Master degree in Oil and Gas Technology
Aalborg University Esbjerg
Reservoir simulation history matching
and forecasting
Author:
Supervisor:
Petya Ivanova Vakova
Muhammad Adeel Nasser Sohal
2014
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Abstract
The project aims at validation of the Nagani oil field. Validation of the reservoir model makes
it reliable source of information for forecasting. The main is to achieve the best possible
match between the simulation and the history data. The history data is the information
received throughout the years of exploitation of the field. History matching as a process
decreases mismatch between the history data and simulation data.
The project work consists of accomplishing the main task – preparation of the history
matched model, and setting a main base case which should represent the future development
of Nagani field.
Software, which are studied and applied in this project for achieving the results are two:
Eclipse black oil simulator and a history matching software called SenEx.
Project theme: Reservoir simulation History matching and Forecasting
Winter Semester 2013
Supervisor: Muhammad Adeel Nasser Sohal
Submission date: 24.April.2014
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Acknowledgements
I would like to express my gratitude to my supervisor Muhammad Adeel Nasser Sohal, for his
guidance through the whole semester work.
I thank him for valuable help with SenEX software to Dr. Husein.
Also I would like to thank to the faculty of the AAU- Esbjerg for providing me with needed
recourses to work.
Petya Ivanova Vakova
…………………………………….
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Table of contents:
Abstract ...................................................................................................................................... 2
Acknowledgements .................................................................................................................... 3
Introduction ................................................................................................................................ 5
Reservoir parameters and transport process into the reservoir .............................................. 7
History matching .................................................................................................................. 11
Gradient Decent Methods................................................................................................. 15
Ensemble Kalman Filters (EnKF) .................................................................................... 16
Forecasting ........................................................................................................................... 17
Software ............................................................................................................................... 17
SenEx ............................................................................................................................... 18
Eclipse .............................................................................................................................. 22
Case study model ..................................................................................................................... 23
History matching process ..................................................................................................... 23
Reservoir model before the history matching process ..................................................... 24
Reservoir model after the history matching process ........................................................ 32
Results .............................................................................................................................. 39
Forecasting of the reservoir model ....................................................................................... 46
Results .............................................................................................................................. 47
Conclusion and discussion ....................................................................................................... 55
Table of figures ........................................................................................................................ 57
References ................................................................................................................................ 59
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Introduction
There are a plenty of reasons to validate (history match) the reservoir simulation model.
Without history matching a robust future production forecasting is not possible. History
matching process is an important part of the reservoir engineering. It can be applied using
manual history matching approach or assisted history matching approach. As a process itself
history matching is time consuming and it needs to be done by experienced reservoir
engineer. Regarding the technology improvement during the years history matching has
become automated. Assisted history matching is widely used because it decreases the used
amount of time and gives relatively good results for improving the quality of the reservoir
model.
Some of the uncertainties related with the history matching process are based on
limitations of simulators (related to its specifications) and others are provoked from limited or
lack of data for validation. Despite all these difficulties reservoir simulation is one of the basic
tools for oil and gas field development. The set of data (input and output) which is
incorporation between data evaluations and interpretation into simulator is the reservoir
model. Geological reservoir model is used as an input for numerical simulator after proper
upscaling of simulation grids. Geomodel grid represents the details like structural features
(surfaces, faults), distribution of petro-physical properties throughout the volume of reservoir.
Upscaled reservoir model keeps the accuracy of geomodel but with upgraded number of grid
cells which make it easier to run simulation. Reservoir model is constructed by implementing
of diverse sources of data, which characterize reservoir and fluid properties. Static parameters
like porosity and permeability, and dynamic variables such as pressure and fluid saturation are
reservoir properties whose authenticity has a great importance for the reservoir model.
Reservoir simulation model should be geologically realistic and also account for a lot of
different measurement scales. Reservoir simulation reflects the transport processes of the
fluids in the reservoir.
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Fig. 1 Relation between input and output data for reservoir simulation
An accurate reservoir model used for production forecasting under existing operating
conditions and it can also be used to optimize recovery efficiency.
Simulation model is constructed based on the assumption that it occupies a three –
dimensional region. This region includes grid and grid blocks which encompass it. Coordinate
system which is used is x-y-z (cartesian system) and the grid blocks are situated in x, y and z
direction respectively. If a grid block is situated out of the region of the reservoir it receives a
zero value for permeability and negligible value for porosity which makes it inactive.
Throughout the years forecasting studies has been developed significantly- from manual
calculations and analogy with the results from other fields to a computer simulation.
Computer simulation is much more accurate because it determines the reservoir in space and
time, which is represented by a complex numerical model. This numerical model integrates
the transport process into the reservoir (the laws of flows in porous media) and the reservoir
data.
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Fig. 2 Mathematical reservoir simulation model (1)
Fig.2 is illustrates the mathematical model of a reservoir which is a mathematical description
of the reservoir with its petrophysical properties. On it can be observed lithology, number of
layers, gas-oil contact and water-oil contact. Also it includes information about porosity and
permeability in x, y and z direction. Based on all this information the reservoir model can be
used for assessing probable future production.
Reservoir parameters and transport process into the reservoir
Reservoir can be characterized as a rock formation, which has particular structure and
petrophysical properties. Those petrophysical properties are porosity, permeability and
saturation.
Porosity characterizes the ability of the rock to accumulate fluid (oil, gas, water) in the
porous space, which reflects the size of the storage capacity. Porous space is the free space
between the rock grains. Porosity depends on both - the distribution of the grain size into the
rock and the depth. Porosity of the rock formation decreases with increasing the depth. This is
because formation has compressibility, due to the decreasing pressure of the fluids in the
pores during the production period. Nevertheless that the compressibility of the solid rock is
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negligible, this leads to the compression of the rock until equilibrium is achieved. It is
expressed in % and can be calculated by the following equation:
(1)
where:
ø – porosity;
Vpores – volume of the pores;
Vtotal – total volume of the rock;
Total porosity (øt) of the formation corresponds to all pores. It is the sum of effective porosity
(øu) and residual porosity (ør).
