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Polymerflood Simulation in a Heterogeneous Idealized
Reservoir with and without Crossflow
by
Oppong Kwame
Submitted in partial fulfillment of the requirements for the Degree of Master of Science in Petroleum Engineering
New Mexico Institute of Mining and Technology Department of Mineral Engineering
Socorro New Mexico December 2009
ABSTRACT A reservoir simulation model predicting polymerflood performance and the potential of
using horizontal well injector in flood operation in stratified (two layers) systems is
presented considering the effect of crossflow and no-crossflow on oil production at varied
mobility ratios. The level of communication (free crossflow) between reservoir layers,
which is characterized by the closeness of the system to vertical equilibrium (VE)
condition, can significantly affect sweep efficiency in heterogeneous reservoirs. In gel
placement as a remedy to early water breakthrough, the details of the gel placement are
strongly affected by the degree of communication between reservoir layers.
The importance of fluid crossflow relative to purely longitudinal convective transport in a
two-dimensional setting depends on several factors. Rock properties such as porosity,
permeability, the ratio of vertical to horizontal permeability and length to thickness ratio
cannot be over looked. Fluid properties such as phase densities, viscosities and interfacial
tension also play important role. Coupled rock-fluid properties, for examples, wettability,
relative permeabilities and capillary pressure are also factors. In this report only the
effects of viscous force on polymerflood performance in a stratified reservoirs is
considered. A fully implicit, black oil reservoir simulation model was used to predict the
displacement efficiency in two-dimensional, fine–grid (x-z) cross-section. At present the
development is limited to two phases and two components injector (water/polymer)
producer (oil) system. These investigations were inspired because planning for efficient
production strategies and targeting unrecoverable oil should be a priority in the petroleum
industry. The conclusion from the simulation model results are comparable to analytical
model results and are directly applicable to similarly scaled viscous-dominated systems at
a reservoir scale.
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ACKNOWLEDGEMENTS I would like to express my gratitude and respect to Dr. Randy Seright at PRRC, New
Mexico Institute of Mining and Technology for his constant support and guidance during
my graduate studies at New Mexico Institute of Mining and Technology Dr. Seright, your
knowledge and experience have been great in our development as engineers; your high
standards and dedication not only make us better professionals but also better individuals.
Thank you always.
I would also like to recognize Dr. Lawrence Teufel and Dr. Thomas Engler for their
contributions, and willingness to serve as part of my thesis committee. Not forgetting
PRRC personnel, the author would like to express his appreciation for the financial
support from the PRRC during the period of this research. Thanks to some of my
classmates and friends with whom I had the opportunity to learn, share and enjoy. It has
been a pleasure. Last but not least, special and infinite thanks to the most important
people in my life, my parents, Mr. Adu and Madam Anna Kyremaah, my daughter
Richlove Oppong, and my best friend and love of my live, Twumwaa Elizabeth; all of
your love, respect, encouragement and support have made me the man I am today. I owe
it to you, thank you.
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TABLE OF CONTENT
1.1 Description of the Problem ................................................................................. 3 1.2 Geological Considerations .................................................................................. 4 1.3 Research Objectives ............................................................................................ 5
CHAPTER2. ....................................................................................................................... 7 GENERAL CONSIDERATIONS AND BACKGROUND ............................................... 7
2.1 Theoretical Foundation ....................................................................................... 7 2.2 Literature Review............................................................................................. 13 2.3 Horizontal versus Vertical Wells ...................................................................... 17
CHAPTER 3 ..................................................................................................................... 24 THEORY AND RESERVOIR MODELS DESCRIPTION ............................................. 24
3.1 Reservoir Simulation ........................................................................................ 24 3.2 Mathematical Model ......................................................................................... 24
3.2.1 Formulation of Single- Phases Flow Equations ........................................ 25 3.2.2 Auxiliary Relations ................................................................................... 27 3.2.3 External Boundary Conditions .................................................................. 28
3.3 Numerical Model .............................................................................................. 28 3.3.1 Discretization of the Flux Term ............................................................... 30 3.3.2 Discretization of the Accumulation Term ................................................. 32
3.4 The Polymer Flood Simulation Model ............................................................. 34 3.4.1 Treatment of Fluid Viscosities .................................................................. 35 3.4.2. Treatment of Permeability Reduction. ...................................................... 36 3.4.3. Treatment of the Shear Thinning Effects .................................................. 36
3.5 Solution Technique ........................................................................................... 38 3.6 Well Models ...................................................................................................... 38
3.6.1 Wells Representation in this Study ........................................................... 39 3 .7 Description of Simulation Models .................................................................... 42
3.7.1 Gridblocks Sensitivity Analysis. ............................................................... 44 3.7.2 Communicating Layers System (crossflow model). ................................. 44 3.7.3 Noncommunicating Layers System (No crossflow model). ..................... 44 3.7.4 Polymer Flood ........................................................................................... 45
CHAPTER 4 ..................................................................................................................... 49 PRESENTATION AND ANALYSIS OF SIMULATION RESULTS ............................ 49
4.1 Validation of the Reservoir Simulator .............................................................. 49 4.1.1 Analytical Solution ................................................................................... 50 4.1.2 Volumetric Material Balance .................................................................... 54 4.2.1 Gravitational Effect (Communicating Layers) ......................................... 55 4.2.2 Rock Compressibility................................................................................ 57 4.2.3 Vertical Permeability Ratio (kv/kh). ......................................................... 57 4.2.5 Permeability Contrast................................................................................ 62 4.2.6 Oil Viscosity ............................................................................................. 63 4.2.7 Polymer Solution Viscosity ...................................................................... 67 4.2.8 Summary of Sensitivity Analysis.............................................................. 70
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4.3 Potential of Horizontal Injector in Waterflooding. ........................................... 70 4.3.1 Results Based on Oil Recovery ................................................................. 70 4.3.2 Results Based on Water Cut ..................................................................... 75
CHAPTER 5 ..................................................................................................................... 77 ECONOMICS ................................................................................................................... 77
5.1 Net Present Value (NPV) .................................................................................. 77 CHAPTER 6 ..................................................................................................................... 84 CONCLUSIONS AND RECOMMENDATIONS ........................................................... 84
6.1 Conclusions ....................................................................................................... 84 6.2 Recommendations ............................................................................................. 85
NOMENCLATURE ......................................................................................................... 87 REFERENCES ................................................................................................................. 89 APPENDIX A ................................................................................................................... 97 The algorithms for the simulation models in this study is presented brief below: ........... 97 APPENDIX B ................................................................................................................. 100 The results from two simulators: Eclipse 100 and POLYGEL-Petro China .................. 100 APPENDIX C…………………………………………………………………………. 103 The calculation of NPV and NET CASH FLOW at oil prices, $ 20 and $ 50 per barrel for Displacing 1000 cp oil with polymer: (crossflow and no-crossflow)…..........103
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LIST OF TABLES
Table 2.1 Effect of Polymer Concentration on water cut ultimate Recovery, and EOR24 .................................................................................................................... 17 Table 2.2 Effect of Polymer Molecular Weight on EOR24. .................................. 17 Table 3.1 Pertinent properties of the reservoir models ......................................... 46 Table 4.1 Comparison, Analytical and Simulation (No-crossflow). .................... 51 Table 4.2 Comparison of waterflooding, Analytical and Simulation (Crossflow)................................................................................................................................ 52 Table 4.3 Polymerflooding, Analytical versus Simulation (Crossflow). ............. 53 Table 4.4 Polymerflooding, Analytical versus Simulation (No-crossflow) ........ 53 Table 4.5 Effect Vertical Heterogeneity on Oil Recovery (Free crossflow). ....... 60 Table 4.6 Effect Permeability Contrast on Oil Recovery (crossflow). ................. 63 Table 4.7 Crossflow versus No-crossflow, Waterflooding. .................................. 65 Table 4.8 Polymerflooding, crossflow versus no-crossflow ................................. 66 Table 4.9 HW Compared to VW, Waterflooding (Free crossflow). ..................... 73 Table 4.10 HW Compared to VW, Waterflooding (No- crossflow). .................... 74 Table 5.1 Net cash flow Tabulated at oil price of $100/bbl: Displacing 1000 cp oil with polymer (No-crossflow). ............................................................................... 80 Table 5.2 NPV tabulated at oil price of $100/bbl: Displacing 1000 cp oil with polymer (No-crossflow) ........................................................................................ 81 Table 5.3 Net cash flow tabulated at oil price of $100/bbl oil: Displacing 1000 cp oil with polymer (Crossflow) ................................................................................ 82 Table 5.4 NPV tabulated at oil price of $100/bbl oil: Displacing 1000 cp oil with polymer (Crossflow) ............................................................................................. 83
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LIST OF FIGURES
Figure 2.1 Molecular Structure of Hydrolyzed Polyacrylamide ........................... 12 Figure 2.2 Molecular Structure of Partially Hydrolyzed Polyacrylamide ............ 13 Figure 2.3 Molecular Structure of Polysaccharide (Xanthan Gum) ..................... 13 Figure 2.4 Cross-sections of a Horizontal Injector and a Production wells. ......... 22 Figure 2.5 Examples of a Multilateral Well .......................................................... 23 Figure 3.1 One-dimensional discretization into blocks.Error! Bookmark not defined. Figure 3.2 Mass balance ....................................................................................... 31 Figure 3.3 Relative Permeability Curves .............................................................. 43 Figure 3.4 Block sizes analysis ............................................................................. 46 Figure 3. 5 Communicating Layers ...................................................................... 47 Figure 3.6 Non- Communicating Layers ............................................................. 48 Figure 4.1 Waterflooding: Analytical versus Numerical (No-crossflow). .......... 51 Figure 4.2 Waterflooding: Analytical versus Numerical (Free crossflow). ......... 52 Figure 4.3 Polymerflooding: Analytical verse Numerical (Free crossflow). ....... 53 Figure 4.4 Polymerflooding: Analytical versus Numerical (No-crossflow). ....... 54 Figure 4.5 Gravitational effect on oil recovery. ................................................... 56 Figure 4.6 Compressility effect on oil recovery ................................................... 57 Figure 4.7 Effect of kv/kh on the Oil Recovery (VW). ........................................ 60 Figure 4.8 Effect of kv/kh on the Oil Recovery (HW). ........................................ 61 Figure 4.9 Effect of kv/kh on the quantity of oil crossflowed (VW). ................... 61 Figure 4.10 Effect of kv/kh on the quantity of oil crossflowed (HW). ................. 62 Figure 4.11 Effect of Permeability Contrast on the Oil Recovery. ....................... 63 Figure 4.12 Crossflow versus No-crossflow, Waterflooding. .............................. 65 Figure 4.13 Crossflow versus No-crossflow, Polymerflooding 1000 cp Oil. ....... 66 Figure 4.14 Effect of oil viscosity on water wct, Polymerflooding. ..................... 67 Figure 4.15 Effect of oil viscosity on water wct, waterflooding (No-crossflow). 67 Figure 4.16 Effect of Polymer viscosity on oil recovery, 1000 cp oil No-crossflow. .............................................................................................................. 68 Figure 4.17 Effect of polymer viscosity on oil recovery, 1000 cp oil free crossflow. .............................................................................................................. 69 Figure 4.18 Effect of polymer viscosity on water cut on 1000 cp oil. .................. 69 Figure 4.19 Polymerflooding: HW Injector versus VW Injector (No-crossflow)............................................................................................................................... 72 Figure 4.20 Waterflooding: HW Injector versus VW Injector (Free Crossflow)................................................................................................................................ 72 Figure 4.21 Waterflooding: HW Injector versus VW Injector (No-crossflow). .. 73 Figure 4.22 Polymerflooding: HW Injector versus VW Injector (Free Crossflow)................................................................................................................................ 74 Figure 4.23 Polymerflooding: Impact of horizontal injector on Water cut. ......... 76 Figure 4.24 Waterflooding: Impact of horizontal injector on Water cut. ............. 76 Figure 5.1 NPV computed at oil price of ($100/bbl oil): Displacing 1000 cp oil with polymer (No-crossflow) ................................................................................ 79
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Figure 5.2 Net cash flow computed at oil price of ($100/bbl oil): Displacing 1000 cp oil with polymer (No-crossflow). ..................................................................... 80 Table 5.2 NPV tabulated at oil price of $100/bbl: Displacing 1000 cp oil with polymer (No-crossflow) ........................................................................................ 81 Figure 5.3 Net cash flow computed at oil price of ($100/bbl oil): Displacing 1000 cp oil with polymer (Crossflow). .......................................................................... 81 Figure 5.4 NPV computed at oil price of $100/bbl: Displacing 1000 cp oil with polymer (Crossflow) ............................................................................................. 82
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CHAPTER 1
INTRODUCTION
Traditionally oil production strategies have followed primary depletion,
secondary recovery and tertiary recovery processes1. Primary depletion, also referred to
as primary production, uses the natural reservoir energy to accomplish the displacement
of oil from the reservoir to the producing wells2. As a general rule of thumb, it is
expected that about one third of the original oil in place can be recovered by this method,
in certain cases these recoveries are much lower, and other sources expect only around
10% of the original oil in place to be produced3. Secondary recovery methods are
processes in which oil is subject to immiscible displacement with injectants such as water
or gas. Lastly, tertiary oil recovery involves injection of miscible gases, the use of
thermal energy or the injection of chemicals into the reservoir to accomplish the
displacement of oil from the reservoirs. These operations are also referred to as enhanced
oil recovery (EOR) methods2. Through the entire life of a reservoir, only about thirty to
fifty percent of the original oil in place is produced under primary and secondary
recovery methods altogether1. Consequently, the oil left in the reservoir can be
substantial. The U.S. DOE Fossil Energy Program3 estimates about 65% of currently
discovered resources will not be produced with the use of current production strategies
and present technologies. Planning for efficient production strategies and targeting
unrecoverable oil should be a priority in the petroleum industry. This has challenged
researchers to come up with viable strategies for optimizing our reservoirs.
