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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
5.21 The oil recovery for Case 1 of 10m . . . . . . . . . . . . . . . . . . . . . 121
5.22 The oil recovery for Case 2 of 1 m . . . . . . . . . . . . . . . . . . . . . . 121
5.23 The oil recovery for Case 2 of 10m . . . . . . . . . . . . . . . . . . . . . 122
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
5.37 The matrix block is 15mhigh . . . . . . . . . . . . . . . . . . . . . . . . 133
5.38 The matrix block is 20mhigh . . . . . . . . . . . . . . . . . . . . . . . . 133
5.39 The matrix block is 25mhigh . . . . . . . . . . . . . . . . . . . . . . . . 134
5.40 The matrix block is 50mhigh . . . . . . . . . . . . . . . . . . . . . . . . 134
5.41 Gravity drainage with an oil viscosity of 103 Pa.s . . . . . . . . . . . . 136
5.42 Gravity drainage with an oil viscosity of 103 Pa.s . . . . . . . . . . . . 136
5.43 Gravity drainage with an oil viscosity of 102 Pa.s . . . . . . . . . . . . 137
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:
Go through the cod