matrix/fracture transfer as a sum of the contributions from these different effects.

In order to improve recovery, it is necessary to maintain pressure - ideally above the

bubble point - by injecting another phase to displace the oil. One possible injectant is

water. Since the fractures have a high permeability, the water will rapidly invade the

fractures, surrounding the matrix block. If the block has water-wet characteristics (that

is a capillary pressure that is positive for some range of saturation), water will enter the

block by capillary imbibition. Oil will be displaced from the block and be recovered from

the fractures. This process is illustrated in Fig. 1.2: it is relatively straightforward to

study this process experimentally by immersing a core sample in brine and using the

weight of the core to monitor recovery. There have been many studies of this process

on different rock types, cores of different shape and wettability. Fig. 1.2 illustrates

a typical recovery profile that has a characteristic shape regardless of the block size,shape, composition or wettability (assuming that there is some imbibition): there is a

rapid initial recovery followed by an approximately exponential relaxation to the ultimate

recovery. We will describe this process physically and mathematically in more detail later

in the thesis. The effectiveness of waterflooding will depend on the amount of water that

will imbibe into the matrix and the rate at which this occurs. We will quantify this

effect and introduce a transfer rate which allows an engineer to assess the effectiveness of

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waterflooding in a fractured field.

water

fracture matrix

Dimensionless time (tD)

numerical solution

experimental data

oil

Oilrecovery-%r

ecoverableoil

Normalized oil recovery vs. dimensionless time for very strong imbibition

Figure 1.2: Connected fracture network that carries the fluid flow in communicationwith a stagnant matrix, showing transfer of fluids from fracture to matrix (left). On theright, experimental data from Zhou et al. are plotted as a function of dimensionless time

(R/R= 1e0.05tD). The recovery can be matched empirically by a simple exponentialfunction (solid line) (Zhou,1997).

It is also possible to inject gas into a fractured reservoir. Like water, the gas will

rapidly invade the fractures, particularly near the top of the formation. Gas injection

is only effective if buoyancy forces allow the gas to enter the lower permeability matrix,

displacing oil from the bottom of the block. Fig. 1.3 shows this gravity drainage behaviour

at different times for an oil/water system. There are two remarkable features to the

displacement pattern: one is that gas rapidly reaches the equilibrium position H beyond

which the column is saturated with the wetting phase - the oil is retained in the matrix

due to capillary forces; another is that the saturation profile is almost uniform with

distance above the distance H, but this saturation slowly decreases over time. The gas

overcomes the capillary pressure and keeps pushing the oil to the bottom of the core

until oil and gas are in capillary/gravity equilibrium. Again, later in the thesis we will

show how to describe this process mathematically and use the information to analyze

gravity-controlled displacement in fractured reservoirs.

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Figure 1.3: Displacement of oil by gas under the influence of gravity . The gas moves infrom the top of the core, since the buoyancy force exceeds the capillary pressure. Thisprocess continues until gas and oil are in gravity/capilary equilibrium (From the PhDthesis by Sahni, 1998).

The final recovery process is diffusion. This will not be discussed in such detail in the

thesis, but is important in miscible gas injection processes. Here the injected gas and

oil can mix to form a single hydrocarbon phase and this hydrocarbon may be swept into

the fracture and produced. The gas and oil reach thermodynamic equilibrium through

the diffusion of individual components of both phases through the relatively stagnant

matrix. This is potentially a slow process but effective is miscibility is reached. Even in

immiscible systems, components of the injected gas may diffuse into the oil and cause it to

swell - such as in carbon dioxide injection - forcing oil out of the matrix and reducing its

viscosity, increasing flow rates. Implicit in this discussion has been the concept of a dual

porosity system which is described in more detail in the next section: we assume that a

fractured medium is composed of connected high permeability fractures that carry the

flow with a relatively low permeability matrix that contains most of the fluids. Recovery

is controlled by the transfer of oil out of the matrix by the physical mechanisms described

above. In this thesis we will develop mathematical models - called transfer functions - to

describe these processes quantitatively. The main concept will be to treat each physical

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effect separately and make the overall transfer the sum of these.

1.2 Dual porosity modelling and shape factors

Fractured reservoir development is one of the most difficult technologies in oil field

exploration. Numerical simulation is more difficult than for unfractured reservoirs since

there is typically greater uncertainty in both the reservoir description and the proper

modeling of the reservoir dynamics.

Numerous papers on single- and two-phase flow in naturally fractured porous media,

have appeared in the literature especially when a medium approach is used. In single-

phase flow, Barenblatt et al.(1960) proposed the dual porosity system to model fractured

media. They represented the reservoir by two overlapping continua which are matrix and

fracture. They then formulated the flow equations for each continuum using conservation

of mass principles; and flow between the matrix blocks and the fractures is accounted for

by source functions. These source or transfer functions are derived using Darcys law

expressed over some mean path between the matrix-blocks, and the density difference

between liquid phases.

Warren and Root(1963) developed an idealized model to study the characteristic behav-

iour of fractured reservoirs. They used Fig. (1.4) to illustrate their model of a fractured

reservoir which is still used in numerical simulation today. This is a dual porosity model

which assumes that the reservoir is composed of two regions by primary porosity and sec-

ondary porosity. The primary porosity represents the matrix with low permeability, andsecondary porosity indicates fracture and high permeability. In their model the matrix

contributes significantly to the pore volume of the system but contributes negligibly to

the flow capacity, while the secondary porosity carries all the fluid flow. They used dual

porosity transfer functions for single-phase flow based on an assumption of quasi-steady

transfer flow:

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=mK

(Pm Pf) (1.1)

where is transfer function with unit s1, m is the ratio of matrix porosity, is the

shape factor which is defined by:

=4n(n+ 2)

l2 (1.2)

n is the number of flow directions and l is a characteristic length. The shape factor

was derived based on an integral material balance on this length. Then they related this

characteristic length to the sides of a cubic matrix block. It assumes a continuous uniform

fracture network oriented parallel to the principal axes of permeability. The matrix blocks

in this system occupy the same physical spaces as the fracture network and are assumed

to be identical rectangular parallelpipeds with no direct communication between matrix

blocks. The matrix blocks are also assumed to be isotropic and homogeneous.

