This thesis focuses on developing static and dynamic reservoir models and predicting properties for a deepwater carbonate reservoir during the early exploration phase when limited data is available. Core, log, and well test data are integrated and used to characterize the reservoir into hydraulic flow units (HFU). Five HFU are identified and upscaled to populate the static model. Well test analysis estimates permeability-thickness product and permeability with less than 20% error. Dynamic simulations of four static models match well test pressure responses and predict a numerical productivity index within 5% of measured. Simulations of the entire oil zone indicate potential recovery of 25% of original oil in place.

1. EXPLORATION PHASE:
DEEPWATER CARBONATE RESERVOIR DATA INTEGRATION
FOR EARLY STATIC AND DYNAMIC MODELING
AND PROPERTIES PREDICTION
A Thesis
by
ALAN MÖSSINGER
Submitted to the Reservoir Geoscience and Engineering
department of IFP School
in partial fulfillment of the requirement for the degree of
MASTER OF SCIENCE
November 2015
Major Subject: Reservoir Engineering

2. ii
ABSTRACT
Exploration Phase: Deepwater Carbonate Reservoir Data Integration for Early Static
and Dynamic Modeling and Properties Prediction. (November 2015)
Alan Mössinger, Petrophysicist Consultant / Geologist – Reservoir Engineer
PETROBRAS
This thesis focuses on assessing and integrating the information available during the
beginning of the exploration phase when an exploratory well has been drilled and has
found a new accumulation. The studied data set consist of cores and fluid laboratory
analysis, logs, well testing and seismic. The main objectives are: (i) develop the static
and dynamic models from the outset of the exploration stage when the geological
framework is still poorly understood; (ii) predict reservoir properties, e.g., permeability-
thickness product, reservoir average permeability and productivity index by applying
analytical and numerical solutions. In order to achieve the goals this thesis proposes an
innovative manner to model the reservoir based on hydraulic flow unit concept. The
workflow generated allows decoupling of geological rock facies from reservoir fluid
flow behavior.
The early data integration and modeling permits to identify and address the key
uncertainties concerning the discovery and to assist and buttress well test design and
interpretations. Moreover, it allows a preliminary understanding of the reservoir and
helps to guide the subsequent steps of the exploration.
The hydrocarbon accumulation consists of a deepwater carbonate reservoir revealing to
be highly heterogeneous and in general composed by thin layers displaying a significant
contrast in flow potential. Despite such heterogeneity the employed methodology
succeeds in adjusting a permeability curve with core measurements. Following the
workflow the entire upscaling process moving from cores to well-logs, from well-logs
to static model and finally, from static model to dynamic model is described.
Furthermore, all the steps to populate the models with petrophysical attributes and the
analytical methods to predict reservoir parameters are also presented.

3. iii
Regarding the well testing interval the analytical solutions can estimate permeability-
thickness product and average reservoir permeability by an overestimation error of 19%.
Using numerical simulations the transient productivity index can be foreseen by an
overestimation error of 5%. The most likely reservoir characteristics and performance
results are also offered in case the entire oil leg is placed into production. Simulation
outcomes points out that the drilling of a horizontal well is discourage due to low
performance.
The dynamic cutoff estimated for the reservoir is 2.14 md and the first approximation of
the recovery factor 25%. Porosity cutoff should not be implemented due to the lack of
correlation between porosity and reservoir dynamic characteristics.
The decision to remove skin in an exploratory/appraisal well is an economic exercise
and particularly in deepwater exploration the expense associated is very high. The
described analytical estimations in order to forecast the production after stimulation
should be previously performed and included in the assessment plan. It might save
company’s expenditures. For a skin of -5 the predicted maximum transient productivity
index would be 10.5 m3/d/bar.

4. iv
DEDICATION
To my wife, Lianara Mössinger,
To my parents, Lúcia and Winfried Mössinger,
To my brothers, Heino, Heiko and Elton Mössinger.

5. v
ACKNOWLEDGMENTS
I would like to thank and acknowledge Petróleo Brasileiro S.A. (PETROBRAS) for the
opportunity to study at IFP School in France - Institut Français du Pétrole - and develop
this thesis at OPC in London. I also would like to thank PETROBRAS for providing me
the data used in this study and for the permission to publish it. I am especially grateful
to Lindemberg Pinheiro Borges for his support and suggestions and Ricardo Pinheiro
Machado and Paulo Sergio Denicol for their support.
I would like to thank Oilfield Production Consultants (OPC) specifically Piers Johnson
for receiving me during the thesis development period in their comfortable working
environment in London, for the hospitality and for allowing me to use their facilities. I
am also grateful to OPC’s staff David Gorsuch for his suggestions, José Morillo for his
assistance with the dynamic simulation, Yatin Suri for his assistance with PIE software,
Greg Bowker for sharing Well Testing reference material, Matthew Thompson for his
assistance with Petrel and Bill Robert for comments and corrections.
I would like to thank PETROBRAS colleagues Paulo Filho for collecting and sending
me the data set and Tiago Rossi for helping me with the software licenses. I would like
to thank Isabella Rios for providing me with technical guide materials from
Schlumberger. I am also grateful to classmates from IFP School Michael Oluchukwu
and John Pinto.
Finally, I want to thank IFP School for teaching me and for sharing the knowledge
which is partly implemented in this thesis and I am grateful to Maria Eugenia Aguilera
for her assistance with PVT analysis. My gratitude also goes to all my colleagues from
the MSc of Reservoir Geoscience and Engineering for making my graduate life
enjoyable.

6. vi
TABLE OF CONTENTS
PAGE
ABSTRACT .................................................................................................................ii
DEDICATION.............................................................................................................iv
ACKNOWLEDGMENTS............................................................................................. v
TABLE OF CONTENTS............................................................................................. vi
LIST OF TABLES.....................................................................................................viii
LIST OF FIGURES .....................................................................................................ix
CHAPTER I INTRODUCTION ............................................................................... 13
1.1 Objectives.................................................................................................... 14
1.2 Data Set Available and Software for Analysis and Interpretation ................. 15
CHAPTER II PETROPHYSICAL INTERPRETATION .......................................... 16
2.1 Permeability Curve Integral and PLT Correlation......................................... 21
2.2 Reservoir Anisotropy ................................................................................... 23
CHAPTER III HYDRAULIC FLOW UNIT CHARACTERIZATION ..................... 24
3.1 Upscaling HFU – From Cores to Well-Logs................................................. 29
3.2 Correlation Between HFU and Mercury Injection Capillary Pressure ........... 31
CHAPTER IV WELL TESTING INTERPRETATION ............................................ 34
4.1 Skin Pressure Drop and Transient Productivity Index................................... 40
4.2 Skin Removal in Exploratory/Appraisal Wells ............................................. 42
CHAPTER V INTEGRATION OF CORES, WELL-LOGS AND WELL TEST....... 48
CHAPTER VI PVT MATCHING............................................................................. 51
CHAPTER VII STATIC RESERVOIR MODELING............................................... 53
7.1 Populating the Static Model ......................................................................... 56
CHAPTER VIII DYNAMIC RESERVOIR MODELING......................................... 62

7. vii
8.1 Well Test Analysis of Simulations ............................................................... 68
8.2 Numerical Productivity Index ...................................................................... 73
8.3 Productivity of the Entire Oil Leg ................................................................ 76
8.4 Horizontal Well Simulation.......................................................................... 78
CHAPTER IX VOLUME OF HYDROCARBON..................................................... 84
9.1 Estimation of Dynamic Petrophysical Cutoff and Recovery Factor............... 84
CHAPTER X SUMMARY OF THE RESERVOIR PROPERTIES PREDICTION... 87
CHAPTER XI CONCLUSIONS AND RECOMMENDATIONS ............................. 89
REFERENCES .......................................................................................................... 93
NOMENCLATURE .................................................................................................. 94

8. viii
LIST OF TABLES
TABLE Page
Table 3.1 – Average and median pore-throat radius for each HFU............................... 33
Table 4.1 – Well test analysis constant........................................................................ 36
Table 4.2 – Reservoir parameters achieved from well testing interpretation................. 40
Table 5.1 – Calibrated alfa exponent and predicted reservoir parameters for the entire oil
leg............................................................................................................................... 49
Table 7.1 – Linear regression equations from upscaled porosity and permeability....... 55
Table 9.1 – Original oil in place calculation for Static and Dynamic Models C ........... 84
Table 9.2 – Statistical analysis of petrophysical properties of cells not contributing or
contributing very little to flow..................................................................................... 85
Table 9.3 – Hydrocarbon volume after applying cutoff of 2.14 md in static Model C. . 86
Table 10.1 – Reservoir properties obtained from well test analysis.............................. 87
Table 10.2 – Forecasted reservoir properties from different methods........................... 87

9. ix
LIST OF FIGURES
FIGURE Page
Fig. 2.1 – Open hole logs. Track 1: Gamma Ray; Track 2: Deep resistivity; Track 3:
Photoelectric factor, Density, Neutron and Sonic; Track 4: Nuclear Magnetic Resonance
(NMR) porosities; Track 5: NMR T2 distribution; Track 6: Acoustic image log..........16
Fig. 2.2 – Calibrated log curves, core petrophysical measurements and average reservoir
petrophysical parameters. Track 1 – Adjusted core porosity curve. Dots represent core
porosity measurements; Track 2 – Water saturation; Track 3 - Core derived permeability
curve in logarithmic scale. Dots represent core permeability measurements ................17
Fig. 2.3 – Acoustic image logs. Notice the layers and the low acoustic amplitude points
were sidewall cores were collected. Track 1: NMR Porosities; Track 2: Static acoustic
image; Track 3: Dynamic acoustic image....................................................................18
Fig. 2.4 – Image log interpretation and seismic correlation defining large stratigraphic
units. Track 1: Stratigraphic units; Track 2: NMR porosities; Track 3: Static acoustic
image; Track 4: Dynamic acoustic image and interpreted bed-boundary sinusoids; Track
5: Tadpole dip-direction; Track 6: Seismic trace .........................................................19
Fig. 2.5 – Image log and core derived permeability correlation. Track 1: NMR
porosities; Track 2: Static acoustic image; Track 3: Dynamic acoustic image and core
derived permeability curve in logarithmic scale...........................................................20
Fig. 2.6 – Production logging correlation with well-log data. Track 1: Well testing
perforation interval; Track 2: Gamma Ray; Track 3: NMR porosities; Track 4:
Integrated permeability for the entire reservoir oil zone; Track 5: Red curve - Integrated
permeability for the perforation interval and, green dashed curve - Cumulative PLT;
Track 6: Core derived permeability in logarithmic scale; Track 7: Core derived
permeability in linear scale; Track 8: PLT interpretation; Track 9: Acoustic image log22
Fig. 3.1 – Porosity versus permeability semi-log crossplot ..........................................24
Fig 3.2 – cumulative frequency curve showing the identification of five HFU .....27
Fig. 3.3 – Log-log plot of ɸ versus displaying the 5 hydraulic flow units......... 299

10. x
Fig. 3.4 – Semi-log porosity versus permeability crossplot showing the HFU
distribution and the linear correlations for each group .................................................29
Fig. 3.5 – HFU statistical analysis. Porosity on the left and permeability on the right.
From top to bottom HFU one to five, respectively....................................................... 30
Fig. 3.6 – Upscaling from HFU defined from plugs to logs in order to generate the HFU
log curve. Track 1 – GR; Track 2 – Photoelectric factor, Neutron, Density and Sonic;
Track 3 – NMR porosity; Track 4 – Acoustic image; Track 5 – Upscaled HFU; Track 7
– HFU from cores and HFU log curve........................................................................31
Fig. 3.7 – Pore-throat size distribution according to specific HFU. The vertical brown
line indicates the average pore-throat radius of the distribution ...................................33
Fig. 4.1 – Well testing pressure and flow-rate history highlighting the buildup periods37
Fig. 4.2 – First, second and third buildup derivative analysis and the matching
parameters. First BU: Brown Triangles; Second BU: Light red squares; Third BU: Blue
crosses; Reservoir model: Blue solid line ....................................................................38
Fig. 4.3 – Superposition on the left and data plot on the right. First BU: Brown
Triangles; Second BU: Light red squares; Third BU: Blue crosses; Reservoir model:
Blue solid line; Flow-rates: Light blue solid lines........................................................ 41
Fig.4 4 – Transient productivity index evolution over time from equation 4.8 ............. 42
Fig. 4.5 – Estimated oil rates due to skin reduction from equation 4.10 .......................46
Fig. 4.6 –Estimated productivity index due to skin reduction.......................................46
Fig. 4.7 – Estimated productivity index due to skin reduction from equation 4.8..........47
Fig. 6.1 – Reservoir fluid phase-envelope ................................................................... 51
Fig 6.2 – PVT match of reservoir fluid physical properties. a) Oil relative volume; b)
Gas-oil ratio; c) Liquid density; d) Liquid viscosity..................................................... 52
Fig. 7.1 – Upscaling results. Track 1 – HFU; Track 2 – Upscaled HFU; Track 3 – Core
adjusted porosity curve; Track 4 – Upscaled porosity; Track 5 – Core derived
permeability curve; Track 6 – Upscaled permeability..................................................54