(1)
Residual porosity characterizes isolated pores, in which the fluid is locked. Effective porosity
accounts for the pores which are connected to each other and to other formation. Effective
porosity can vary in the range from less than 1% up to over 40%. Based on this porosity can
be stated to be:
-
Low: ø < 5% ;
-
Mediocre: 5% < ø < 10% ;
-
Average: 10% < ø < 20% ;
-
Good: 20% < ø < 30% ;
-
Excellent: ø > 30%.
Porosity can be determined by core analysis or by well logging.
Permeability characterizes the ability of the fluid to flow through rock pores. Based on the
porous medium fluid can flow with greater or lesser difficulty. There are absolute, specific
and relative permeability which relies on the ability of the rock to sustain the movement of the
saturated fluid. Relative permeability is the ratio between effective permeability of a
particular fluid (oil) and absolute permeability of that fluid.
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(1)
Relative permeability of one phase (oil) decreases due to the increasing saturation of the other
phase (water).
Fig. 3 Oil - water relative permeability curves (2)
On the Fig. 3 are depicted the typical relative permeability curves for the oil and water in a
sandstone reservoir.
Absolute permeability measurement is based on the presence of a single fluid or phase present
in the rock. It depends on the direction in the rock – vertical and horizontal. Horizontal
permeability is characterized by the parallel flow toward the well. Vertical permeability
characterizes the segregation problems of the fluids based on the difference in densities.
Regarding stratification of the formation, the horizontal permeability is much higher than the
vertical permeability.
Effective permeability is the ability to transmit a particular fluid in presence of other
immiscible fluids in the reservoir. Effective permeability is affected by the relative saturation
of the fluid and the nature of the reservoir. It can be calculated by Darcy`s law, which is based
on experiments.
(1)
Where:
Q - Flow rate – m3/s;
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A - Cross- section – cm2;
k - Permeability coefficient - mD;
µ - Viscosity - cP;
dP - Downstream pressure - atmospheres;
dx - Length of the sample - cm.
There is dependence between porosity and permeability. In some cases it can be established
through equation based on a particular sample. Basically those two properties (porosity and
permeability) are related and they are proportional. There is no particular unique equation for
calculating the dependence between porosity and permeability.
Saturation is a parameter which characterizes the relative amount of oil, gas and water in
place in the reservoir. It is expressed in percent of volume and can be calculated by applying
the following equations:
(1)
Where:
SW - water saturation;
VW – water volume;
SO - oil saturation;
VO – oil volume;
SG - gas saturation;
VG – gas volume;
VP – pore volume.
Transportation of the fluid in the reservoir structure is based on all of the already mentioned
parameter and pressure difference between the bottomhole pressure (BHP) and the pressure at
the reservoir boundary. On Fig.4 is presented graphically the pressure change throughout the
reservoir during the period of exploration.
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Fig. 4 Change into the pressure in the reservoir with increasing the distance between wellbore and the reservoir
boundary
From Fig.4 can be observed that with increasing the distance between from the wellbore
pressure is increasing. The most significant is the drop in pressure across skin zone. This is a
zone which refers to a wellbore damage. It is characterized with altered permeability due to
the drilling, completion or workover operations. Skin zone also alter the pressure distribution
around the wellbore. Thickness of the zone cane vary between a few inches up to several feet
History matching
History matching is a process which requires an interdisciplinary approach. It can be
characterized like a trial-and-error approach. This approach is based on solving problems by
trying out various means for achieving a satisfactory result. It demands a wide scale of
information (reservoir and flow characteristics). History matching is an important part of the
reservoir engineering. A reliable production forecasting and future EOR scheme for a
particular reservoir based on history matched simulation model. It is important to mention that
there are dynamic models which lead to history match but yet the changes might not be
consistent with the static interpretations. History matching should be geologically consistent
and the numerical simulation model should be properly calibrated. This will lead to better
results and confidence in predictions of future performance of the reservoir. Calibration
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includes series of simulation runs. An output from each of the simulation runs is the response
of the simulator. The simulation runs analyses describe how closely model replicates
historical data. The most altered, through the calibration (the match) is the uncertain
information about the reservoir.
It could be determined by sensitivity analysis, which parameters have high impact on the
responses of the model during the series of runs.
Fig. 5 General reservoir modeling workflow (3)
During the history matching process some of the parameters should be modified. These
parameters can be split into two groups – volumetric and flow parameters. These groups and
the included parameters are shown on the Figure.6.
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Fig. 6 Parameters that affect history matching can be modified during the process (4)
The volumetric parameters generally affect material-based model. Some of them can be
addressed during the data preparation stage by using analytical approaches. Such analytical
approaches are material- balance models. Material- balance is not able to resolve volumetric
parameters satisfactorily in case of strong aquifer drive which affects the reservoir-fluid-flow
mechanism. Because of that these parameters are often modified within the range of
uncertainty. The volumetric parameters are adjusted during the pressure match in case that the
reservoir-pore-volume compressibility and fluid PVT properties are reasonably accurate.
Pressure matching process involves the validation of static pressure by compartment.
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The flow parameters have impact on the fluids within the reservoir model (global and local
flow). They are concerned with the fluid movement through the reservoir and the factors.
If not adjusted both two types of parameters can affect negatively on the results from the
process of history matching. If not corrected they will affect the reliability of the reservoir
model for further studies.
For first time brief graphical image of the manual history matching process is provided by
Saleri and Toronyi (1988). This approach is still used for history matching with slight
modifications
(4).
It consist on two basic process – Pressure matching and Saturation
matching. First the pressure match should be achieved and then the saturation match. At the
end of the saturation match, depending on the results it is possible to check pressure match
and repeat it, if it is needed. The basic steps involved in these two processes are mentioned in
Fig.4.
Fig. 7 Graphical depiction of the history matching process (4)
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At every step of these two processes mismatched parameters can be adjusted. Magnitude and
shape of average pressure vs. time (step two of pressure matching process – Fig.7.) can be
affected by adjusting volumetric parameters like aquifer size. Pressure gradient (step three of
pressure matching process – Fig.7.) can be improved by adjusting the horizontal permeability
regionally. Well pressure (step four of pressure matching process – Fig.7.) can be improved
by adjusting horizontal permeability locally or well connection.