Reservoir Model

Vugs

Matrix

Fracture

Matrix

Fracture

Actual reservoir

on grid block

Idealized model reservoir

one gridblock

One matrix block

surrounded by fracture

z

x y

z

x yLyLx

Lz

x

y

z

x

xy

Figure 1.4: Idealization a fractured heterogeneous porous medium. The figure on theright presents the geological model which is represented by a cubic block on the left(Warren, 1963)).

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Kazemi et al.(1976) extended Warren and Roots (1963) single-phase flow model to

simulate multi-phase flow. They solved the dual porosity system in three dimensions

numerically. As with the Warren and Root model, two sets of different equations are

required to define the complete system, which define a flow in the fracture and in the

matrix. The transfer function for a black-oil system was given by:

p=mKmKrp

(Ppm Ppf) (1.3)

In addition, Kazemi et al. also gave a new definition for the shape factor for paral-

lelepiped and isotropic matrix blocks as:

= 4 1

L2mx+

1

L2my+

1

L2mz

(1.4)

derived based on a direct material balance on a rectangular parallelpiped matrix block

by assuming a pseudo-steady state.

Since then, many researchers have attempted to improve the dual porosity model pro-

posed by Kazemi et al. from different aspects: (1) shape factor calculation; (2) physical

modelling of multi-phase flow in naturally fractured reservoirs.

The shape factor is always hard to define since there is no clear definition of the value, it

is frequently used as a tuning factor for history matching. Several authors have proposed

different expressions for the shape factors.

Thomas et al.(1983) studied fine-grid single porosity upscale to a single-block dual-

porosity model. They gave 25/L2

for a three-dimensional oil and water model with

near unit mobility ratio displacement where L is block side length. Udea et al.(1989)

studied the shape factor for one and two-dimensional models . They also estimated that

the Kazemi shape factor needs to be multiplied by 3. Coats (1989) on the other hand

obtained a shape factor that is exactly twice the Kazemi model.

Lim and Aziz (1995) derived the shape factor by applying analytical solutions for the

single phase pressure diffusion equation for different parallelepiped geometries for the

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matrix blocks:

= 2

(KxKyKz)1/3Kx

L2x+

Ky

L2y+

Kz

L2z (1.5)

This shape factor is only used for single phase flow of parallelepiped geometries. It is

not suggested to use it for other geometries.

Chang et al.(1993) avoided the assumption of pseudo-state by combining the geomet-

rical aspects of the system with analytical solutions of the pressure diffusion equation for

flow between matrix and fracture. They proposed a time-dependent shape factor. Using

the complete solution to the diffusion equation, they derived for one-dimensional flow:

= 2

L2x

exp

(2m+ 1)2tD 1

(2m+1)2exp

(2m+ 1)2tD m= 1, (1.6)where tD is the dimensionless time.

Although this equation counts for the transient flow and gives an analytical form of the

transient shape factor, the expression is too complicated to be used in simulators. It can

not be validated for non-orthogonal systems.

Van Heel et al.(2006) studied the shape factor for a both convection and diffusion

processes for a steam-enhanced gravity drainage model:

th(t) = dmdt

/th(m f) (1.7)

where th is the thermal diffusivity of the rock, and is temperature.

They found that shape factor is not just determined by the geometric form of a matrix

block, for the same matrix different processes require different shape factors. The shape

factor is not always a constant as assumed in current simulators.

From these studies we can conclude that the shape factor can not be used by an engineer

reliably since the definition is not clearly related to any physical processes. There are

lots of arguments about how to use the shape factor in certain circumstances, but it is

hard to say which one is the standard.

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1.3 Transfer functions

Litvak (1985) presented a formulation for the simulation of natural fractured reservoirs

for a matrix block immersed in water:

p= m

krppBp

m

Km(Pm Pf) + CGamf (1.8)

The termCGamf includes the effects of different forces. In his transfer functions, Litvak

(1985) shows a gravity term that involves a product of the difference in density between

phases and the difference in saturation heights between matrix and fracture.

However, Litvak (1985) did not show how to calculate the saturation height in matrix

and fracture. He gave a procedure to implement capillary and gravity forces (CGamf) in

the equation by single matrix block simulations, considering the number of matrix blocks

contained in a grid cell and the water (gas) level in the grid block.

There are several assumptions in Litvaks work (1985): (1) water can move rapidly

through the highly permeable fractures and water imbibes over the entire height in non-

fractured reservoirs; (2) the dual porosity treatment of capillary and gravity forces as-

sumes that imbibition of water (oil drainage in the gas case) in the matrix can occur only

in a portion of the oil zone invaded by water (displaced by gas); (3) water saturation in

the matrix blocks is not related to the oil-water contact due to the separation of matrix

blocks by fractures (matrix discontinuity). This is defined only by the properties of the

matrix rock. Tighter matrix may have higher water saturation because of higher capillary

pressure. As a result, higher initial water saturation can be observed in zones above a

low water saturation zone.