2
Polymer flooding is a chemically augmented waterflood in which chemicals, (polymer)
such as polyacrylamide or polysaccharides, are added to injected water to increase the
effectiveness of the water in displacing oil.
Formation heterogeneity affects the performance of most flooding operations.
Unfortunately, most oil formations usually exhibit random variations in their
petrophysical properties. In such reservoirs, statistical as well as geological criteria 4
usually are used to divide the pay zone between adjacent wells into a number of
horizontal layers each with its own properties and homogeneous in itself. Such reservoirs
are usually referred to as ‘stratified ’, ‘layered’ or ‘heterogeneous’ reservoirs.
Heterogeneity plays a dominant role in predicting waterflooding performance in stratified
reservoirs. Typically, heterogeneity in reservoir may be present both in the vertical and
the horizontal directions. Throughout the development of this thesis, we assumed
heterogeneities only in the vertical direction of the reservoir. Under this assumption, the
reservoir fluid tends to flow from one layer to the other if there is sufficient
communication between the layers. This is referred to as fluid crossflow between layers
or vertical communications between layers. In some cases, the depositional sequence may
be such that during successive depositions, an impermeable shale layer is sandwiched
between successive reservoir layers to isolate the layers completely form each other, such
that there is very little or no vertical communication between the layers. This is referred
to as no vertical crossflow. Fluid crossflow may be the result of any or all of the four
driving forces: viscous forces, capillary forces, gravity forces and dispersion. These
driving forces interact with each other in the displacement. Our objective will be to
investigate crossflow caused by gravity forces and viscous forces on the sweep efficiency
3
polymer flood process in a stratified reservoir. This is accomplish by simulating cases
with crossflow (no vertical communication) and then crossflow using a black oil model in
Eclipse, a general purpose reservoir simulator.
The concept of vertical equilibrium has been used extensively in the petroleum literature,
mainly as a way of collapsing simulations to the lower dimension5. Vertical equilibrium
(VE) is simply; an assumption that the sum of the driving forces in the vertical direction
is zero for all fluids components at all positions so that pressure will be the same on any
vertical line in each layer. Assuming (VE) implies perfect vertical communication, which
is equivalent to assuming infinite vertical permeability. VE will be good assumption for
reservoirs with aspect ratio (RL) of 10 more5. RL is expressed as:
h
VL K
K
H
LR (1.1)
Where, kV and kh represent vertical and horizontal permeabilities, H and L represents
total thickness and the length of the reservoir respectively.
This thesis evaluates polymer flood potential for an idealized two-layered reservoir by
simulation of waterflood and polymer floods under varying conditions and comparing the
results.
1.1 Description of the Problem
Reservoir heterogeneity plays a major role in oil recovered by
waterflooding/polymer flooding through its influences on fluid crossflow. Simulation
runs were conducted to assess the extent to which the vertical heterogeneity (degree of
fluid crossflow) affects the displacement efficiency. Surprisingly, simulation results using
4
Eclipse100 black oil simulators, agrees well with the results from two other simulators,
VIP and POLYGEL (Petro China), and suggests that fluid crossflow is not a factor to be
consider in waterflooding/polymer flooding operations. Accepted reservoir engineering
(Craig, Lake, Coats) claimed that as the mobility ratio becomes increasingly unfavorable
(high), recovery efficiency worsens more rapidly for the crossflow cases than the non-
crossflow cases. This discrepancy requires further investigations. The Eclipse simulation
results are presented in appendix B figures (B-1 and B-2) and results for POLYGEL
(Petro China) is shown in the same appendix figures (B-3 and B-4).
Furthermore, earlier screening criteria developed for polymer flooding6,7 indicated
that the conventional polymer flooding should be limited to reservoirs with oil viscosity
not exceeding 150 cp for economical recovery. However, a significant number of
reservoirs have been identified with crude oil viscosities above 1000 cp.
1.2 Geological Considerations
Among the factors that influence the success of EOR operations is the
heterogeneity of the oil formation. Reservoir heterogeneities occur both in vertical and
horizontal directions. For heterogeneity in vertical direction, the degree of vertical
communication between the layers is of prime concern of this work. Vertical
heterogeneity depends on the depositional environment of the formation and geologic
time in which it occurs8. Vertical trends are formed and can be monitored with
characteristics of the formation, such as porosity, grain size distribution, and
permeability. Stratigraphy of a reservoir may be identified with logging techniques or
direct measurements on cores. Natural radioactivity measurements can be used to identity
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depositional environment, and the type and age of the formation9. Gamma ray log
responses are widely used to estimate stratigraphy trend of formation. The identified
vertical trends in most oil-bearing formation are fining upwards and downwards. Fining
upward describes formations that consist of increased grain sizes and permeability in the
downward direction of the depositional sequence of the formation. Fining downward
cases are just the reverse.
In some cases, the depositional sequence may be such that during successive
depositions, an impermeable shale layer is sandwich between successive reservoir layers
to isolate the layers completely form each other, such that there is very little or no vertical
communication between the layers, the layers only communicate through the wellbore
this create a condition of no vertical crossflow between the layers. On the other hand,
vertical crossflow refers to the case where there is direct communication between the
layers. The objectives of this thesis are further outlined in the next section.
1.3 Research Objectives
The main objective of this research focuses on use of water-soluble polymers to
provide greater sweep efficiencies in multi-layered unconventional reservoirs with and
without crossflow. Specifically, this study consists of polymer injection simulation
studies for this work consists of polymer injection simulation studies for chemical EOR
processes, immiscible displacement operations. Also we will spread our wings to
examine the potentials of the use of horizontal well injectors to improve sweep efficiency
of heterogeneous viscous oil reservoirs at varied mobility ratio. This is accomplish by
simulation for the cases model in Eclipse, a general purpose reservoir by:
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1. Developing reservoir simulation models for waterflood and polymer flood at
varied mobility ratio,
2. Comparing the performance of horizontal well injectors with vertical well
injectors in improving sweep efficiency during polymer flood,
3. Examine the impacts of the degree of fluid crossflow (gravity and viscous),
Layering, permeability contrast, and mobility contrast between the displacing
fluid and the injectant on the levels of the sweep improvement and compare the
results to analytical results
4. Investigate the impact of the variation in polymer viscosity on the polymer
performance,
5. Economic analysis.
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CHAPTER 2.
GENERAL CONSIDERATIONS AND BACKGROUND
This section focuses on important theoretical aspects waterflooding and immiscible
displacement operations, which are the basis of these simulation studies. Also a literature
review is presented on past work, to examine how polymers provide mobility control and
also compare the success of using horizontal wells versus vertical wells to enhance
displacement efficiency of waterflooding and polymer flooding operations.
2.1 Theoretical Foundation
Many factors influence the success of waterflooding operations and immiscible
displacement processes. These factors can be separated into two categories, one that refer
to characteristics of the reservoir fluids and one that referred to the formation8, 10.
Reservoir characteristics that influence the efficiency of waterfloods may include depth,
porosity, fluid saturation distribution, rock structure and type, and the degree of
formation heterogeneity. This last reservoir characteristic, the degree of formation
heterogeneity, is a primary focus of this study10. The heterogeneity effect on immiscible
displacement and waterflooding processes depends on horizontal and vertical non-
uniformities that allow fluids to move preferentially through the high permeability porous
medium. This flow allows for part of the oil in place to be bypassed in lower permeability
areas10 Many prediction methods have been created for this type of process, where fluid
flow, well patterns, and vertical heterogeneity are considered. Most of these methods
8
assume formations with homogeneous areal rock properties and include heterogeneities
only in the vertical direction8, 10, 11. These techniques originate from Buckley and
Leverett’s work10, and consist of prediction methods for waterfloods in stratified
formations. The earliest group of prediction methods in which heterogeneity of the
formation was considered includes works by Dykstra and Parsons12, Stiles4, and Yuster-
Suder-Calhoun13. These methods have been modified and have become the basis of other
methods, such as, Higgins and Leighto14, Craig-Geffen-Morse15, and Prats-Matthews-
Jewett-Baker16. These methods are among the most accepted although the use of
reservoir simulation has diminished the use of these prediction techniques17.
Some results published on stratified reservoirs show that the variation of reservoir rocks
is mainly controlled by specific factors such as depositional environment, grain size
distribution, and formation mineralogy. Consequently, fractures, re-deposition, and
compaction could also be factors to consider in the origin and variation of the reservoir
arrangement18. All of these parameters and formation characteristics influence oil
recovery efficiency (ER), which measures the fraction of oil in place at the start of a
secondary or tertiary displacement process that can be recovered during displacement
operations10, 19 ER can be expressed as:
DVR EEE (2.1)
In this equation, (ED) represents the microscopic displacement efficiency, which can be
defined as the fraction of the total oil present in the reservoir that has been displaced by
the injecting fluids. ED is control by the wettability of the formation rock, as well as, the
pore size distribution of the reservoir volume contacted by the displacing fluid. In
equation (2.1), the volumetric sweep efficiency (EV) represents the portion of the
9
reservoir that is contacted or swept by the injected fluids with respect to the total volume
of the reservoir. This parameter is affected mainly by the degrees of formation
heterogeneity and the mobility ratio between the displacing and the displaced fluids
during the displacement process19. The three-dimensional volume sweep efficiency can
be separated into two-dimensional areal sweep efficiency term and a vertical sweep
efficiency term8, 11, 19. The volumetric sweep efficiency can be express in terms of the
areal sweep efficiency (EAS) and the vertical sweep efficiency (EVS) as:
VSASV EEE (2.2)
Thus the overall oil displacement efficiency is express as:
DVSASR EEEE (2.3)
The areal sweep efficiency (EAS) depends on the mobility ratio, inter- walls spacing, and
the well arrangement36. In the literature, most prediction methods designed to examine
the behavior of waterflooding operations, combined the effect of the microscopic
displacement efficiency and the areal sweep efficiency. The vertical sweep efficiency
component describes the volumetric sweep efficiency dependency on the vertical
stratigraphy or heterogeneity of the formation and mobility ratio. Mobility ratio
measures the relative velocity of the phases in the reservoir. During the displacement
process, the fluid with higher velocity break through first in the production wells. This
further suggests that viscosity of the phases in the reservoir is the primary fluid
characteristic that affects waterflooding performance, as viscosities of the displaced and
displacing phases affect the mobility ratio in immiscible displacement operations1, 2, 8.
Many techniques have been developed to improve the recovery of oil when
mobility ratio and formation heterogeneity cause adverse effects on the waterflooding
10
operations. One such method is the mobility control technique. This method uses
chemical agent such as polymers to enhance the volumetric sweep efficiency11 by altering
the relative fluid flow in favor of the displaced fluids. According Muskat20 mobility ratio
(M) is the ratio of the mobility of the displacing fluids to that of the displacing fluids in
the regions of the reservoir contacted by the injected fluid2, 11, 12. Equations (2.4) and (2.5)
present the mobility and mobility ratio respectively.
i
iri
kk
(2.4)
In this equation, λi is the mobility of the phase i, where i is the displacing or the displaced
fluid, (k) is the absolute permeability, and (kri) is the relative permeability to the ith phase
and, (µi) is the viscosity of the ith phase. The mobility ratio is presented next.
d
DM
(2.5)
Equation (2.5) presents an expression for the mobility ratio as a function of the mobility,
relative permeability and viscosities of fluid phases. The displacing phase is represented
with the D subscript and the displaced phase distinguished with the d subscript. For
situations in which the displaced fluid (oil) has a higher viscosity (viscous oil), the
mobility ratio is unfavorably (M>1). For such cases, the displacing fluid finger through
the porous medium, leaving oil behind in the unswept regions of the reservoir10. Because
of the understanding of this important concept, EOR processes have incorporated the use
of higher viscosity injection fluids, most commonly accomplish with large
macromolecules called polymers1, 19.
Polymer flooding has been referred to as an improved waterflood in which water-
soluble polymers are added to the injection water to improve the efficiency of the
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displacement process19. Polymer flooding improves oil recovery by increasing the
viscosity of the displacing fluid. A polymer flood would improve recoveries where
mobility ratios between the displaced and the displacing fluids are unfavorable (greater
than one) and in formations where the heterogeneity is moderate.
