Integrated Master in Chemical Engineering Development of a reduced-order model for oil and gas reservoirs Master Thesis of Helena Isabel Monteiro Barranha Developed in the context of the Dissertation course carried out in Process System Enterprise Ltd. Supervisor at FEUP: Dr. Alexandre Ferreira, Prof. José Miguel Loureiro Supervisor at PSE: Dr. Adekola Lawal Department of Chemical Engineering July 2015
75
Embed
Integrated Master in Chemical Engineering … Master in Chemical Engineering Development of a reduced-order model for oil ... 1 Introduction ... 3.1 gPROMS ® ModelBuilder ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Integrated Master in Chemical Engineering
Development of a reduced-order model for oil
and gas reservoirs
Master Thesis
of
Helena Isabel Monteiro Barranha
Developed in the context of the Dissertation course
carried out in
Process System Enterprise Ltd.
Supervisor at FEUP:
Dr. Alexandre Ferreira, Prof. José Miguel Loureiro
Supervisor at PSE:
Dr. Adekola Lawal
Department of Chemical Engineering
July 2015
“Science is a delight; evolution has arranged that we take pleasure in
understanding – those who understand are more likely to survive.”
- Carl Sagan, in Cosmos
Development of a reduced-order model for oil and gas reservoirs
Acknowledgements
First of all I would like to thank my family for all the support, not just through this stage,
but during my whole life as a student. Most importantly, I have to mention my parents
who have always encouraged me to be a better student and a better person.
I want to acknowledge Dr. Adekola Lawal, Dr. Alexandre Ferreira and Prof. José Miguel
Loureiro for the support and contribution to this project, and Prof. Costas Pantelides for
this opportunity, in such a great and welcoming company, and for the financial support.
To my “neighbours”, Mariana Gomes and Ana Morgado, I want to show my appreciation for
encourage me to accept this challenge, and for accompanying me in this path.
I want to thank all my professors at FEUP for all that they have taught me, and especially
Prof.Luís Miguel Madeira for providing such great opportunities for academic internships.
Last but not least I would like to mention all my friends at FEUP, who made the last five
year become unforgettable.
Development of a reduced-order model for oil and gas reservoirs
Resumo
Este projeto consiste no trabalho desenvolvido no âmbito da unidade curricular
Dissertação, durante a realização de um estágio académico na Process System Enterprise
Limited, no contexto de modelação de reservatórios de petróleo na área de captura e
armazenamento de carbono.
Este estudo é motivado pela importância da previsão do comportamento de reservatórios
na produção de petróleo e na avaliação da sua propensão para armazenamento de dióxido
de carbono.
Neste relatório é realizada uma revisão das mais importantes propriedades do reservatório
e dos fluídos nele contidos. Quanto ao reservatório são também analisados e avaliados os
diferentes tipos de produção, tendo por base a recuperação de petróleo.
A análise nodal é apresentada como o tipo de modelo utilizado e é feita uma descrição
detalhada da fase de modulação, na qual são explicadas as equações implementadas.
Seguidamente, são utilizados dois casos de estudo para validação e análise de
sensibilidade do modelo.
Após a execução de simulações, foi possível concluir que apesar das simplificações
aplicadas, o modelo e capaz de prever corretamente o comportamento de um reservatório
com mecanismo de gás em solução. Contudo, a baixas pressões o desempenho do
reservatório não é bem descrito devido a grande quantidade de gás que torna o
reservatório mais semelhante a um reservatório de gás.
Palavras-chave: reservatório de petróleo, análise nodal, mecanismo de
gás em solução, modulação.
Development of a reduced-order model for oil and gas reservoirs
Abstract
This project consists on the work developed in the Dissertation course during an academic
internship at Process System Enterprise Ltd, in the context of reservoir modelling within
the carbon capture and storage (CCS) field.
The motivation for this study was the importance of predicting reservoir’s performance on
the oil production side and on evaluating the reservoir’s propensity to injection of carbon
dioxide.
This monography comprises a review on reservoir and fluids’ most important properties
and, on the reservoir side, the types of drive mechanisms are analysed and evaluated,
based on the ultimate oil recovery.