Sonier et al. (1986,1988) also proposed the following transfer function for oil, gas, and

water respectively:

o= m

kmkrooBo

m

Pom Pof+ og(zwf+ zgf zwm zgm)

(1.9)

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g =m

kkrggBg

m

Pom Pof (Pcgof Pcgom) gg

zgf zgm

(1.10)

w =m

kkrwwBw

m

Pom Pof (Pcowf Pcowm) wg

zwf zwm

(1.11)

wherez is the height of fluid and is calculated by the corresponding fluid saturation:

zwf= swf swf i1 sorwf swf ih (1.12)

Quandalle and Sabathier (1989) defined a transfer function which separates viscous,

capillary and gravity within a grid block. The model defines flow toward all six faces

of a three-dimensional parallelepiped-shaped block. They used coefficients for each force

acting in each flow direction:

p=

Vb

VKkrpcpp

p

m

pm pf (1.13)The second term in parenthesis is defined for different faces of the parallelepiped, in

the x+ direction, it can be written as:

pm pf=Pom Pof v(p+f z Pf) c(Pcpof Pcpom) (1.14)

and the in the z+ direction,

pmpf=PomPofvp+f zPf+g

z

2

g(pmg)z2c(PcpofPcpom) (1.15)

The coefficients (v, g, c) were used to match fine grid simulations; they do not

necessarily have a physical meaning.

Gilman and Kazemi (1983) updated the earlier dual porosity model of Kazemi et al.

(1992) by refining the treatment of mobility. They alternated the transfer functions to

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include fracture relative permeability when fluid is flowing from the fracture to the matrix.

Birks (1955), Mattax et al. (1955) proposed practical ways of calculating oil recovery

from the matrix blocks. Birks formulated the mechanics of oil displacement from the

matrix blocks first by an idealized capillary model, and second by a simple relative-

permeability model. Mattax et al. (1955) were concerned with imbibition oil recovery

from matrix blocks in water-drive reservoirs. They developed an oil-recovery prediction

technique based on the semi-empirical relation that the time required to recover a given

fraction of oil from a matrix block is proportional to the square of the distance between

fractures.

The dual permeability model, proposed by Hill et al. (1985) differs from the dual

porosity approach by considering matrix block-to-block flow. This model is useful where

there is significant permeability in the matrix.

1.4 Streamline-based simulation

The coupling between fracture and matrix generally leads to dual porosity models

being more time consuming and memory intensive than single porosity (unfractured)

simulations. This frequently means that dual porosity simulations are restricted to very

coarsely gridded approximations that fail to capture the geological complexity of the

fracture network (Sonier, 1986).

To overcome these run-time limitations, the use of streamline-based simulation has

been considered (Di Donato et al., 2003,2004; Huang et al. 2004; Lake et al, 1981; Thieleet al., 2004). The streamline simulator solves a three-dimensional problem by decoupling

it into a series of 1-D problems, each one solved along a streamline (Pollock, 1988). The

transfer between fracture and matrix is represented by source or sink terms in the one-

dimensional transport equations, making the method both fast and elegant see Fig. (1.5)

(Di Donato et al., 2003, 2004, 2007).

Streamlines were first applied in the study of well patterns and total recovery by Muskat

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Matrix Streamline

Fracture fluid transfer between the flowing and

stagnant regians along streamlines

(a) (b)

Figure 1.5: Streamline dual-porosity model for fractured reservoirs. (a) The real fieldcontains both a fracture network and a relatively low permeability matrix. Effectivetransport properties are defined at the grid block scale (dashed lines). (b) Streamlinesfollow the flow field computed at the grid block scale. This captures the movementthrough the fracture network and high permeability matrix: the flowing fraction of thesystem.

et al. (1935). Higgins et al. (1962) used stream tubes, which carry a fixed fluid flux to

treat two-phase flow in a homogeneous medium. The method is composed of dividing

the stream tubes into elements of equal volume. Average mobility and geometric shape

factors were calculated for each element and the total resistance along each stream tube

was used to calculate the total flow rate for each stream tube.

Gelhar et al. (1971) introduced the concept of the time-of-flight along the streamline

by modeling a reservoir with anisotropic permeability. Time-of-flight was defined as the

time which a particle travels from an injection point to a sink or production point, and

it was different for each streamline. It is related to the reservoir heterogeneity directly,

and accounts for the reservoir driving forces when the velocity field is calculated -see Fig.

(1.6).

Bommer et al. (1962) solved the transport equations along streamlines to account for

chemical reactions and physical diffusion for modeling in-situ uranium leaching. Lake et

al. introduced the concept of decoupling the vertical response from the areal one in a

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Permeability field Pressure solve Saturation along SL

Initial saturation SL tracing

Saturation for

the next time step

Figure 1.6: The principle of streamline-based simulation is: (1) at the beginning of eachtime step the saturation, permeability and porosity are defined on an underlying grid.Just as in conventional grid-based methods, total mass balance in each grid block is usedto construct an equation for pressure and this is solved on the grid with known boundaryconditions at wells; (2) then, from Darcys law, the total velocity is found at each cellface and these velocities are used to trace streamlines from injectors to producers; (3)Saturations are mapped from the grid to streamlines and the conservation equation issolved along each streamline ignoring gravity; (4) The saturation is then mapped down

onto the grid and including only the gravity terms is solved on the grid; (5) The simulationreturns to step (1).

polymer/surfactant displacement (Lake, 1981).

Pollock (1988) used a linear interpolation of the velocity field vector within each grid

block to trace three-dimensional streamlines. His algorithm has been widely used in

most streamline simulation models since it is simple and accurate. Batycky et al. (1997)

introduced a three-dimensional streamline-based fluid flow simulator. It accounted for

the effects of changing well conditions as well as gravity for incompressible multi-phase

flow. An operator splitting technique was used to separate the calculation of the viscous

and gravity flow components (Bratvedt,1996). Ponting (2004) extended the streamline

technique to handle compressibility and depletion by a hybrid method.

Di Donato et al.(2003,2004) presented a dual porosity streamline-based model for frac-

tured reservoirs. They applied this methodology to study capillary-controlled transfer

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between fracture and matrix and demonstrated that using streamlines allowed multi-

million cell models to be run using standard computing resources. They showed that the

run time could be orders of magnitude smaller than equivalent conventional grid-based

simulation (Belayneh, 2004). This streamline approach has been applied by other au-

thors that have extended the method to include gravitational effects, gas displacement

and dual permeability simulation, where there is also flow in the matrix (Di Donato,

2003; Lake,1981).