11. xi
Fig 7.2 – Upper figure, upscaled porosity and permeability crossplot. Lower figure,
upscaled porosity and permeability crossplot highlighting the four trends used to
generate the linear regressions..................................................................................... 55
Fig. 7.3 – Fitted probability distribution function for upscaled porosity for individual
HFU............................................................................................................................57
Fig. 7.4 - J-function curves calculated for the HFU .....................................................59
Fig. 7.5 – Model C quality check using histograms. Each histogram shows the property
defined from cores and well-logs, upscaling procedure and assigned for the static 3-D
model. Notice that the property distribution is maintained throughout the process.......60
Fig. 7.6 – East-West models cross-section cutting through the well. Four different
models according to the specified major = minor variogram range. The property shown
is HFU with no vertical exaggeration..........................................................................61
Fig. 7.7 – 3-D petrophysical properties of static Model C. No vertical exaggeration....61
Fig. 8.1 – Match between core relative permeability and Corey’s method ...................63
Fig. 8.2 – Dynamic reservoir petrophysical parameters upscaled from static Model A.64
Fig. 8.3 – Dynamic reservoir petrophysical parameters upscaled from static Model B.65
Fig. 8.4 – Dynamic reservoir petrophysical parameters upscaled from static Model C.66
Fig. 8.5 – Dynamic reservoir petrophysical parameters upscaled from static Model D.67
Fig. 8.6 – Simulated bottom hole pressure and oil rate versus time for the four dynamic
models ........................................................................................................................68
Fig. 8.7 – Third BU pressure derivative comparison between the four dynamic models
...................................................................................................................................69
Fig. 8.8 – BU derivative analysis and the matching parameters for simulated Model C
bottom hole pressure................................................................................................... 70
Fig. 8.9 – Model C pressure analysis match of superposition plot on the left and data
plot on the right...........................................................................................................70
Fig. 8.10 – Models cross-section displaying pressure disturbance at the final timestep of
the third buildup..........................................................................................................72

12. xii
Fig. 8.11 – Model C well cross-section. Notice the good agreement between the well
testing cumulative PLT and the simulated cumulative PLT. Track 1 – Porosity; Track 2
– Permeability in logarithmic scale; Track 3 – Water saturation; Track 4 – Completion
and perforation interval; Track 5 – Well testing cumulative PLT; Track 6 – Simulated
cumulative PLT for Model C ......................................................................................73
Fig. 8.12 – Simulated numerical productivity index over time for Model C during the
final buildup ...............................................................................................................75
Fig. 8.13 – Simulated numerical productivity index over time for Model C based on 9-
point pressure for the three flowing periods................................................................. 75
Fig. 8.14 – Model C bottom hole pressure, rate and 9-point pressure numerical
productivity index for 20 days of simulation for the entire oil section. Control set for
bottom hole pressure and equivalent to 200 bar...........................................................76
Fig 8.15 – Derivative pressure analysis for the entire oil zone ..................................... 77
Fig. 8.16 – Model C well cross-section; PLT response for the entire oil section. An
increase of 27% should be expected when comparing well testing cumulative PLT with
the simulated PLT and the upper and lower part of the reservoir are not contributing to
production. Track 1 – Porosity; Track 2 – Permeability in logarithmic scale; Track 3 –
Water Saturation; Track 4 – Completion and perforation interval; Track 5 – Well testing
cumulative PLT; Track 6 – Simulated PLT .................................................................78
Fig. 8.17 – Bottom hole pressure, rate and numerical productivity index for a horizontal
well simulation in Model C......................................................................................... 79
Fig. 8.18 – Derivative pressure match for horizontal well simulation ..........................80
Fig. 8.19 – Match of Superposition plot on the left and data plot on the right for the
horizontal well simulation...........................................................................................81
Fig. 8.20 – Simulated PLT for a horizontal well. Only 30% of the total horizontal length
is effectively contributing to production......................................................................82
Fig. 8.21 – Model cross-section showing pressure as property and the horizontal well at
the final drawdown timestep .......................................................................................83

13. 13
CHAPTER I
INTRODUCTION
The greatest challenge reservoir geoscientists and reservoir engineers face is performing
the reservoir description from which crude oil or gas is produced and apply the results
achieved on reservoir modeling in order to ultimately generate numerical simulations of
reservoir performance. There are mainly two difficulties which have to be overcome to
obtain a reliable description of the reservoir:
(i) Reservoir heterogeneity;
(ii) The limited amount of data set available, i.e., number of wells at which
observations can be made.
In the oil and gas industry the standard practice is to start the reservoir modeling after
finishing the exploration phase when at least one exploratory well and a few appraisal
wells have been drilled. The idea behind is to gather a minimum information required to
start modeling the field with at least some degree of certainty.
This thesis proposes a workflow to accomplish the reservoir description and modeling
by starting assessing, correlating and integrating the available information from the
outset of the exploration stage when only one exploratory well has been drilled and its
data set accessible. The suggested process could allow an initial understanding of the
reservoir and it could help to guide the next steps of the exploration, for instance, what
the major uncertainties are and what the key aspects are that need to be addressed in
order to reduce these uncertainties? Furthermore, it could permit to easier transfer a new
discovery from the exploration department to the production department simply due to
the fact that the static and dynamic modelings have already been started and the main
uncertainties recognized and addressed.
The reservoir geological framework is the cornerstone of static modeling, however
during the exploration phase it is usually poorly understood and this is one of the main
complications preventing companies to develop the reservoir model. In an attempt to
detach the geological rock facies from the dynamic reservoir behavior and overcome

14. 14
this issue, this thesis proposes an innovative manner to build the static reservoir model
based on hydraulic flow units.
The permeability-thickness product ( ℎ) and the related average reservoir permeability
( ) are the most important reservoir parameters to be measured during the exploratory
phase. The well productivity is mainly controlled by ℎ and skin; even though the skin
factor is noteworthy, during the exploration stage the wells are generally not optimized
for production and the skin factor might be ignored.
The ability to predict reservoir parameters is another vital aspect of reservoir assessment
because it can support how to design and interpret the well testing and how to optimize
and place the drilling in forthcoming wells. The predictability could also lead managers
to find what should be the best strategy for the exploratory area or field by giving a
wider picture of the current scenario and not only buttress, but also simplify crucial
decisions that many times have to be made in short notice. Moreover, in case the
predictions turn out to be reliable it might save company’s efforts and investments in
acquiring new data. In this regard, an additional important aspect that is approached in
this thesis is the ability to predict reservoir attributes, e.g., reservoir permeability-
thickness product, average permeability and productivity index. Two methodologies are
attempted, analytical solutions based on cores and well-logs and numerical solutions
based on simulations.
1.1 Objectives
This thesis has two mains objectives:
1) Generate the static and dynamic models based on hydraulic flow units through
assessment and integration the data set available for an exploratory area;
2) Predict reservoir parameters, i.e., permeability-thickness product, average
reservoir permeability and productivity index from analytical solutions using
cores and well-logs and from numerical solutions applying simulations;

15. 15
1.2 Data Set Available and Software for Analysis and Interpretation
The full data set of a deepwater exploratory area with one drilled well responsible for a
significant oil discovery is available for this study and it constitutes of:
a) Sidewall and whole cores routine and special laboratory analysis – porosity,
capillary pressure, absolute permeability and relative permeability;
b) Logs from Wireline including Formation Testing;
c) Transient Pressure Testing and Production Logging;
d) Fluid laboratory analysis – Flash liberation and Pressure-Volume-Temperature
certificates;
e) Seismic;
The following software were used to assist in the data analysis and interpretations:
a) Petrel® - Schlumberger;
b) Eclipse® 100 - Schlumberger;
c) PVTi® - Schlumberger;
d) Interactive Petrophysics® - Senergy;
e) MATLAB® - The Mathworks, Inc;
f) PIE® - Well Test Solutions Ltd;
g) Office Package® - Microsoft;
It is important to point out that some information, such as, reservoir depths, precise well
location, legends and figures are not fully displayed in order to preserve the confidential
aspect of the studied region and play.

16. 16
CHAPTER II
PETROPHYSICAL INTERPRETATION
The open hole logs acquired are shown on Fig. 2.1 and it reveals in almost the entire
section a carbonate reservoir saturated with hydrocarbons, excluding the bottom part
where a water leg and some non-reservoir segments are identified. There is no direct
oil-water contact recognized in the logs, but the free water level depth is known from
RFT pressure data.
Fig. 2.1 – Open hole logs. Track 1: Gamma Ray; Track 2: Deep resistivity; Track 3:
Photoelectric factor, Density, Neutron and Sonic; Track 4: Nuclear Magnetic Resonance
(NMR) porosities; Track 5: NMR T2 distribution; Track 6: Acoustic image log.
The reservoir net to gross, average porosity and average water saturation calculated are
0.94, 12% and 25%, respectively. The reservoir is statistically well represented from
cores and all porosity and permeability core analyses were performed under reservoir
FWL

17. 17
conditions. The absolute permeability was corrected to Klinkenberg effect and no
equations were used to modify from absolute permeability to effective permeability.
The porosity and permeability log curves were adjusted to respect core porosity and
permeability data. The permeability curve calibration methodology is described in
details in CHAPTER III - Hydraulic Flow Unit Characterization. Fig. 2.2 shows the
calibrated reservoir porosity, water saturation and permeability curves and the average
reservoir petrophysical parameters.
Fig. 2.2 – Calibrated log curves, core petrophysical measurements and average reservoir
petrophysical parameters. Track 1 – Adjusted core porosity curve. Dots represent core
porosity measurements; Track 2 – Water saturation; Track 3 - Core derived permeability
curve in logarithmic scale. Dots represent core permeability measurements.

18. 18
The image log interpretation discloses a reservoir with layers varying from a few
centimeters to a few meters of thickness. Many intervals show laminations with less
than 10 centimeters of thickness. In such highly heterogeneous reservoirs the precise
depth of the sidewall cores are crucial to perform a reliable reservoir characterization
and because the sidewall cores recovery depth are seen as points with low acoustic
amplitude it was possible to define the exact depth position from where the cores were
collected. All the reported cores depth were corrected to the depth encountered in the
image log and the error range varied from a few centimeters to 2 meters, upward or
downward (Fig 2.3).
Fig. 2.3 – Acoustic image logs. Notice the layers and the low acoustic amplitude points
were sidewall cores were collected. Track 1: NMR Porosities; Track 2: Static acoustic
image; Track 3: Dynamic acoustic image.
Regarding the image log interpretation the entire section is dominated by bed-
boundaries and no faults or fractures are present. The bed-boundaries dip are in general

19. 19
smaller than 20o and by analyzing the reservoir section in a large scale four wide
stratigraphic units can be recognized due to the layers dip and strike variation. The wide
stratigraphic units also coincide with the more significant impedance reflectors observed
on the seismic trace which was crossed by the well. These stratigraphic units were
named from top to bottom Alpha, Beta, Theta and Gamma (Fig. 2.4).
Fig. 2.4 – Image log interpretation and seismic correlation defining large stratigraphic
units. Track 1: Stratigraphic units; Track 2: NMR porosities; Track 3: Static acoustic
image; Track 4: Dynamic acoustic image and interpreted bed-boundary sinusoids; Track
5: Tadpole dip-direction; Track 6: Seismic trace.
The acoustic image log also exposes key information about reservoir quality. The high
acoustic amplitude layers tend to exhibit inferior reservoir properties when compared to
the low acoustic amplitude layers and the calibrated permeability curve is the
petrophysical attribute which indicates the best correlation. The core derived
permeability agreement with the acoustic image log is shown on Fig. 2.5.

20. 20
The high degree of correlation between the core derived permeability and the acoustic
image log was only possible to be achieved due to the hydraulic flow unit
characterization method which is described in details in the next chapter. This method
allowed the permeability curve to move from high permeability to low permeability
layers in just a few centimeters of vertical scale and such assignment is impossible to be
achieved using only the basic well logging data due to vertical resolution limitations.
Fig. 2.5 – Image log and core derived permeability correlation. Track 1: NMR
porosities; Track 2: Static acoustic image; Track 3: Dynamic acoustic image and core
derived permeability curve in logarithmic scale.