Modern approaches for history matching are based on the integrated workflows. That is
because it is necessary for the adjustments to be consistent with and constrained by original
interpretations (geological, petrophysical, and geophysical). It is hard to make history
matching completely automatic process because it is not an isolated task which can be easily
put into linear process. Assisted history matching (AHM) is the name of software- enabled
methods which can calibrate a reservoir model using known data. The set of parameters which
is needed for successful AHM process are called history matching parameters (HMP). Results
of AHM not always lead in the right direction, because of the possibility of nonlinear
interactions. AHM is a numeric approach, whose purpose is to define an objective function
(OF) and reduce it until its level is acceptable. OF is the difference between the history and
the simulated- calculated values. It is calculated by taking under consideration a scale of
importance of the different parameters (pressure difference, water cut, gas flow, ect.) for the
particular case.
Methods which are included in AHM are result of decades of study. Because of that big part
of the approaches are hybrid methods. They are based on improving the efficiency of the
history matching by automatizing the process as much as possible. Below are presented two
of the AHM methods which can introduce the strong and the week points of that approach.
Gradient Decent Methods
These methods are easy to understand. The gradient (change in OF with respect to the varying
parameters) is used to determine the change in the parameters that lead to minimization of the
OF. One of the limitations of those methods is that depending on the start point they might
find local minimum, which will not give the correct value for the sensitivity parameter which
is the global minimum. (As shown in - Fig.8.) The gradient is the slope of the function with
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respect to the sensitivity parameter. The slope gives the magnitude and direction of change the
sensitivity parameter which is required in order to reduce the OF.
Fig. 8 Error of mismatch (left side) vs. the value of history matching sensitivity parameter (4)
Other important limitation is that the method offers only one solution, which contradicts the
principle of history matching for nonunique solutions (a number of solutions may be
acceptable).
Ensemble Kalman Filters (EnKF)
KnKF is a useful tool in the history matching process. It can handle large parameter sets. And
also it is capable of solving emerging issues of combined state- and parameter-estimation.
―This is a method, in which an ensemble of reservoir state variables (e.g., pressure and
saturation) is generated on multiple reservoir models and kept up to date as new data are
obtained.‖ (4)
As every method EnKF has its own advantages and limitations. As an advantage it can be
considered to modify the inter-well petrophysical properties and afterwards automatically
account for correlations between reservoir parameter and simulation response which lead to
satisfactory history matching results. As a limitation it can be pointed out that it is perfect for
linear models (the response of the reservoir model is non-linear).
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Forecasting
Forecasting is the final phase of the reservoir simulation study which encompasses predicting
the future behavior (performance) of the reservoir under different operational scenarios. Some
of these scenarios might include well simulations, abandonment of existing wells and/or
drilling of new one, injecting of fluids (water) and implementing EOR methods. These
scenarios aim to support and increase the hydrocarbon recovery. Because of the significant
capital investments which are required in order to fulfill some of the scenarios it demands
economical evaluation, which is also part of the evaluation and comparison between the
operational scenarios.
Even if the validated reservoir model has a good quality there is a possibility of receiving not
so realistic prediction. This is because it is needed a lot of additional information, which is not
always available. There are parameters which are mandatory to be included and properly
calibrated during the process of history matching. Some of those parameters are:
-
Bottomhole pressure (BHP) or tubing-head pressure (THP) data. If these values are
not available during the history matching, it is highly possible that the well indices
(WIs) were not tuned;
-
Liquid rates;
-
Water cut;
-
Gas-oil ratio (GOR).
Generally the forecasting refers to transition from phase constrained rated rate to total or
group rate limit, or individual well- pressure control. If the productivity is properly tuned, a
simple base prediction from a history match to a forecast model can be done Transition to the
new constraints, which concerns all wells should be smooth.
Software
History matching and forecasting of Nagani oil field is achieved by SenEx and Eclipse black
oil simulator, product of FirmSoft Technologies and Schlumberger respectively.
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SenEx
SenEx (Sensitivity Explorer) is for reservoir history matching. It uses numerical approach.
The workflow is typical for the history matching and it contains three main components- Preprocessor, Simulation and Post-processor. (Fig.9.)
Fig. 9 SenEx workflow
Simulation component of the workflow, unlike other two components, is performed
independent of SenEx. It is performed by Eclipse and it is very important part of the history
matching workflow. Without simulator it is not possible to complete history matching
process.
Each component of history matching workflow comprises a number of steps that is required
to accomplish the required results. The important steps involved in history matching
procedure are explained in Fig.10. The important steps mentioned in given diagram are
repeated until a good match is achieved.
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Fig. 10 Steps included in the history matching procedure (5)
SenEx is different from other history matching software because it validates the reservoir
model by adjusting the grid block parameters without combining them in to boxes. In other
words it works on grid block level instead of using ―multiplier boxes‖. Despite the fact that
box multipliers are widely used in history matching process but it is an unnatural way to
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validate a reservoir model. Multiplier boxes are ―unnatural‖ and also hard to define. SenEx
calculates the effect of the every grid block on the history matching and compute its
sensitivity. It is possible to find which grid block properties needed to be changed to improve
the history matching process. Changes in the grid properties are smooth and continuous. Also
those changes are introduced only for the grid blocks where they are needed. As a result from
sensitivity analyze some of the cells get positive and other negative value for sensitivity. A
cell with positive sensitivity value means that the current value is high and has to be
decreased to improve the match. Improving the match for a cell with negative sensitivity is
done by increasing the current value of the cell.
History matching is an optimization process in which objective function (OF) value is
decreased to its minimum limit. Quality of the match improves with decreasing the value of
the OF.
Generally OF characterizes the mismatch between the history and simulated data of the
reservoir. Maximum value of the OF is 1(corresponds to full mismatch) and minimum value
is 0 (corresponds to full match). All of the values in between maximum and minimum are
used to characterize the size of the difference between the simulation and real production
history.
Objective function for the reservoir is obtained by a weight sum of the mismatch scores of all
existing wells.
∑
(5)
Where
Mi – mismatch at well i;
Wi – a user-specified non-negative weight for well i;
i – has a value from 1 to the total number of the wells.