Then Nodal Analysis was chosen as the model-type and a full description of the modelling
stage is made, in which the model equations are explained. Subsequently, two case
studies are used to the model’s validation and sensitivity analysis.
From the simulations executed, it was possible to conclude that despite the simplifications
applied, the model is capable of correctly predict the behaviour of a solution-gas drive
reservoir. However, at low reservoir pressures the reservoir’s performance is not well
described as the high amount of gas present makes it similar to a gas reservoir.
Development of a reduced-order model for oil and gas reservoirs
Background 4
The effect of these changes on rock properties may range from negligible to substantial,
depending on the characteristics of the formation and property of interest.
The main properties are porosity, permeability, saturation, overburden pressure, capillary
pressure, wettability, surface and interfacial tension. These property data are essential
for reservoir engineering calculations as they directly affect both the quantity and the
distribution of hydrocarbons and, when combined with fluid properties, control the flow
existing phases within the reservoir (Ahmed 2006).
2.2.1 Porosity
The porosity of a rock � is a measure of the storage capacity that is capable of holding
fluids. It must be estimated for the entire reservoir and it is affected by compactness,
character and amount of cementation, shape and arrangement of grains and by uniformity
of grain size and distribution (Reinicke et al. 2014, David Martin and Colpitts 1996).
As sediments were deposited during rocks’ formation, some void spaces developed became
isolated from the others by excessive cementation. This result on the existence of two
distinct types of porosity: absolute and effective porosity (Ahmed 2006, David Martin and
Colpitts 1996).
The absolute porosity is defined as the ratio of the total pore space in the rock to that of
the bulk volume. A reservoir may have high porosity but it might not be accessible to
fluids’ flow for lack of pore interconnection.
The effective porosity, used in all reservoir engineering calculations, represents the
interconnected pore space that contains the recoverable hydrocarbon fluids (Ahmed
2006).
The reservoir rock may generally show large variations in porosity vertically but does not
show significant variations in porosity parallel to the bedding planes. Porosity of oil-
bearing sandstones is 15% to 30%, and it is higher than limestones and dolomites’ porosity
(0 to 20%) (Ahmed 2006, Reinicke et al. 2014, David Martin and Colpitts 1996).
2.2.2 Permeability
The permeability, , is expressed in units of Darcy and represents a resistance to flow
caused by the tortuosity of the pore network. It is given by the Darcy equation for the flow
rate, which is represented in equation 2-2, where is the flow rate, ∆ is the pressure
Development of a reduced-order model for oil and gas reservoirs
Background 5
difference, is the length and is the area, has the same meaning as previously
mentioned.
= ∆ (2-2)
The rock permeability controls the directional movement and the flow rate of the
reservoir fluids in the formation (Ahmed 2006, Reinicke et al. 2014).
An adequate knowledge of permeability distribution is critical due to the prediction of
reservoir depletion by any recovery process. It is rare to encounter a homogeneous
reservoir in actual practice. In many cases, the reservoir contains distinct layers, blocks,
or concentric rings of varying permeability.
Where smaller-scale heterogeneities exist, permeability must be averaged depending on
how its values are distributed in the reservoir (David Martin and Colpitts 1996).
2.2.3 Rock compressibility
A reservoir, situated thousands of feet underground, is under an overburden pressure
triggered by the weight of the overlying formations. Overburden pressures vary with
region depending on factors such as depth, nature of structure, consolidation of the
formation, and possibly the geologic age and history of the rocks. However, depth of the
formation is the most important factor to consider (Ahmed 2006).
The compressible force (pressure) applied to the reservoir by the weight of the
overburden does not approach the overburden pressure. The difference between
overburden and internal pore pressure is called effective overburden pressure.
Throughout the pressure depletion processes, the internal pore pressure decreases, and so
effective overburden pressure increases causing the reduction of the bulk volume of the
reservoir, and expansion of the sand grains within pore spaces (Ahmed 2006).
These two volume changes tend to reduce the pore space and, therefore, the porosity of
the rock. Compressibility typically decreases with increasing porosity and effective
overburden pressure (Ahmed 2006, Reinicke et al. 2014).