Thiele et al. (2004) have described a commercial implementation of a streamline dual

porosity model based on the work of Di Donato that efficiently solves the one-dimensional

transport equations along streamlines (Pollock, 1988).

1.5 Overview

In this work I will present a physically-motivated approach to modeling displacement

processes in fractured reservoirs. This main contribution of this thesis is the development

of a general transfer function that accurately captures the average recovery from the ma-

trix due to fluid expansion and the combined effects of gravitational and capillary forces.

A analytical expression for the average recovery for gas gravity drainage and counter-

current imbibition are used to derive the transfer functions. For capillary-controlled

displacement the recovery tends to its ultimate value with an approximately exponential

decay (Barenblatt, 1960). When gravity dominates the approach to ultimate recovery is

slower and varies as a power-law with time. I will apply transfer functions based on theseexpressions for core-scale recovery in field-scale simulation using streamlines (Di Donato,

2004).

On the basis of this work, I extended the formulation for the matrix-fracture trans-

fer function in dual permeability and dual porosity reservoir simulation. The current

Barenblatt-Kazemi approach uses a Darcy-like flux from matrix to fracture, assuming a

quasi steady-state between the two domains (Barenblatt,1960; Warren, 1963, Kazemi,

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1992). However, this does not correctly represent the average transfer rate in a dynamic

displacement. Based on one-dimensional analytical analysis in the literature, expressions

for the transfer rate accounting for both displacement and fluid expansion are found

(Vermeulen, 1953; Zimmerman, 1993).

The resultant transfer function is a sum of two terms: a saturation-dependent term rep-

resenting displacement and a pressure-dependent term representing fluid expansion. The

expression reduces to the Barenblatt (1960) form for single-phase flow at late times, but

more accurately captures the pressure-dependence at early times. The transfer function is

validated through comparison with one-dimensional fine-grid simulations and compared

with predictions using the traditional Kazemi et al. (1976) formulation. I will show

that our method captures the dynamics of expansion and displacement accurately, giving

better predictions than current models, while being numerically more stable.

I also extend a model of fracture/matrix transfer in dual porosity and dual permeability

systems to mixed-wet media, where there can be displacement due to imbibition when

the capillary pressure is positive combined with gravity-controlled displacement. This can

lead to a characteristic recovery curve from the matrix, with a period of rapid imbibition

followed by slower recovery where gravitational effects dominate. The general transfer

function model is refined to accommodate such cases by including transfer due to hori-

zontal and vertical displacement separately (Lu et al. 2006). The model is tested against

fine-grid simulation in one and two dimensions and accurate predictions are made in all

cases; in contrast the conventional Kazemi et al. (1976) model gives poor predictions of

rate and ultimate recovery.

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Chapter 2

Streamline-based dual porosity

simulation

2.1 Background

Streamline-based simulation is now established as an attractive alternative to conven-

tional grid-based techniques for simulating displacements in highly heterogeneous reser-

voirs (Batycky, 1997; King, 1998). For incompressible or nearly incompressible flow

simulated through more than around 100, 000 grid blocks, streamlines have been shown

to be generally faster than grid-based methods and have been successfully used to study

several field cases in recent years (Baker, 2002; Grinestaff et al., 2000; Samier et al.,

2002).

Streamline-based models have recently been generalized to model fluid flow in fracturedreservoirs including matrix-fracture interactions. This methodology assumes that the

fluid flow occurs primarily through the high permeability fractured system and the matrix

acts as fluid storage (Barenblatt, 1960; Dean, 1988; Kazemi et al., 1992). A matrix-

fracture transfer function is used to calculate the fluid exchange between the matrix and

fracture.

As discussed in the previous chapter, many different transfer functions have been pro-

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posed to address different problems such as water counter current imbibition, gravita-

tional segregation and viscous displacement (Bratvedt, 1996; Di Donato, 2003, 2004;

Sahni, 1998). The main characteristic of these transfer functions is that they are an ex-

tension of Warren and Roots model that assume quasi-static Darcy-like flow and neglect

saturation and pressure gradients in the matrix blocks.

Di Donato et al. (2003, 2004) developed a physically-motivated approach to modeling

displacement process in fractured reservoirs. They used expressions of matrix/fracture

transfer functions that match capillary imbibition experiments. These transfer functions

were used in field-scale simulation. They proposed a multi-rate model to account for

heterogeneity in wettability, matrix permeability and fracture geometry within a single

grid block. This allows the matrix to be composed of a series of separate domains in

communication with different fracture sets with different rate constants in the transfer

function. This is the main methodology I am going to use to investigate fractured reser-

voir recovery mechanisms in my project. I will summarize the streamline-based model

methodology in the subsequent sections.

2.2 Streamline-based dual porosity formulation

Conceptually, in a dual porosity simulator the flowing (fracture and high permeability

matrix) and stagnant (low permeability matrix) domains are each defined by different

blocks with specific porosity, permeability, depth, etc. It assumes that there is no signif-

icant fluid flow between matrix grid blocks. Transport occurs through exchange of fluidfrom the matrix to the fractures and in the fracture network itself. In the streamline-

based dual porosity model, high permeability matrix is combined into the flowing fraction

conceptually by considering its contribution to the effective permeability of the flowing

fraction (Di Donato, 2003, 2004).