Weight represents the importance of the parameter for a particular history matching case.
Weight can be changed in the scale from 0 to 1.
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Every well have five mismatch parameters: Mgas ,Moil ,Mwater ,MBHP
and MSIP . Total
mismatch for the well is calculated by following equation:
(5)
Where:
M – mismatch;
w – a user-specified non-negative weight.
Other more detail equation for calculating the mismatch for well i is defined as follows:
∑ [
(
)
(
)
(
)
Where:
– non-negative weight factor for the oil;
– non-negative weight factor for the water;
– non-negative weight factor for the gas;
– oil production rate [stb/d] for the well;
– water production rate [stb/d] for the well;
– gas production rate [Mscf/d] for the well;
– average well-block pressure [psia] for the well;
– average daily liquid rate for the well.
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]
(6)
(
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History matching is achieved through set of iterations, each of which finish with computing
sensitivities of porosity and permeability (in x, y and z direction). Sensitivity is described as
the partial derivatives of OF with respect to each parameter x at every grid block:
(6)
Sensitivity (partial derivatives) indicates how the change in parameters would affect the
match. Based on the partial derivatives of OF new set of property arrays have been generated
for the next simulation run. Derivatives of OF are used as the guide to whether increase or
decrease each property at each block, in order to improve the match. The process completes
when the best match (the lowest value of the OF) is achieved.
Eclipse
Eclipse is a reservoir simulator which is used for building and executing simulation models. It
solves material and energy balance equations. Through these equations subsurface of
petroleum reservoir is modeling. They are expressed in a three- coordinate system (x-y-z) for
the total number of grid blocks. Regarding those equations mass balance for each of the
component of the system can be defined for each of the grid blocks. Simulation model
input/output structure is explained in Fig.11.
Fig. 11 Input and output data structure for Eclipse software (7)
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In order to operate with Eclipse in its simulation mode it is necessary to apply a specific
approach for creating a data file. The data file is a source of information which contains the
reservoir model. The data file is created based on the specific file syntax.
Case study model
Nagani oil field model is used to perform history matching and forecasting studies in this
research project. This reservoir has been in exploration for six years (from 01.11.2005 to
18.12.2011). It includes three production wells and one injection well. Proximately 50 000
STB/DAY of water has been injected from the beginning of the field production. The history
data consist of oil and water production rates through the years. There is no data on gas flow
and pressure during the period of exploration. The reservoir simulation model is matched to
the production history of the field to validate it for reliable future forecasting and EOR
strategies. This simulation step of SenEx workflow is achieved by Eclipse black oil.
History matching process
The history matching procedure is exactly performed as explained in Fig.10.
The first step in the history matching process was to prepare a SenEx compatible simulation
deck. This is to ensure that the required data to perform the essential steps is available to the
software. After preparation deck/check deck, model is ready for history matching process. In
order to receive the plots which illustrate the mismatch, it is mandatory to choose an OF first,
which is used throughout the history matching process. First received plots illustrate the
output level of mismatch in cumulative production, flow rates and water cut ratio. This level
of mismatch has to be diminished as much as it is possible to achieve high quality of the
results. Each group of plots shows the mismatch for a particular parameter regarding
production wells (PROD1, PROD2 and PROD3). Each of the plots introduces the change of
those parameters with time (in days). Dotted line represents the history production data and
the bold line represents the simulation data. Green color expresses the oil production rate and
the blue color express water production rate.
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Reservoir model before the history matching process
Well Cumulative Oil Production
Plots below (Fig.12) illustrate the size of the mismatch regarding cumulative oil and water
production in the case before history matching process. Cumulative production reflects the
amount of produced oil up to a given date.
a) PROD1;
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b) PROD2;
c) PROD3;
Fig. 12 Cumulative oil and water production a) PROD1; b) PROD2; c) PROD3;
On the plot (Fig.12-a) is depicted cumulative production of the oil and water for the period of
production. Production is measured in stock tank barrels per day (STB/DAY). Values
received during the first 750 days of exploration are showing almost complete match with the
simulated data. After 750 day the mismatch starts to increase gradually. This is because of an
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error in water oil contact (WOC). This error is related with inaccurate output parameter
estimated for WOC.
On the plot (Fig.12-b) is depicted the mismatch in the cumulative production of oil and water
from PROD2. Until day 800 there is almost perfect match between the curves for history and
simulated data. After day 800 history data shows deviation from simulated data which
increases gradually with the time. According to the history data after day 2100 water
cumulative production becomes higher than oil cumulative production. This is a result from
changes in reservoir behavior during the production.
On the plot (Fig.12-c) is depicted the cumulative production of oil and water. Until day 500
there is a very good match between history and simulated data. History data shows higher
production of oil and lower production of water compared with the simulated data. Similar
picture of cumulative production is observed for PROD1 well. This deviation is reflected
from error related with inaccurate output parameter estimated for WOC.
Sensitivity for WOC is introduced on Fig.13.
Fig. 13 Sensitivity for WOC at case_1
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Sensitivity is represented on the plot as Delta Depth vs. Objective Function.
Delta depth = New Contact Depth – Current Contact Depth
Delta OF = New expected OF value – Current OF
(5)
Values for Delta Depth higher than 0 indicates lowering the contact depth and values lower
than 0 indicates raising the contact depth. The negative value of Delta OF indicates that there
will be improvement in the mismatch.
Well Rates of oil and water
Plots below (Fig.14) illustrate the size of the mismatch regarding rates of oil and water
production in the case before history matching process.
a) PROD1;
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b) PROD2;
c) PROD3
Fig. 14 Oil and water rates vs. time: a) PROD1; b) PROD2; c) PROD3;
On the plot (Fig.14-a) are depicted the rates of water and oil production during the years for
PROD1. Here the deviation between history and simulated data starts from the 500 day and
increases fast. According to the plot PROD1 has produced significantly higher amount of oil
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and lower amount of water during its production period compared with the simulated data.
This is an example of the importance of reservoir validation (history matching). History
matched model of the reservoir reflects the behavior of the reservoir throughout the years
which is the closest to the reality. Only validated model can be a reliable source of
information for further studies.