For most petroleum reservoirs, the rock and bulk compressibility are considered small
when compared with the pore compressibility, . Accordingly, the common term used to
describe the total compressibility is the formation compressibility and it is set equal to
. In general, the formation compressibility is the same order of magnitude as the
Development of a reduced-order model for oil and gas reservoirs
Background 6
compressibility of the oil and water and, therefore, cannot be regulated (Ahmed 2006,
David Martin and Colpitts 1996).
2.2.4 Saturation
Saturation is defined as the fraction of pore volume that a particular fluid occupies. The
densities of the fluids define the way fluids are separated, i.e. oil overlain by gas and
underlain by water. Connate or interstitial water also exists throughout the oil and gas, as
it is retained by forces, called capillary forces because they are only significant in pore
spaces of capillary size.
Critical saturation is what the fluids must exceed to flow, and at this particular
saturation it does not flow. After the displacement operation of the oil from the pores by
water or gas injection, the remaining oil is characterized by the residual oil saturation.
According to the previous types of saturation, movable saturation can be defined as
the fraction of pore volume occupied by movable oil, given by equation 2-3, in which
and are the critical saturations of water and oil, respectively (Ahmed 2006). = − − (2-3)
2.2.5 Wettability
Wettability is the tendency of one fluid to adhere to a solid surface in the presence of
other immiscible fluids. This property is expressed by measuring the angle of contact at
the liquid-solid surface. This angle, termed contact angle �, is always measured through
the liquid to the solid (see Fig. 2-2).
Fig. 2-2 – Rock wettability (Reinicke et al. 2014).
Surface and interfacial tensions � are the surface free energy resulting from molecular
interactions, which affects the capillary pressure. When temperature increases or
dissolved gas is present, surface tension of crude oil decreases (Reinicke et al. 2014, David
Martin and Colpitts 1996).
Development of a reduced-order model for oil and gas reservoirs
Background 7
2.2.6 Capillary Pressure
The capillary forces result from the combined effect of the surface and interfacial
tensions of the rock and fluids, the pore size and geometry, and the wetting
characteristics of the system (Ahmed 2006).
When two immiscible fluids are in contact, a discontinuity in pressure appears between
the two fluids, which depends upon the curvature of the interface separating the fluids.
This pressure difference is the capillary pressure and it is expressed by equation 2-4
where the subscripts and are for the non-wetting and wetting phases, respectively
(Ahmed 2006, Reinicke et al. 2014, David Martin and Colpitts 1996). = − (2-4)
In order to uphold a porous medium partially saturated with non-wetting fluid and while in
presence of the wetting fluid, maintaining the pressure of the non-wetting fluid at a value
greater than that in the wetting fluid is essential. That is, the pressure excess in the non-
wetting fluid is the capillary pressure, and this quantity is a function of saturation (Ahmed
2006, Reinicke et al. 2014).
2.3 Flow geometries
Although the real path of the fluids in the porous medium is irregular, the average paths
may be represented by three flow geometries: linear, radial and spherical, from which the
first two have the greatest practical interest (see Fig. 2-3).
The linear flow consists on parallel flow lines, in a cross section with constant flow. In the
radial flow, straight flow lines converge toward the centre which represents the well.
Finally, the spherical flow is represented by straight flow lines that converge toward a
common centre in three dimensions (Craft and Hawkins 1991).
Fig. 2-3 – Illustration of the three flow geometries (Craft and Hawkins 1991).
Development of a reduced-order model for oil and gas reservoirs
Background 8
2.4 Primary Recovery
The first fraction of crude oil is recovered from the reservoir by fluids expansion, as it is
trapped under pressure in the rock. When pressure starts to drop, the oil’s movement
through the wellbore decreases requiring the installation of pumps to lift the oil to the
surface (Craft and Hawkins 1991, David Martin and Colpitts 1996).
As production continues, pressure declines and it is required that a fluid enters the
reservoir to maintain pressure. The amount of oil that can be produced by the natural
reservoir energy depends on the reservoir type which can be water-drive, solution-gas
drive, gas-cap drive or gravity drainage drive reservoir (Ahmed 2006, Reinicke et al. 2014,
David Martin and Colpitts 1996, Craft and Hawkins 1991).