21. 21
2.1 Permeability Curve Integral and PLT Correlation
Correlating permeability curve with production logging (PLT) is a good manner to
verify the confidence of the adjusted permeability curve and it also allows accessing the
flow potential of a limited reservoir interval. The calibrated permeability curve ( )
can be integrated for a defined interval from to as follows:
ℎ = ( ) ……………………………………………………(2.1)
and to simplify the result is also called ℎ. It is acknowledged that the core
permeability used to calibrate the curve represents absolute permeability and it will be
compared to effective reservoir permeability, therefore the calculated ℎ should be
multiplied by the oil relative permeability ( ). However, during the exploration phase
the relative permeability to hydrocarbon is little understood and it was decided to do a
direct association without applying any correction to the permeability curve integral
result.
The production well logging was acquired during the well testing and as reported by the
PLT 100% of the perforated interval is contributing to flow. Fig. 2.6 displays the well
testing perforation interval and correlation between PLT, core derived permeability
curve, integrated permeability and well-logs. Adjusting the permeability curve in linear
scale enhances the recognition of the higher and lower production zones. The higher and
lower production zones according to the core derived permeability curve shows a decent
agreement with interpreted production logging.
Mainly in the middle and upper parts, the integrated permeability curve relates very well
with the cumulative production logging. A perceptible difference is at the bottom part of
the perforated interval where the integrated permeability shows three sudden changes in
the inclination, suggesting a significant increase in production coming from thin layers
while in the same interval the cumulative production logging shows a smoother
inclination (Fig. 2.6). However, it should be pointed out that there is a vertical
resolution limitation for PLT and it might not be able to precisely distinguish
laminations with only a few centimeter thicknesses. Another point is that PLT is
measured in a cased hole and the fluid flows through the gun perforations and the

22. 22
perforations might not exactly coincide with the depth of the higher production
laminations.
Fig. 2.6 – Production logging correlation with well-log data. Track 1: Well testing
perforation interval; Track 2: Gamma Ray; Track 3: NMR porosities; Track 4:
Integrated permeability for the entire reservoir oil zone; Track 5: Red curve - Integrated
permeability for the perforation interval and, green dashed curve - Cumulative PLT;
Track 6: Core derived permeability in logarithmic scale; Track 7: Core derived
permeability in linear scale; Track 8: PLT interpretation; Track 9: Acoustic image log.
The integral of the core derived permeability curve for the perforation interval yields a
permeability-thickness product ( ℎ) of 953 md.m and using 125 meters corresponding
to the perforation interval as reservoir effective thickness the average reservoir
permeability ( ) is 7.6 md. When taking into account the entire oil zone and applying
the NTG previously computed results in 1208 md.m x 0.94 (NTG) = 1135.5 md.m and
for the whole oil leg drops to 3.7 md (Fig. 2.6).
Analyzing statistically the core permeability data in the perforation interval, the
arithmetic, geometric and harmonic average reservoir permeabilities are 16.26 md, 1.32
md and 0.000625 md, respectively. Using 125 meters which represents the perforation

23. 23
interval as reservoir effective thickness yields an estimated ℎ of 2032 md.m for
arithmetic average permeability, 165 md.m for geometric average permeability and
0.078 md.m for harmonic average permeability. It is normally expected the reservoir
permeability-thickness product lies between arithmetic and geometric average values.
2.2 Reservoir Anisotropy
The reservoir anisotropy ( ) can be calculated from:
= ..……..……………………………………………….……………(2.1)
where is vertical permeability and is horizontal permeability. An approximation
to calculate the reservoir anisotropy is to assume the horizontal permeability as the core
arithmetic average and the vertical permeability as the core harmonic average. This
yields reservoir anisotropy of 3.8 * 10-5 and since all cores were used in the
computation the low value means that in a reservoir scale it is unlikely to occur any
significant cross-flow between large stratigraphic zones and the reservoir flow might be
controlled by pressure drop spreading laterally and not moving upward or downward
through another stratigraphic unit.
Another approach which unveils a different result is to calculate reservoir anisotropy
using vertical permeability measured on vertical cores and use the nearest horizontal
core permeability to calculate the anisotropy ratio. A few vertical cores permeability
measurements are available and the depth they were collected correspond to the
perforation interval depth. This method produces an anisotropy ratio extending from
0.001 to 0.6 and the arithmetic average of 0.1, meaning that in a metric to centimetric
scale it might be possible to find cross-flow between some layers.
This calculations regarding the reservoir anisotropy highlights that there is a significant
degree of uncertainty associated with the results and in forthcoming wells this point
should be addressed, perhaps by acquiring mini-DSTs after fluid sampling and from the
pressure derivative analysis define a more accurate reservoir vertical permeability in
case the spherical flow regime is identified.

24. 24
CHAPTER III
HYDRAULIC FLOW UNIT CHARACTERIZATION
Knowledge of permeability distribution is critical to reservoir effectively description
and core data is a fundamental factor to its determination. In heterogeneous reservoirs
the traditional porosity versus permeability crossplot cannot be used to reliably estimate
permeability from porosity data. Fig. 3.1 displays the ɸ − semi-log crossplot and the
correlation is very poor. For the same porosity value the permeability measured
incredibly fluctuates eight log-cycles, more specifically, an order of magnitude of 108.
Fig. 3.1 – Porosity versus permeability semi-log crossplot.
In order to improve the ɸ − correlation, perform rock-typing and assist in the
modeling process, the cores were classified in different hydraulic flow units (HFU)
groups. “A hydraulic flow unit is a zone that is continuous over a defined volume of the
reservoir and has similar average properties which affect fluid flow with similar bedding
characteristics”.(2) A slightly different definition is “Volume of the total reservoir rock

25. 25
within which geological or petrophysical properties that affect fluid flow are internally
consistent and predictably different from properties of other rock volume”.(10)
Only one hydraulic flow unit could be present in several depositional facies or within a
single facies several HFUs could be found.(10) This means that there is no direct relation
between facies boundaries and HFU boundaries. In some geological situations the
distribution of HFU could be connected to facies distribution, but this is not always the
case. For instance, in carbonate reservoirs the early or late diagenesis process might be
able to modify completely the rock petrophysical properties and transform an expected
high porosity and/or permeability facies into a low porosity and/or permeability facies.
The opposite could also happen, a low porosity and/or permeability facies become a
high porosity and/or permeability facies. In other words, a “good” facies does not
necessarily mean a high HFU number with high flow potential.
Hydraulic flow unit could be called dynamic rock-typing and its classification allows
detaching the fluid flow behavior from the rock facies giving an alternative manner to
execute rock-typing classification. This is a great advantage when performing reservoir
characterization studies in exploratory areas, because at the exploration stage the
geological model is usually poorly understood and facies organization have not been
defined.
A HFU is primarily controlled by pore geometrical attributes, such as pore geometry
and pore-throat distribution. The pore geometry is controlled by mineralogy, grain size,
grain shape, sorting and packing and determines the rock storativity capacity. Besides
the macro-sedimentary structures, such as bedding, laminations, etc., the pore-throat
distribution is the major parameter determining the rock flow capacity.(10)
Hydraulic unit theory is based on porosity, permeability and capillary pressure
correlation. Firstly combining Darcy’s law for flow in porous media and Poiseuille’s
law for flow in tubes results:(10)
= ɸ ……...…………………………………………………………….(3.1)
and the equation shows that permeability depends on pore characteristics, precisely the
pore-throat radius.

26. 26
The generalized Kozeny-Carman(12) equation for porous media is more realistic because
the connected pore structure is not straight and the permeability is related to tortuosity
factor ( ) and the surface area per grain volume ( ):
=
ɸ
( ɸ )
………..……………………………………….……...(3.2)
where is μ and ɸ is a fractional volume. The term is known as the Kozeny
constant and varies from 5 to 100 for most reservoir rocks. The term is a
function of the geological characteristics of porous media and is related to pore
geometry. This term is the focal point of the HFU classification technique. Dividing
both sides of equation 3.2 by porosity and then taking the square root of both sides
results in:(2)(10)(11)
0.0314
ɸ
=
ɸ
( ɸ )
………..……………………………………..(3.3)
where the constant 0.0314 comes from the square root of the conversion from μ to
. The term ( /ɸ ) proves to be a very powerful correlation parameter for various
measured petrophysical properties such as formation factor.
The flow zone indicator term ( ) is defined as:(2)(10)(11)
= ………………..……………………………………………..(3.4)
and reservoir quality index term ( ) is given by:
= 0.0314
ɸ
………..………………………………………………....(3.5)
The pore volume to grain volume ratio term (ɸ ) is defined as:
ɸ =
ɸ
( ɸ )
………..………………………………………….……………..(3.6)
and substituting equations 3.4, 3.5 and 3.6 into equation 3.3 yields:
= ɸ ………..……………………………………………………...(3.7)

27. 27
Taking the logarithm of both sides of equation 3.7 results:
log( ) = log(ɸ ) + log( ) ………..………………………….………(3.8)
where and are given in μ . On a log-log plot of ɸ versus all samples
with similar values will cluster on the same straight line with unity slope and
samples with different values will lie on other parallel lines. Samples with the same
classification tend to have similar pore-throat attributes; therefore, will constitute a
unique hydraulic flow unit (HFU). Each defined line on the log-log plot is an HFU and
the intercept of the straight line with ɸ = 1 will give the value for that group of
data which defined the HFU.
In order to correctly identify the numbers of HFU a cumulative frequency histogram
was created using the calculated values for the samples. The number of HFU is
defined by a change in the cumulative histogram slope and five HFU were recognized
(Fig.3.2). It is acknowledged that many more hydraulic flow units could have been
defined; however the definition of a too large number would turn the workflow
impractical.
Fig 3.2 – cumulative frequency curve showing the identification of five HFU.

28. 28
The log-log plot of ɸ versus is shown on Fig. 3.3. The upper and lower limits of a
given HFU cluster were defined by half distance between straight lines. Fig. 3.4
illustrates the ɸ − semi-log crossplot with the HFU distribution and the ɸ − linear
correlation equation for each group. Clearly the porosity versus permeability correlation
is significantly improved by analyzing each group individually. Fig. 3.5 summarizes the
HFU porosity and permeability statistical analysis. It is noticed that the porosity
histograms for the HFU overlap and individualizing the HFU using only porosity is an
impossible task.
3.1 Upscaling HFU – From Cores to Well-Logs
In order to upscale the defined HFU using cores to well-log scale a manual
interpretation procedure was executed assisted mainly by the acoustic image log. In
general, different HFU recognized in the cores are well correlated to image layers
according to its acoustic amplitude. As a result, the top and bottom of each HFU could
be determined using the top and bottom of the rock layer were the sample was collected.
In other words, the HFU limits were defined by the layer bed-boundary visualized on
the image log. The final output is that a continuous HFU log curve was created for the
entire reservoir section where cores were available and it has the vertical resolution of
the image log (Fig. 3 6).
It is important to highlight that the reservoir is extremely heterogeneous and due to
basic logs vertical resolution limitations it is not possible to accurately fine-tune the
permeability curve with cores using the standard process. This core-log upscaling
workflow allowed creating a core derived permeability curve using the HFU as a
constrain, i.e., HFU 1: correlation 1, HFU 2: correlation 2 and so on. The consequence
is that now it is possible to move from high permeability layers to low permeability
layers in just a few centimeters scale because of the high vertical resolution of the image
log (Fig. 2.5).

29. 29
Fig. 3.3 - Log-log plot of ɸ versus displaying the 5 hydraulic flow units.
Fig. 3.4 – Semi-log porosity versus permeability crossplot showing the HFU
distribution and the linear correlations for each group.

30. 30
Fig. 3.5 – HFU statistical analysis. Porosity on the left and permeability on the right.
From top to bottom HFU one to five, respectively.