There are two principal types of polymer being used in field applications to accomplish
displacement processes: hydrolyzed polyacrylamides (HPAM) and polysaccharide
biopolymer or xanthan gum. Polyacrylamides are produced synthetically through
polymerization of the acrylamide monomer19. The hydrolyzed Polyacrylamides are
usually hydrolyzed to reduce the adsorption property of the original polymer when
injected into the formation. Through hydrolysis, some of the reactive acrylamide are
converted carboxylate groups with negative charges1. The degree of hydrolysis of the
polymer is usually within the ranges of 20% to 40%. In saline water, the electrolyte in
solution causes the molecule to coil. This reduces the viscosity. The hydrolyzed
polyacrylamide solutions are salt sensitive. Other susceptibilities of HPAM solutions are
caused by the presence of oxygen and divalent ions, which are the sources of instability
and chemical degradation by temperature and mechanical degradation. The HPAM
molecules’ long chain may be broken, especially at high velocity and temperature
conditions when the injected solution passes through the well’s perforation interval and
flow through the porous spaces of the formation near the wellbore. Being less expensive
and providing higher residual resistance to drive water injection, polyacrylamide is more
widely used in the field than the polysaccharide as a mobility control agent. Biocide such
as formaldehydes, need to be used to prevent the viscosity loss cause by microbes. On the
12
other hand, polysaccharide biopolymer is obtained from sugar in a fermentation process
caused by the bacterium, Xanthomonas campestris.
The polysaccharide molecular structure gives the molecules a greater stiffness13,
their behavior being like a semi-rigid-rod molecule13. In contrast to polyacrylamide, the
viscosity of a xanthan gum is not affected by salinity, and shearing can be tolerated.
Despite, the advantages, the polysaccharide biopolymer is expensive, and its stability
decreases with temperatures of about 160οF. Biodegradation of polysaccharide by
enzymes is very common. Biocides are always added to the Xanthan biopolymer before
injection to the formation to protect the integrity of the polymer from bacterial attack and
aerobic degradation19. Xanthan biopolymers have low retention on reservoir rock surface.
Figure 2.3 shows the molecular structure of xanthan gum and Figures 2.1 and 2.2 both
forms of polyacrylamides.
Figure 2.1 Molecular Structure of Hydrolyzed Polyacrylamide
13
Figure 2.2 Molecular Structure of Partially Hydrolyzed Polyacrylamide
Figure 2.3 Molecular Structure of Polysaccharide (Xanthan Gum)
2.2 Literature Review
As mentioned in Chapter 1, the objectives of this research are to examine the
effectiveness of using water-soluble polymers and horizontal well to provide a more
sweep efficiencies of multi-layered unconventional reservoirs with and without
crossflow. This was done with the help of a reservoir simulator, developed for this study.
An extensive review of the literature was performed to understand the procedure in
14
building a simulator with proper representation of the horizontal well in a reservoir
simulator. The following is an overview of the literature.
Waterflooding is the most common secondary recovery operations in the
petroleum industry8. The method has gained lot of acceptability in the industry since the
mid 1890s. However, the method has a limited applicability when the displacement is
characterized by a remarkably unfavorable mobility ratio. Kumar et al22 examined
waterflood performance using unfavorable mobility ratios. They concluded that viscous
fingers dominate high mobility ratio floods, that mobile water can significantly reduce oil
recovery and that thief zones accentuate poor displacement performance. They strongly
suggested that any improvement in mobility ratio (e.g., polymer flooding) could improve
recovery and sweep efficiency.
Polymer floods have been applied during several occasions13, 17, 20, 23. These
include polymer floods applied at the Daqing oil field24, 25, the world’s largest polymer
flood field, Marmul26, Oerrel7, and Courtenay 27. Field tests have proved that the method
has potential to provide superior oil recovery. In formations where long fractures
dominate the formation, and cause severe channeling, gel treatments or other types of
“profile modification” methods before polymer flooding can greatly enhance reservoir
sweep24. Polymer flooding has been widely used as an improved waterflooding operation;
its mechanism of oil displacement and the chemistry of the polymers are not being
questioned in this work. Polymer injection studies conducted in this thesis will focus on
how the degree of crossflow (vertical heterogeneities), permeability contrast, polymer
slug size, adsorption, and concentration affects polymer flooding recoveries. Some of
these components will be examine at varied mobility ratio to establish a limit of mobility
15
ratios within which polymer flooding will be more advantageous over the conventional
waterflooding operations. Several authors7, 28, 29, 31, 32 have examined polymer flooding
operations and have published about the degree of crossflow, molecular weight of
polymer, polymer concentration, viscosity, degradation, brine salinity and cost
effectiveness. These researchers affirmed that that the aforementioned properties are
critical during polymer flood operations.
Zhang and Seright33 examined the degree of crossflow in polymer flooding.
They concluded that if crossflow can occur between adjacent strata, sweep in the less-
permeable zones can be almost as great as that in the high-permeable zones if the product
of mobility ratio and the permeability differential is less than unity. However, if no
crossflow occurs, sweep in the less- permeable zones will not be better than the square
root of the reciprocal of the permeability differential. Wu et al34 devised a separate-layer
injection technique to improve the sweep efficiency when crossflow does not occurs. The
technique was found to improve flow profile, reservoir sweep efficiency and also
minimize water cut in the production wells. Numerical simulation studies conducted by
Wu et al34 revealed that efficiency of polymer flood depends on permeability contrast
between the adjacent layers and at the time at which separate –layer injection occurs.
Recent work published24 on polymer systems, where properties such as molecular
weight viscosity have been modified to create a system that could effectively improve the
mobility ratio between the displacing and the displaced fluids and improve sweep
efficiency in returns. The effectiveness of the system was found to increase with
increased polymer viscosity Table 2.1 shows these results. At a giving set of conditions,
polymer viscosity increases with increasing polymer molecular weight. Wu et al35
16
performed laboratory tests with affixed volume of polymer solution injected but varied
the molecular weight of the polymer. They confirmed that oil recovery increases with
increasing polymer molecular weight.
Table 2.1 shows the results of this study24. For a giving polymer, chemical retention
increases and the rate of polymer propagation decreases, as the rock permeability is
decreases. High-molecular weight polymers usually experience high retention and low
propagation rate for lower rock permeablities6, 36.
Properties of hydrolyzed polyacrylamide solutions are salt sensitive, the solution
viscosity decrease drastically with slight increase in salinity21. Thus, for high-salinity
formations, HPAM solution is fairly ineffective during a polymer flood. Maitin7 studied
polymer flooding of high- salinity reservoirs using HPAM. He injected fresh water of low
salinity before injection of HPAM solutions. Maitin7 suggested that pre-conditioning a
high-salinity formation with fresh water of low salinity can effectively reduce the
formation salinity and improve the performance of the polymer flood.
Jennings et al38 presented numerical variables defined as the “resistance factor
(Fr)” and “Residual resistance factor (Frr)” to account for the mobility reduction of the
injected polymer solution and measure of reduction in rock’s permeability to water after
polymer injection. The resistance factor (Fr) express the ratio of mobility of the water in
place compared to the polymer injected, and the Frr, the change of the mobility of the
water in place before and after the polymer injection has taken place38.
Residual resistance factor (Frr) and resistance factor (Fr) are express mathematically as:
p
wrF
(2.4)
17
)after(
)before(F
w
wrr
(2.5)
In equations (2.4) and (2.5), λW and λP denotes mobility of water and polymer
respectively.
Table 2.1 Effect of Polymer Concentration on water cut ultimate Recovery, and EOR24
Polymer Concentration
(mg/L)
Minimum
water cut, %
Ultimate
Recovery, %
EOR
%
600 87.1 50.58 7.69
800 85.0 52.52 9.64
1000 83.1 52.83 9.95
1200 82.4 52.89 10.01
1500 81.0 53.03 10.15
Table 2.2 Effect of Polymer Molecular Weight on EOR24. Molecular Wight Waterflood Polymer Ultimate
106, Daltons Recovery (%) Recovery (%) Recovery (%)
5.5, Daltons 32.7 10.6 43.3
11, Daltons 32.9 17.9 51.8
18.6, Daltons 32.2 22.6 54.8
2.3 Horizontal versus Vertical Wells
The main objective of using horizontal wells (injectors and producers) is to
increase the well's contact with the reservoir, thereby improving the injection and
18
production process. The increased surface area of contact enables the horizontal well
injector to invade parts of the reservoir that are not accessed by the vertical well injector.
Applications of horizontal wells in water and chemical flooding projects as both injectors
and producers continue to grow. It has been used in waterflood and in polymer flood
applications to improve sweep efficiency39. The advantages of horizontal wells in
waterflood/enhanced oil recovery (EOR) applications are to enhanced injectivity and
productivity. Another advantage lies on their ability to reduce the number of vertical
injection and production wells without sacrificing injectivity or productivity39. The cross-
section of injector and producer is shown in Figure 2.4. The progress in the field of
directional drilling in the recent years have immerged a multilateral well technology. A
multilateral well is defined as one vertical wellbore draining from two or more horizontal
wells as shown in Figure 2.5. This is very useful in the cases where one or more vertical
permeability barriers are present in the reservoir or the surface is environmentally
sensitive. The horizontal portion of the wells can be drilled form a single vertical
wellbore to access different parts of the reservoir, hence, bypassing the permeability
barriers.
Joshi39, 41 evaluated the production performance of horizontal and vertical wells.
He suggested not using horizontal wells in uniform formations whose thickness exceeds
200 feet. This is because the advantage of a horizontal well in a thick formation
diminishes as compared to a fully penetrating vertical well. Taber and Seright45 reported
the benefits of horizontal wells over vertical wells, in waterflooding, based on analytical
equations numerical simulation studies. Their study showed that horizontal wells showed
better areal sweep efficiencies of about 25% to 40%, higher flooding rates, and lower
19
injection pressures as compared to vertical wells. The above-mentioned properties of
horizontal wells make them very beneficial for all EOR processes. The analytical
equations used in their study, however, do not account for after water breakthrough
performance, capillary pressure or geologic layering.
Kossack, Kleppe, and Aasen42 published the investigation of oil production from
the Troll Field in the Norwegian North Sea. This field comprises of a thin formation in
deep water environments. A problem identify with the Troll Field is that the conventional
wells cone gas and/or water within 2-3 years of production. If too few vertical well
producers are drilled, the wells will be shut in due to a high gas-oil ratio before the oil
recovery is satisfactory. If too many vertical wells are drilled, they interfere with each
other and gas coning problems may arise earlier. As a result of this reasoning, drilling of
horizontal wells was proposed in their study. Kossack et al42 also compared horizontal
and vertical wells for the Troll Field in the North Sea. These researchers reported that
horizontal wells performance was much better in production of thin oil zones than
vertical wells.
A significant number of numerical simulation studies are available on use of
horizontal well for waterflooding projects. Pieters and Al-Khalifa43, using a three-
dimensional reservoir simulation model, investigated the use of horizontal and vertical
wells in waterflooding for a layered heterogeneous carbonate reservoir. They showed that
horizontal and vertical wells recovered the same amount of oil in tight reservoirs
provided the vertical well penetrates the entire reservoir. Dykstra and Dickinson44
calculated the gravity drainage oil recovery from vertical and horizontal wells. They
stated that for flat formations (no gravity effect), with thickness less than 0.85 times the
20
well spacing, a horizontal well produces better than the vertical well whereas, at
formation thickness greater than this, a vertical well performs better. Also, for flat
formations, the formation thickness affects the ratio of horizontal/vertical well flow rates.
But for dipping formations, formation thickness has no effect on the ratio of
horizontal/vertical well flow rates. Joshi et al45 uses two-dimensional reservoir simulation
studies to showed that the use of horizontal wells as producers or injectors do not provide
a significant increase in areal sweep efficiency over vertical wells. However, horizontal
wells have higher productivity as producers and higher injectivity as injectors. Moreover,
the reservoirs with high permeability, horizontal wells may not provide a significant
advantage.
Two- and three-dimensional simulation studies performed by
Ferreira et al46 showed that vertical to horizontal permeability ratio, injection and
production rate, and reservoir thickness have little effect on waterflood oil recovery for a
particular mobility ratio. They observed that waterflood performance is better with a
horizontal well as compared to a conventional vertical well. They developed a correlation
that expresses the volumetric sweep efficiency as a function of mobility ratio and is
useful to predict the waterflood recovery. Gharbi et al47 used three dimensional chemical
flood simulator, investigated the performance of immiscible displacement with horizontal
and vertical wells in heterogeneous reservoirs. They studied the sensitivity of the
displacement performance to the horizontal well length and the ratio of horizontal to
vertical permeability using various well combinations. They showed that the degree and
structure of the heterogeneity of the reservoir have a significant effect on the efficiency of
immiscible displacement with horizontal wells. Long horizontal wells in highly
21
heterogeneous reservoir do not necessary guarantee improved oil recovery. In subsequent
work, Gharbi et al48 showed that the performance of enhanced oil recovery processes
with horizontal wells is strongly affected by the permeability variation and the spatial
correlation of the reservoir heterogeneity. Algharaib and Ertekin49 studied the effect of
various waterflooding fluid parameters together with some operational design parameters.
The numerical analysis showed that the combination, in which one horizontal and one
vertical well are utilized, performs similar to the combination of two horizontal wells.