On the second plot (Fig.14-b) are depicted the rates of oil and water during the period of
production of PROD2. After day 450 until the end of the observed period it can be seen that
the size of the mismatch, which is not significantly big in the different parts of the curves.
Those two negative peaks on the plot (each if which is covering a period of 50 days) are
related with well shut-down. The aim of the well shut-down could be to increase the pressure
in the reservoir.
On the third plot (Fig.14-c) oil and water rates are depicted throughout the years for PROD3.
The mismatch begins to increase rapidly from day 325. Until that day we observe almost full
match between simulated and history rates for PROD3 well. Both history data and simulated
data curves are following almost the same recovery. According to the simulated data the cross
point after which the well is starting to produce more water than oil is achieved earlier. From
the history data can be observed that this point is achieved at day 1750.
Well Water cut ratio
Plots below (Fig.15) illustrate the size of the mismatch regarding water cut ratio in the case
before history matching process. Water cut is an important parameter which measures the
ratio of water produced compared to the volume of total produced liquids. In reservoirs which
are water drive, water cut can reach high values.
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a) PROD1;
b) PROD2;
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c) PROD3;
Fig. 15 Water cut ratio: a) PROD1; b) PROD2; c) PROD3;
On the plot (Fig.15-a) is depicted the ratio of the water cut with the time. There is no
information related with gas-oil ratio (GOR). Again the mismatch between history and
simulated data starts from day 500. The size of the mismatch observed on the plot if not
diminished will affect in a negative on the future studies of the reservoir. According to
simulated data the amount of the water in the reservoir has started to increase fast after
relatively short production period. This will affect the accuracy of further calculations related
with the existing amount of the oil in the reservoir.
On the plot (Fig.15-b) is depicted the ratio of water cut vs. time. As it is showed on the plot,
the mismatch starts from day 400. According to history data water cut increases more rapidly
for a shorter period of time. On this plot also can be seen the negative peak which marks the
shut-down period of the well.
On the plot (Fig.15-c) is depicted the ratio of the water cut vs. time. The curve which
characterizes changes in the water cut according to the simulated data starts to increase
rapidly from day 325. This contradicts to the history data whose dotted curve starts to increase
gradually from day 600.
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As it is seen from the plots above there is a significant difference between history and
simulated data in different wells. The above plots are computed after first sensitivity analysis
that point out which grid cells properties needed to be adjusted. To achieve quick results some
constraints could be used. The constraints provide the chance to manipulate the ratio in which
values of porosity, permeability, initial water saturation and initial oil saturation can vary.
Also the minimum and maximum values of these properties can be manipulated. It is also
possible to choose a particular well or a single parameter to practice in next iteration. It is
performed to achieve the best match. In order to diminish the mismatch in this project forty
cases (iterations) are constructed.
Reservoir model after the history matching process
Due to the process of history matching the quality of match has increased as OF has decreased
with increasing number of iterations.
At the final iteration – sase_40, is accomplished the best possible match based on the history
data. In final plots it can be observed that the mismatch has decreased to its minimum level
regarding the observed parameters for each well.
Well Cumulative Oil Production
Plots below (Fig.16) illustrate the size of the mismatch regarding cumulative oil and water
production in the case after history matching process.
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a) PROD1;
b) PROD2;
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c) PROD3;
Fig. 16 Cumulative oil and water production in the case after history matching process: a) PROD1; b) PROD2;
c)PROD3;
On the plots above are depicted the cumulative production rates of oil and water. The
deviation between history and simulated data is diminished to the lowest possible level. As a
result from the process it is achieved almost perfect match for PROD1, OPROD2 and
PROD3.
Well Rates of oil and water
Plots below (Fig.17) illustrate the size of the mismatch regarding rates of oil and water
production in the case after history matching process.
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a) PROD1;
b) PROD2;
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c) PROD3;
Fig. 17 Rates of production at the end of the history matching process: a) PROD1; b) PROD2; c) PROD3;
On the plot are depicted oil and water rates during the production period. Those are the final
results from the validation process for the wells. Although that there is still some level of
mismatch, the reservoir model is significantly improved compared with case_1. The observed
result on the plots is the best result which can be received at the end of the history matching
process.
Well Water cut ratio
Plots below (Fig.18) illustrate the size of the mismatch regarding water cur ratio in the case
after history matching process.
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a) PROD1;
b) PROD2;
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c) PROD3;
Fig. 18 Water cur ratio at the end of history matching process: a) PROD1; b) PROD2; c) PROD3;
It is observed a significant improvement in the mismatch for the wells.
On the plot for PROD2 (Fig.18-b) is illustrated that the minimum level of mismatch in some
sections is higher than in others. Nevertheless PROD2 has the highest minimum level of the
mismatch and at the end it shows a significant improvement in the match between history and
simulated data.
On the plot for PROD3 (Fig.18-c) there is still some level of mismatch which cannot be
reduced more but it is not so significant. Received results reflect high quality of the model,
achieved due to the history matching process.
It is obvious from the plots so far that the quality of the match has been increased
significantly. PROD2 has the worst match among all the matched wells. There is still
relatively high mismatch related to water cut and the oil and water production rates. Also
there is no change in pressure with time. This is because the data related to pressure change
with time is not available. During history matching process pressure did not change.
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Achieved level of mismatch is the best possible because of many reasons. Some of them are
related with the fact that the measured well data cannot be free of noises. Also other reason
can be that some of the structural characteristics of the reservoir are not included in the model
(faults, fractures). In this case it is not possible to reach 100% match regardless of how
accurate the properties are adjusted.
Results
The reservoir model has been validated by achieving a good match between the history and
simulation data at field and well scale level. The size of the mismatch was decreased to the
lowest possible level. The change of the mismatch throughout the process is presented on the
field plots.
Field cumulative oil and water production
On the Fig.19 are introduced the plots which show the level of mismatch for the field in the
beginning of the history matching process.
a)Field cumulative oil and water production at case_1
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b) Field cumulative oil and water production at case_40
Fig. 19 Mismatch for the field in the beginning of the history matching process: a) Field cumulative oil and water
production at case_1; b) Field cumulative oil and water production at case_40;
On the plot (Fig.19-a) is depicted the cumulative production of the field for the period of
production. It summarizes the mismatch related with cumulative production from the
production wells (PROD1, PROD2 andPROD3). From the plot can be concluded that there is
an error related with oil-water contact which during the process of history matching has been
diminished.