2.4.1 Water-drive reservoir
In this type of reservoir there is a connection between the oil and a porous, water
saturated rock called aquifer. It is the pressure caused by this compressed water that
forces the oil to the surface. Fig. 2-4 shows a sketch of an idealized system of a reservoir
and an edge aquifer (Craft and Hawkins 1991, Ahmed 2006).
Fig. 2-4 – Top view of an idealized reservoir and an edge aquifer (Sureshjani and Gerami 2011).
As pressure is reduced gradually during oil and gas production, a natural water-flood is
created, displacing the oil in the reservoir almost volume by volume.
This natural displacement maintains pressure, stopping gas from evolving from solution. As
a result, the producing gas-oil ratio suffers little change, particularly if there is no free
gas initially present in the reservoir.
Development of a reduced-order model for oil and gas reservoirs
Background 9
The ultimate oil recovery achieved in this type of reservoir is usually much larger than
under any other mechanisms. However, it depends upon the encroachment efficiency of
the water, which decreases with the heterogeneity increase of the rock, because of the
unevenly spreading of the water (Ahmed 2006, Sureshjani and Gerami 2011, Reinicke et
al. 2014, Craft and Hawkins 1991).
The ultimate oil recovery ranges from 35% to 75% of the original oil in place (Ahmed 2006).
2.4.2 Solution-gas reservoir
Crude oil under high pressure can contain a significant amount of dissolved gas. When oil
is produced, pressure in the reservoir decreases and in some regions it can drop below the
bubble-point pressure, which leads to gas escape (Craft and Hawkins 1991).
In this type of reservoir the pressure drops rapidly and continuously as there are no
extraneous fluids or gas caps to displace the oil removed until the bubble point is reached.
When the reservoir pressure reaches the bubble point, the gas evolves from solution
throughout the reservoir, and once the critical gas saturation is exceeded, the free gas
flows towards the wellbore and gas-oil ratio increases.
The formation of gas saturation along the reservoir contributes for this type of drive
mechanism to be the least efficient method when it comes to ultimate recovery. It can
vary from 5% to about 30% (Ahmed 2006, Craft and Hawkins 1991).
2.4.3 Gas-cap drive reservoir
The displacement of oil is due to the expansion of compressed gas on the top of the
reservoir called gas-cap, when pressure decreases during oil production (see Fig. 2-5).
Fig. 2-5 – Gas-cap expansion (Reinicke et al. 2014).
Development of a reduced-order model for oil and gas reservoirs
Background 10
The reservoir pressure decreases slowly, tending to being maintained higher than in a
solution-gas drive reservoir. The degree at which pressure can be maintained depends on
the volume of the gas cap compared to the oil in the reservoir.
The pressure wave from the gas cap expansion, combined with the fact that no gas
saturation is being formed makes this type of drive achieve a recovery that ranges from
20% to 40% (Ahmed 2006, Reinicke et al. 2014).
2.4.4 Gravity drainage drive reservoir
This drive mechanism is a result of differences in densities of the fluids in the reservoir.
The action of the gravitational forces in the fluids determines the relative positions of
fluids: gas on top, oil underlying the gas, and water underlying the oil.
During the long periods of time of petroleum accumulation and migration processes, it is
assumed that the reservoir fluids are in equilibrium, which means that gas-oil and
oil-water interfaces are essentially horizontal (Ahmed 2006).
Gravity drainage of fluids is present in all reservoirs, but it may have larger contribution to
oil production in some reservoirs.
The rate of pressure decline on this type of mechanism depends mainly upon the amount
of gas conservation. If the reservoir operates only under drainage drive, pressure will
decline rapidly.
The evolved gas migrates to the top of the field due to gravitational segregation of the
fluids, which leads to low gas-oil ratio when producing from low wells. On the other hand,
high wells will involve increasing gas-oil ratio.
Ultimate recovery will vary widely, depending on the extent of depletion of gravity
drainage alone. In this type of reservoir it is important that the oil saturation near the
well is maintained as high as possible, because high oil saturation means a higher oil
flowrate and lower gas flowrate.
Gravity drainage mechanism is best exploit if the wells are located as low as possible to
avoid any gas near the well. Also, permeability, oil viscosity, and producing rates are
major factors affecting the ultimate recovery (Ahmed 2006).