Popa et al50 analyzed the overall efficiency of a waterflooding process that is influenced
by well pattern using horizontal/multilateral injectors and producers in different
configurations. They showed that main parameters, such as breakthrough time, oil
recovery at breakthrough, sweep efficiency, injection-production pressure, etc. are
strongly affected by the type of configuration considered.
Recently, Algharaib and Gharbi51 investigated the performance of non-
conventional wells in water flooding projects under different operating reservoir
conditions using numerical simulation techniques. Their results show that the well pattern
used for waterflooding has a significant effect on the displacement performance of non-
conventional wells. Moreover, long horizontal/multilateral wells do not automatically
guarantee improved oil recovery. Very limited experimental studies have been conducted
to investigate the effect of horizontal well on waterflooding oil recovery.
In 1991, Peaceman52 provided guidelines regarding the representation of
horizontal wells in a numerical reservoir simulator. He suggested that for a horizontal
well, it is sufficient to interchange ∆y and ∆z, as well as ky and kz in his previous
equations for calculating the equivalent wellbore radius (ro) for vertical wells. The reason
22
h
is that, the horizontal well is located along the horizontal plane and its performance is
affected by permeability and block dimensions in the z-direction and x-direction or y-
direction depending on the orientation of the well.
Shirif et al54 with the help of experimental study, examined the effect of vertical
and horizontal injection and production well combinations and found that the use of
horizontal wells showed slightly better oil recovery over vertical wells in a waterflood of
reservoirs under bottom water conditions. From the existing literature, it can be
concluded that horizontal wells are advantageous in EOR over conventional vertical
wells. An objective of this work is to investigate, by numerical simulation studies, the
performance of horizontal wells (production well and injection) in unconventional
reservoirs
Figure 2.4 Cross-sections of a Horizontal Injector and a Production wells.
23
Figure 2.5 Examples of a Multilateral Well
Bilateral Well
Cross-Section of Reservoi rr Formation
Multilateral Well
24
CHAPTER 3
THEORY AND RESERVOIR MODELS DESCRIPTION
3.1 Reservoir Simulation
The main purpose of reservoir simulation is to understand the fluid flow behavior
of a petroleum reservoir in order to optimize the hydrocarbon recovery. There are four
major stages to the modeling process for reservoir simulation.
First, a physical model is constructed to represent the physics of fluid flow processes in
the reservoir. Second, a mathematical model is built based on the physical model; this
involves nonlinear partial differential equations. The third stage constitutes the
transformation of the mathematical model into the discretized numerical model capable
of producing solutions representing the basic physical features in the reservoir. Finally,
an algorithm followed by a computer program is developed to perform the necessary
computations for solving the discretized numerical model. In order to understand the
complexities of the entire modeling process, it is important to comprehend the physical
behavior of the recovery process.
3.2 Mathematical Model
The physical laws that govern fluid flow in a porous media are based on
conservation of energy, mass, and momentum in addition to Darcy's law. The flow
equations are partial differential equations, which model the basic processes that occur
within the reservoirs. The flow equation for each phase present in the reservoir is referred
25
to as a single-phase flow equation. The flow equations will be discussed in the next
section.
3.2.1 Formulation of Single- Phases Flow Equations
The following assumptions were made in order to formulate single-phases the flow
equations in a two-layered system.
1. Two phases exist: oil and water,
2. No capillary effects,
3. Fluids are incompressible,
4. Oil and water are immiscible,
5. Reservoir temperature remains fairly constant throughout the study due to
continual injection.
The two- phases flow equations for a black oil model used in this study for both oil
phase and water phase are given by:
.StB
1q
B
V.
(3.1)
Where ℓ= o, w, (subscript ℓ will be used in the rest of the chapters, consistently to
represent oil or water) and del ( ) is the divergence operator, which can be expressed in
terms of the Cartesian Coordinates (x, y, and z) as:
zyx
(3.2)
Equations (3.1) and (3.2) represent three-dimensional flow via the del ( ) operator. In
order to explain the different terms in these equations, a reservoir in the shape of a block
is shown in Figure 3.1, where the flow is in the x-direction only. The first term;
26
B
V. (3.3)
is the flux term and represents a change in mass flow across the cross-sectional area of
the faces x, and x+Δx.
The del operator in this term represents partial differentials with respect to the space
coordinates (x, y, z). The (q) term in these equations is the source/sink term or the net rate
of withdrawal/injection from the reservoir. The last term;
.StB
1
(3.4)
is the rate of accumulation or change in amount of mass inside the reservoir. Again, the
third term is a partial differential with respect to time. Darcy’s velocity, v, in the
equations (3.1) for the two phases are formulated as:
(3.5)
Combining equations (3.1) and (3.2) with equation (3.5) results in a single-phases flow
equations for a black oil model for two phases (oil and water). These are partial
differential equations, formulated as:
.StB
1qZP.
(3.6)
B
kkr
(3.7)
In addition to the above partial differential equations, certain other supporting relations
are required to complete the mathematical model. These will be treated next.
ZPkk
V r
27
3.2.2 Auxiliary Relations
These are the supporting relations required to support the partial differential
equations to complement the mathematical model. For two phases reservoir (oil and
water) the saturations of all phases present in the reservoir sum up to 1, thus,
So + Sw = 1. (3.8)
The formation volume factor of a phase (oil or water) is defined as the volume of that
respective phase at the reservoir pressure and temperature required to produce one cc of
that same phase at the stock tank pressure and temperature. The formation volume factor
is given by the equation:
ST
SCo V
VB (3.9)
Where RC represents reservoir conditions of pressure and temperature, and
ST stands for stock tank conditions of pressure and temperature.
The porosity ( ), which measures the storability of a porous medium is the ratio of the
pore volume (void space) in a rock, to the total or bulk volume of the same rock. It is
related to the rock compressibility and pressure within the medium according to the
relation:
.refr.ref PPC1 (3.10)
In equation (3.10), .ref is the porosity at reference pressure Pref., the pressure at which the
porosity is measured, usually, at the initial pressure of the reservoir.
For incompressible flow in a medium, Bo and Bw are constants and not necessary equal to
1, but for equal densities53:
Bo = Bw =1 (3.11)
28
3.2.3 External Boundary Conditions
The external boundary condition for the reservoir needs to be specified for a
system in order to complete the mathematical model. The external boundary conditions
relate to any external flow or no-flow into the reservoir. A closed boundary or (no-flow)
is used as external boundary condition in this work. In that case, there is no flow of any
phase (oil or water) across the external boundary of the reservoir model and the reservoir
assumed to a closed system. In other words, the component of velocity perpendicular to
the external boundary is zero. This can be expressed by a dot product shown below:
Vℓ · n = 0 (3.12)
0nZPkk r
(3.13)
Where v is the macroscopic velocity given by equation (3.3) and (n) is the unit vector
normal to the boundary of the reservoir.
3.3 Numerical Model
The next step after the selection of the mathematical model is the selection of the
formulation technique to be incorporated in the numerical model. Basically, there are two
methods for the formulating multiphase flow equations:
1. Implicit pressure-explicit saturation method (IMPES),
2. Fully implicit method.
The IMPES technique assumed that there is no change in the capillary pressure over a
time step. The capillary pressure is updated after each time step. The IMPES technique
combines the single-phase equations (for oil and water) into one multiphase equation that
represents both oil and water. The multiphase equation is then solved implicitly for the
29
pressure and calculated oil and water saturations explicitly for each spatial point in the
discretized reservoir model. The fully implicit method involves two, single-phase
equations (for oil and water) in a form where the saturation derivatives with respect to
time are converted to pressure derivatives. The two, single-phase equations are then
solved implicitly for pressures in the oil and water phases. The saturations are then
calculated implicitly using capillary pressure relations. A fully implicit finite difference
technique was used in this study because it is recommended for modeling and
computational reasons because it solves for both pressure (oil and water) and saturations
(oil and water) implicitly as compared to IMPES technique which solves for pressures
implicitly and then calculates the saturations explicitly. The finite difference form of the
single-phase equations in section 3.1 is as follows:
1,,
1
,
1
n
kitkin
ki qB
S
tVZPT
(3.14)
The superscript ‘n’ represents the previous time step and ‘n+1’, the current time step. T is
the transmissibility, and defined in the x, and z-directions as:
x
AT x
xx (3.15)
z
AT z
zz (3.16)
Alternatively, equations (3.14) and (3.15) can be simplified as:
1,,
1
,
1
n
kikin
ki qB
S
tVT
, (3.17)
The Φ is the phase potential, and defined as:
ZP (3.18)
30
The equation (3.14) is the same but different forms of equations (3.1) and (3.2) given in
section 3.1, the first term in equations (3.14) represents the flux term, the second term on
the right hand side is the change of amount of mass inside the reservoir or the rate of
accumulation, and the q term represents the source/sink term.
3.3.1 Discretization of the Flux Term
Discretization is the process of obtaining a finite –difference equations that
approximate a given differential equations. The discretized forms of the oil and water
flux terms in equations (3.17) for a two-dimensional model in Cartesian coordinates is
given below:
kiki
kikiki
kix
n
ki ZZPPTZpT ,,,1,
2
1,
,,,1,,
2
1,
1
,
kiki
kikiki
kixZZPPT ,,1
,2
1,
,,,1,,
2
1,,
kiki
kikiki
kizZZPPT ,1,
2
1,,
,,1,,
2
1,,,
.,1,
2
1,,
,,1,,
2
1,,,
kiki
kikiki
kizZZPPT
(3.19)
In equation (3.19), 'i+1/2' represents the boundary between gridblock i+1, and i
whereas 'i-1/2' stands for the boundary between gridblock i-1 and i. Pressure and the cell
depth (Z) are independent variables so their individual values for different gridblocks are
used for subsequent blocks. Whereas the transmissibility is a dependent variable, which
depends on rock and fluid properties, so transmissibility is calculated by averaging the
31
properties of two gridblocks, across which the flow is being calculated. This means that
the transmissibility is not for a respective gridblock but rather at the boundary between
two gridblocks. This explains the use of '1/2' subscript in the transmissibility. For One-
dimensional discretization into blocks shown in Figure 3.1, the transmissibility between
the i-1 and i blocks calculated at i-1/2.
Figure 3.1 One –dimensional discretization into blocks
xi-1 xi+1
xi
xi+1/ 2 xi-1/ 2
Δx
32
Figure 3.2 Mass balance
3.3.2 Discretization of the Accumulation Term
The discretizations of the accumulation term in equation (3.14) are as follows:
wotn
w
n
o
nn
kiookiki PPSbV
BStV '
1 2
1
2
1
,
,, (3.20)
wotn
w
n
w
nn
kiwwkiki PPSbt
VBStV '1 2
1
2
1
,,, (3.21)
B
1b (3.22)
The independent variables are oil pressure (Po), water pressure (Pw), and water saturation
(Sw). The pressure dependent variables are the expansion terms (bℓ) and its derivatives,
viscosity (µℓ), and porosity (Φ). The saturation dependent variables are relative
permeability (krℓ). The previous time step, 'n' is the time where the dependent variables
Y
X
Mass in
Z
Face X +ΔX Face X
Face X +ΔX Face X
Mass Accum
Mass Out
q (Withdraw/Injection)
33
have already been calculated and are to be evaluated at the current time step,'n+1', where
the dependent variables in the flow equations (3.31) and (3.41) are to be calculated
Usually, it is assumed that the change independent variables (oil pressure (Po ), water
pressure (Pw), and water saturation (Sw) are very small over a small time step and thus, Sw
is evaluated at the previous time step (n).
The pressure dependent parameters are evaluated at the averaged value of
pressure at n+1/2, which is the average value of the pressure between n and n+1. The
saturation dependent parameters krw and kro are evaluated at n+1/2. The oil pressure (Po),
water pressure (Pw), and water saturation values are averaged in time between the
previous timestep (n) values and are extrapolated in the current timestep (n+1) values.
Hence, the averaged values of time, at time step n+1/2.The extrapolation is performed in
order to solve for the values of independent variables at the current time step, the values
of the dependent variables at the current time step are required. The extrapolated values
of water saturation, oil pressure and water pressure are calculated using the following
equation:
1nn1n
n1n uu
t
tu
(3.23)
Where u represent the independent variables (pressure in the oil and water phases, and
water saturation). The values at 'n+1/2' time level are evaluated as follows:
2
uuu
n1n2
1n
(3.24)
So the values of the independent variables are extrapolated to the timestep (n+1) using
equation (3.23) and then are averaged in time to n+1/2 using equation (3.24). The next
section will be devoted for the presentations of polymer simulation model in this study.
34
3.4 The Polymer Flood Simulation Model
The flow of polymer solution through the porous medium is assumed to have no
influence on the flow of hydrocarbon phases. The standard black –oil equations are used
to describe the hydrocarbon phases in the model. A modification is required to the
standard water equation and additional equations are needed describe the flow of polymer
and brine within the finite difference grid. The water, polymer and brine equations used
in the model are as follow55:
wwwFew
rw
r
w
w
qZPRB
Tk
B
VS
tB
1.
(3.25)
pwwwFew
prwar
r
p*
w
w
CqZPRB
CTk1CV
t.
