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Dispersed Flow
Reservoir Fluid Displacement Core
Learning Objectives
By the end of this lesson, you will be able to:
Recognize immiscible fluid displacement and sources of dataused for calculations
Describe the assumptions and limitations of the Buckley-Leverett theory
Describe the derivation of the dispersed fractional flow equation
Recognize the impact of mobility, gravity, and capillary pressureon the dispersed fractional flow equation
Identify how the Welge solution can be applied before, during,and after breakthrough
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1
In This Section
Immiscible Fluid Displacement
Where does immiscible fluid displacement occur?
Buckley-Leverett Theory
Fractional Flow
Capillary Pressure
Fluid Mobility
Gravity Term
Frontal Advance and Welge Solution
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What is the Buckley-Leverett Theory?
Calculates pore level displacement efficiency
Some moveable oil is bypassed, due to: Viscosity contrast Relative permeability Capillary pressure
Assumes “piston-like” displacement and linear flow
Immiscible displacement characterized as: Uniformly dispersed Segregated
This module uses an oil-water system but analogous equations apply for a gas-oil system and a water displacing gas system, these will be covered in the fundamental module of this series.
Buckley-Leverett Theory
Buckley-Leverett only applies to linear systems• Most injection patterns and real formation displacement
systems are not linear
Problem B, however, will cover a linear displacementsystem
• Sweep efficiency
• Horizontal and non-horizontal reservoirs
Linear Systems
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Buckley-Leverett Theory
Front Boundary
Rapid saturation change
Ahead of the Front
Displaced oil bank
Assume that a stabilized flood front develops, including:
Behind the Front
Two-phase flow of oil and water
Buckley-Leverett Theory Assumptions
Front
Horizontal Reservoir
Variable Saturations Initial Conditions
Assumptions
1. Incompressiblein-situ and injectedfluids
2. Constant averagesystem pressure
3. Steady state flow
• Same reservoirflow out as in
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Fractional Flow at Reservoir Conditions
Where:
qo = oil flow rate per unit cross-sectional area, at any point along the core
qw = displacing phase (here water) flow rate per unit cross-sectional area, at the same point in the core as qo
qt = total flow rate per unit cross-sectional area, at any point in the system
Assuming water as the displacing phase:
qin qout= = qo + qw = qt
Fractional Flow at Reservoir Converted to Surface Conditions
fw = qw / qt = qw / (qw + qo)
Where:
fw = fraction of the total flow rate at a given point that is the displacing phase at reservoir conditions of temperature and pressure
Assuming water as the displacing phase:COPYRGHT
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Fractional Flow at Reservoir Converted to Surface Conditions
Where:
Wc = the amount of water produced at surface, as a fraction of the total production of oil and water
Bo = reservoir volume of oil (at reservoir temperature and pressure)/surface volume of oil at standard conditions
Bw = Reservoir volume of water/surface volume of water
))1)1((*/( owc BfBBW wo
Fractional Flow – Reservoir Conditions
Where:u = the linear direction of flow (measured from the inlet end),∂po / ∂u = the pressure gradient in the oil phase∂pw / ∂u = the pressure gradient in the displacing phaseα = the angle of the fluid flow with respect to the horizontal (updip flow is assigned to be "+"; downdip is "-")qo = displaced fluid (oil) flow rate per unit of cross-sectional area normal to uqw = displacing fluid flow rate per unit of cross-sectional area normal to uρ = fluid densityµ = fluid viscosityo = subscript indicating displaced or oil phase, and
w = subscript indicating displacing phase
sino
o
o
oo gu
p
k
q
sinw
w
w
ww gu
p
k
q
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Fractional Flow – Reservoir Conditions