31. 31
Fig. 3.6 – Upscaling from HFU defined from plugs to logs in order to generate the HFU
log curve. Track 1 – GR; Track 2 – Photoelectric factor, Neutron, Density and Sonic;
Track 3 – NMR porosity; Track 4 – Acoustic image; Track 5 – Upscaled HFU; Track 7
– HFU from cores and HFU log curve.
Excluding acoustic image no other well-log response showed a good relationship with
HFU.
3.2 Correlation Between HFU and Mercury Injection Capillary Pressure
Capillary pressure curves for rocks can be determined by mercury injection and
withdrawal. The data can be used to obtain the pore size distribution, to study the
behavior of capillary pressure curves and to infer characteristics of pore geometry.(8)

32. 32
The determination of the pore-throat radius ( ) distribution of rocks based on capillary
pressure curve ( is an approximation and after conversion from laboratory
conditions to reservoir conditions can be calculated by:
= 0.1450377 ………..…………………………………………….(3.9)
where is in micrometer, is in dynes/cm and is in psi.
Fig. 3.7 displays the pore-throat size distribution from mercury injection measurements
from the same cores used to define the hydraulic flow units and Table 3.1 shows the
average and median pore-throat values for individual HFU. It is verified that there is a
very good correlation between the HFU and the pore-throat of the rocks. Greater the
HFU, larger the FZI value associated, larger the pore-throat size of the rock and it is
known that the major parameter controlling the rock flow potential is the pore-throat
size and its distribution. Therefore, the HFU classification method distinguish the rock
samples according to its flow behavior.
Ideally, in order to perform the static and dynamic modeling each HFU group should be
represented by its intrinsic petrophysical parameters, e.g., porosity, water saturation and
permeability and the laboratory core measurements are the main input to calculate these
attributes. An observed advantage of implementing HFU methodology from the outset
of the exploration stage is that it promptly allows to improve the selection of cores
which should be submitted to special core analysis and also identify which HFU are not
well characterized from cores. As a result, in the forthcoming wells the collection of
cores can be arbitrary driven and not random. It also means that the methodology can
avoid unregulated core laboratory measurements and reduce company’s effort and
expenditure in this matter.
Table 3.1 – Average and median pore-throat radius for each HFU.

33. 33
Fig. 3.7 – Pore-throat size distribution according to specific HFU. The vertical brown
line indicates the average pore-throat radius of the distribution.

34. 34
CHAPTER IV
WELL TESTING INTERPRETATION
Well test analysis is an inverse problem and in general cannot be solved uniquely.
Hence, more than one mathematical model will generate the output to represent the
measured data. A consistent description of the reservoir can only be achieved by
assessing and correlating all the available information from different sources and
adjusting a comprehensible physical model of the system.(10) The well testing
interpretation usually focuses on the transient pressure response and the pressure
variation response is a function of well geometry and reservoir properties, e.g.,
permeability and heterogeneity. Transient responses are observed before constant
pressure or closed boundaries effects are reached.(3)
The objective of testing exploratory or appraisal wells have been summarized as
follows:(10)
 Acquire information on reservoir description (permeability, heterogeneity,
boundaries, compartmentalization);
 Measure well productivity;
 Obtain fluid samples for laboratory PVT analysis;
 Measure reservoir temperature and pressure – static and dynamic;
 Estimate completion efficiency, e.g., skin factor ( );
However, the completion of an exploratory/appraisal well is not optimized for
production and the skin factor might not be representative. When an exploratory or
appraisal well is drilled, the mud system will not be adjusted with respect to formation
damage and these wells are drilled with a substantial overbalance which will result in
mud filtrate loss promoting damage. In this scenario the permeability-thickness product
and average reservoir permeability are the important factors which most of the attention
should be focused on and the completion efficiency – skin factor – might be ignored.(10)
A foremost important feature of transient well testing is that the interpretation allows
the independent assessment of intrinsic formation average permeability and near

35. 35
wellbore effects - skin factor. Consequently, even if the skin effect is high in the
exploration/appraisal well, the productivity index of the future development wells may
be predicted analytically or numerically.
Beforehand starting the well test interpretation the following items should be
incorporated in the analysis and are known from petrophysical interpretation and
correlation, i.e., open hole logs, cores and seismic:
 The reservoir is highly heterogeneous and stratified. Thin layers with just few
centimeter thickness are present and highly differ in flow potential;
 One of the main difficulties associated with the design of well testing is the
requirement for a permeability estimate. From cores, the arithmetic, geometric
and harmonic reservoir permeability average for the perforation interval is 16.26
md, 1.32 md and 0.000625 md, respectively. When applying 125 meters as
effective reservoir thickness the permeability-thickness product is 2032 md.m
for arithmetic reservoir permeability, 165 md.m for geometric reservoir
permeability and 0.078 md.m for harmonic reservoir permeability. It is
customary that the well testing reservoir ℎ lies between arithmetic and
geometric average values;
 From the integral of the core derived permeability curve the perforated interval
yield a ℎ = 953 . and = 7.6 md;
 The total perforation interval should contribute to production;
 The flow might be dominated by some high permeability thin layers;
 The permeability anisotropy ( / ) for the reservoir as a whole is very low
(3.8 * 10-5). No significant cross-flow between large stratigraphic layers is
expected; as a result, the pressure disturbance is unlikely to move downward
and/or upwards in the reservoir and might be concentrated around the perforated
top and bottom boundaries. In a metric or centimetric scale the average
permeability anisotropy calculated from cores is 0.1;
 The reservoir permeability seems to be matrix dependent. No fractures or faults
are interpreted on the image logs and there is no evidence of geological
boundaries from seismic interpretation, e.g., faults (seismic interpretation will
not be discussed in this thesis, but the information is known in advance);

36. 36
The perforation interval comprehends 125 meters and the static constants applied in the
well test analysis are shown on Table 4.1.
Table 4.1 – Well test analysis constants.
Fig. 4.1 displays the well testing pressure and flow-rate history. The test sequence
consists of three drawdowns and three buildups. After 16 hours of cleanup flow starts
the first buildup and it lasts 43 hours. Then 10 hours of flowing period for the second
flow followed by 8 hours of shutin and lastly 14 hours of flow followed by the third
buildup which lasts 43 hours. For the first and third buildup a hard shutin maneuver was
applied whereas for the second buildup a soft shutin maneuver. The consequence of the
two different shutin methods can clearly be identified from evaluating the wellbore
storage effect (Fig 4.2).

37. 37
Fig. 4.1 – Well testing pressure and flow-rate history highlighting the buildup periods.
The third buildup period was designated to perform the pressure test analysis and the
derivative plot shows in early-time a first stabilization after 3 minutes of shutin and
subsequently, corresponding to mid-time an upturn, which reflects a transitional period.
Then, in late-time after three hours of shutin a second stabilization is detected (Fig 4.2).
The second straight line is 45% smaller than the first straight line. In order to solve and
match this derivative behavior a linear composite model was selected and it is formed
by two distinct media in the reservoir. Initially the region around the well is producing
alone and the pressure behavior corresponds to a homogeneous reservoir. When the
linear interface is reached the two regions are producing together and a second
homogeneous behavior is observed. The equivalent homogeneous system is defined by
the average properties of the two regions, because during the late homogeneous regime
the two regions are contributing to production and this is obtained by the average of the
two regions mobility.(3)

38. 38
Fig. 4.2 – First, second and third buildup derivative analysis and the matching
parameters. First BU: Brown Triangles; Second BU: Light red squares; Third BU: Blue
crosses; Reservoir model: Blue solid line.
The comparison between region 1 and 2 is expressed by the Mobility ratio ( ) and
Storativity ratio ( ):(3)
= ………...…………………………………………………………...(4.1)
=
(ɸ )
(ɸ )
………...…………………………………………………………..(4.2)

39. 39
The level of the plateaus indicates the apparent mobility [( /μ) ] of the total
homogeneous regime:(3)
= = 0.5 1 + ………..…………………….(4.3)
The interpreted reservoir model unveils a second region 7.8 meters away from the
wellbore exhibiting poorer reservoir quality when compared to the first region. The total
system reservoir permeability-thickness product, ℎ, is 775 md.m and using 125 meters
as effective reservoir thickness, the average reservoir permeability, , is 6.2 md. The
radius of investigation reached by the pressure disturbance is 136 meters.
The mobility ratio and storativity ratio attained by modeling are both equivalent to 0.1
and because both ratios change proportionally implies that the effective reservoir
thickness was reduced. However, it is important to point out that using the linear
composite model it would be possible to match the derivative behavior with other
mobility and storativity combinations, since a decrease in reservoir quality is expressed
for the second region. In other words, several mathematical solutions for mobility and
storativity ratio are feasible. Apart from this non-unique mathematical solution, what is
essential to be noticed is that there is a decrease in reservoir quality away from the
wellbore.
From the well-logs it is possible to observe that the carbonate reservoir is vertically
greatly heterogeneous and this heterogeneity is also present in the cores when porosity
and permeability crossplot is analyzed. From a geological point of view it is very likely
that this heterogeneity is going to spread laterally and what is probably varying when
moving away from the wellbore is the permeability-thickness product. As a result, not
only the mobility, but also the storativity ratio will be affected and with the current
information available is not practical to precisely quantify the real mobility and
storativity ratios. Nevertheless, it is plausible to confirm that the reservoir flow potential
is reduced by 45% when moving 7.8 meters away from the wellbore. Another
interesting point is that besides the reservoir being highly heterogeneous, a
homogeneous radial flow is reached, meaning that concerning fluid flow the system as a

40. 40
whole behaves homogeneously. The increase in fluid viscosity could also cause similar
derivative behavior, nonetheless is unlikely to have occurred.
When comparing the first, second and third buildup derivatives there is no noteworthy
change existent, apart from a slight decrease in skin from the first to the third derivative
during early-time, what is perfectly normal and it reflects the release of some
supercharge effect after the cleanup flow. It is interesting to notice that the second
buildup wellbore storage hides the early-time stabilization present in the first and third
buildup. The matching model and reservoir parameters are summarized on Table 4.2
and on Fig. 4.3 the superposition and data plot results are shown. From the
superposition plot the extrapolated reservoir initial pressure ( *) is 609 bar at gauge
depth and no depletion can be identified from the first BU extrapolated pressure to the
third BU extrapolated pressure.
Other models could also match the pressure and rate of the well testing, such as
homogeneous reservoir with boundary effect. However, there is no enough evidence to
support the presence of a geological boundary, for instance, a sealing fault close to the
wellbore.
Table 4.2 – Reservoir parameters from well testing interpretation.

41. 41
Fig. 4.3 – Superposition on the left and data plot on the right. First BU: Brown
Triangles; Second BU: Light red squares; Third BU: Blue crosses; Reservoir model:
Blue solid line; Flow-rates: Light blue solid lines.
4.1 Skin Pressure Drop and Transient Productivity Index
The total skin obtained by well testing interpretation is 0.55 and can be classified as
mechanical skin. The corresponding pressure change due to skin ( ) can be
expressed in oilfield units as:(3)
=
.
………..…………………………………………...……(4.4)
resulting in 8.3 bar (120.4 psi).
During the infinite acting period the productivity index is decreasing with time and can
be referred as transient productivity index. The actual and ideal productivity index can
be related to rate as:
=
( )
......................................................................................(4.5)
=
( )
……....………………………………………..(4.6)
The computed using the extrapolated reservoir pressure, the final flowing
pressure and the flow-rate measured during the third flowing period is 2.77 m3/d/bar

42. 42
(1.20 stb/d/psi). When correcting for the pressure change due to skin ( ) the ideal
productivity index is equal to 2.95 m3/d/bar (1.27 stb/d/psi).
The flow efficiency ( ) can be computed from:(3)
= ………...………………………………………………………...(4.7)
resulting in 0.94, meaning that the reservoir is slightly damaged.
The analytical solution in oilfield units for the transient productivity index ( ) is:(1)
=
.
ɸ
. . ]
………..………………..………(4.8)
In order to calibrate equation 4.8 with the well testing the transient productivity
index evolution with time was calculated using the derivative model reservoir
parameters as input and skin equivalent to zero. The results are shown on Fig. 4.4 and
because time enters as a logarithmic term the is quite insensitive to it. Only after
500 hours of shutin the productivity index reaches 2.99 m3/d/bar (1.29 stb/d/psi) and in
this period the decay function of over time is nearly asymptotic. In forthcoming
wells in the field in case equation 4.8 is used to estimate the productivity index it is
recommended to apply shutin time of 500 hours to better approximate the ideal
productivity index.
Fig.4 4 – Transient productivity index evolution over time from equation 4.8.