The streamline code that I use assumes incompressible two-phase flow of oil and wa-

ter (Batycky, 1997; Di Donato, 2003, 2004; Sahni, 1998). While gravitational forces

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are included in the transport of fluid in the flowing regions, the only mechanism for

fracture/matrix transfer that we allow is capillary-controlled imbibition (extensions to

gravity-controlled flow are discussed later). The volume conservation equations are:

fSwf

t + vt fwf+ gG = T (2.1)

mSwm

t =T (2.2)

where T is the transfer function. Define a time-of-flight flight as the time taken for neutral

tracer to move a distance salong a streamline (Gelhar, 1971):

(s) =

s0

fVt

ds (2.3)

Then we can transform Eq. (2.1) into an equation along a streamline as follows:

Swf

t +

fwf

+

1

f gG =

T

f(2.4)

Single porosity streamline simulation ( equivalent to setting T = 0;m= 0 in Eq. (2.1)

and (2.2)) can be described in the following five steps: (1) At the beginning of each time

step the saturation, permeability and porosity are defined on an underlying grid. Just

as in conventional grid-based methods, total mass balance in each grid block is used to

construct an equation for pressure and this is solved on the grid with known boundary

conditions at wells. (2) Then, from Darcys law, the total velocity is found at each cellface and these velocities are used to trace streamlines from injectors to producers. (3)

Saturations are mapped from the grid to streamlines and the conservation Eq. (2.4) is

solved along each streamline ignoring gravity. (4) The saturation is then mapped down

onto the grid and Eq. (2.4) including only the gravity terms is solved on the grid. (5)

The simulation returns to step (1). In our dual porosity simulator, we follow exactly

the same approach. In computing the pressure field and tracing streamlines we only

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use information on the flowing fraction saturation, porosity and permeability. The only

change is to modify step (3). Both the flowing and stagnant fraction saturations are

mapped onto streamlines and we solve the following two conservation equations along

each streamline:

Swft

+fwf

= T

f(2.5)

Swmt

= T

m(2.6)

Swf and Swm are then mapped back to the grid and the simulation follows the single

porosity methodology. Note that the effects of gravity in the fracture flow are accounted

for in step (4) (as in single porosity streamline methods), while gravity affecting frac-

ture/matrix transfer is included in the form of the transfer function T.

2.3 Single and multi-rate transfer functions

2.3.1 Capillary-controlled imbibition

Ma et al. presented an analysis of capillary-controlled imbibition, where water is im-

bibed into the matrix by capillary pressure and oil comes out into the fractures and high

permeability matrix (Ma, 1997). This mechanism is in line with the streamline conceptual

model which captures movement through the flowing fraction while the transfer of fluid

from flowing to stagnant region is modeled as a source/sink term in the one-dimensional

transport equation along each streamline.

Fig. (1.2) illustrates an idealized representation of a fractured reservoir. During water-

flooding of a fractured reservoir, most of the water flow is in the high permeability chan-

nels called fractures. The water from the fractures imbibes into the matrix by capillary

action and the oil comes out provided the matrix is water-wet. In this chapter we will

make this assumptions later (Chapter 4) this will be related.

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A number of authors have performed counter-current imbibition experiments where

water-wet cores have been surrounded by water (Aronofsky, 1958; Morrow, 2001, Ding,

2005). The recovery of oil can be matched by a simple exponential function of time (Ding,

2005). Zhang et al. (1996) proposed an expression that matched a range of imbibition

experiments on samples with different geometry and fluid properties (Di Donato, 2003).

We will use a transfer function based on a semi-analytical solution for counter-current

imbibition proposed by Barenblatt et al. (1963). Since this solution has been ignored in

the petroleum literature, it is reviewed in Appendix A. The recovery, R, is:

R= R(1 et) (2.7)

where R is the ultimate recovery. The rate constant is defined by:

= 3

Kmm

owL2c

Jow

t

Swm=Sw

(2.8)

where ow is the oil/water interfacial tension. The mobilities = kr/ are defined at

Sw, the maximum saturation reached in the matrix during imbibition; we assume that

the system is not strongly water-wet and so this saturation is lower than 1 Sorm, whereSorm is the residual oil saturation in the matrix after waterflooding, including forced

displacement due to viscous r gravitational forces. The case whereSw = 1Sorm, giving= 0 in Eq. (2.8), has been considered by Tavassoli et al. (2005). J is the dimensionless

gradient of the capillary pressure at Sw, defined by:

dPcmdSwm

Swm=Sw

=

mKm

ow dJ(Swm)dSwm

Swm=Sw

=

mKm

owJ (2.9)

where we assume Leverett J-function scaling of the capillary pressure:

Pcm=ow

mKm

J(Swim) (2.10)

The definition of the rate constant is similar to that derived by Zhou et al. (2000),

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although we evaluate the mobility at the end of imbibition.

Only a linear system of length L was considered in the analytical analysis of Barenblatt

et al. (1963). To account for more complex fracture geometries we define Lc as an

effective length given by (Ma, 1997):

L2c =V

ni=1

Aili

(2.11)

where V is the matrix block volume,Ai is the area open to flow in the ith direction and

li is the distance from the open surface to a no-flow boundary. If we assume Swm as the

average saturation in the matrix, then we can write:

R

R=

Swm SwmiSw Swmi

(2.12)

where Swmi is the initial water saturation in the matrix. Thus:

Swm = Swmi+ (Sw Swmi) (1 et) (2.13)

then from Eq. (2.12):

mSwm

t =T=m(S

w Swm) (2.14)

This assumes that the transfer function is independent of the flowing saturation, as

long asSwf>0. This is consistent with the assumption that the capillary pressure in the

low permeability matrix is much higher than in the fractures. The imbibition continues

until the capillary pressure reaches its equilibrium between matrix and fracture, when

Swf= 0 and Swm =Sw. Thus we take:

T =

m(Sw Swm) Swf>0,

0 Swf= 0

(2.15)

The transfer function varies linearly with matrix saturation. We will call this the

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linear transfer function. This linear transfer function represents the similar physics to

the conventional functions: transfer controlled solely by the matrix capillary pressure

with negligible fracture capillary pressure, so the results should be similar.