Development of a reduced-order model for oil and gas reservoirs
Background 11
2.5 Secondary and Enhanced Oil Recoveries
When the rate of oil production declines, it can be increased by injecting secondary
energy as gas or water, in order to maintain pressure in the reservoir.
The injection of water is in some cases designed to disposal of brine water or to
implement a water-drive, after primary recovery. If permeability is too low, gas injection
is preferred as the rate of water injection may be low (Reinicke et al. 2014).
The enhanced oil recovery (EOR) processes are the techniques which allow a higher
recovery than primary or secondary recovery. These techniques include miscible
processes, chemical oil flooding, thermal recovery and microbial processes (Reinicke et al.
2014, David Martin and Colpitts 1996).
2.5.1 Mobility control
The mobility of any fluid, , is given by equation 2-5, which represents the ratio of the
fluid’s permeability to its viscosity. = (2-5)
The mobility ratio is calculated by equation 2-6. = � � � � �� � � � (2-6)
To improve the displacement efficiency, the mobility ratio should be reduced to one or
less, which is called mobility control (Ahmed 2006, Dake 1978, Sureshjani and Gerami
2011).
2.5.2 Water flooding
Water flooding is the most common method of secondary recovery, but before undertaking
this process it is necessary to consider factors such as reservoir geometry and depth, fluids
and rock properties and fluid saturations (Ahmed 2006).
As the oil is moving in head from the injected water front, its permeability must be
evaluated at the initial water saturation.
The water’s mobility before breakthrough will be constant, as water permeability is
characterized by average water saturation. After breakthrough, the water average
saturation increases, which increases the mobility ratio (Ahmed 2006, Dake 1978).
Development of a reduced-order model for oil and gas reservoirs
Background 12
The determination of the optimum time to water-flood is based on oil recovery prediction,
production flowrates and on the costs of maintenance and monetary investment (Ahmed
2006).
2.5.3 Carbon dioxide flooding
Carbon dioxide, CO2, is injected in the reservoir as gas, and its high solubility in oil has
favourable effects on oil recovery. When CO2 is dissolved, oil saturation increases above
the residual saturation, increasing oil’s permeability. Also, oil’s viscosity is reduced,
improving the mobility control (Ahmed 2006, Dake 1978).
This type of enhanced oil recovery also enables the CO2 storage, for which it is required
that the reservoir is situated at depths below 800 m, where it is in a liquid or supercritical
state.
Once injected in the reservoir, the fraction of CO2 retained depends on physical and
geochemical trapping mechanisms, such as an impermeable layer (“cap rock”), and
capillary forces, respectively (Metz et al. 2005).
Development of a reduced-order model for oil and gas reservoirs
Materials and Methods 13
3 Materials and Methods
3.1 gPROMS® ModelBuilder
gPROMS® ModelBuilder 4.1.0 was the platform used to develop the reservoir model. This
advanced modelling and flowsheeting tool is the heart of the gPROMS® products (see
Fig. 3-1).
ModelBuilder is used to build, validate and execute steady-state and dynamic process
models of any complexity. It combines industry-leading custom modelling competences
with a process flowsheeting environment, to offer the process industries the most
powerful advanced process modelling tool (PSE).
The conception of a new model entity enables the user to write the model equation in the
language tab, build the input window, by defining the required inputs in the interface
language tab, and select the icon of the model in the interface tab.
For this project, after the implementation of the equations, a Process Entity was created
for each simulation to define how it should be performed. The simulations can be
executed for a chosen period of time, or it can be selected a condition which will
determine the end of the simulation. This is defined on the Schedule tab of the Process.
Fig. 3-1 – gPROMS ModelBuilder appearance.
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 14
4 Mathematical Description
4.1 Nodal Analysis
The nodal analysis procedure consists of selecting a division point in the producing well
and dividing the system into a reservoir dominated component and a piping system
component.
All the components upstream of the node compromise the inflow section (reservoir), while
all of the components downstream influence the outflow section (pipes).
The method’s implementation requires that the flow into the node equals the flow out of
the node, and that only one pressure exists at the node (Beggs 2003).