43. 43
4.2 Skin Removal in Exploratory/Appraisal Wells
The well productivity for the same drawdown pressure can be increased by reducing
fluid viscosity, increasing the drainage area or reducing the skin. The skin reduction can
be attained by stimulation. The skin estimation is a nearly impossible task and the only
way to obtain a reliable value is by measuring and interpreting it from well testing
analysis during a radial flow and after ℎ is determined.
The skin is commonly referred to as “wellbore damage” and it is caused by a
modification in permeability zone around the wellbore. This modified zone can extend
from a few centimeters to several meters. During drilling, completion or workover some
materials such as mud filtrate, cement slurry, or clay particles can enter the formation
and reduce permeability, resulting in a positive skin. The permeability around the
wellbore could also be increased by acidizing or fracturing, resulting in a negative
skin.(1)
Analytically the skin is used to describe a deviation of the diffusivity equation from
these base conditions:(3)(5)
(i) Infinite acting and homogeneous reservoir;
(ii) Undipping;
(iii) Open hole;
(iv) Vertical well and penetrating the complete reservoir thickness;
The skin measured from well testing is the total skin ( ) and it includes:(3)(5)
= + + + …….………………………………….……(4.9)
where
= ℎ
=
= − ( ):
=
(

44. 44
The skin factor in exploratory/appraisal wells many times is a controversial issue in
terms of whether it should or not be altered by stimulation once it is determined from
well testing analysis. The following points should be analyzed holistically before
deciding for a stimulation operation in an exploratory/appraisal well:
(a) The decision to proceed with the stimulation should involve an economic
exercise and a compromise must be reached such that the cost of the operation is
compatible with the value of the information gained;
(b) In general the minimum skin attainable by acidification varies from -1 to -2 and
the minimum skin attainable by fracturing is -5;(5)
(c) The only skin that will be affected by stimulation is the mechanical skin ( );
thus it is crucial to firstly identify the type of skin which is present in the
reservoir and sometimes this is not a trivial task;
(d) Is the exploratory/appraisal well going to be used as a producer later on? In a
positive answer the stimulation operation might be worthwhile, since item (a) is
fulfilled.
(e) Analytical solutions regarding production (described below) should be
performed and verified if the solutions fulfill item (a). For instance, in case the
minimum economic value of the area is known and even with the lowest skin
achievable the area is still non-economical, the stimulation might not be
worthwhile.
(f) In case the well is going to be abandoned due to impossibility to be equipped to
production, should be investigated whether it is worthwhile to just rely on the
analytical solutions to estimate the new rates and avoid stimulation costs.
Two methods to estimate the production due to skin reduction are described and in the
analyzed scenario both represent transient solutions.
In oil reservoirs a simplified and practical analytical approximation of the new oil rate
achieved by reducing the skin factor independent of reservoir attributes is from:(5)
=
( )
( )
……….…………………………………………………..…(4.10)
where

45. 45
=
= ℎ
=
=
The graph on Fig. 4.5 displays an example of the expected rates reached through
reducing the skin factor of the reservoir using the rates from the third drawdown [380
m3/d (2390 stb/d)] and applying equation 4.10. The graph on Fig. 4.6 shows the
productivity index resulting from equation 4.5 according to the increased rates due to
reduced skin. The outcomes indicate that for the same reservoir pressure drawdown and
assuming the well is stimulated the maximum oil rate after a skin of -5 is obtained is
1435 m3 (9023 stb) resulting in a maximum productivity index of 10.5 m3/d/bar (4.5
stb/d/psi).
In this example, if the least economic value of the area is 11 3/ / , it might
not be worthwhile to invest in stimulation in an attempt to reduce the skin. However, it
might be worthwhile to try to reduce skin in case the gained information is applicable in
forthcoming wells. This example points out that the decision to proceed with the
stimulation have to involve an economic evaluation. The term 7 in equation 4.10 should
be calibrated according to the progress of knowledge about the field. It is recommended
to replace the term 7 by 8 in gas reservoirs as a first approach.(5)
The second method to estimate the transient productivity index in terms of reduced skin
is from equation 4.8 and it yields a maximum transient productivity index of 7.5
m3/d/bar (3.20 stb/d/psi) for a skin of -5 and applying the reservoir parameters
interpreted from the well test analysis for 500 hours of shutin (Fig. 4.7).

46. 46
Fig. 4.5 – Estimated oil rates due to skin reduction from equation 4.10.
Fig. 4.6 –Estimated productivity index due to skin reduction.

47. 47
Fig. 4.7 –Estimated productivity index due to skin reduction from equation 4.8.

48. 48
CHAPTER V
INTEGRATION OF CORES, WELL-LOGS AND WELL TEST
The integration of well-logs, cores and well test data essentially revolves around
geostatistical average permeability techniques in which cores permeability
measurements and logs permeability curves attempt to represent the well testing results
and allows predicting reservoir parameters in wells where no well testing is available. It
also permits not only to assist the well testing analysis, but also to assess the
consistency of the well testing interpretation. Moreover, it could guide the optimization
and positioning of the perforation depth and extension.
The main problem when comparing core or log permeability to well testing
permeability is the scale of the investigated zone. Well testing will provide an average
permeability according to the intrinsic reservoir diffusivity and it can reach dozens of
meters, or sometimes, thousands of meters of investigated area, while the core and log
permeability will provide permeability values reflecting just a few centimeters of the
area surrounding the well location. Strictly speaking, the rock volume probed by well
testing is significantly higher than the rock volume probed by cores. Based on this
uncertainty many experts accept an error of 10 times as a good approximation of core or
logs permeabilities compared to well testing permeability.
An useful formulae to calculate average permeability is the power average ( ) and it is
expressed by:(7)
= ∑
/
………...……………………………………………...(5.1)
where the alfa ( ) term will reflect the equivalent average. The arithmetic average
corresponds to = 1, the harmonic average corresponds to = −1, while the
geometric average corresponds to → 0.
Using only the core permeability data which lies between the perforation interval
equation 5.1 was used to calibrate the alfa exponent which match the well testing
average permeability of 6.2 md and then the well testing permeability-thickness product

49. 49
of 775 . . It resulted in = 0.518 and the calculated using only cores points out
that the reservoir average permeability stays between arithmetic and geometric.
Once the exponent is calibrated the same can be used to predict the ℎ in other
wells in the field or in other zones of the reservoir in the same well, since there are core
measurements available and it is assumed that the same is still valid after altering the
spatial position. In order to apply the described technique, a larger reservoir zone which
covers the entire oil leg was chosen and the same equation 5.1 with = 0.518 was used
to calculate for this larger reservoir oil section and it resulted in 10.22 md.
The next step was compute ℎ and because it is for the entire oil zone it is necessary to
take into account the net to gross value, so [ ℎ = Total thickness x NTG (0.94) x 10.22
md = 2443 md.m]. After calculating ℎ the final step was estimate the productivity
index from equation 4.8. Table 5.1 summarizes the reservoir parameters resolved by
using the fine-tuned as input. The predicted was computed applying skin factor
equal zero and shutin time of 500 hours.
Table 5.1 – Calibrated alfa exponent and predicted reservoir parameters for the entire oil
leg.
Assuming that the NTG value is correct, this procedure allows predicting that if a the
entire oil leg was perforated and put into production the reservoir permeability-
thickness product would be 2843 md.m and the transient productivity index would reach
10.63 m3/d/bar (4.61 stb/d/psi). The result states that, when compared to the well
testing, the reservoir transient productivity index would have been increased by a factor
of three. Another important point which deserves attention is that according to the
applied methodology the predicted average reservoir permeability of 10.22 md is greater
than the average reservoir permeability obtained from well testing analysis (6.2 md),
meaning that perhaps the chosen perforated interval might not have been optimized.

50. 50
The encountered above results are more optimistic than the results calculated from the
integral of the core derived permeability curve which shows ℎ = 1135.5 . for
the entire oil leg. In that case = 3.7 md and calculating the transient productivity index
from equation 4.8 results in 4.52 m3/d/bar (1.96 stb/d/psi).
When applying the ℎ = 953 . and = 7.6 md calculated from the integral of the
core derived permeability curve for the perforated interval in equation 4.8 the ideal
productivity index is 3.63 m3/d/bar (1.57 stb/d/psi) and represents and overestimation
error of 19% when compared to the well test analysis. The average reservoir
permeability calculated from the core derived permeability (7.6 md) is very close to the
6.2 md obtained from well testing. Since the permeability curve was calibrated only
with cores and the measured permeability in plugs is essentially matrix permeability, it
is reasonable to mention that the reservoir permeability is indeed matrix dependent and
there is no significant flow contribution coming from natural fractures. This last
supposition is supported by the lack of natural fractures observed on the image log
interpretation.

51. 51
CHAPTER VI
PVT MATCHING
In order to perform simulations an adequate Pressure-Volume-Temperature (PVT)
matching is crucial to describe a mathematical model for the reservoir fluid. A PVT
laboratory analysis from a reservoir fluid sample taken under reservoir conditions was
used as input for the PVT matching assignment. Fig. 6.1 shows the reservoir fluid phase
diagram and the critical pressure and temperature are 175.11 bar and 564.19o C,
respectively. It portrays that according to the reservoir pressure and temperature the
reservoir fluid is under-saturated oil and situated to the left-side of the critical point.
Assuming a pressure drop over a constant temperature the reservoir fluid reach bubble
point pressure at laboratory measured 397 bar and the initial reservoir pressure at a
reference depth is 620.5 bar.
Fig. 6.1 – Reservoir fluid phase-envelope.
Probably due to some content of non-hydrocarbon components in the reservoir fluid it
was not possible to match exactly the reservoir saturation pressure. The matching
saturation pressure calculated is 7 bar smaller than the reservoir saturation pressure

52. 52
measured in the laboratory. However, for the simulation purpose of the proposed
workflow, which is simulate a well testing and estimate reservoir parameters, the error
found will not affect the results because the flowing pressure will not reach the reservoir
bubble point pressure.
The physical properties and the PVT matching results of the reservoir fluid are
displayed on Fig. 6.2. Regarding the described fluid characteristics a black-oil
hydrodynamic model can well represent the reservoir fluid behavior in dynamic
simulations.
Fig 6.2 – PVT match of reservoir fluid physical properties. a) Oil relative volume; b)
Gas-oil ratio; c) Liquid density; d) Liquid viscosity.

53. 53
CHAPTER VII
STATIC RESERVOIR MODELING
Reservoir characterization is defined as the construction of a realistic three-dimensional
image of petrophysical properties to be used to predict reservoir performance.(6) In order
to build the 3-D reservoir model two steps must be accomplished, first build the static
reservoir model which should reflects the geology and then the dynamic reservoir
model which should reflects reservoir fluid behavior.
Facies modeling is the cornerstone of the static reservoir model and it means to populate
the model with discrete facies into cell grids. To build a realistic facies model the
conceptual geological processes during and after deposition should be understood and
the reservoir connectivity as well. The input to the model should honor the different
descriptive facies information and level of heterogeneity. The ultimate target is to build
a 3-D model that captures the reservoir architecture with flow units and barriers.
However, usually during the exploration phase the geological model is poorly
understood and due to the lack of drilled wells uncertainties are very high and in many
cases not even analogues can represent the reservoir properly. In order to overcome this
problem the proposed 3-D model will be based on hydraulic flow units. This approach
allows disconnecting the reservoir geological facies from the reservoir dynamic
behavior. Thus, the proposed 3-D model does not have as a target to represent reservoir
geology, but predict reservoir parameters such as reservoir permeability-thickness
product, average permeability, transient productivity index and mimic the observed well
testing flow behavior, i.e., pressure derivative and production logging.
Firstly a fine static model grid was build which is later upscaled to a reservoir
simulation model. The fine grid size chosen is X = 5m, Y = 5m and Z = 1m over an area
of 1500 x 1500 meters surrounding the exploratory well. The four large stratigraphic
units detected on the image log interpretation, Alpha, Beta, Theta and Gamma, were
mapped using the equivalent seismic reflectors. In order to allocate layers a follow top
method is applied using the mapped seismic surfaces as reference and the layers
thickness were defined as 1 meter. The created hydraulic flow units, core adjusted

54. 54
porosity and core derived permeability curves were upscaled into the 5 x 5 x 1 meter
blocks (Fig. 7.1). The upscaling process is assisted by quality checks using histograms
(Fig. 7.5). The fundamental idea is to preserve the property distribution after upscaling.
HFU was upscaled using the “most of” method, porosity using “arithmetic average”
method and permeability using “mid-point pic” method.
Fig. 7.1 – Upscaling results. Track 1 – HFU; Track 2 – Upscaled HFU; Track 3 – Core
adjusted porosity curve; Track 4 – Upscaled porosity; Track 5 – Core derived
permeability curve; Track 6 – Upscaled permeability.
Due to the very low fraction of HFU 5, only 0.4%, was decided to group it together with
HFU 4. Fig.7.2 shows the upscaled porosity versus upscaled permeability crossplot and
the equations resulting from the linear regressions are displayed in Table 7.1. Each
trend reflects a hydraulic flow unit and the correlation coefficient was improved
significantly when compared to the other presented Phi x K crossplots (Fig. 3.1 and Fig.
3.4).

55. 55
Fig 7.2 – Upper figure, upscaled porosity and permeability semi-log crossplot. Lower
figure, upscaled porosity and permeability semi-log crossplot highlighting the four
trends used to generate the linear regressions with the specific HFU.
Table 7.1 – Linear regression equations from upscaled porosity and permeability.