To check the validity of the approach, Di Donato et al. (2003) first single porosity

simulations on the same reservoir model with all properties the same as the SP E 10th

Comparative Solution project (Christie et al., 2001). As well as the fine grid model

60 220 85, they generated a series of coarser grids: 20 55 17, 20 55 85,60 55 85. Permeability was uspcaled using geometric averaging. The results and runtime were compared with conventional grid-based simulation using Eclipse. The results

are identical to those obtained from Christie and Blunts work (2001). Both streamline

and grid-based methods gave very similar predictions of oil recovery for the same sized

grid. They then ran the streamline-based dual porosity model (see Chapter 3.1) for the

different grids using the linear model with = 5 109 s1 they obtained very similarpredictions of oil recovery for the same sized grid compared to conventional grid-based

simulation. Fig. (2.1) shows the oil production rate. For the same transfer function,

grid-based and streamline dual porosity models gave virtually identical results.

Figure 2.1: Comparisons of oil production rate for grid-based and streamline-based dualporosity models for different grid sizes. In both cases a conventional transfer functionwith the same shape factor was used. Both simulation methods give very similar results

for the same grid size (Di Donato et al., 2003).

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Fig. (2.2) shows comparisons of the run times for the various simulations. All the

times are for a 3GB RAM, 2.4 GHz Pentium Xeon workstation. For both single and

dual porosity models the run time of the streamline code scales approximately linearly

with the number of grid blocks. The dual porosity model is slower because of the more

complex one-dimensional transport solver, in particular the iterative procedure for the

conventional model makes it significantly slower than the other models. The grid-based

code gives run times that are approximately proportional to the number of grid blocks

squared. This leads to run times that are one to two orders of magnitude slower for the

finest grids.

Number of grid blocks

1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

1.E-011.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

CPU

time

(min)

Linear model,streamline

Conventional model, streamline

Dual porosity model

Single porosity,streamline

Single porosity,Eclipse

Figure 2.2: Comparisons of run times as a function of grid block numbers for a simulationtime of 2,000 days (Di Donato et al., 2003).

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The transfer rate constant is a function of the fracture spacing, interfacial tension

and matrix permeability. It is also a strong function of wettability and viscosity contrast

through the introduction of mobilities in Eq. (4.8) (Birks, 1955). In mixed-wet systems,

the transfer rate can be 100 10, 000 times lower than that in similar strongly water-wetmedia since the water mobility is low (Behbahani, 2005; Morrow, 2001).

It is important to input a proper rate constant in fracture modeling, sincedepends

on the matrix, fracture and fluid properties. It is different from the traditional approach

of using a shape factor, which is physically opaque, to represent fracture geometry since

it does not relate easily to a transfer rate (Gilman, 1983).

The matrix relative permeabilities and capillary pressure can be adjusted to account for

wettability and mobility in a conventional model. However, the impact of such changes on

recovery is obscured by the unnecessarily complex non-linearity of the resultant transfer

function that simply adds computational difficulty while failing to represent the correct

physics (Di Donato, 2007).

2.3.2 Gravity drainage

The displacement of oil by gas under gravity in the presence of water is an important

recovery process in oil reservoirs (Ding, 2005). Compared with water flooding, the gas

gravity drainage process can be very slow, and the ultimate recovery is very high. Sahni

et al. (1998) used CT scanning to image saturations directly in a long vertical column.

Fig. (1.3) shows the behaviour at different times for air/water gravity drainage. There

are two remarkable features to the displacement pattern: one is that gas rapidly reaches

the equilibrium positionHbeyond which the column is saturated with the wetting phase;

another is that the saturation profile is almost uniform with distance above the distance

H, but this saturation slowly decreases over time. The gas overcomes the capillary

pressure and keeps pushing the oil to the bottom of the core until oil and gas are in

capillary/gravity equilibrium.

In our streamline model gas injection is also considered. The gas is assumed as in-

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compressible and water is everywhere at its irreducible saturation. Each matrix block is

treated separately and there is no capillary continuity between blocks allowed (Por, 1989;

Horie, 1990). In the governing transport Eq.(2.5) and (2.6) we simply substitute gas for

the water phase.

As for water flooding, we base our transfer function on analytical solutions to the flow

equations in an idealized geometry - in this case vertical gravity drainage of oil in the

presence of gas. The ratio of gravitational to capillary forces in each matrix block,r, is

defined as:

r= LH

=gLgoJ

Kmm

(2.16)

whereL is the block height,His the amount of capillary rise of oil in the presence of gas,

andJ is the dimensionless entry pressure for gas invasion into oil. The density difference

= o g.For r 1 gas cannot enter the block and there is no gravity drainage. For r 1,

gravitational forces dominate and we can write a solution for the average recovery at late

times due to Hagoort (1980):

R

R= 1 (t)

1a1

Sg(2.17)

where the rate constant is:

=

aaKmKmaxrom g

(a 1)(a1)oL (2.18)

The exponent a (it is assumed that a > 1) is defined such that at low oil saturation

the oil relative permeability is:

kro = kmaxrom (Som Sorgm)a (2.19)

where Sorgm is the residual oil saturation in the presence of gas. Note that we define

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the ultimate recovery by finding the maximum gas saturation Sg at infinite time from

capillary/gravity equilibrium, not simply by assuming that the oil saturation eventually

reaches Sorgm everywhere:

Sg = 1 Swmi L0

P1cgo (gh)dh (2.20)

Then following the same approach as for capillary-controlled imbibition:

R

R=

SgmSg

(2.21)

Sgm =Sg (t)

1a1 (2.22)

and hence writing the transfer as a function of gas saturation:

mSgm

t =T =

ma 1 (S

g Sgm)a (2.23)

Then we define:

T =

ma1

(Sg Sgm)a Sgf>0,

0 Sgf= 0

(2.24)

Note that while the original expression for gas saturation, Eq. (2.21), diverges at t = 0,

the transfer function itself is always well behaved. This is one reason why the transfer

function is written as a function of saturation and not of time explicitly.