B
CVS
tB
1.
(3.26)
nwwwFew
nrw
r
nw
w
CqZPRB
CTk
B
CVS
tB
1.
(3.27)
dpVw*
w SSS (3.28)
In equations (3.26)-(3.28), the terms in the left hands side represent the mass rate of
accumulation. The subscripts (p) and (n) stand for polymer and brine, (r) for rock and (V)
denotes pore volume, the others have their usual meaning.
The model makes the assumption that the density and formation volume factor of the
aqueous phase are independent of local polymer and brine concentrations. The polymer
solution, reservoir brine and the injected water are presented in the model as miscible
components of the aqueous phase the degree of mixing is specified through the viscosity
term in the conservation equations. The principal effects of polymer and brine on the
35
flow of the aqueous phase are presented by the equations 3.25 and 3.28. The effective
fluid viscosities (ue) are dependent on the local concentration of salt and polymer in the
solution. The polymer adsorption is presented the additional mass accumulation term on
the left hand side of the equation. The effect of pore blocking and adsorption on the
aqueous phase relative permeability is treated through the reduction factor (RF) term that
requires the residual resistance factor of each rock type. The equations solved by Eclipse
polymer model are a discretized form of the differential equation 3.25-3.28.
3.4.1 Treatment of Fluid Viscosities
The viscosity terms used in the fluid flow equations contains the effects of change
in the viscosity on the aqueous phase due to the presence of the polymer and salt
solutions however, to incorporate the effects of physical dispersion at the leading edge of
the slug and also the fingering effects at the rear edge of the slug the fluid components
are allocated the effective viscosity values that are calculated using the Todd-Longstaff
technique.
The effective polymer viscosity is expressed as:
1ppmep uCuu (3.29)
Where uep is effective polymer viscosity, um is polymer viscosity at maximum
concentration, and is Todd-Longstaff mixing parameter. The mixing parameter is used
in modeling the degree of segregation between the water and the injected polymer
solution. In this study, =1 is used to indicate that the polymer solution and water are
completely mixed in each block. In order to calculate the effective viscosity of water to
36
use in equation 3.8, the total water equation is written as the sum of contribution s from
the polymer and the pure water. The effective viscosity of water is expressed:
epewew
cc11
(3.30)
max,pc
cc (3.31)
And c is the effective saturation for the injected polymer solution within the total
aqueous phase in the gridblock.
3.4.2. Treatment of Permeability Reduction.
The adsorption process causes a reduction in the permeability of the rock to the
passage of the aqueous phase and is directly correlated with the adsorbed polymer
concentration. To compute the reduction in rock permeability the residual resistance
factor (Frr) for each rock type is used. The actual resistance factor can be calculated by
the equation:
max,a
arrF C
C0.1F0.1R (3.32)
3.4.3. Treatment of the Shear Thinning Effects
The shear thinning of polymer has the effect of reducing the polymer viscosity at
higher flow rates. Eclipse assumes that shear rate is proportional to the flow viscosity.
This assumption is not valid in general, as for example a given flow in a low permeability
rock will have to pass through smaller pore throats than the same flow in a high
permeability rock, and consequently the shear rate will be higher in the low permeability
37
rock. However for a single reservoir this assumption is probably reasonable. The flow
velocity (Vw) is calculated as:
A
qBV w
ww (3.33)
A is the flow area between two cells.
However, Seright et al37 studied polymer rheology and mechanical degradation of two
polymer solutions, xanthan and HPAM in porous medium using a Berea Sandstone core
of about 14.5 cm long and cross sectional area of 11.34 cm2 with wide range of flux
values (u = 0.035-2222 ft3/ft2/d). They showed that xanthan solutions exhibit
pseudoplastic behavior in porous medium that parallels that in viscometers and xanthan
solution resistance factor in porous medium can best be modeled by the expression:
5.0r u205.2F (3.34)
With HPAM solution, these researchers published that though, HPAM solutions in
viscometers exhibit pseudoplastic behavior, it is consistently revealed Newtonian or
pseudodilatent behavior (resistance factor increases with increase flux) in porous media,
they decried the common notion by most commercial and academic simulators that
HPAM solutions exhibit pseudoplastic behavior in porous rocks. They concluded that in
modeling HPAM rheology in porous medium equations below gives a close
approximation of the HPAM residence factor (Fr) for flux, (u= 0.017-7 ft/d).
75.0r u9065F (3.35)
38
3.5 Solution Technique
The solution to the numerical models can be solved by using either direct methods
or iterative methods. The direct methods are not suitable for solving the system of partial
differential equations described in section 3.1. These equations are usually solved for a
large number of gridblocks and the round off error incurred during the execution of the
final iteration is the compilation of the round off errors inherent in the subsequent
iterations. This is due to the fact that the round off error in direct methods is added for
each iterating step towards the final result. For the iterative methods, the round off error
is only limited to one iteration. This is because the round off error at the end of each
iteration results in a different estimate of the next iteration and hence, the final result only
has the runoff error incurred during the final iteration. The example of the iterative
methods is the Point Successive Over Relaxation (PSOR) method, which is an
improvement of the Gauss-Seidel method, also an improvement of the Point Jacobi
iterative method. In this study we employed the Point Successive Over Relaxation
method in solving the finite difference equations (3.19) because it is simple iterative
method, easy to use, and more suited for solving a large system of equations (for
example, the single-phase equations described in section 3.1). The solution calculated by
PSOR is faster as compared to other similar iterative methods.
3.6 Well Models
In Kossack et al’s42 model, reviewed in the Chapter 2, they assigned a very high
permeability to the well grid blocks in the perforated section by multiplying the rock
permeability by a large factor (104-107). This presentation of a horizontal well is
39
unrealistic assumption about the physical nature of the system because in actuality their
simulator is still using a vertical well representation model. The only difference between
representation of a vertical well and a horizontal well in their study is that, they assign
high permeability to the gridblock which is supposed to have a horizontal well. These
researchers42 acknowledge that their presentation of the horizontal well as a row of very
high permeability and porosity gridblocks is only an approximation of the physical
situation. The largest uncertainty in their model is the multiplication factor used to
increase the permeability of the gridblock where the horizontal well is located. It should
be noted that injection processes usually create steady state flow conditions due to
pressure maintenance. Therefore, ideally the well model should be based on steady state
flow for injection processes. Peaceman's model52 was used for both vertical wells and
horizontals well in this study because it based on steady state flow assumption, and
incorporates a correction for anisotropy in rw. It is also simple to implement.
3.6.1 Wells Representation in this Study
Given the total liquid flow rate, qT, for a production well, the oil and water flow
rates are calculated using the following equations:
TT
TooT qq
, (3.36)
TT
wTwT qq
. (3.37)
Where λℓT in equations (3.34) and (3.35), represents the total oil and water mobility at the
wellbore which are given by the following equations:
40
k
1k kT
ooT , (3.38)
k
1k kT
wwT , (3.39)
wToTT , (3.40)
B
kr
. (3.41)
Where k is the well nodes (number of gridblocks in which the well is completed), λwT and
λoT are oil and water mobility at each well node. The oil and water flow rates in the case
of a production well in each well node are calculated using the following relations:
k
1kko
kooTk,o
WI
WIqq (3.42)
ko
wokk,w qq
(3.43)
The water rates for the individual well nodes in the case of an injection well are given by:
k
1kkw
kw.inj,Tk,w
WI
WIqq (3.44)
Where WI in equations (3.41) and (3.43) is the well index, for both vertical and the
horizontal wells the well index (WI) can be express as:
w
o
2
1
yxV
r
rln
zkk2WI (3.45)
41
w
o
2
1
zyH
r
rln
xkk2WI (3.46)
The subscript V and H used in equations (3.35) and (3.36) represent vertical and
horizontal wells respectively, and ro is the equivalent well block radius defined for the
both wells as:
2
1
4
1
y
x4
1
x
y
22
1
y
x22
1
x
y
Vo
k
k
k
k
yk
kx
k
k
28.0r
(3.47)
2
1
4
1
z
y4
1
y
z
22
1
z
y22
1
y
z
Ho
k
k
k
k
yk
kz
k
k
28.0r
(3.48)
The orientation of the vertical well is along the z-direction of the reservoir and that of the
horizontal well is in the x-direction, this is evidence of the present of ∆z and ∆x in
equations (3.43) and (3.44). Depending on the orientation of the well (vertical or
horizontal) WI can be calculated by using either equation (3.43) or (3.44). After these, the
next step is to discussion of the solution of the flow equations.
42
3 .7 Description of Simulation Models
The ultimate goal of this research is to determine the optimum production scheme
of an unconventional reservoir. This accomplished by a simulation studied conducted
with black-oil option in Eclipse100; a general-purpose reservoir simulator was employed
to model the performance predictions. The fully implicit solution method was used to
solve the governing equations for the simulation results presented in this report. It
includes options, which models secondary displacement and polymer flooding for a
variety of reservoir geometry. The simulation cases developed in this study were
designed to capture the impact of the degree of fluid crossflow (viscous and gravity
crossflow) and the variation in some polymer properties on recoveries from several
combinations of injector producer pairs during the waterflooding and polymer flooding
operations.
This research consisted of two studies, which focus on mobility control and the
use of horizontal well injector and or producer. In the polymer flooding and
waterflooding operations studies, vertical stratification is taken into accounts through the
use of non-communicating layers (Dykstra-parsons12 assumption) and the use of uniform
vertical pressure drop, communicating layers Zapata and Lake5. Reservoir under
consideration is assumed with no-flow boundaries on all sides. Two wells (injector and a
producer pair) ware employed in all the simulation cases studied. The injector well
(vertical/horizontal) is located in the center of cells (1, 1, 1) and the producer
(vertical/horizontal) in the cell (50, 1, 1) of the grid. The vertical wells were set to
perforate through the entire thickness of the formation.
43
The injector wells were constraints to operate at maximum injection pressure of
78.6 atm and injection rate of 100 cc per hour. At the same time, the production well was
set to be constraint at bottomhole pressure of 12 atm and 100 cc per hour for cases with
waterflooding. All simulation cases were modeled with 50 x 1 x 2 gridblocks. This
conclusion was arrived as a result of the block sizes sensitivity analysis conducted. Other
variables including the initial reservoir conditions, PVT properties data, and relative
permeability curves are presented in Table 3.1 and in Figure 3.3. These parameters were
common to all simulation cases developed for this study.
Figure 3.3 Relative Permeability Curves
The relative permeabilities were computed using a power law model with an index of 2
for oil and water relative permeability curves. Water relative permeability endpoint value
of 0.1 and oil relative permeability endpoint of value of 1.0 was used.
No gravity crossflow was obtained by modifying the fluid properties of the oil and water
so that the densities of the oil and water phases at reservoir conditions were the same.
Rel at i ve Per meabi l i t y Cur ve
0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
0. 8
0. 9
1
0 0. 2 0. 4 0. 6 0. 8 1Sw
Rela
tive
Per
meab
ilit
y
k r wkr o
44
The algorithms for the simulation models in this study is presented brief in Appendix B.
The block sizes sensitivity analysis will be presented next.
3.7.1 The Gridblocks Sensitivity Analysis.
Blocks sensitivity analysis was performed to determine the optimum size of the
gridblocks that will ensure maximum oil recovery. This was done by dividing system
into number of gridblocks (20, 50, 100, and 150) and run the model successively, one
after the other for 24 days with water viscosity of 1cp and oil viscosity of 100cp. Figure
3. 4 show the result of the sensitivity analysis.
3.7.2 Communicating Layers System (crossflow model).
In this model, the layers were assumed completely connected in the vertical
direction and the crossflow between the layers is instantaneous such that no vertical
pressure drop exists. This implies high vertical flow conductivity because of the large
lateral area for crossflow. Switching on to the vertical equilibrium (VE) model ensures
maximum crossflow (gravity or viscous or both). Impact of vertical heterogeneity was
examined by varying the horizontal to vertical permeability ratio within 0.05 to 0.35.
Setting the oil-water capillary pressures to zeros controlled the capillary
crossflow. Figure 3.5 shows the schematic diagram for the crossflow model
3.7.3 Noncommunicating Layers System (No crossflow model).
With noncommunicating layers system, the layers assumed completely separated
from each other by an impermeable thin strata such that no crossflow takes place between
the reservoir layers. To ensure no fluid crossflow, we set the vertical permeability to zero
45
or inserted permeable strata between the two layers. Figure 3.6 shows the schematic
diagram for no-crossflow model.
3.7.4 Polymer Flood
The polymer flood simulation is accomplished with the polymer option in the Eclipse
simulator. This section of the entire work is developed to define the limit oil viscosity at
which the polymer flood remains cost effective. The polymer viscosities of 1, 10, 100,
and 1000 cp were used in displacing oil with viscosities 10, 102, 103, 104, and 105 cp. The
injection and the production wells were constrained at the same pressures same as that of
the waterflooding cases but controlled at a rate of 100 cc per day. This change in
operating constraints is to avoid oscillations and instability in the numerical solution
provided by Eclipse simulator. Apart from the oil and polymer viscosities that were
varied, the other sensitivity parameters include: polymer concentration and slug size. The
discussion on the sensitivity parameters will be elaborated in detail in chapter 4.