56. 56
7.1 Populating the Static Model
Once the grid was defined the next step was populate it. The cells which compose the
grid have the size of 5 x 5 x 1 meter and a hydraulic flow unit, porosity, permeability
and water saturation value were assigned to each cell.
The first step to assign discrete values to the model was distributing spatially the
hydraulic flow units and the Sequential Gaussian Simulation algorithm was chosen. It is
a stochastic method based on kriging and it honors variograms, input trends and
specified fractions. No trend was employed, however the vertical HFU fractions for
individual stratigraphic zones were respected.
The variograms aims at capturing the regional organization of data which is not purely
random. Four different models were created using four different major equal minor
ranges, i.e., 10, 100, 500 and 1000 meters and named Models A, B, C and D,
respectively. The additional variogram parameters were preserved as constant for all
models and the established parameters are listed below:
 Variogram type: Spherical;
 Nugget effect = 0.1;
 Sill = 1;
 Major range = Minor range = 10, 100, 500 and 1000 meters, respectively,
Models A, B, C and D.
 Vertical range: Zone Alpha: 9 meters; Zone Beta: 14 meters; Zone Theta: 20
meters; Zone Gamma: No variogram adjusted because only HFU 1 is present.
 Fractions:
Zone Alpha: HFU 1= 51%; HFU 2= 19%; HFU 3= 19%; HFU 4= 11%;
Zone Beta: HFU 1= 2%; HFU 2= 25%; HFU 3 = 67%; HFU 4= 6%;
Zone Theta: HFU 1= 55%; HFU 2= 29%; HFU 3= 15%; HFU 4= 1%;
Zone Gamma: HFU 1= 100%;
The second step was distribute porosities and in order to undertake that initially the
upscaled porosity data for each HFU was analyzed using frequency histograms and a

57. 57
probability distribution function was adjusted. For HFU 2, 3, 4 and 5 a Gaussian
distribution function fitted the data and for HFU 1 a log-normal distribution function
was the best option (Fig 7.3). Then a Gaussian random function simulation algorithm
conditioned to hydraulic flow units was used to distribute porosities spatially and the
fitted probability distribution functions were applied as input data.
Fig. 7.3 – Fitted probability distribution function for upscaled porosity for individual
HFU.
In order to assign permeability value to each cell the linear regressions of permeability
as a function of porosity were employed (Table 7.1). The final step of the petrophysical
modeling was allocate water saturation to the blocks. Since membrane capillary
pressure measurements from cores are available a dimensionless J-function versus water
saturation could be computed.

58. 58
The dimensionless Leverett J-function in oilfield units is defined as:(5)(8)
( ) = 0.21645
ɸ
……..……………………………………….…(7.1)
and for a set of samples a least squares regression was built using J values as the
independent variable. Then a power law equation correlation of the form:
= ( ) ……...…………………………………………………………..(7.2)
was adjusted in order to compute and index. In oilfield units converted capillary
pressures to the height domain can be written as:
=
( )
……..…………………………………………………………(7.3)
Finally by combining equations 7.1, 7.2 and 7.3 water saturation as a function of height
above the free water level ( (ℎ)) can be calculated in oilfield units from:
(ℎ) =
. ( )
ɸ
……..…………………………....(7.4)
Fig. 7.4 illustrates the J-curves computed for the hydraulic flow units. Due to J-curve
similarities between HFU 1 and HFU 2 both families were represented by the same
function. No threshold pressure was measured in the capillary pressure experiments;
therefore, the oil water contact and the free water level share the same depth. This
indicates that the rock wettability is likely to be mix. In order to compute the water
saturation the permeability and porosity factors applied in equation 7.4 are the discrete
values defined for each cell. The water and oil densities ( , ), wettability contact
angle ( ) and interfacial tension ( ) were kept as constant throughout the model. The
water saturation curve computed from petrophysical interpretation was not only used as
quality control to verify the consistency of the saturation height function approach, but
also to adequately adjust it in order to better represent the irreducible water saturation.
In order to preserve the petrophysical properties distribution while upscaling and
modeling, in all the steps the results were quality check using histograms and Fig. 7.5
displays the Model C histogram distribution for HFU, porosity and permeability,

59. 59
defined from cores and logs, calculated by the upscaling procedure and assigned for the
static 3-D model. Important to notice that the distribution of the initial histograms which
are derived from cores are preserved throughout the workflow, meaning that laboratory
core measurements are the foundation of the static modeling.
Fig. 7.4 - J-function curves calculated for the HFU.
Fig. 7.6 highlights the contrast of the four models created by different X and Y
variogram ranges. It is possible to recognize that larger the range, larger the lateral
continuity of the property and smaller the anisotropy. Fig. 7.7 exemplifies the HFU,
porosity, permeability and water saturation for the 3-D Model C, which is based on 500
meters major = minor variogram range. The created 3-D static models seem to repeat
depositional sequences and it reminds depositional environments based on cycles. When
the geological 3-D model is build it is recommended to compare it to the static models
based on HFU in order to identify similarities.

60. 60
Fig. 7.5 – Model C quality check using histograms. Each histogram shows the property
defined from cores and well-logs, upscaling procedure and assigned for the static 3-D
model. Notice that the property distribution is maintained throughout the process.

61. 61
Fig. 7.6 – East-West models cross-section cutting through the well. Four different
models according to the specified major = minor variogram range. The property shown
is HFU with no vertical exaggeration.
Fig. 7.7 – 3-D petrophysical properties of static Model C. No vertical exaggeration.

62. 62
CHAPTER VIII
DYNAMIC RESERVOIR MODELING
Reservoir simulation based on dynamic reservoir model uses finite difference method to
simulate the flow of fluid within the reservoir. In order to reduce the computational
time, firstly the 3-D static models were upscaled to a coarser grid with 20 x 20 x 5
meters cell size. To quality check the upscaling process, the hydrocarbon volume in
place of the static and dynamic models were compared (CHAPTER IX Volume of
Hydrocarbon). Porosity and water saturation were upscaled by arithmetic averaging –
volume weighted technique and permeability was upscaled by directional averaging
power-arithmetic technique using the alfa exponent of 0.518 calculated from cores,
well- logs and well testing integration.
Through the analyses of the production logging data 5 meters is the smallest vertical
size which a change on the slope of the cumulative production curve can be identified
(Fig. 2.6). As a result, 5 meters thickness for vertical layers in the dynamic model to
represent fluid flow seems to be a good approximation. It is interesting to highlight that
as reported by the PLT in 5 vertical meters the flow behaves homogeneously, despite
the fact that the reservoir is organized by many thin layers, some of with just a few
centimeter thickness as interpreted on the image logs and also showing a significant
contrast in intrinsic permeability as observed in permeability core measurements.
According to the reservoir fluid properties a black-oil simulation model can well
represent the reservoir dynamic behavior. The water-oil relative permeability curves for
the dynamic model were adjusted using Corey’s method(4) based on a whole core
laboratory relative permeability measurement and Fig. 8.1 displays the curves match
and the defined Corey’s exponents. The same relative permeability curve was applied to
all hydraulic flow units. According to Corey’s exponents the reservoir has a mixed
wettability and by analyzing the graph it is recognized that oil relative permeability is
almost zero for water saturation higher than 0.7 (Fig. 8.1); thus 30% of the hydrocarbon
accounts as residual oil saturation.

63. 63
It is recognized that, ideally, each HFU should be represented by petrophysical
properties, such as, a relative permeability curve. However, this detailed information is
still not available due to the fact that this study is based on an exploratory area. Hence,
it is recommended to selective collect cores in the forthcoming wells to perform relative
permeability measurements to represent all HFU groups and improve the dynamic fluid
behavior characterization.
Fig. 8.1 – Match between core relative permeability and Corey’s method.
Fig. 8.2, 8.3, 8.4 and 8.5 depict the four dynamic models upscaled from the four static
Models A, B, C and D, respectively. In order to perform the dynamic simulations the
perforation interval was set to represent the same perforation interval as executed in the
well testing. The prediction depletion strategy was also set to correspond to the rates,
drawdown and buildup period observed in the well testing. The only alteration is that
the third BU period was increased from 43 hours to 216 hours in order to represent
better the reservoir pressure recovery. The vertical permeability was adjusted as 10% of
the horizontal permeability.
The simulated bottom hole pressure and rates versus time for the four dynamic models
is shown on Fig. 8.6. The pressure drop is the largest in Model A and the lowest in
Model D. It means that the lateral continuity of the hydraulic flow units plays the main
role in order to control the reservoir pressure support, giving an enhanced support for
models based on a larger major equal minor variogram distance.

64. 64
Fig. 8.2 – Dynamic reservoir petrophysical parameters upscaled from static Model A.

65. 65
Fig. 8.3 – Dynamic reservoir petrophysical parameters upscaled from static Model B.

66. 66
Fig. 8.4 – Dynamic reservoir petrophysical parameters upscaled from static Model C.

67. 67
Fig. 8.5 – Dynamic reservoir petrophysical parameters upscaled from static Model D.

68. 68
Fig. 8.6 – Simulated bottom hole pressure and oil rate versus time for the four dynamic
Models.
8.1. Well Test Analysis of Simulations
The simulated bottom hole pressure and rates were analyzed following the same
procedure as a physical well testing interpretation. Fig. 8.7 displays the pressure
derivative for the final BU for the simulated models. Model A derivative shows a half-
wave cycle pattern, moving from steep upturn followed by a short stabilization, then a
downturn. Model B derivative shows a complete wave cycle pattern, however the
wavelength is larger and the amplitude is smaller when compared to Model A. Model C
derivative displays first a stabilization then an upturn and a tendency for a second
stabilization in late time. Model D shows a stabilization in early and middle time and a
slight downturn tendency in the end of the late time. The model’s derivative behavior
also reflects the spatial distribution of the HFU, decreasing anisotropy from Model A to
Model D.
Amid all models’ derivative, the one which represents the most similar shape when
compared to the physical well testing derivative is Model C; consequently, Model C
was chosen to perform a derivative match interpretation. Fig. 8.8 portrays the derivative

69. 69
match and a linear composite derivative model was selected, which is the same
derivative model observed in the well testing. The permeability-thickness product
calculated is 300 md.m, resulting in an average reservoir permeability of 2.4 md. The
same way as observed in the well testing the upturn in the derivative indicates a second
region showing poorer reservoir properties when compared to the first region and the
second region is located 55 meters away from the wellbore. The superposition and data
plot match are shown on Fig. 8.9.
Fig. 8.7 – Third BU pressure derivative comparison between the four dynamic models.
The wellbore storage and skin obtained by the derivative of the simulated pressure data
should be neglected, because a numerical simulator cannot handle the case of nonideal
wellbore storage and any proposal to analyze early time response using derivative
methods have to be regarded with some skepticism.(10) Another point is that the
horizontal grid size is equivalent to a square of 20 meter side and the skin occurs near
the wellbore, as a result the skin factor cannot be measured with confidence.

70. 70
Fig. 8.8 – BU derivative analysis and the matching parameters for simulated Model C
bottom hole pressure.
Fig. 8.9 – Model C pressure analysis match of superposition plot on the left and data
plot on the right
10-1
100
101
102
10-2
10-1
Delta-T (hr)
DP&DERIVATIVE(BARS/M3/D)
02-JAN-2015 02:00
04-JAN-2015 07:00
05-JAN-2015 05:00
Base Case
Base Case
2015/01/05-0500 : OIL
Linear-Composite 2-Zone
** Simulation Data **
well. storage = .9033E-03 M3/BAR
skin = -1.6000
permeability = 2.4000 MD
Areal Ky/Kx = 1.0000
X-Interface(1) = 55.000 METRE
Mob.ratio(1) = 0.40000
Stor.ratio(1) = 0.40000
Perm-Thickness = 300.00 MD-METRE
Initial Press. = 615.957 BARS
Static-Data and Constants
Volume-Factor = 1.600 vol/vol
Thickness = 125.0 METRE
Viscosity = 1.030 CP
Total Compress = .8215E-04 1/BAR
Rate = 380.0 M3/D
Storivity = 0.001232 METRE/BAR
Diffusivity = 83.99 METRE^2/HR
Gauge Depth = N/A METRE
Perf. Depth = N/A METRE

71. 71
Fig. 8.10 depicts an E-W cross-section of all models cutting the wellbore at the end of
the third drawdown timestep. The pressure disturbance of Model A shows an oval
shape, with the pressure disturbance moving upward and downward around the
wellbore. Models B, C and D the pressure disturbance tends to move more laterally,
following the higher permeability layers and in Model D reaches the largest investigated
zone. This distinct behavior can also be explained by the different spatial distribution of
the HFU, with the continuity increasing from Model A to D.
The behavior of the pressure disturbance in Model C is consistent with the large scale
very low reservoir anisotropy ( = 3.8 ∗ 10 ) , because the pressure disturbance is
essentially distributed inside a large stratigraphic unit, mainly unit Beta, and it has not
any preference to move to other stratigraphic units after moving a few meters from the
wellbore.
In order to verify the consistency of the dynamic Model C the PLT curve acquired
during the second drawdown period of the well testing was compared to the simulated
PLT and Fig. 8.11 displays a well cross-section showing the very good agreement
between the acquired and simulated PLT curves.