2.3.3 Extensions to the approach

It is possible to extend this approach to study other physical processes, such as transfer

due to fluid expansion (implicitly included in the traditional formulations), displacement

due to a combination of capillary and gravitational forces and interaction between blocks

due to capillary continuity (Barenblatt, 1863; Warren, 1963; Gilman, 1983; Por, 1989).

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We treat each matrix block as independent source term. Some of these extensions will

be discussed in Chapter 4.

2.3.4 Multi-rate transfer function

Realistically, the transfer function is not a constant, even within a single grid block,

because of different fracture sets with different spacing, or sub-grid block variability in

wettability and matrix permeability (Barenblatt, 1963).

Fig. (2.3) illustrates the complex interaction between fracture and matrix of a single

grid block in field-scale simulation; a single grid block tens to hundreds of meters in extentwould include many fractures. Capturing the transfer from these different fracture sets

is the key to solving this kind of problem. Di Donato et al. (2007) derived a multi-rate

transfer function to account for heterogeneity in a single grid block.

frcoFigure 2.3: A single grid block in a field-scale simulation, there are different spacingfractures in one grid block, so it is not accurate to set the fracture property by one value.The multi-rate transfer model will describe the complexity physically by different transfer

rates within a single block (Matthai, 2005).

2.3.5 Multi-rate model

The single-rate model assumes that there is just a single set of fracture and matrix

properties within each grid block and the matrix permeability may have small-scale vari-

ations (Daly, 2004). Lumping all the transfer between fracture and matrix in a single

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rate may therefore be inaccurate, and, as we show later, will tend to over-predict oil re-

covery at late time, since a single rate model will not adequately account for the regions

of the matrix that are in poor communication with the fractures. As an example of this

complexity, Fig. (2.4) illustrates intersecting fracture sets in a limestone outcrop near

the Bristol Channel in England (Belayneh, 2004).

BEBc dFigure 2.4: Intersection fracture sets in a limestone and shale outcrop in the Bristolchannel (Belayneh, 2004). The fracture/crack spacing is between 0.3 and 1m. We use a

multi-rate model to accommodate transfer from different fracture sets.

In dual porosity simulation, a single grid block tens to hundreds of meters in extent

would include many fractures. A single rate is insufficient to capture the transfer from

these different fracture sets. Although it is more realistic to capture the geometry explic-

itly, there is no proper computing resource to cope with this complicated problem.

To account for sub-grid-block heterogeneity we follow the approach of Ponting (2004)

and propose a multi-rate model where the matrix is assumed to be composed of N

domains each with different transfer rates. The transfer into each of the domains is

handled separately:

T=N

k=1

Tk; m=N

k=1

mk; mSwm =N

k=1

mkSwmk; (2.25)

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Tk=

kmk(Swk Swmk) Swf>0,

0 Swf= 0

(2.26)

Swmkt

= Tkmk

(2.27)

Mathematically, Eq. (2.24)-(2.26), with two or three domains are similar to models

proposed by other authors to describe multiple transfer rates in a single block, although

the physical interpretation is very different (Terez, 1999; Civan, 2001).

The same equations can be used for gas injection with water substituted by gas andthe transfer function is:

Tk =

kmkak1

(Sgk Sgmk) Swf>0,

0 Sgf= 0

(2.28)

2.3.6 Coupling with the fracture fractional flow

The multi-rate model leads to the regions of the matrix in contact with fractures having

the largest surface area and consequently the highest transfer rates being drained first,

while the regions with the lowest transfer rates are drained last. The lowest transfer rates

may come from the largest aperture fractures. Hence, during waterflooding the largest

fractures become saturated by water first and the smaller aperture fractures, with the

largest surface area, are only contacted by water later. Thus the multi-rate model does

not capture the proper sequence of displacement within a grid block.

We propose another model where in the upscaled fracture fractional flow curve different

average saturation regions represent the flooding of different fracture sets. Corresponding

to each fracture set would be transfer into the matrix - in this scenario the smaller matrix

blocks, with the highest transfer rates, may be recovered last, since imbibition will not

start until their average fracture saturation was sufficiently high. Fig. (2.5) illustrates

this model schematically. This formulation allows for a more realistic multi-rate model,

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where which portions of the matrix are recovered is controlled by the average fracture

saturation that in turn represents which fracture sets are flooded.

w

Swf0Swf1SwflargefractumedfractsfrFigure 2.5: Schematic of the conceptual model used to develop our dual porosity model.The left hand figure would represent a single grid block in a field-scale simulation. Theaverage fracture fractional flow (right-hand figure) has contributions from all fracturesets - the larger aperture fractures are filled first at low average saturation with smalleraperture fractures filled at higher saturations. Associated with each fracture set is adifferent matrix/fracture transfer rate. We represent the complex interaction betweenfracture and matrix by a series of linear functions representing transfer from differentfracture sets at different rates (Di Donato et al., 2007)

Mathematically, the fractional flow curve is divided into N sections Swf k (by defin-

ition Swf N = 1 and Swf0 = 0, assuming no residual saturation in the fractures) with

corresponding matrix porosities and transfer rates. The expressions are only a simple ex-

tension of Eq. (2.24)-(2.26) above, but now have the structure to represent very complexinteractions between fracture and matrix. We also allow for reduced transfer if a frac-

ture set is not completely saturated, assuming now that locally fractures are either fully

saturated or dry and that fractures at the small scale are rarely partially saturated-the

single-rate version of this model then reduces to the empirical model of Kazemi et al.

(1992).