46
Figure 3.4 Block sizes analysis Table 3.1 Pertinent properties of the reservoir models Reservoir thickness, cm 10
Reservoir length, cm 100
Permeability (k1& k2), D 0.1 & 1.0
Reservoir pressure, atm 78.6
Oil density, g/cc 0.808264
Oil formation volume factor, rcc/scc 1
Oil viscosities, cp 1, 10, 102, 103, 104, and 105
Oil compressibility, atm-1 0
Oil saturation, fraction 0.7
Oil production rate, cc/day 2400
Water density, g/cc 0.999125
0
0.2
0.4
0.6
0.8
1
0 0.4 0.8 1.2
Mob
ile O
il R
ecov
ered
(fr
acti
on)
Injected Pore Volume
Injected pore Volume vs Mobile oil Recovered (fraction)
150 blocks
100 blocks
50 blocks
Water Visc = 1cpOil Visc = 10cp
47
Water compressibility, atm-1 0
Water formation volume factor, rcc/scc 1
Water viscosity, cp 1
Initial connate water saturation, fraction 0.3
Water injection rate, cc/day 2400
Number of grid blocks 50 x 1 x 2
Grid block size, cm 2 x 5 x 5
Porosity, % 30
Rock compressibility, atm-1 2.0 E-8
Figure 3. 5 Communicating Layers
Injection Well (Vertical /Horizontal)
Production Well (Vertical /Horizontal)
1 cm
K1 = 0.1 D Ф1 = 30 %
K2 =1 D Ф2 = 30 %
Crossflow
48
Figure 3.6 Non- Communicating Layers
Injection Well (Vertical /Horizontal)
Production Well (Vertical /Horizontal)
Impermeable strata
1 cm
K1 =0.1 D Ф1 = 30 %
K2 =1 D Ф2 = 30 %
49
CHAPTER 4
PRESENTATION AND ANALYSIS OF SIMULATION RESULTS
This chapter presents graphical and numerical results of simulations runs from this study.
Similar to the previous chapter, this section is divided into several parts, which presents
analysis and discussion of several simulation cases, and a parameter sensitivity study
under different approaches. Cases compare the oil recovery
efficiency from the use of waterflooding, polymerflooding and horizontal (injector/
producer) pair, where cumulative oil production is used as a performance indicator. In
order to find meaningful conclusion it is necessary to validate the model results. This is
accomplish by comparing the results generated by the simulator to an analytical
fractional-flow solution and a volumetric material balance solution. Additionally, it is
necessary to check if these results are dependent on the specific values of different input
parameters (rock and fluid properties) used in the simulator. A sensitivity analysis on the
input parameters must be performed.
4.1 Validation of the Reservoir Simulator
To validate the simulator developed in this study, the results generated by the
simulator were compared with two different solutions assuming no gravitational effects,
no capillary forces, and incompressible fluid.
1. Two-phase analytical fractional flow solution,
2. Volumetric material balance solution.
50
4.1.1 Analytical Solution
The analytical and simulation results for both the waterflooding and the polymer
flood are compared to validate the simulator. The analytical model selected to perform
this task is the convectional fractional flow modeled by Randy Seright of New Mexico
Tech. Petroleum Recovery Research Center (PRRC). The simulator results are plotted
against the analytical results in Figure 4.1 through Figure 4.4 for the polymer and
waterflooding, each on under the two scenarios; free crossflow and no-crossflow. Figure
4.1 and Figure 4.2 show of the communicating and no -communicating layers cases of the
water flooding and Figure 4.3 and Figure 4.4, the polymerflooding. The results show a
very good match between the simulator and analytical at late injection but not at early
injection. We attribute this to the injectivity (injection rate per unit dropdown) lost at
early periods of the injection. In wells that are not fractured, initial injectivity will
decrease up to a time where fractures are initiated. This phenomenon is more particular
when injecting viscous solution. Also for incompressible fluids in simulation studies, we
expect a sharp rise in the reservoir pressure during the initial injection but this was not
seen during the simulation, suggesting that though we assumed incompressible fluids in
the simulator, it may not be so. It is well established within the industry that water
injection mostly takes place under induce fracture condition56. Table 4.1 though Table 4.4
below show oil recovery response at early injection 1 PV and 5 PV of the floods
operations.
51
Table 4.1 Comparison, Analytical and Simulation (No-crossflow). uo/uw Recovery (%) at 1 PV Recovery (%) at 5 PV
Analytical Simulation Analytical Simulation
1 81.80 87.00 99.46 99.35
10 59.00 70.84 92.66 92.28
102 48.00 49.21 74.16 72.89
103 32.00 34.99 48.84 46.99
104 17.00 14.46 25.94 23.82
105 8.00 7.83 13.75 12.08
Figure 4.1 Waterflooding: Analytical versus Numerical (No-crossflow).
52
Table 4.2 Comparison of waterflooding, Analytical and Simulation (Crossflow). uo/uw Recovery (%) at 1 PV Recovery (%) at 5 PV
Analytical Simulation Analytical Simulation
1 98.80 99.97 99.16 99.97
10 59.00 68.40 71.31 99.64
102 39.00 44.07 49.43 63.99
103 23.00 18.89 33.63 37.55
104 12.00 6.52 19.26 14.32
105 6.00 2.40 9.85 5.17
Figure 4.2 Waterflooding: Analytical versus Numerical (Free crossflow).
53
Table 4.3 Polymerflooding, Analytical versus Simulation (Crossflow). up cp Recovery (%) at 1 PV Recovery (%) at 5 PV
Analytical Simulation Analytical Simulation
10 28.88 37.21 47.40 49.10
102 47.93 54.69 65.79 64.83
103 99.04 99.98 99.80 99.98
Figure 4.3 Polymerflooding: Analytical verse Numerical (Free crossflow). Table 4.4 Polymerflooding, Analytical versus Simulation (No-crossflow)
up cp Recovery (%) at 1 PV Recovery (%) at 5 PV
Analytical Simulation Analytical Simulation
1 31.66 34.98 48.80 46.97
10 45.62 51.42 72.78 75.93
102 56.96 64.82 84.20 86.33
103 60.39 69.07 86.64 94.20
54
Figure 4.4 Polymerflooding: Analytical versus Numerical (No-crossflow).
4.1.2 Volumetric Material Balance
To compare the results from the volumetric material balance with the results from
simulation model, the pressure and formation volume factor data and other relevant data
used for the volumetric material balance calculations are given in Table 3.1.
Using this data, the original oil in place (N) is calculated by the following material
balance equation:
N= oiSVb (4.1)
Where Vb is the bulk volume, � is the porosity, and Soi is the initial oil saturation. The
simulation results is generated with, a 50 x1x 2 cm fine grid with dimensions, 2 x 5 x 5
cm. Two wells, the producer and the injector placed in opposite direction at the extreme
55
ends of the reservoir. The reservoir is assumed to be “No-flow” in all its boundaries. The
initial oil in place for both material balance and simulation calculations is kept at 1050 cc
4.2 Sensitivity Analysis
This section presents and identifies trends to determine the impact that the
permeability ratio and permeability contrast, and other selected parameters have on oil
production and recoveries. Cases compare the oil recovery efficiency from the use of
waterflooding and polymer injection operations, where cumulative oil production is used
as a performance indicator to determine a successful flood. The water flooding and the
mobility control analyses were performed considering two different scenarios:
Communicating and non-communicating layer reservoir. In some cases the advantage
horizontal injector well as against vertical well injector was examined.
1. Gravitational effects,
2. Rock compressibility,
3. Vertical permeability,
4. Mobility contrast,
5. Permeability contrast.
6. Polymer solution viscosity.
4.2.1 Gravitational Effect (Communicating Layers)
This section is presented to examine the impact that the gravity have on oil
production and recoveries. Figures 4.5 shows the simulation results as gravity is
incorporated into the simulation model, compared to the base case scenario (no
56
gravitational effect). It shows clear from Figure 4.5 that gravity effect is a very slow
recovery process in this study to effectively aid oil production. This is attributed to the
low density difference between the displaced and the displacing fluids (Δρ=0.104 g/cc).
This is very small to induce gravitational crossflow necessary to affect oil recovery. The
oil displacement is dominated purely by viscous forces. To strengthen this line reasoning,
the dimensionless gravity number (NG) was computed to help locating the flow regime.
NG is express mathematically as:
Lu
HkkN
w
orwv
G
(4.2)
In the equation (4.1) above, u (cm/s) is velocity of the aqueous phase. The NG was
computed to be 0.16 this shows that the strength of the gravitational effect is very weak.
Figure 4.5 Gravitational effect on oil recovery.
57
4.2.2 Rock Compressibility
The rock compressibility was increased from 2E-08 atm-1 to 2E-06 atm-1 .The oil
recovery response is shown in Figure 4.6. It can be seen that the oil recovery does not
show any significant change with this change. The higher rock compressibility of a
reservoir rocks acts as a drive mechanism when the reservoir pressure decreases, but in
this study the overall reservoir pressure is sustained through continual injection. Hence,
the higher compressibility does not show any significant affect on the oil recovery.
Figure 4.6 Compressility effect on oil recovery
4.2.3 Vertical Permeability Ratio (kv/kh).
The permeability in the vertical direction (kv) is generally less than the
permeability in the horizontal direction (kh) due to the overburden pressure of the rock
and the depositional sequence. In most cases, kv is not a measureable quantity but set
after successful historical study. Kz is also referred to as the permeability in the z-
58
direction and kh is also known as the permeability in the x-direction (kx) and the
permeability in the y-direction (ky). This analysis was conducted under two scenarios:
communicating and no- communicating layer systems. The kh for the layer 2 is 2000 md
and that of layer 1 is 100 md. For the communicating layer system, the vertical to
horizontal permeability ratio (kz/kh) was pegged at 0.35, 0.1, and 0.05. These correspond
to the effective length to thickness ratio (RL) defined in chapter 1 of 11.89, 6.74, and
4.68. The RL is the criteria to approximate a system to vertical equilibrium model:
maximum crossflow. Vertical Equilibrium VE concept has been used extensively, mainly
as a way of collapse simulation to a lower dimension. Generally, vertical equilibrium
means, that the sum of the driving forces in the vertical direction is zero for all
components. For viscous crossflow only, VE means that the vertical pressure drop is zero
at all time and position in the reservoir. This implies that the horizontal pressure gradients
are equal at all vertical positions. It is not generally recognized that assuming VE in a
displacement process implies perfect vertical communication this is the basis for the
claim that VE implies the maximum degree of possible. VE will be a good assumption
for reservoirs with effective length- to- thickness ratios (RL) of 10 or more5. The (kz/kh)
was set to 0 (RL=0) for the no-communicating layers system. These values were used to
determine the kz values for both layers in the simulation study. The fraction of oil
recovered as a function of pore volume injected for different values of kv/kh ratio for VW
and HW injectors are shown in Figure 4.7 and Figure 4.8. The increase in permeability
in vertical directions increases the recovery response for both HW and VW injector
increases but to a lesser extent as expected. This is due to the fact that the reservoir has a
lager lateral extension than vertical and also the gravity is not properly operating
59
effectively in the system to cause more vertical fluid movement. Permeability is the
measure of the ability of the reservoir to transmit fluid, so it is expected that when the kv
/kh ratio increases, more oil would be produced from the HW-configuration then the VW-
configuration. This argument is also strengthened by the fact that the injected water from
the horizontal injector invades the reservoir in the z-direction and in x-directions whereas
vertical injector injects only in the z-direction with much lesser degree. This is evident
from the trend of cumulative volume of oil crossflow from the less permeable layer (layer
1) into the more permeable layer (layer 2) as shown in Figure 4.10 and Figure 4.9 for the
HW-configuration and the VW- configuration respectively. Also the oil recovery
response for the HW configuration is better than that of the VW-configuration for each kv
/kh value. The better recovery from the HW-configuration is also anticipated because of
the large contact of the horizontal well with the formation. Similar trend is also shown in
the Table 4.5, the quantitative comparison oil recovery as a function of pore volume of
water injected for (kv/kh) values of 0.35, 0.1, and 0.05 at 10 pore volume of injection for
two well configurations. When the kv/kh ratio increased from the base case value of 0.1
to 0.35 oil recovery from the HW- configuration increased to 2.68 % while VW-
configuration increased only to1.42 %. On the other hand, when the ratio is reduced from
0.1 to 0.05, the oil recovery for the HW–configuration reduced to 0.78 % but that of VW-
configuration reduced only to 0.28 %. Furthermore, it is also clear from this analysis that
the HW –configuration is more sensitive to the vertical permeability distribution than that
of the vertical VW- configuration. Therefore, a horizontal well injector proved to cause
higher recovery than a vertical well injector with the increase in vertical permeability.
60
Table 4.5 Effect Vertical Heterogeneity on Oil Recovery (Free crossflow). Kv /Kh % Oil Recovered at 10 PV, 1000cp oil.
VW-configuration HW-configuration
0.05 51.33 52.03
0.1 51.61 52.81
0.35 53.03 55.49
Figure 4.7 Effect of kv/kh on the Oil Recovery (VW).