The pressure drop in the reservoir varies with flowrate, and if the node’s pressure is
plotted against oil flowrate , two curves will be produced, i.e. one for each section,
which result from the pressure losses in the respective component. The intersection of the
two curves will give the conditions that satisfy the requirements of the method (see
Fig. 4-1).
Fig. 4-1 – Nodal analysis plot (Beggs 2003).
At a particular time of the systems’ life, two pressures are fixed, i.e. they are not a
function of the production flowrate. These pressures are the reservoir average pressure
and the system’s outlet pressure which usually is the separator pressure. However, if
either pressure suffers changes, the curves will change, and the intersection will be
shifted. This leads to a new flowrate and a new node’s pressure (Beggs 2003).
pn
od
e
qo
Inflow
performance
Outflow
performance
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 15
4.1.1 Reservoir: Solution-gas drive modelling
This model considers the node at the bottom hole of the production well, and the
separator pressure as the outlet pressure of the whole system.
As oil is produced, the average reservoir pressure decreases, but it is also important to
analyse the pressure profile inside the reservoir. Equation 4-1 relates the pressure inside
the reservoir with the radius , and a typical pressure profile is shown in Fig. 4-2 (Reinicke
et al. 2014).
− = . ℎ �n � (4-1)
Where is the pressure at radius , is the wellbore flowing pressure, ℎ is the reservoir
thickness, and the radius of the well. This profile considers radial and horizontal flow of
oil.
Fig. 4-2 – Illustration of a reservoir and its pressure profile.
The pressure inside the reservoir decreases as it gets near the production well, and during
oil production the pressure profile is altered because of the drop in the average reservoir
pressure. This alteration in the pressure profile is illustrated in Fig. 4-3.
Fig. 4-3 – Reservoir’s pressure profile at different production times.
pr
(psi
a)
r (ft)
0 days
5 days
10 days
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 16
The inflow performance relationship (IPR), which relates the wellbore flowing pressure
with the average reservoir pressure to obtain the oil flowrate, is represented by Darcy’s
law (equation 4-2) (Chen 2007, Beggs 2003).
= ℎ( � − � )[ �⁄ − . ] (4-2)
Where is the radius of the drainage area of the reservoir.
The IPR curve is affected by the changes in oil properties as relative permeability and
viscosity. Moreover, it is important to consider gas evolving from solution if pressure drops
below the bubble point pressure.
Both equation 4-1 and equation 4-2 are obtained from Darcy’s law, but the first one was
rearranged to calculate pressure, while the second is used to calculate the flowrate,
based on the pressures at the end and beginning of the system. The combination of the
two ways of writing the same equation allows to model oil’s behaviour inside the
reservoir, assuming a constant production flowrate flowing in the reservoir and entering
the well.
The presence of water, when above its irreducible saturation will also affect the
production flowrate, which will now contain water, and consequently the pressure profile
will be changed.
The water flowrate is expressed by equation 4-3, and the reservoir pressure profile will
be, in this case, written as a function of the total flowrate and of the two fluids’
properties (Craft and Hawkins 1991).
= �ℎ( � − � )w w[ �⁄ − . ] (4-3)
As water and oil are produced, both quantities will be important to evaluate the decrease
in average reservoir pressure. Equation 4-4 relates the initial average reservoir pressure � with the cumulative productions of oil and water (Ahmed 2006).
= � − + ��� (4-4)
The cumulative productions are calculated by establishing that their derivatives are equal
to the flowrate of the correspondent fluid.
Production of fluids leads to pressure drop, and consequently to a decrease in total
flowrate as seen in Fig. 4-4.
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 17
Fig. 4-4 – Pressure and oil’s flowrate evolution.
Pressure can also be evaluated considering both its changes with time and distance from
the well. An example of pressure behaviour inside a reservoir above the bubble point is
shown in Fig. 4-5. The pressure profile’s shape is evident throughout the reservoir, and so
is the pressure drop with time, considering that the radius zero is the centre of the
production well.
Fig. 4-5 – Pressure evolution in time and with distance from the production well.
The oil’s saturation is calculated by a material balance (equation 4-5) which relates the
initial oil in the reservoir with the cumulative productions of oil and dissolved-gas .