o
w
w
o
c
to
o
w
kkup
qk
f
1
sin)(434.0001127.0
1
Thanks to:
Capillary pressure
(Pressure difference across the interface of two fluids in a confined space)
and Algebra, you get:
Units: md, cp, Bbl/D/ft2, feet, psi; (=w‐o)
Pc = Po- Pw
Absolute permeability, grain size, rock type
Saturation, wettability, interfacial tension, pore structure
Fractional Flow – Reservoir Conditions
1
2
3
4
Flow rate and angle of dip
Fluid viscosity and density
Capillary pressure
Effective fluid permeabilities
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Define Fluid Mobility
w = kw / w = k krw / w
o = ko / o = k kro / o
Where:
λW = mobility of the displacing phase (water), darcy/cp
kW = effective permeability of the displacing phase, darcies
µW = viscosity of the displacing phase, cp
λo= mobility of the displaced phase (oil), darcy/cp
ko = effective permeability of the displaced phase, darcies,
µo = viscosity of the displaced phase, cp
Mobility Ratio
Where:
krw (Sw) = the relative permeability of the displacing phase evaluated at the average displacing-phase saturation behind the front
kro (Swi) = relative permeability to oil evaluated at the initial water saturation in the oil bank
wwiro
owrw
wwio
oww
aheado
behindw
Sk
Sk
Sk
SkM
)(
)(
)(
)(
)(
)(
Of greater importance is the mobility ratio – how the displacing or behind fluid moves with respect to the displaced or ahead fluid:
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Mobility Ratio
Mobility Ratio
Compares fluid mobilities at two points in the system
Favorable displacement
M < 1
Oil has greater mobility
Unfavorable displacement
M > 1Displacing fluid moves
more easily
Fractional Flow Influences
Sw
fw
0
0.5
1
0.2 0.5 0.8
-30º
30º
1 - SorSwirr
Dip angle,
Displacementupwards
Displacementdownwards
In a waterflood:
Displacement efficiency is increased when angle of dip is increased
• Water injected low on structure
Water injection rate is decreased, slower displacement is better
Density difference is greatest
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9
Fractional Flow Influence on Wettability
o
w
w
o
c
to
o
w
kkup
qk
f
1
sin)(434.0001127.0
1
Kw will be lower for a water-wet system. Thus, the mobility term (denominator) will be lower.
Water-wet reservoirs yield a higher displacement efficiency than comparable oil-wet reservoirs.
Fractional Flow Viscosity Influences on Mobility Term
Displacement efficiency is improved by• Increasing water viscosity with polymers
• Decreasing oil viscosity using hot water orsteam
Viscosity Mobility RatioCOPYRGHT
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Gravity Term
sin)(434.0
001127.0
u
p
q
k c
ot
o
Maximum displacement of oil will resultwhen fw is kept to a minimum.
Thus, you would like the Gravity term,
to be as large as possible.
Fractional Flow Influence on Gravity Term
o
w
w
o
c
to
o
w
kkup
qk
f
1
sin)(434.0001127.0
1
o
w
rw
ro
to
roa
w
k
k1
sinq
Akk000488.01
f
Neglect ∂pc /∂u since it is only significant at the front Front is region of rapid
saturation change
Neglecting derivativeresults in shock front
Field Units: k (md)μ (cp)qt (RB/D)A(ft2)α (degrees) = (water - oil)
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11
Fractional Flow – Capillary Pressure Pc and Gravity Ignored
(1-Sorw)Swirr
1.0
0
0
fw
Sw
Neglecting capillary pressure and the effects of gravity:
o
w
w
ow
kk
f
1
1
This is the most common form of the fractional equation
The resulting curve:
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Why Calculate Fractional Flow Curves?
Used to compute the saturation distribution in a linear waterflood system as a function of time
Saturation distribution is a function of the slope of thefractional flow curve
Used to get the oil recovery behind the front at breakthrough
Can be used to predict oil recovery and required water injection vs time
Shock Front Travels through Reservoir
Time 1 32 4
Each saturation plane travels at its own constant speed andis proportional to gradient of the fw curve;
The displacing fluid saturation moves at the same velocity atthe leading edge of the front.