61
Figure 4.8 Effect of kv/kh on the Oil Recovery (HW).
Figure 4.9 Effect of kv/kh on the quantity of oil crossflowed (VW).
62
Figure 4.10 Effect of kv/kh on the quantity of oil crossflowed (HW).
4.2.5 Permeability Contrast
In this study, we defined permeability contrast as the ratio between the
permeabilities of the adjacent layers (higher permeability k2 =1000 md to the lower
permeability k1=100 md). In this analysis, the permeability ratio of the base case (k2
=1000 md and k1= 100 md) was reduced to halve its original value of ten by increasing
k1 to 200 md. This change is carefully considered such that the length to thickness ratio
(RL) defined in Chapter 1 still hold. Figure 4.11 shows the numerical results of the two
well configurations under the waterflooding case. As the permeability ratio between
layers increases, the recovery increases for both well configurations at very close
matching. Numerical results are presented in table 4.6. It can be observed in the Table 4.6
that at 1 pore volume, the oil recovery response for VW- configuration increased by 1.42
% while that of the HW- configuration increased by 1.71 % as the result of reducing the
permeability contrast layer 1 and 2 by half its original value of 10.
63
Table 4.6 Effect Permeability Contrast on Oil Recovery (crossflow). K2/K1 Recovery (%) at 1 PV Recovery (%) at 3 PV
VW HW VW HW
5 20.29 22.20 33.26 36.40
10 18.87 20.49 30.77 33.28
Figure 4.11 Effect of Permeability Contrast on the Oil Recovery.
4.2.6 Oil Viscosity
The oil viscosity values of 1, 10, 102, 103,104 and 105 cp were used to check the
sensitivity of the results due to this parameter. Figures 4.13 and 4.14 show a graphical
comparison of the oil recovery response for communicating and no- communicating
layers (kv/kh = 0) for values of oil viscosity tested for both the polymer and
waterflooding. It is evident from Figure 4.13 and Figure 4.14 that when the viscosity is
64
increases, the oil recovery decreases. The water potential or velocity of water becomes
higher than the oil potential or velocity due to higher value of oil viscosity. When the oil
viscosity increases, its velocity decreases the injected water bypasses most of oil to
breakthrough at the production well earlier as shown in the water cut plot in Figure 4.15.
When oil viscosity increase less oil is produced regardless of whether there is a vertical
communication between the reservoir layers or not. Furthermore, as the mobility ratio
becomes between the displaced and the displacing fluids increasingly unfavorable high,
M >>1.0 recovery efficiency worsen rapidly for the crossflow then no-crossflow4, 5, 11. At
favorable displacement, M <1.0 the crossflow adds advantage to the displacement
performance. For a favorable displacement M<1.0 the direction of the crossflow is from
the low to the high velocity layer at the leading water front and in the reverse direction
at the trailing front Thus, crossflow cause the leading and the trailing fronts to be
receded and advance, respectively over their no-crossflow position. This in turn causes an
improvement in the vertical sweep efficiency over that would be expected from
conductivity change in the absence of crossflow.
For an unfavorable displacement, M>1.0 the crossflow directions are reversed causing
the leading and the trailing fronts to become farther apart relative their no-crossflow
positions. Very importantly, for M<1 .0 , the crossflowing fluid is either all oil (leading
front ) or water (trailing front) but for M>1.0, the crossflowing fluid is an oil-water
mixture at both fronts , this causes a mixing zone to develop which generally improve
vertical sweep efficiency compare to the crossflow case5. At M=1.0, crossflow does not
occur thus, no-crossflow and crossflow cases give almost the same recovery as shown in
Figure 4.12 and in Figure 4.13.This tabulated in tables 4.7 and 4.8. It’s obvious from
65
these tables that at favorable displacement, crossflow cases recover 99.98 and 99.97 % of
the mobile oil for polymer and waterflooding while, no-crossflow cases recover 69.07
and 87.00 % for polymer and waterflooding at 1 PV. As displacement becomes
unfavorably higher, recovery from crossflow is impaired and no-crossflow takes
advantage.
Figure 4.12 Crossflow versus No-crossflow, Waterflooding.
Table 4.7 Crossflow versus No-crossflow, Waterflooding. uo/uw Recovery (%) at 1 PV.
No-crossflow Free Crossflow
1 87.00 99.97
10 70.00 68.40
102 49.21 44.07
103 34.99 18.89
66
104 14.46 6.52
105 7.83 2.40
Figure 4.13 Crossflow versus No-crossflow, Polymerflooding 1000 cp Oil.
Table 4.8 Polymerflooding, crossflow versus no-crossflow up cp Recovery (%) at 1 PV Recovery (%) at 5 PV
Free crossflow No-crossflow Free crossflow No- crossflow
10 37.21 51.42 47.40 75.93
102 54.69 64.82 65.79 86.33
103 99.98 69.07 99.98 94.20
67
Figure 4.14 Effect of oil viscosity on water wct, Polymerflooding.
Figure 4.15 Effect of oil viscosity on water wct, waterflooding (No-crossflow).
4.2.7 Polymer Solution Viscosity
Similar studies analogous to the oil viscosity were analyzed for the polymer
solution viscosity used in the polymerflooding under the assumption that polymer
68
solution is a Newtonian, and no polymer retention. The polymer used in this analysis has
viscosities of 10, 100 and 1000 cp displacing oil of 1000 cp. Figure 4.17 and 4.16 show
the oil recovery responses for the free crossflow and no-crossflow cases. It can be seen
that, at higher polymer viscosity, oil recovery is higher, and decrease as the polymer
viscosity decreases. The reason is that, high polymer viscosity reduces the mobility
contrast between injectant and the displacing fluid (oil) thus, increases oil recovery. Also,
high polymer solution viscosities delay the breakthrough of the injected fluid. It also
slows the velocity of the water by increasing its viscosity. Figure 4.18 show the plot of
water cut as the function of pore volume injected. It is clear from the plot that the
breakthrough time is longer at high polymer viscosity and decrease as the viscosity
decreases.
Figure 4.16 Effect of Polymer viscosity on oil recovery, 1000 cp oil No-crossflow.
69
Figure 4.17 Effect of polymer viscosity on oil recovery, 1000 cp oil free crossflow.
Figure 4.18 Effect of polymer viscosity on water cut on 1000 cp oil.
70
4.2.8 Summary of Sensitivity Analysis
It is evident from Figure 4.5 through Figure 4.11(discussed earlier) that gravity
and rock compressibility have virtually no appreciable impact on the oil displacement
performance. Apart from these, all the other parameters study under sensitivity analysis
has slight influence on the oil recovery performance. The HW-configuration immerged to
have shown a slight sensitivity or advantage for the range of the parameter studied than
the VW-configuration. Now that the sensitivity analyses have been discussed, further
results generated by the simulator to see the potential of horizontal well injector in
polymer injection and waterflooding can be presented.
4.3 Potential of Horizontal Injector in Waterflooding.
This section is presented to determine the advantage that the horizontal well
injector pair has over vertical well injector pair in the polymer and in waterflooding. This
is done by comparing the performance of horizontal well injector pair with the
performance of the vertical well injector pair in both the polymer and waterflooding. This
comparison will help determine the conditions under which it is beneficial to use
horizontal well injector pair in the flooding operations. The performance of the horizontal
well injector pair based on oil recovery and water cut as a function of pore volume
injected will be discussed next.
4.3.1 Results Based on Oil Recovery
The oil recovery as a function of the pore volume injected for a range of oil
viscosities 1, 10, 102, 103, 104 and 105 cp for water flooding cases and 102, 103, 104 and
71
105 cp for polymerflooding cases under the assumptions of infinite vertical permeability
(vertical communication between layers) and non-communicating layers systems is
plotted in Figure 4.19 through Figure 4.22. These figures show the comparison of the
performance of the HW-configuration and VW -configuration for different oil viscosity
under assumptions stated above. Figure 4.20 and Figure 4.21 shows the oil recovery as a
function of the pore volume injected for the HW- configuration compared with the VW-
configuration for waterflooding case; crossflow and no-crossflow cases. Figure 4.22 and
Figure 4.19 show the communicating layers case and no communicating layer case of the
polymerflooding. It can be seen in Figure 4.19 through Figure 4.22 that the oil recovery
from HW-configuration is slightly higher than that of VW-configuration for the
waterflooding and mobility control cases for the range viscosities study. This advantage
is attributed to the increased contact area of the HW- configuration with the reservoir
formation as against the VW- configuration. These comparisons are further simplified
quantitatively in Table 4.23 though 4.26. It is openly from these tables that the horizontal
well injector has a slight advantage of the vertical well injector. For example, in table
4.26 (free crossflow , waterflooding), at 1 pv of injection, HW – configuration recovers
75.66 % of the mobile oil(10 cp) but, the VW- configuration recovers only 68.40 %. This
trend is seeing in all the tables. It must be noted that the length of the horizontal wells
used in this studies remains unchanged for all the simulation cases studied.
72
Figure 4.19 Polymerflooding: HW Injector versus VW Injector (No-crossflow)
Figure 4.20 Waterflooding: HW Injector versus VW Injector (Free Crossflow).
73
Table 4.9 HW Compared to VW, Waterflooding (Free crossflow). uo/uw Recovery (%) at 1 PV. Recovery (%) at 5 PV.
VW HW VW HW
1 99.87 99.99 99.16 99.99
10 68.40 75.66 99.64 99.86
102 44.07 47.03 63.89 66.00
103 18.89 20.39 37.55 40.40
104 6.52 8.36 14.32 16.42
105 2.40 3.69 5.17 6.42
Figure 4.21 Waterflooding: HW Injector versus VW Injector (No-crossflow).
74
Table 4.10 HW Compared to VW, Waterflooding (No- crossflow). uo/uw Recovery (%) at 1 PV. Recovery (%) at 5 PV.
VW HW VW HW
1 87.00 88.51 99.35 99.97
10 70.00 67.05 92.28 92.32
102 49.21 53.78 72.89 74.58
103 34.99 38.51 46.99 52.47
104 14.46 23.97 23.82 30.69
105 7.83 7.10 12.03 12.53
Figure 4.22 Polymerflooding: HW Injector versus VW Injector (Free Crossflow).
75
4.3.2 Results Based on Water Cut
Another parameter used to compare the performance of a horizontal well injector
with the performance of a vertical well injector is water cut. Water cut measures the
fraction of water in the total flow stream (oil plus water). Water cut as a function of pore
volume injected is plotted in Figure 4.23 for the polymerflooding and Figure 4.24
waterflooding. It can be seen in these figures 4.23 and 4.24 that water breakthrough times
are slightly higher for the horizontal injector than that with vertical injector. We believed
that because horizontal injector invades the reservoir stronger (Figure 4.34) in the vertical
region, the vertical fluid flow uniforms the front thereby delaying the water breakthrough.
This advantage weakens as the injection progresses. Summarizing the results presented
in this chapter, the horizontal well injector in polymer and in waterflooding operations is
compared to a vertical well injector. The horizontal well injector proved advantageous of
over the vertical well injector. This advantage of a horizontal well deceases as if the
vertical permeability is not very favorable for the horizontal well to invade in the vertical
zone of the reservoir. Furthermore, the advantage of horizontal injector on water cut over
vertical injector is pronounced within the range of the parameter studied. The viscosity
controls the recovery performance in the flooding operations.
76
Figure 4.23 Polymerflooding: Impact of Horizontal Injector on Water Cut.
Figure 4.24 Waterflooding: Impact of Horizontal Injector on Water Cut.
77
CHAPTER 5
ECONOMICS
The objective function used in this project is the net present value of the
polymerflood operation for a given production period. The objective is to design the best
production and injection strategy for the unconventional reservoir. This chapter explains
the concept of net present value and how it can be determined.
5.1 Net Present Value (NPV)
Present value of money compares the value of a certain amount of money today to
the value of that same amount in the future and vice versa, taking into consideration
inflation and returns. Net present value (NPV) is actually the present value of the net cash
flow (the present value of cash inflows less the present value of cash outflows). Given an
investment opportunity, NPV is used by an organization to analyze the profitability of the
project or investment and to make decisions with regards to capital budgeting. It is
sensitive to the future cash inflows that an investment or project will yield. NPV can be
computed b y the relation below54.
NPV =
T
tot
t Cr
C
1 1 (5.1)
Where t = Time of cash flow (time step)
Ct = Net (after tax) cash inflow (cash inflow-outflow) after time t, (PV) $
r = Annual (or periodic) discount rate, fraction
T = Cumulative investment (or production) period or pore volume injected (PV),
78
Co = Initial investment cost, $
A conservative annual discount rate of 10% was used in this study in the estimation of the
present value of money and is based on the current rates at which eligible institutions are
charged to borrow short-term funds directly from a Federal Reserve Bank (approximately
6.5%). Also, most oil companies use this rate for evaluating the viability of proposed
investments.