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Tk =

kmk(Swk

Swmk) Swf

Swf k

kmk

SwfSwfk1

SwfkSwfk1(Swk Swmi) + Swmi Swmk

Swf k > Swf> Swf k1

0 Swf Swf k1

(2.29)

Swmkt

= Tkmk

(2.30)

For gravity drainage, Eq. (2.29) becomes:

Tk=

kmkak1

(Sgk Sgmk)ak Sgf Sgf kkmkak1

SgfSgfk1SgfkSgfk1

Sgk SgfkSgfk1SgfSgfk1Sgmk

akSgf k > Sgf> Sgf k1

0 Sgf Sgf k1

(2.31)

The numerical implementation of these transfer functions in the one-dimensional trans-

port equations along streamlines is described in Appendix B.

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Chapter 3

Oil field case studies

Three different cases are studied in this chapter: 1. A non-fractured sandstone oil field

with synthetic fractures; 2. A Chinese fractured dolomite oil field; and 3. The Clair field

with high matrix permeability.

3.1 Synthetic fractured reservoir simulation

3.1.1 Reservoir description and computing comparison

The reservoir model we use is derived from the 10th SPE Comparative Solution Project

(Christie, 2001). The model is a two-phase (oil and water) model with no dipping or

faults. The dimensions of the reservoir are given by 366m 670m 52m; the numberof grid cells used for the simulations was 1, 122, 000, given by 60 220 85 in the x, y

and z directions respectively. It is a non-fractured sandstone fluvial reservoir based on

a North Sea oilfield with high permeability meandering and channels surrounded by low

permeability shale.

The original data set includes a set of heterogeneous permeability data, a set of het-

erogeneous porosity data, compressible two-phase PVT data and relative permeability

data. We turn the original model into quasi-incompressible data set by changing the

compressibility for both rock and fluid to nearly zero to satisfy the needs of testing the

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new streamline-based code. This adjustment has no big impact on the oil recovery pre-

dictions, since the field is operated above the bubble point.

Fig. (3.1) (a) shows the porosity for the whole model, and Fig. (3.1) (b) shows part of

the Upper Ness sequence, with the channels clearly visible.

Figure 3.1: The porosity distribution of North Sea reservoir model: (a) the porosity forthe whole model; (b) part of the Upper Ness sequence well distribution (Christie, 2001).

We made an artificial dual porosity model to test the streamline-based research code

by doing the following: (1) use a constant porosity for both matrix and fracture; (2) the

original heterogeneous permeability data is treated as defining the fracture permeability

distribution, with the exception that we set a minimum fracture permeability of 1 mD,

while a single uniform permeability of 1 mDis assigned to the matrix; (3) take the original

relative permeability curve as the one in the matrix. For the fracture, we assume a linear

fractional flow for water, even if the oil and water viscosities are not equal. From this

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assumption, we derived the following expressions to calculate the relative permeabilities

in the fracture:

krwf=

1 +

1 SwfSwf

ow

(3.1)

krof=

1 +

Swf1 Swf

wo

(3.2)

The capillary pressure in the fracture impact is ignored by setting it to zero. The

reference pressure is 41.3

106 P a at a depth of 3657.6 m. For all models, one injector

is located at the center of the model; four producers are at the corners of the model - see

Fig. (3.1). All wells are vertical and perforated throughout the formation. In addition,

we use a constant well bore rate for the injector and a constant well bore pressure for

the producers as the inner boundary conditions as well as no flow on the outer boundary.

The parameters used in the simulations are listed in Table 3.1.

Table 3.1: The parameters used in dual porosity simulation

m= 0.2 f=0.02Km= 1.0 mD Kf= 1 20, 000 mDSwim =Sorm = 0.2 Swif=Sorf= 0.0krom(Swi) =krwm(Sor) = 1.0 krof(Swi) = 1.0w = 0.3 103 cpo= 3 103 cpw = 1026 kg.m

3

o= 1026 kg.m3

Bw= 1.01Q= 300 m3.day1

Bottom Hole Pressure =27.6 106 P a

3.1.2 Transfer rates and different transfer functions comparison

As described earlier, this model was tested by Di Donato et al.(2003) to compare

the difference between streamline-based and grid-based dual porosity simulators. They

designed several different cases to compare the difference between a streamline-based

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model and a conventional grid-based model and found that the streamline-based model

can easily cope with incompressible displacement in a highly heterogeneous reservoir,

and streamline-based simulation gave almost identical results to commercial grid-based

simulation for single and dual porosity models.

Based on the work of Di Donato et al. (2003, 2004), we design two cases to study

which are transfer rate sensitivity studies. This research give us a more comprehensive

understanding of fractured reservoir recovery mechanisms, especially where the multi-rate

model represents the complexity in one grid block in a field scale model.

3.1.3 Permeability-dependent transfer rates

Mathematical model used to assign the transfer rate

The transfer rate is a parameter which is related to the geometry and fluid properties -

see Eq. (2.8). We have tested the one-, two- and three-rate models for the field. We also

assigned different rates to each grid block dependent on the fracture spacing and matrix

permeability. The aim of this section is to explore the impact of using a multi-rate model.We use Eq. (2.8) to find the average transfer rate for the whole reservoir av. is

proportional to

Km and 1/L2c . Kf is proportional to the fracture spacing Lc, so we

used:

=av

KmKma

KfKfa

2(3.3)

to assign the transfer rate for each grid block where Ka is the average permeability. The

range of the transfer rate is from 1.7 1013 s1 7.0 105 s1. av is the averagetransfer rate for the whole reservoir calculated by Eq. (2.8) to be 5109 s1. Kma=Kmwhile Kfvaries from 1 mD to 2000 mD and Kfa is 170 mD.

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Complement the equation into the streamline-base code

The work to code the permeability dependent transfer rate equation into the streamline-

based model is composed of three parts:

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Investigation of Recovery Mechanisms in Fractured Reservoirs

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