0
1-Sor
Swc
0% 100%Distance through reservoir
Capillary trapped (residual) oil
Capillary trapped (connate) water
Time 1 2 3 4COPYRGHT
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13
Frontal Advance and the Welge Solution
1.0
0
1.0
0
Capillary Trapped (Residual) Oil
Capillary Trapped (Irreducible) Water
Injection Production
Wat
er S
atu
rati
on
Distance
1- Sor
Swirr
1- Sor
Swirr
WaterAccumulation
Flow ofWater In
Flow ofWater Out
(∆Αφx) dSw/dt = Qt∆fw
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Efficiency of Displacement
1.0
0
1.0
0
Capillary Trapped (Residual) Oil
Capillary Trapped (Irreducible) Water
Injection Production
Wat
er S
atu
rati
on
Distance
1- Sor
Swirr
1- Sor
Swirr
Piston Versus Non-piston Displacement
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Mobility Ratio
Where:
x (ft) = distance traveled by a fixed saturation. For a linear system, the front is traveling perpendicular to the cross-sectional area A (ft2), refer to the lavender Time bars on our displacement diagram.
Wi = cumulative water injected, bbls
t = time, days
dfw / dSw = derivative of the fractional flow curve at the saturation of interest, and must be equal for all saturations in the stabilized zone
Via substitution and integration:
x = (5.615 Wi / A) * dfw / dSw
Welge Graphical Solution
Sw
fw
0
0.2
0.4
0.6
0.8
1
0.2 0.3 0.4 0.5 0.6 0.7 0.8
fw,bt
SwfSwirr
wS
wcwSw
w
SSdS
df
fw
1
Simple graphical solution atwater breakthrough
Shock front saturation
Average saturation behindfront
Water saturation at flood frontis determined from point oftangency on fractional flowcurve (greatest slope) change(Part IV of Problem)
Average water saturationbehind the flood front isdetermined from where thetangent line intersects fw = 1.0
Problem A Parts II & III – Calculate the Sw at breakthrough and the average Sw
behind the front for each Reservoir in Part I
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15
Recovery Efficiency at Water Breakthrough
wi
wiwBTD
S
SSE
1 oi
oBToiD
S
SSE
or
Where:
ED = displacement efficiency or fractional recovery of the total original oil in place, at pore level (fraction)
Sw = mean displacing-phase saturation behind the front at breakthrough (found by extending tangent of fw curve to fw = 1.0), fraction
Swi = initial (irreducible) displacing-phase saturation, fraction
Problem A Part IV – Using Part III, the average Sw, calculate the ED at breakthrough for each of the three reservoirs
Surface Water Cut
Where:
Wc = Surface fraction of water of total oil and water (qw/(qo + qw)
Bw = Reservoir volume of water/surface volume of water
Bo = Reservoir volume of oil (at reservoir temperature and pressure)/surface volume of oil at standard conditions
Convert to a surface water cut by converting reservoir volumes to surface volumes using:
Problem A Part V – Using Part III and reading the Fw at breakthrough, calculate the corresponding Wc for each oil that has a different Bo
)1)/1((*(
ww
oc
FB
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Welge Solution Results
Swf provides the fractional flow at the front, fwf, which is used to calculate the surface water cut at breakthrough
Sw provides the average oil saturation behind the front which is used to calculate the pore displacement efficiency, ED
Breakthrough time (days) can be estimated from ED, OOIP and the producing oil rate o
D
q
EOOIPt
Problem A Parts VI and VII – Calculate mobility ratio and, using Part IV as well as the data on OOIP and qo, calculate the t (days) for each reservoir to breakthrough.
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17
Welge Solution after Breakthrough
1
0
1 - Sor
Swc
Capillary trapped (residual) oil
Capillary trapped (connate) water0% 100%Distance through reservoir
Swe
Swf
0.8
0.85
0.9
0.95
1
0.55 0.6 0.65 0.7Sw
fw
SwBT
fwe
Swe
Sw
fw
0
0.2
0.4
0.6
0.8
1
0.2 0.3 0.4 0.5 0.6 0.7 0.8
BTwS Sw
Welge Procedure after Breakthrough
1
2
3
4
Construct tangents to fractional flow curve at increasing values of water saturation.