72. 72
Fig. 8.10 – Models cross-section displaying pressure disturbance at the final timestep of
the third buildup.

73. 73
Fig. 8.11 – Model C well cross-section. Notice the good agreement between the well
testing cumulative PLT and the simulated cumulative PLT. Track 1 – Porosity; Track 2
– Permeability in logarithmic scale; Track 3 – Water saturation; Track 4 – Completion
and perforation interval; Track 5 – Well testing cumulative PLT; Track 6 – Simulated
cumulative PLT for Model C.
8.2 . Numerical Productivity Index
The numerical productivity index in a Cartesian grid is a function of the connection
factor ( ) and can be represented in oilfield units from:(5)(9)
=
.
……...…………………………………………………..(8.1)

74. 74
for a vertical well the permeability used is the geometric mean of X and Y direction
permeabilities and is the pressure equivalent radius of the grid block and for an
anisotropic reservoir in a rectangular grid is given by:(5)(9)
= 0.28
( ) ∆ ( ) ∆
( ) ( )
……..…………………………………(8.2)
Lastly, the numerical productivity index attached to one connection ( ) can be
computed from:
= ………..…………………………………………………....(8.3)
and a summation of all the connections belonging to the producing interval of the well
will give the total numerical productivity index ( ). To simplify the
depends on the connection factor, which depends on the geometry of the grid block, the
well bore radius and rock permeability.
It is usually found that the numerical productivity index is higher than the natural
productivity index. Fig. 8.12 illustrates the numerical productivity index calculated by
the Model C simulation resulting in 3.44 m3/d/bar (1.49 stb/d/psi) for the final period of
the third drawdown.
A second approach to calculate the is using the 9-point pressure ( ), which is
the average pressure given by 9 cells around the wellbore and can be calculated by:
=
( )
……...……………………………………………………(8.4)
Fig. 8.13 shows the numerical calculated for Model C using equation 8.4 resulting in
3.12 m3
/d/bar (1.35 stb/d/psi) at the final period of the third BU and it can be compared
to the obtained by the well testing [2.95 m3/d/bar (1.27 stb/d/psi)], representing
an overestimation error of only 5%.

75. 75
Fig. 8.12 – Simulated numerical productivity index over time for Model C during the
final buildup.
Fig. 8.13 – Simulated numerical productivity index over time for Model C based on 9-
point pressure for the three flowing periods.

76. 76
8.3 . Productivity of the Entire Oil Leg
A scenario was simulated using Model C with a larger perforation zone covering the
entire oil section and changing the control from rate to bottom hole pressure. The
simulated period was set for 20 days, 10 days of drawdown and 10 days of buildup and
applying a delta pressure equivalent to 200 bar. In this scenario the 9-point pressure
numerical productivity index reaches 4.25 m3/d/bar (1.84 stb/d/psi) at the end of the
drawdown period, representing an increase of 27% when compared to the previous
simulation for the perforated interval. Fig. 8.14 displays the pressure behavior, rate and
while Fig. 8.15 illustrates the derivative pressure analysis and a linear composite
model was used to match the simulated data. The permeability-thickness product
calculated is 403.2 md.m with an average reservoir permeability of 1.68 md. According
to the simulated PLT a cumulative rate increase of 27% should be expected when
comparing to the well testing PLT and mainly the upper and lower part of the reservoir
are not contributing to production. (Fig. 8.16)
Fig. 8.14 – Model C bottom hole pressure, rate and 9-point pressure numerical
productivity index for 20 days of simulation for the entire oil section. Control set for
bottom hole pressure and equivalent to 200 bar.

77. 77
Fig 8.15 – Derivative pressure analysis for the entire oil zone simulation.
100
101
102
10-2
10-1
Delta-T (hr)
DP&DERIVATIVE(BARS/M3/D)
2015/01/10-1000 : OIL
Linear-Composite 2-Zone
** Simulation Data **
well. storage = .9085E-05 M3/BAR
skin = -1.8020
permeability = 1.6800 MD
Areal Ky/Kx = 1.0000
X-Interface(1) = 40.052 METRE
Mob.ratio(1) = 0.40000
Stor.ratio(1) = 0.40000
Perm-Thickness = 403.20 MD-METRE
Initial Press. = 609.167 BARS
Smoothing Coef = 0.,0.
Static-Data and Constants
Volume-Factor = 1.600 vol/vol
Thickness = 240.0 METRE
Viscosity = 1.030 CP
Total Compress = .8215E-04 1/BAR
Rate = 411.8 M3/D
Storivity = 0.002366 METRE/BAR
Diffusivity = 58.80 METRE^2/HR
Gauge Depth = N/A METRE
Perf. Depth = N/A METRE
Datum Depth = N/A METRE

78. 78
Fig. 8.16 – Model C well cross-section; PLT response for the entire oil section. An
increase of 27% should be expected when comparing well testing cumulative PLT with
the simulated PLT and the upper and lower part of the reservoir are not contributing to
production. Track 1 – Porosity; Track 2 – Permeability in logarithmic scale; Track 3 –
Water Saturation; Track 4 – Completion and perforation interval; Track 5 – Well testing
cumulative PLT; Track 6 – Simulated PLT
8.4 Horizontal Well Simulation
In order to investigate the performance a horizontal well scenario was simulated using
Model C and it was decided to project the well in the highest permeability layers which
were the most affected by pressure drop according to Fig. 8.10. The well horizontal
section was set for 1200 meters with 89o of inclination and the entire horizontal interval

79. 79
completed for production. The simulated period of time was adjusted to 10 days of
drawdown and 35 days of buildup.
Fig. 8.17 shows the simulated bottom hole pressure, rates and numerical productivity
index which reached 6.66 m3/d/bar (2.88 stb/d/psi) at the end of the buildup.
Fig. 8.17 – Bottom hole pressure, rate and numerical productivity index for a horizontal
well simulation in Model C.
The derivative analysis of the horizontal well pressure simulation shows at early times a
stabilization followed by an upturn, then a second stabilization. The second derivative
stabilization is at a level twice the first. A horizontal well model was used to match the
derivative behavior and Fig. 8.18 displays the results. The horizontal well is not
centered in the zone thickness and it is closer to the upper boundary than the lower
boundary of the reservoir. The first stabilization reflects a vertical radial flow and
because only the upper limit of the reservoir is reached a hemi-radial flow regime
develops afterwards resulting in a ¼ slope in the derivative. At late time the flow lines
converge from all reservoirs directions toward the well producing a horizontal radial
flow regime. The second derivative stabilization corresponds to the infinite acting radial

80. 80
flow and gives the horizontal ℎ = 36.9 . and = 0.61 md. Fig 8.19 shows the
superposition and data plot match.
Fig. 8.18 – Derivative pressure match for horizontal well simulation.
100
101
102
103
10-2
10-1
Delta-T (hr)
DP&DERIVATIVE(BARS/M3/D)
2015/01/10-1000 : OIL
Horizontal Well and Homogeneous Reservoir
** Simulation Data **
well. storage = .1078E-04 M3/BAR
Skin(mech.) = -3.2434
permeability = 0.61590 MD
Areal Ky/Kx = 1.0000
Kv/Kh = 0.12166
Drain Length/2 = 160.00 METRE
Zw/H = 0.30000
Skin(Global) = -5.5762
Skin(geom.) = -3.8327
Perm-Thickness = 36.954 MD-METRE
Initial Press. = 616.031 BARS
Smoothing Coef = 0.,0.
Static-Data and Constants
Volume-Factor = 1.600 vol/vol
Thickness = 60.00 METRE
Viscosity = 0.1030 CP
Total Compress = .8215E-04 1/BAR
Rate = 876.2 M3/D
Storivity = 0.0005914 METRE/BAR
Diffusivity = 215.5 METRE^2/HR
Gauge Depth = N/A METRE
Perf. Depth = N/A METRE
Datum Depth = N/A METRE
Analysis-Data ID: GAU004
Based on Gauge ID: GAU004

81. 81
Fig. 8.19 – Match of Superposition plot on the left and data plot on the right for the
horizontal well simulation.
As discussed previously the early time response should be treated with some skepticism
due to the fact that a numerical simulator cannot handle the case of nonideal wellbore
storage and it is a coarse grid, thus the near wellbore skin cannot be measured with
confidence. However, for horizontal wells the infinite late time horizontal radial flow is
used to compute the geometrical skin, therefore the value of = −3.83 can be
assumed as accurate.
The simulated PLT for the horizontal well (Fig. 8.20) reveals that only 30% of the
horizontal length is effectively contributing to production and Fig. 8.21 displays the
model cross-sections showing the well position and the pressure disturbance at the final
drawdown timestep. The pressure disturbance has a tendency to concentrate in the high
permeability layers and the pressure drop spreads mainly laterally.
The under-performance of the horizontal well can be attributed to the low vertical
permeability and low flow contribution over the total horizontal length. A large number
of cells are not contributing to flow.

82. 82
Fig. 8.20 – Simulated PLT for a horizontal well. Only 30% of the total horizontal length
is effectively contributing to production.

83. 83
Fig. 8.21 – Model cross-section showing pressure as property and the horizontal well at
the final drawdown timestep.

84. 84
CHAPTER IX
VOLUME OF HYDROCARBON
Original oil in place ( ) and original gas in place ( ) refers to the total volume
of hydrocarbon stored in a reservoir prior to production. An approach to quality check
the upscaling process from static model to dynamic model is comparing the volume of
hydrocarbon in place before and after upscaling. In oilfield units the original oil volume
in place ( can be computed from:
=
ɸ ( )
..……………………………………………..(9.1)
and Table 9.1 shows the results calculated for Model C. This method calculates the
hydrocarbon volume in each cell and the results shown are the sum for all cells. A
decrease of 6% in OOIP from static model to dynamic model is observed and according
to this outcome it is acknowledged that after upscaling the dynamic grid retained the
spatial petrophysical properties distribution of the static grid.
Table 9.1 – Original oil in place calculation for Static and Dynamic Models C.
Usually in E&P Hydrocarbon Companies the Exploration and Production departments
calculate the volume of hydrocarbon in place in different manners. Once a new
accumulation is discovered the Exploration department compute OOIP using the
petrophysical parameters obtained by well-log interpretation which were acquired from
the drilled well responsible for the finding and then run a Monte Carlo simulation to

85. 85
create a probability distribution of hydrocarbon in place, whereas the Production
department will later compute from a 3-D model using cutoffs and in many cases
there is a substantial volume contrast when comparing both calculated volumes in place.
The early reservoir modeling from the outset of the exploration stage allows the
Exploration department to compute in a similar way as the Production department
and the contrast between hydrocarbon volumes could be reduced.
9.1 Estimation of Dynamic Petrophysical Cutoff and Recovery Factor
Due to the fact that some cells are showing none to very little contribution to flow the
simulated PLT can be used to estimate the reservoir petrophysical cutoff that influence
the fluid behavior according to the set drawdown pressure. The total oil section PLT
simulation (Fig. 8.16) and the horizontal well PLT simulation (Fig. 8.20) were analyzed
and Table 9.2 displays the statistics of oil saturation, porosity and permeability for the
cells which are not contributing or contributing very little to flow. The maximum delta
pressure drawdown implemented was 200 bar and it should be pointed out that this
analysis is valid only for this drawdown because the flow potential contribution is
dependent on the delta pressure.
Table 9.2 – Statistical analysis of petrophysical properties of cells not contributing or
contributing very little to flow.
The maximum permeability value (K= 2.14 ) was applied as cutoff in Model C for
computing and Table 9.3 displays the results. According to the relative
permeability curve (Fig. 8.1) the residual oil saturation is equivalent to 30%,
consequently, the moveable oil is 70%. Applying this factor over the ratio between the
volume of oil in place before cutoff and after cutoff and assuming that the result might

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be the recovery volume, an approximation of the possible Recovery Factor ( ) for the
field would be:
= 0.35 ∗ 0.7 = 0.25 ……..……………………………………...…(9.1)
or 25%.
Table 9.3 – Hydrocarbon volume after applying cutoff of 2.14 md in static Model C.
It is important to mention that in order to have a good approximation of the oil recovery
factor is necessary to have knowledge of the driving mechanisms that control the
behavior of fluid within the reservoir. The overall performance of reservoir is largely
determined by the nature of the energy, i.e., driving mechanism, available for moving
the oil to the wellbore and the final recovery factor will be also strongly dependent on
secondary and tertiary recovery methodologies later employed in the field life-time.
This = 25% should be seem as a first rough estimation of the recovery factor for
the field.
It should be pointed out that no porosity cutoff for the studied reservoir should be
establish due to the fact that none coherent correlation between porosity and dynamic
reservoir characteristics was encountered. Cells with good permeability exhibit low
porosity values and vice-versa. Therefore, in case a porosity cutoff is implemented it
will cut producible payzone intervals.