On the plot (Fig.19-b) is depicted the history matched cumulative production of the field. It
can be seen that the deviation is not significant as it was in the beginning of the process and
that there is almost perfect match between history and simulated data.
Field Rates of oil and water
Plots below (Fig.20) illustrate the size of reduced mismatch regarding oil and water rates due
to the history matching process.
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a) Field oil and water rates at case_1
b) Field oil and water rates at case_40
Fig. 20 Reduced mismatch regarding oil and water rates due to the history matching process: a) Field oil and water
rates at case_1; b) Field oil and water rates at case_40;
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On the plot (Fig.20-a) are depicted oil and water rates of production of the field. The observed
size of the mismatch reflects the mismatches of the production wells in the beginning of the
history matching process. The negative picks on the plot can be related with closing of the
well in order to build-in pressure in the reservoir which will help to support or/and increase
the production. The mismatch has been decreased to the lowest possible level at the end of the
process.
On the plot (Fig.20-b) is depicted the final result of the history matching related with oil and
water rates during the production period of the field. There is a high level of match between
the history and simulated data.
Field Water cut ratio
Plots below (Fig.21) illustrate the size of reduced mismatch regarding water cut ratio due to
the history matching process.
a)Field water cut ratio at case_1
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b) Field water cut ratio at case_40
Fig.21 Reduced mismatch at water cut ratio due to the history matching process: a) Field water cut ratio at case_1; b)
Field water cut ratio at case_40;
On the plot (Fig.21-a) is depicted the ratio of water cut vs. time for the period of the
production of the field. This plot reflects the summarized mismatch from the production
wells.
On the plot (Fig.21-b) is depicted the ratio of water cut vs. time for the period of production
of the field. It reflects the history matched model of the reservoir.
Depicted plots represent the lowest possible level of mismatch, achieved at the end of the
history matching process at case_40 compared to the output data. Each of them reflects the
remaining level of the mismatch for the field which has been significantly decreased. During
the validation process the quality of the reservoir model has been improved, which increases
the reliability of the model for further studies.
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Fig.22 depicts the change in porosity for a particular grid cell, as a result of the history
matching process. Used for this purpose cell with coordinates 22:9:1 is located in the top layer
of the reservoir in x-y direction, next to the PROD2 well.
a) Porosity in case_1
On the figure (Fig.23-a) is depicted how porosity is spread on the top layer of the reservoir in
the beginning of the history matching process. Value which the observed cell has is 0,164649.
During the sets of iterations this value has been changed in a certain ratio in order to increase
the match between simulated and history data, and to receive the closest to the real value for
porosity for that part of the reservoir.
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b) Porosity in case_40;
Fig. 22 Changes in porosity during the process of history matching in a grid cell 22:9:1 : a) porosity value in case_1; b)
porosity value in case_40
On the figure (Fig.23-b) is depicted the final history matched view of the porosity of the top
layer of the reservoir. Final porosity value which grid cell 22:9:1 has received is 0.158962.
Decreasing the value of porosity with 0.005687 improves the quality of the match. This value
falls in the range of average porosity.
Along with the changes in porosity there are also changes in permeability. For the same grid
cell (22:9:1) permeability (K) in x, y and z direction was: Kx = Ky = Kz = 627.196 mD in the
beginning of the history matching process. Permeability values received at the end of the
process are: Kx = 857.01 mD; Ky = 100.632 mD; Kz = 761.484 mD.
The quality of the match is relatively high. The mismatch level is also visible in the well bar
chart. Well bar chart depicts the size of mismatch for each of the wells by comparing them.
(Fig.23-a and -b)
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Fig. 23-a Comparison of the size of the mismatch between
case_1, case_10, case_20, case_30 and case_40
Fig. 23-b Comparison of size of the mismatch between case_10,
case_20, case_30 and case_40
From the Fig. 23 above it is seen that the mismatch has decreased significantly. In Fig.23-a,
well mismatch level and how it is changed from case_1 (at the beginning of the history
matching process) to case_40 (at the end of history matching process) is explained. It can be
observed that the received results for case_40 cannot be seen on the figure. This is because
further received results compared to case_1 are showing a negligible mismatch.
Fig.23-b excludes case_1 which allows following the trend of decreasing mismatch because it
becomes more gradual. Here it can be observed that there is still a level of mismatch. There is
a certain level to decrease the mismatch between history data and simulated data. After that
limit, increase in number of iterations cannot produce significant change into result. High
level of match was achieved at case_40.
Forecasting of the reservoir model
Eclipse is used to create the forecast of further development of the reservoir in the base case.
There are some parameters which should be included as target of the process. It is important
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to have specified well production/injection targets as an output for the forecasting.
Forecasting can be accomplished by restarting an Eclipse run. The simulation run which is
restarted is the history matched simulation model.
Restarting of the simulation run is done by creating a ‗restart file‘ for Eclipse. The restart file
is used to ensure that the forecasting will not affect the history matched model by changing
some of the parameters. Preparing of the restart file has to be done by following a particular
scheme to make it appropriate for execution.
Two methods can be applied to restart an Eclipse run:
-
A fast restart – This method uses a ‗save file‘ as a base case. The data which is stored
in the saved file is in processed form, which makes the restarting run quick to
initialize. A fast restart method is used mainly for restarts related with data
modifications only in Schedule section.
-
A flexible restart – It is slower method than a fast restart because the data have to be
processed again. The advantage of this method is that it allows changes to be made in
the data which provides the flexibility of the process. And also this method makes
possible restarts produced by early version of Eclipse.
For forecasting the future behavior of Nagani reservoir model is used a flexible restart
method. Construction of the restart file was made according to the specific file syntax
illustrated in ECLIPSE Reference Manual 2008.1 (8)
Both of the mentioned earlier methods for restarting an Eclipse run were tried. Base on that
experience it was chosen a flexible restart method.