= � − − (� �⁄ )�� (4-5)
0
200
400
600
800
1000
1200
1400
1600
1800
2400
2600
2800
3000
3200
3400
3600
0 500 1000 1500 2000 2500
qt
(stb
/d)
pr
(psi
a)
t (d)
Average pressure
Total flowrate
2000
1849
1698
1547
1396
1245
1094
943
792
641
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 18
The cumulative produced dissolved-gas is multiplied by the ratio of the fluids’ densities to
convert it from volume of gas to the correspondent volume of oil in which it is dissolved.
The water solubility is obtained by applying the principle that the sum of all of the fluids’
volume in the reservoir is equal to the pore volume (equation 4-6). = + (4-6)
Once pressure reaches the bubble point, solution-gas starts evolving from solution. This
division point is characterized by evaluating the average reservoir pressure; however,
some regions of the reservoir may have already dropped below the bubble point before
the considered moment.
This new stage can also be divided into two periods: below and above the critical gas
saturation , which determines the moment when gas starts to flow.
First, gas comes out of solution but it is unable to flow as its saturation in the reservoir is
less than its critical saturation. While gas is accumulated in the reservoir, pressure drops
slowly, improving oil’s production.
For the reservoir pressure evaluation not only the oil and dissolved gas produced need to
be considered, but also the gas evolved from the solution. The volume of gas in the
reservoir � is calculated by the material balance in equation 4-7. � = � × � − � − − (4-7)
Where � is the original oil in place, � and are the initial and current dissolved-gas
oil ratio, respectively.
Thus, this expression says that the volume of free gas in the reservoir is the difference
between the initial dissolved-gas and the current and produced dissolved-gas.
When gas is formed, volume of oil is reduced, and this volume reduction ∆ is given by
equation 4-8, considering that the gas’ mass is conserved. � × � = ∆ × � (4-8)
As a result, the average reservoir pressure variations is now expressed by equation 4-9,
which considers the cumulative formation of gas.
= � − + �+ −� � −���� (4-9)
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 19
As soon as gas saturation exceeds the critical saturation, gas is able to flow and it is
produced with oil.
The oil’s flowrate in saturated reservoir conditions is estimated by equation 4-10, the gas
flowrate by equation 4-11 and finally the water flowrate by equation 4-12 (Beggs 2003,
Craft and Hawkins 1991).
= . ℎ� [ln / � − . ] ( − ) (4-10)
= . ℎ� [ln / � − . ] ( − ) (4-11)
= . �ℎ� � �[ln / � − . ] ( − ) (4-12)
In this period, the equation for determining the pressure profile is obtained by summing
the three previous equations, and then it is rearranged to calculate pressure in each point
of the horizontal flow.
Moreover, the change in reservoir pressure has to consider the cumulative volume of gas
leaving the reservoir , which will increase the pressure drop (equation 4-13).
= � − + �+ + −� � −���� (4-13)
Whether the critical gas saturation was exceeded or not, the saturations’ calculation has
to consider the existence of free gas in the reservoir, which means that gas saturation
is greater than zero under the bubble point. The oil saturation is now estimated by
equation 4-14.
= � − − (� �⁄ )−�(� �⁄ )�� (4-14)
The water saturation can also be calculated by a material balance, if its production is
considered in this stage, and so the gas saturation is estimated by equation 4-15. = + + (4-15)
Fig. 4-6 shows the evolution of the average reservoir pressure under the considered
mechanism, differentiating the pressure’s behaviour as production proceeds.
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 20
Fig. 4-6 – Evolution of reservoir pressure under a solution-gas drive mechanism.
The calculation of oil’s recovery must consider the gas in solution, as it has been proved
that a large part of the liquid hydrocarbon produced is obtained from the gas that enters
the well. This approach tries to avoid the false assumption that all the free gas entering
the well remains in the gaseous phase as it is produced (Cook, Spencer, and Bobrowski
1951).