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CHAPTER X
SUMMARY OF THE RESERVOIR PROPERTIES PREDICTION
Table 10.1 displays the reservoir parameters obtained from well testing interpretation
and Table 10.2 displays the forecasted reservoir parameters using the different methods
described in this thesis.
Table 10.1 – Reservoir properties obtained from well test analysis.
Table 10.2 – Forecasted reservoir properties from different methods.

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Regarding reservoir permeability-thickness product and average reservoir permeability
predictions for the perforated interval the integral of the core derived permeability
shows the closest approximation when compared to the well testing results and the
predicted parameters are overestimated by 19%. In terms of productivity index the
Model C simulation using the 9-point pressure and equation 8.4 approach displays the
most similar result when compared to the well testing and the foreseen is
overestimated by only 5%.
In case the entire oil leg was tested the power averaging technique forecasted the most
optimistic results whereas the permeability integral and simulation the most pessimistic.
Since the permeability integral method showed the best approximation for ℎ and for
the perforated interval, it seems reasonable to assume that in case the entire oil zone was
tested the best estimation should be the permeability curve integral method as well.
Likewise, the best estimation for the productivity index would be the 9-point pressure
approach from equation 8.4.
According to the simulation and well test analysis results, despite increasing the
production by a factor of two the drilling of a horizontal well is discouraged because it
is considered a low performance for a horizontal well length of 1200 meters.

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CHAPTER XI
CONCLUSIONS AND RECOMMENDATIONS
The following conclusions and recommendations are derived from this study:
1. The early assessment and integration of all available information from different
sources and from the outset of the exploration phase allows to identify uncertainties,
forecast reservoir properties and to start the 3-D reservoir model.
2. It also permits an initial understanding of the reservoir and could give managers a
broader perspective of the exploratory area or new field and support taking decisions in
short notice. Moreover, it can optimize the design of well testing and not only assist, but
also validate well test interpretations.
3. Building the static and dynamic models using hydraulic flow unit concept is the key
to overcome the lack of geological knowledge typically present during the exploratory
stage. The proposed workflow decouples the reservoir fluid behavior from the
geological facies and provides an alternative manner to create reservoir models when
the geological framework is still poorly understood.
4. Laboratory core measurements are the foundation of the created 3-D models and the
early data integration could save company’s investment and time in this regard, because
it can avoid unrestrictive and unsupervised core laboratory analysis. The proposed
workflow allows to promptly identify which cores should be selected and, in
forthcoming wells, in which interval cores have to be collected to perform laboratory
analysis in order to fully represent the reservoir with petrophysical attributes. It also
simplifies the identification of the relevant seismic reflectors that should be mapped.
5. The described workflow permits to work with highly heterogeneous reservoirs and it
succeeds in building a derived core permeability curve which honors low and high
permeability core data in just a few centimeters of vertical scale and reflects the
reservoir flow potential. The workflow also permits to significantly improve the
correlation factor of permeability as a function of porosity through the upscaling steps.

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6. Model C pressure derivative based on an X = Y = 500 meters range variogram
exhibited the most similar pattern compared to the well testing analysis.
7. The results achieved for the perforated interval using well-logs and cores as source
and permeability curve integral as method predicted reservoir permeability and
permeability-thickness product by an overestimation error of 19%. Based on
simulations and 9-point pressure the productivity index could be foreseen by an
overestimation error of 5%.
8. In case the entire oil leg was tested the most likely ℎ , and would be
1135.5 . , 3.7 md and 4.52 m3/d/bar, respectively.
9. Considering a scenario of 200 bar of producing drawdown, the first estimation of the
dynamic cutoff for the discovered field is 2.14 md and the first rough approximation of
the recovery factor 25%. Porosity should not be used as cutoff because of the lack of
correlation between porosity and dynamic reservoir characteristics.
10. For the dynamic model cells with 5 meters of vertical thickness seems to be a good
estimation.
11. The average permeability of the carbonate reservoir examined lies between
arithmetic and geometric and it is matrix dependent; no natural fractures or dissolution
channels are observed. Thus the presented workflow should be replicated in similar
cases. In upcoming studies it is recommended to first assess the type of reservoir
permeability present and in case permeability distinct than the analyzed is identified, the
workflow has to be adapted to take into account the discrepancies.
12. In forthcoming wells in the field in case equation 4.8 ( ) is used to estimate the
productivity index it is recommended to extend the results until 500 hours of shutin to
better approximate the ideal productivity index.
13. According to the simulations and well test analysis of the simulated pressure,
despite doubling the productivity the drilling of a horizontal well is discourage because
of the low performance. However, in case a better estimation of macro is obtained,
new simulations should be performed and the horizontal well scenario reassessed.

91. 91
14. A decent core statistical representation of the reservoir is crucial to develop the
suggested workflow and early reservoir modeling. In reservoirs with similar features a
minimum of one sidewall core each 0.5 meter is recommended.
15. The HFU classification shows a good agreement with pore-throat size distribution
from mercury injection. It means that the HFU methodology indeed separate families
according to its flow potential. Apart from acoustic image no other well-log response
shows relationship with hydraulic flow units.
16. The anisotropy ratio of 0.1 seems to be a reasonable estimation, however, in the
studied case the reservoir vertical permeability is an uncertain factor and it is suggested
that mini-DSTs after sampling should be acquired in the next wells in order to confirm
or better approximate .
17. When the reservoir 3-D geological model is built, it should be compared to the HFU
based model in order to identify similarities and the core descriptions and rock facies
should also be compared to the defined hydraulic flow units.
18. The early reservoir modeling from the onset of the exploration phase allows the
Exploration department compute in a similar way which is calculated by the
Production department and the contrast between hydrocarbon volume calculation
usually found could be reduced. Furthermore, the early modeling permits to easier
transfer a new discovery from the Exploration department to the Production department
simply due to the fact that the static and dynamic modelings have already been started
and the main uncertainties recognized and addressed.
19. The decision to remove skin in exploratory/appraisal wells after well testing is an
economical exercise and the described analytical solutions should be included in the
assessment plan. It might save company’s efforts and investments. Particularly in the
offshore situation, the cost associated with rig time is very high and the adage is “time is
money”. What is always recommended to be investigated are strategies to avoid and
minimize formation damage in appraisal and development wells after using the
exploratory well as a learning case and apply refining drilling and completion actions.

92. 92
20. In general the minimum skin attainable by acidification varies from -1 to -2 and the
minimum skin attainable by fracturing is -5. In the analyzed reservoir if stimulation is
carried out and a skin of -2 is achieved the estimated transient productivity index would
be around 4 m3/d/bar and in case a skin of -5 is achieved it might stay between the
range of 7.5 to 10.5 m3/d/bar.
21. For future developments another option that should be implemented in the proposed
workflow in order to create and populate the 3-D model with hydraulic flow units is
applying co-kriging with seismic as a second correlation factor. In that case the 3-D
model might also be useful to define new well prospects.
22. Geological facies and static geological models are subjected to personal
interpretations and can widely vary according to the interpreter point of view and degree
of information available. An advantage of the proposed modeling approach is that it is
pragmatic and purely based on evidences, i.e., well testing, logs, cores and fluid
laboratory measurements. Therefore, it intends to reduce the exposition to individual
interpretations. In other words, geological models based on rock facies are customarily
built with little data and a lot of motivation and one of the main inputs, the creativity
will always have to be exposed to critics and updates, meaning that a defined “Facies
A”, in the future might become “Facies B” or “Facies C”. Whereas figures can be
processed and manipulated analytically or numerically, the raw input data will remain
unalterable, meaning that a defined “HFU A”, in the future will continue to be “HFU
A”. Moreover, the comparison and correlation between different wells in the field is
much more straightforward using the HFU based model.
23. Personal interpretations are important to create different scenarios and they should
be further investigated and assimilated, but hydrocarbon exploration decisions and main
conclusions should be taken grounded on facts.
24. Finally, this thesis underline the importance to involve specialists with specific and
diversified backgrounds since the beginning of hydrocarbon exploration, for instance,
reservoir geologists, reservoir engineers and petrophysicists; essentially when exploring
complex formations, low-permeability and/or unconventional reservoirs.

93. 93
REFERENCES
1. Ahmed, T., Mckinney Paul D.; “Advanced Reservoir Engineering”, Elsevier
(2015).
2. Amaefule, J.O., Altunbay, M. Tiab, D.,Kersey, D. G., ad Keelan, D.K.:
“Enhanced Reseservoir Description Using Core and Log Data to Identify
Hydraulic Flow Units and Predict Permeability in Uncored Intervals or Wells,”
SPE 26436, (68th Annual Technical Conference, Houston (Oct-1993)).
3. Bouldet, D.: “Well Test Analysis: The Use of Advanced Interpretation Models”,
Elsevier, Amsterdam, The Netherlands (2002).
4. Corey, A. T.; “The Interpretation Between Gas and Oil Relative Permeabilities,”
SPEJ. (April-1984).
5. IFP School - Institut Français du Pétrole - Masters of Reservoir Geoscience and
Engineering notes; 2014/2015.
6. Lucia, F. Jerry; “Carbonate Reservoir Characterization – An Integrated
Approach”, Second Edition, Springer, 2007.
7. Noetinger B., Haas A.: “Permeability Averaging for well tests in 3D Stochastic
Reservoir Models”, SPE paper 36653-MS, SPE Annual Technical Conference
and Exhibition, Denver, Colorado. (1996).
8. Schön, Juergen; “Physical Properties of Rocks”, Elsevier, 2011.
9. Schlumberger; “Eclipse Technical Description Manual”, Version 2012.1.
10. Stewart, G.: “Well Test Design and Analysis,” Penn Well Corporation, Tulsa,
Oklahoma (2011).
11. Tiab D., Donaldson C. E.: Petrophysics: Theory and Practice of Measuring
Reservoir Rock and Fluid Transport Properties,” Elsevier, Gulf Professional
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12. Wylie, M.R.J. and Gardner, G.H.F.: “The Generalized Kozeny-Carmen
Equation,” World Oil (March-April 1958).

94. 94
NOMENCLATURE
= reservoir area
= power average permeability
= alfa factor
= reservoir anisotropy
= surface area per unit grain volume
= formation volume factor
= initial formation volume factor
= total compressibility
= connection factor
= flow efficiency
= shape factor
= flow zone indicator
= reservoir thickness
ℎ = height above the free water level
= Leverett J-function
= average reservoir permeability
= log permeability curve calibrated with core data
ℎ = permeability-thickness product
= relative permeability
= vertical permeability
= horizontal permeability
= permeability in the X direction
= permeability in the Y direction
= mobility ratio
= corey exponent for water relative permeability curve
= corey exponent for oil relative permeability curve
= net to gross
= capillary pressure
= productivity index
= numerical productivity index
= transient productivity index
= actual productivity index
= ideal productivity index
= initial reservoir pressure
= average pressure given by 9 blocks surrounding the well
( ) = reservoir flowing pressure as a function of time
= pressure change due to skin

95. 95
= reservoir rate
= total rate
= capillary radius or pore-throat radius
= grid block pressure equivalent radius
= recovery factor
= reservoir quality index
= wellbore radius
= correlation coefficient
= storativity ratio
= skin factor
= geometrical skin caused by flow convergence
= mechanical skin due to permeability alteration near wellbore
= total skin
= pressure drop caused by tubing friction between gauge and sandface
= non-Darcy (Turbulent skin): variation from linear flow
= water saturation
= irreducible water saturation
= shutin time
= limited reservoir interval
∆ = X dimension of the grid block
∆ = Y dimension of the grid block
Greek Symbols
= water density
= oil density
μ = fluid viscosity
= wettability contact angle
ɸ = pore volume to grain volume ratio
ɸ = porosity
ɸ = effective porosity
= tortuosity
= interfacial tension