Reservoir forecast was accomplished. Its aim is to foresee how the reservoir is going to
behave in the next ten years under the same circumstances of production. Number of the wells
is kept the same in the forecasting model of the reservoir.
Results
Results received from the history matching process reflect the reservoir behavior for the past
period of production. History matched model is a base for further studies of the reservoir.
Forecasting is applied in order to check, what would be the future behavior of the reservoir if
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the conditions are kept the same. This will provide important information about the necessity
of improving the production by adding new wells (for production or/and water injection) or
implementation of enhanced oil recovery (EOR) methods.
As a result from the forecasting the base case, it has been the following results for each of the
wells - well oil production rate (WOPR), well water production rate (WWPR) and well water
cut (WWCT) - for period of ten years.
Well oil production rate (WOPR)
a)PROD1;
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b)PROD2;
c)PROD3;
Fig. 24 Well oil production rate (WOPR): a) PROD1; b) PROD2; c) PROD3;
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On the plot (Fig.24-a) is depicted well oil production rate (WOPR) for PROD1. From day
3400 the production rate starts to decrease rapidly. For a period of 1100 days it drops to 0.
On the plot (Fig.24-b) is depicted WOPR for the next ten years for PROD2. Oil production
rate starts to decrease faster from day 3400. The slope of the curve is steeper between 3400
and 4000 day. Afterwards is continues to decrease gradually until it gets to 85 Sm3/day
(534.6339 STB/day).
On the plot (Fig.24-c) is depicted WOPR for PROD3. Between day 2200 and day 3400
production rate is relatively stable. After day 3400 the production rate starts to decrease
rapidly until it gets the value of 150 Sm3/day (943.47 STB/day).
Well water production rate (WWPR)
a)PROD1;
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b)PROD2;
c)PROD3;
Fig. 25 well water production rate (WWPR): a) PROD1; b) PROD2; c) PROD3;
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On the plot (Fig.25-a) is depicted well water production rate (WWPR) for PROD1. As it is
seen from the plot rate will increase fast in the first 1200 day until it achieves its maximum
value. Decrease in the production after day 3400 is related with the decreasing well
production.
On the plot (Fig.25-b) is depicted WWPR for PROD2. From day 2200 to day 3400 water
production rate increases gradually. After day 3400 the rate decrease rapidly until it gets the
value of 250 Sm3/day (1572.4527 STB/day) at day 6000.
On the plot (Fig.25-c) is depicted the WWPR for PROD3. Water production rate increases
gradually until it gets its maximum value (4400 Sm3/day) at day 3400. After that day it starts
to decline with steep slope, until it drops to 0 Sm3/day at day 6000.
Well water cut (WWCT)
a)PROD1;
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b)PROD2;
c)PROD3;
Fig. 26 Well water cut (WWCT): a) PROD1; b) PROD2; c) PROD3;
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On the plot (Fig.26-a) is depicted well water cut (WWCT) vs. time for PROD1. WWCT
increases gradually until day 4000 after which it decreases and at day 4400 it drops rapidly.
This drop is related with the decreasing well production. Fluctuation in between day 4400 and
day 4600 reflects a simulator noise.
On the plot (Fig.26-b) is depicted the WWCT for PROD2. Water cut increases up to day
3400. After that day water cut stars to decline gradually until it get the value of 0.38.
On the plot (Fig.26-c) is depicted WWCT for PROD3. Water cut increases gradually until it
gets the highest value – 0.75 at day 4000. After that day water cut starts slowly to decrease.
Between day 5950 and day 6000 there are some fluctuations in the well water cut which
reflect a simulation noises.
Based on the plots for all three wells can be concluded that the production of oil is going to
decrease fast throughout years. Production of water will increase until some level and
afterwards it will start to decrease fast. The same is the situation in regard to water cut. On the
plot which reflects the pressure drop for the field is also observed the same tendency– Fig.27.
Fig. 27 Pressure forecast for the next ten years of production
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Decreasing pressure in the reservoir, depicted on the plot (Fig.27), affects on the production
of the well. This leads to the conclusion that the difference between the BHP and the reservoir
pressure is significantly decreased. This means that there is not enough pressure to support the
production. Pressure can be optimized by increasing the water injection rate. If the manner of
exploration is kept the same the field is going to stop production in a short period of time –
around day 4000.
Conclusion and discussion
In this project all procedures of history matching and forecasting were applied. Also it was
underlined how important they are.
The final history match was achieved by 40 iterations. Due to them size of the mismatch
decreased significantly. This result has been achieved by improving the reservoir properties in
order to meet the history data. Adjustment of the parameters is related to sensitivity analysis
which detects the incorrect value of the grid cell. Changes into the values are in a small ratio
which prevents from radical changes into the parameters of the reservoir. From case_1 to
case_40 the size of the objective function and respectively the size of the mismatch had been
declining gradually. The level of the mismatch at the end (at case_40) has achieved its
minimum size. Further iteration after case_40 does not give significant results about
improving the match between history and simulated data. This conclusion has been
established based on observation. The error in water cut was corrected successfully. It was
diminished based on the change in the objective function. As a result, the quality of the
reservoir model was also increased and it can be used for further reservoir studies.
Validated model used for forecasting has been received as a result of history matching
process. Forecasting proses has been done in Eclipse. It was simulated for next 10 years of
production. The aim of the forecast is to depict the reservoir future behavior under the same
circumstances: number of wells and rates of water injection. Also it will show what could be
the best solution in order to support or/and increase the production. Forecast has been done by
creating a restart file in Eclipse. A flexible restart method was applied for creating the restart
file. Result of the forecasting was received under the form of plots which illustrate changes in
the parameters (water cut, flow rates, injection rate, pressure, etc.). In the project only some of
them are explained.
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Final result of the accomplished forecasting shows that changes are needed to be done in
order to prolong the lifetime of the field. Water injections which are part of secondary
recovery and were implemented from day one for supporting the pressure and increasing the
production are not enough at that point of reservoir development. Adding a new water
injection well, shutting down some of the production wells and adding new wells cane help to
sustain and increase the production. Also further implementation of the EOR methods can be
a reasonable solution in order to prolong exploration period of the reservoir. Otherwise, based
on the received result the wells have to be shut down.