4.1.2 Piping System
The outflow performance depends on the separator’s pressure and on the pressure loss in
the well, an illustration of the system is represented in Fig. 4-7. Equation 4-16 represents
the application of Bernoulli’s equation between the bottom of the well and the separator
(Ahmed 2006, Campos 2012). = + ∆ + ∆ (4-16)
The friction losses ∆ are given by equation 4-17, where is the friction factor, and the
gravitational losses ∆ described by equation 4-18.
∆ = � � (4-17)
∆ = � (4-18)
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1000 2000 3000 4000
pr
(psi
a)
t (d)
Above BP
Below BP and Sgc
Below BP; Above
Sgc
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 21
Fig. 4-7 – Illustration of the pressure losses in the piping system from the reservoir to the separator (Beggs
2003).
The density and velocity of the mixture must be evaluated at the well’s condition, so it
has to be considered that inside the well phase behaviour is altered.
The oil and free gas flowrate entering the separator can be expressed by equations 4-19
and 4-20 respectively, considering the change in gas solubility from the bottom hole to the
separator.
| = | − | ( | − | ) �� (4-19)
| = | + | ( | − | ) (4-20)
4.2 Reservoir and Reservoir Fluids’ properties
A good prediction of the rock and fluids’ properties is vital to correctly estimate the oil
production of a reservoir.
4.2.1 Compressibility
Fluids can be divided in incompressible, slightly compressible and compressible. From
these categories, the water and oil are considered slightly compressible and gas is
compressible. Compressibility calculation can be made using correlation, but for this
model it is considered constant for each fluid.
Δp
pr
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 22
The slightly compressible components have compressibilities of around 10-6 psia-1, while
gas’ compressibility varies from 10-3 to 10-4 psia-1, for pressures from 5000 to 500 psia,
respectively (IHS 2014).
For the model estimations the total compressibility of the system is required, and it is
obtained by equation 4-21, considering that the rock is also slightly compressible.
= + + + (4-21)
Although all compressibilities are considered constant through time, the total
compressibility will suffer changes because of its dependence on fluids’ saturation, which
change as production progresses (Ahmed 2006, Craft and Hawkins 1991).
4.2.2 Porosity
The assumption that the reservoir rock is slightly compressible determines that the pore
volume will suffer alterations during production. This variable can be predicted with
equation 4-22 (Ahmed 2006, Craft and Hawkins 1991).
� = ��� (4-22)
This relationship indicates that when pressure drops, the pore space decreases, decreasing
consequently the volume available for fluids’ storage.
4.2.3 Permeability
Absolute permeability is crucial to determine the fluids’ flow through the porous medium,
and, if no data is available, it can be predicted by equation 4-23 (Timur 1968, Ahmed
2006).
= . × � .�� (4-23)
Although absolute permeability is the same for every fluid, their flow is different because
it depends on fluids’ relative permeability. This type of permeability is calculated based
on fluid’s saturation.
Moreover, it is important to know the conditions of the reservoir to define what type of
relative permeability has to be used. When oil and gas are present, the relative
permeabilities can be calculated by equations 4-24 and 4-25, which were proposed by
Corey (1954) for gas-displacing oil processes.
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 23
= − − �� (4-24)
= − �� − − �� (4-25)
However, the Corey’s method is only valid to well-sorted homogenous rocks, and the high
dependence of this property on the rock type requires a set of equations which can be
used to any type of rock (Ahmed 2006). Table 4-1 shows the different equations for various
types of rocks.
Table 4-1– Gas-oil relative permeabilities for various types of reservoir rocks (Ahmed 2006).
Type of reservoir rock � �
Unconsolidated sand, well
sorted ( − � ) ( − − � )
Unconsolidated sand, poorly
sorted ( − � ) .
( − − � ) − ( − � ) .
Cemented sandstone, oolitic
limestone, rocks with vulgar
porosity
( − � ) ( − − � ) − ( − � )
For a successful estimation of oil’s production, it is important that these equations are
available in the model, so that they can be used based on the type of reservoir that is
being studied.
On the other hand, if gas does not exist, and there is water in the reservoir, oil and water
relative permeabilities are estimated by the equations in Table 4-2, depending on the type
of reservoir rock.
Development of a reduced-order model for oil and gas reservoirs
Mathematical Description 24
Table 4-2 – Oil-water relative permeabilities for various types of reservoir rocks (Ahmed 2006).