Cash inflow is calculated from the oil, water production and injection rates obtained from
of the filed or from the cumulative production from the reservoir. The price of oil is
pegged at $20, $50, and $100 per barrel for the entire three-week production period while
the cost of water handling is $0.25 per barrel of water and $2/lb of polymer (specifically
for the polymerflood section in this study). The total cash inflow for the floods operations
for the entire production periods are given by,
Cw = (fopt × $/bbl) − (Fwt × $wat) (5.2)
Cp = (fopt × $/bbl) − Fwt × ($wat+$Fp) (5.3)
Where, C = Net cash inflow, (w, p represent water and polymer respectively), $
$/bbl =Price of Oil per bbl, $
$wat= Cost of water handling per bbl, $
$Fp= Cost of polymer treatment per bbl, $
Fopt = Cumulative oil production, SCC
Fwt = Cumulative water production and injection, SCC
Figures 5.1 and Figure 5.2 show the net cash flow (revenue) for the free crossflow and
no-crossflow cases for oil price of $100 per barrel of oil and used to compute their
corresponding NPV plot in figure 5.3 and figure 5.4. The NPV plots and the net cash
79
flow plots show economically attractive project but this advantage diminishes after about
5 PV of injection. Also, as the oil price dropped, example $ 50, and $ 20 per barrel, and
the viscosity of polymer solution reduces, the project (polymer injection) becomes less
and less economically attractive. Tables 5.1 through 5.4 show the net cash flow and NPV
computed at 5 and 15 PV for the crossflow and no-crossflow to further aid in the
explanations/interpretations of the graphs. The net cash flow and NPV plot corresponding
to the oil prices of $ 20 and $ 50 per barrel are presented in appendix C. It must be noted
that the NPV is computed at Co = 0 (no initial investment cost) and the injection rate was
constant for all the cases study. If the injection is inversely proportional to polymer
viscosity, that could affect the results.
Figure 5.1 NPV computed at oil price of ($100/bbl oil): Displacing 1000 cp oil with polymer (No-crossflow)
80
Table 5.1 Net cash flow Tabulated at oil price of $100/bbl: Displacing 1000 cp oil with polymer (No-crossflow). Viscosity NPV at $100/bbl of oil, (NoX), $ Relative profit, $
5 PV 15 PV 5 PV 15 PV
1 cp water 2.641 7.942 - -
10 cp, pol. 7.270 15.432 4.629 7.490
100 cp, pol. 26.482 38.881 23.831 30.939
1000 cp, pol. 30.501 42.187 27.870 34.245
Figure 5.2 Net cash flow computed at oil price of ($100/bbl oil): Displacing 1000 cp oil with polymer (No-crossflow).
81
Table 5.2 NPV tabulated at oil price of $100/bbl: Displacing 1000 cp oil with polymer (No-crossflow) Viscosity Revenue at $100/bbl of oil, (NoX), $ Relative profit, $
5 PV 15 PV 5 PV 15 PV
1 cp water 0.165 0.187 - -
10 cp, pol. 0.270 0.275 0.105 0.088
100 cp, pol. 0.338 0.330 0.173 0.143
1000 cp, pol. 0.348 0.338 0.183 0.151
Figure 5.3 Net cash flow computed at oil price of ($100/bbl oil): Displacing 1000 cp oil with polymer (Crossflow).
82
Table 5.3 Net cash flow tabulated at oil price of $100/bbl oil: Displacing 1000 cp oil with polymer (Crossflow) Viscosity Revenue at $100/bbl of oil, (crossflow), $ Relative profit, $
5 PV 15 PV 5 PV 15 PV
1 cp water 0.118 0.140 - -
10 cp, pol. 0.173 0.182 0.055 0.042
100 cp, pol. 0.232 0.232 0.114 0.092
1000 cp,
pol.
0.364 0.350 0.246 0.210
Figure 5.4 NPV computed at oil price of $100/bbl: Displacing 1000 cp oil with polymer (Crossflow)
83
Table 5.4 NPV tabulated at oil price of $100/bbl oil: Displacing 1000 cp oil with polymer (Crossflow)
Viscosity NPV at $100/bbl of oil, (crossflow), $ Relative profit, $
5 PV 15 PV 5 PV 15 PV
1 cp water 5.034 9.848 - -
10 cp, pol. 7.940 14.440 2.906 4.592
100 cp, pol. 11.847 20.409 6.813 10.561
1000 cp, pol. 27.902 62.019 22.868 52.171
84
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions
The simulation models developed in this study was used to investigate the scheme
of polymer injection in waterflooding. The conclusions are as follows.
1. The important factors that affect the recovery of oil when a horizontal well is used
as injector are vertical to horizontal permeability ratio. The permeability ratio
(kv/kh) was varied from 0.05 to 0.35.
2. The results of this study showed that the use of horizontal well injectors in
waterflooding is advantageous as compared to the vertical well injectors for the
range of kv/kh, considered in this study.
3. The use of horizontal well injector results in more oil recovered at the producer as
compared to a vertical well injector. Specifically, this advantage of horizontal
well injector was more pronounced when the vertical to horizontal permeability
ratio was 0.3 and above.
4. This study showed that the water cut at the producer is less when a horizontal well
injector is used as opposed to when a vertical well injector is used, below the pore
volume injected value of about 0.40. This water cut for horizontal well injector
case becomes more than the water cut for vertical injector case above pore
volume injected value of about 0.5.
85
5. At stable displacement, Crossflow adds advantage to displacement with crossflow
then the corresponding no-crossflow case. As the displacement become
increasingly unstable, the performance of crossflow model is worsen more for the
crossflow case then the no-crossflow case.
6. The economic gain from the polymer injection operations is higher for the
displacement with no–crossflow then the crossflow at any giving oil price and
polymer viscosity, but this advantage reduces for both models beyond about 4 PV
of injection
6.2 Recommendations
The development of this thesis is base on idealized geological models from which
many ideals may be derived for further investigate the advantage of using HW-
configuration and the impact vertical heterogeneities have on polymerflooding and
waterflooding. The recommendations for the future work using the simulation model,
developed in this study, are as follows.
1. The horizontal well length, which was kept constant in this study, can be varied to
see if an increase in horizontal well length always increases horizontal well's
advantage over a vertical well or if there is an optimum horizontal well length
beyond which there is no increase in advantage.
2. A vertical well was used as a producer in this study. Several simulation runs can
be performed to evaluate if the use of a horizontal well producer would
significantly increase the oil recovery.
86
3. The thickness of each reservoir layer was kept constant in this study. An opposite
scenario would be to vary the layer thickness to evaluate the potentials of a
horizontal well injector.
4. Further work can also be carried out on reservoir geology while considering
uncertainties in the reservoir model parameters and also on full field scale
including reservoir heterogeneity.
5. The study should include capillary curves developing at low injection rates to
study the effect capillary force have on the oil recovery.
6. The crossflow model fails to match the results of the analytical model the subject
of discrepancy should be investigated.
87
NOMENCLATURE
Bo = Oil formation volume factor, rcc/SCC
Bw = Water formation volume factor, rcc/SCC
Ct = Net cash flow
Cf = Rock compressibility, atm-1
Co = Oil compressibility, atm-1
Cw = Water compressibility, atm-1
EAS = Areal sweep efficiency, fraction
ED = Microscopic displacement efficiency, fraction
EOR = Enhance Oil Recovery
ER = Oil recovery efficiency factor, fraction
EV = Volumetric sweep efficiency, fraction
EV S = Vertical sweep efficiency, fraction
Fopt = Field oil production, CC
Fwpt = Field water production, CC
Fr = Resistance factor, dimensionless
Frr = Residual resistance factor, dimensionless
H = Total formation thickness, cm
HW = Horizontal well injector
L = length of the reservoir, cm
kh = horizontal permeability, md
kv = Vertical permeability, md
NPV = Net present value
88
Pol. = Polymer
Pin = initial reservoir pressure, atm
Ue = Effective
89
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APPENDIX A
The algorithms for the simulation models in this study is presented brief below:
A.1 Case Definition:
This is section where we define the task for the simulator: study the flow within
and from the reservoir over time (Black-Oil-Model) or the effects of phase composition
on the flow behavior (Compositional model) .It give an options to specify the phases
present in our reservoir (water and oil), solution techniques use the study (IMPES) and
the type of geometry use to represent reservoir in question: Cartesian, Block Centered.
A.2 Grid
The basic geometry of the simulation grid and various rock properties (porosity,
absolute permeability, etc) in each grid cell are specified in the grid section. From these
properties, the simulator calculates the pore volumes of the grid blocks and the inter-
block transmissibilities. The keywords used in this section usually depend on the
geometry option selected in the initialization section. In this case, we used fine grid,
Cartesian and block-centered geometry options. The porosity distribution in the reservoir
is assumed to be homogeneous with a porosity of 0.30 while the permeability is
homogeneous within each layer with values of 100md for layer 1and 1000md, layer.
98
A.3 PVT Properties of the Reservoir Fluids
This section of the input data contains pressure and saturation dependent
properties of the reservoir fluids and rocks. The reservoir fluids are oil and water which
are incompressible. The oil contains a very little or no concentration of dissolved gas. At
a reference pressure of 78atm the oil has a viscosity ranging 1 to105 cp specify in table
3.1 cp. The oil formation volume factor, water (Bw) and oil (Bo) both equal to 1. The bulk
compressibility of the rock was set at 2 x E-8 atm-1. This value was picked from the work
done from similar studies. The relative permeability curve used is shown in figure 3.1. A
summary of the reservoir properties is shown in Table 3. 1
A.4 Scal (saturation Section)
Corey correlation was used duplicate the relative permeability curve in figure 3.1
with Corey exponent with respect oil and water both equal to 2. The vertical equilibrium
option was used to modify the relative permeability curve into pseudo –relative
permeability for the crossflow modes.
A.5 Initialization
This section contains input data for the initial of the reservoir. The datum depth
and water-oil contact depth (WOC) were set at 500 cm and 540cm.The pressure at the
datum depth was set at 78.6atm with oil- water capillary pressure set to 0 atm. There is no
gas cap.
99
4.6 Schedule
This contains information regarding scheduling producers and injector for
production and injection. The wells (producer and injector) were scheduled to operate for
a three week period with constant control settings. The injector well (vertical/horizontal)
is located in the center of cells (1, 1, 1) and the producer (vertical/horizontal) in the cell
(50, 1, 1) of the grid. The vertical wells were set to perforate through the entire thickness
of the formation and horizontal well completed in blocks X1 – X8. The injector wells
were constraints to operate at maximum injection pressure of 78.6 atm and injection rate
of 100 cc per hour. The injector well schedule had controlled modes with an injection rate
of 100 cc/hr and injection pressure control of 280 atm. Same time, the minimum
allowable bottom hole pressure for the production well was set at 12 atm but same rate
control. These controls were slightly varied in the polymer injection operation and also in
the horizontal well injection for the sake of injectivity and as a means of reducing
dispersion.
A.7 Section
This contains information and keywords use to specify various
compartments/regions in the reservoir. It has facilities that models sector-sector flow,
which we used to track the crossflow in the model.
100
APPENDIX B
The results from two simulators: Eclipse 100 and POLYGEL-Petro China
Figure B-1 Free crossflow result from Eclipse 100 Simulator
Figure B-2 No-crossflow result from Eclipse 100 Simulator
101
mobi l e oi l r ecover y f or f r ee cr ossf l ow case by POLYGEL
01020304050
60708090
100
0 0. 5 1 1. 5 2 2. 5I nj ect ed Por e Vol ume
oil
reco
very
\%
dl f cf ov1dl f cf ov10dl f cf ov100dl f cf ov1000dl f cf ov10000dl f cf ov100000
Figure B-3 Free crossflow result from POLGEL Simulator by PetroChina
mobi l e oi l r ecover y f or no cr ossf l ow case by POLYGEL
01020304050
60708090
100
0 0. 5 1 1. 5 2 2. 5I nj ect ed Por e Vol ume
oil
reco
very
\%
dl ncf ov1dl ncf ov10dl ncf ov100dl ncf ov1000dl ncf ov10000dl ncf ov100000
Figure B-4 No-crossflow result from POLGEL Simulator by PetroChina
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APPENDIX C The calculation of NPV and NET CASH FLOW at oil prices, $ 20 and $ 50 per barrel for Displacing 1000 cp oil with polymer: (crossflow and no- crossflow).
Figure C-1 NPV computed at oil price of $ 20/bbl: Displacing 1000 cp oil with polymer (Crossflow)
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Figure C-2 NPV computed at oil price of ($ 20/bbl oil): Displacing 1000 cp oil with polymer (Crossflow)
Figure C-3 Net cash flow computed at $ 50/bbl of oil: Displacing 1000 cp oil with polymer (Crossflow)
Figure C-4 NPV computed at $ 50/bbl oil: Displacing 1000 cp oil with polymer (Crossflow)
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Figure C-5 Net cash flow computed at $ 20/bbl of oil: Displacing 1000 cp oil with polymer (No-crossflow)
Figure C-6 NPV computed at $ 20/bbl of oil: Displacing 1000 cp oil with polymer (No-crossflow)
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Figure C-7 Net cash flow computed at oil price of $ 50/bbl: Displacing 1000 cp oil with polymer (Crossflow)
Figure C-8 NPV computed at oil price of ($ 50/bbl oil): Displacing 1000 cp oil with polymer (No-crossflow)
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