There were also a number of limitations regarding different aspects of the workflow. Some of
these limitations are related with insufficient information - not enough data available.
Despite the limitations some good results for history matching and forecasting were obtained.
During the calibration process the level of mismatch decreased to its minimum value and the
model was successfully validated.
Most of the difficulties in this project were related with the forecasting part. Nevertheless
forecasting was also successful and it can be used for improving the well production
throughout the years.
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Table of figures
Fig. 1 Relation between input and output data for reservoir simulation .................................... 6
Fig. 2 Mathematical reservoir simulation model (1) .................................................................. 7
Fig. 3 Oil - water relative permeability curves (2) ..................................................................... 9
Fig. 4 Change into the pressure in the reservoir with increasing the distance between wellbore
and the reservoir boundary ....................................................................................................... 11
Fig. 5 General reservoir modeling workflow (3) ...................................................................... 12
Fig. 6 Parameters that affect history matching can be modified during the process (4) .......... 13
Fig. 7 Graphical depiction of the history matching process (4) ............................................... 14
Fig. 8 Error of mismatch (left side) vs. the value of history matching sensitivity parameter (4)
.................................................................................................................................................. 16
Fig. 9 SenEx workflow............................................................................................................. 18
Fig. 10 Steps included in the history matching procedure (5) .................................................. 19
Fig. 11 Input and output data structure for Eclipse software (7) .............................................. 22
Fig. 12 Cumulative oil and water production a) PROD1; b) PROD2; c) PROD3; .................. 25
Fig. 13 Sensitivity for WOC at case_1 ..................................................................................... 26
Fig. 14 Oil and water rates vs. time: a) PROD1; b) PROD2; c) PROD3; ................................ 28
Fig. 15 Water cut ratio: a) PROD1; b) PROD2; c) PROD3; .................................................... 31
Fig. 16 Cumulative oil and water production in the case after history matching process: a)
PROD1; b) PROD2; c)PROD3; ............................................................................................... 34
Fig. 17 Rates of production at the end of the history matching process: a) PROD1; b) PROD2;
c) PROD3; ................................................................................................................................ 36
Fig. 18 Water cur ratio at the end of history matching process: a) PROD1; b) PROD2; c)
PROD3; .................................................................................................................................... 38
Fig. 19 Mismatch for the field in the beginning of the history matching process: a) Field
cumulative oil and water production at case_1; b) Field cumulative oil and water production
at case_40; ................................................................................................................................ 40
Fig. 20 Reduced mismatch regarding oil and water rates due to the history matching process:
a) Field oil and water rates at case_1; b) Field oil and water rates at case_40; ....................... 41
Fig.21 Reduced mismatch at water cut ratio due to the history matching process: a) Field
water cut ratio at case_1; b) Field water cut ratio at case_40; ............................................... 43
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Fig. 22 Changes in porosity during the process of history matching in a grid cell 22:9:1 : a)
porosity value in case_1; b) porosity value in case_40 ............................................................ 45
Fig. 23-a Comparison of the size of the mismatch between case_1, case_10, case_20, case_30
and case_40 .............................................................................................................................. 46
Fig. 24 Well oil production rate (WOPR): a) PROD1; b) PROD2; c) PROD3; ...................... 49
Fig. 25 well water production rate (WWPR): a) PROD1; b) PROD2; c) PROD3; ................. 51
Fig. 26 Well water cut (WWCT): a) PROD1; b) PROD2; c) PROD3; .................................... 53
Fig. 27 Pressure forecast for the next ten years of production ................................................. 54
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References
1. COSSE, R. Oil ans gas development techniques Basics of reservoir engineering. s.l. :
Editions technip, 2005.
2. Hawes, R.I. DARCY'S LAW. THERMOPEDIA. [Online] February 9, 2011. [Cited:
February 11, 2014.] http://www.thermopedia.com/content/683/.
3. Streamsim Technologies. [Online] [Cited: Sept 20, 2013.]
http://streamsim.com/technology/history-matching.
4. Ozgen, James R. Gilian and Chet. Reservoir Simulation: History Matching and
Forecasting. s.l. : Society of Petroleum Engineers, 2013.
5. Inc., FirmSoft Technologies. SenEX User`s Manual. 2013.
6. History-Matching With Sensitivity-Based Parameter Modifications at Grid-Block Level. H.
Almuallim, SPE, FirmSoft Technologies, Inc., K. Edwards, SPE, Barrick Energy Inc., L.
Ganzer., SPE, TU Clausthal. s.l. : Society of Petroleum Engineers, 2010. SPE 131627.
7. Schlumberger. Eclipse 100 UserCourse. 1999.
8. —. Simulation SoftwareManuals 2008.1.
9. Wikipedia. Wikipedia. [Online] [Cited: 11 20, 2013.]
http://en.wikipedia.org/wiki/Finite_volume_method.
10. Dean S. Oliver, Albert C. Reynolds, Ning Liu. Inverse Theory for Petroleum Reservoir
Characterization and History Matching. s.l. : Cambridge University Press, 2008.
11. Romeu, Régis Kruel. History Matching and Forecasting. [Online] april 2010. [Cited:
october 15, 2014.] www.onepetro.org.
12. Developing New Fields Using Probabilistic Reservoir Forecasting. C.S. Kabir and A.
Chawathe, ChevronTexaco. s.l. : SPE Reservoir Evaluation and Engineering, february 2004.
13. ASSISTED HISTORY MATCHING FOR PETROLEUM RESERVOIRS IN THE SOCIAL
COMPUTING ERA. Michel Cancelliere, Dario Viberti and Francesca Verga. 10 , s.l. :
American Journal of Applied Sciences : , 2013. 1546-9239.
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Oil and Gas Technology K9og-7-E13
14. Ahmed, Tarek. Reservoir Engineering Handbook Third Edition. s.l. : Elsevier, 2006.
15. Goal programming: a new tool for optimization in petroleum reservoir history match.
Sayyouh, M. H. Riyadh : Appl. Math. Modelling, 1981, Vol. vol. 5.
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