Average water saturation behind the front is determined from intersection of tangents with fw = 1.0
Calculate recovery efficiency
Calculate cumulative water injected
wi
wit
S
SSER
1
..
w
w
i
S
fQ
1 Where:
Qi = cumulative water injected, pore volumes
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Welge Solution Formulas
After breakthrough, Sw = Swe
Water saturation at the production well
Where:
Wid = the Cumulative water injected in the displacement process in bblor m3
PV = pore volumes in reservoir bblor reservoir m3
qi = the rate of injection of water in bblsper day or m3 per day
t = time in days
Npd = Cumulative oil produced in thedisplacement process in bbl or m3
Welge Solution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 5 10 15
Water injection displacing oil, Wid
Cu
mu
lati
ve o
il re
cove
red
fr
om
dis
pla
cem
ent,
Np
d
After breakthrough, an extended period of high water cut production takes place to recover remaining mobile oil. If a tangent line cannot be constructed, then a stabilized front or oil bank will not form.COPYR
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19
Map of Reservoir
The reservoir being studied is a thin, shallow, under-saturated oil accumulation, it is under-going line drive dispersed flow water flood, as shown below.
3000
3100
3200
3300
3400
3500
OWC 3530 3600
0 1000' 2000'
DEPTH: ft ss
Cross Section (Part 1)
The LD-A reservoir consists of four patterns with dimensions as follows:
1870'
1000'
15.5°
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Understanding the Fractional Flow Equation (Part 2)
Set up a table in Problem B.xls as shown below. Fill it in using Table 1, Figure 2, and your knowledge of the Welge solution to the fractional flow equation.
Plot of Fractional Flow (Part 3)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
Fra
ctio
nal
Flo
w o
f W
ater
Water Saturation (fraction)
At Breakthrough
Calculate the frontal saturation and the average saturation of water behind the front.
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21
Oil Recovery at Breakthrough (Part 4a)
Use either of the equations below to calculate Part 4. You should also prove to yourself that they are the same.
wi
wiwBTD
S
SSE
1 oi
oBToiD
S
SSE
Using your units of choice, calculate oil recovery at breakthrough and time to breakthrough.
Water Injection and Front Maintenance (Part 4b)
Review the reservoir rate of water to maintain reservoir pressure and surface rate from the total field of 2000 bopd. How much is it?
2400 bwpd (381 m3wpd)
The fractional flow equation has to calculated at reservoir conditions and the difference between oil reservoir volume and water reservoir volume is 1.2/1 (or 20%).
Refer to the problem statement Bo and Bw
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Water Drive Recovery Efficiency, in Real Reservoir
Where:
RE = water drive recovery efficiency, percent
ED = pore level recovery from displacement
EV = recovery vertically
EA = recovery areally
Boi = initial oil formation volume factor, rb/STB
Bo = reservoir volume of oil (at reservoir temperature and pressure)/surface volume of oil at standard conditions
Water drive oil recovery efficiency (above pb)
Typical range ofrecovery efficiency:under 40% to over60%
Statistical estimate forwater drive yields anaverage recoveryefficiency of 51%
Water Injection and Front Maintenance (Part 4c)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
Fra
ctio
nal
Flo
w o
f W
ater
Water Saturation (fraction)
At Breakthrough
Calculate the fractional flow of water at the time of breakthrough from the curve on the left.
What is it?
After breakthrough, do you think the required water rate will go up or down?
77%
Down, because of the 20% volume difference between oil and water.
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23
Oil Recovery at Breakthrough (Part 4d, Part 5)
Finish the problem using the equation below for water injection rate after breakthrough to maintain reservoir pressure.
Water Component = (0.77*2000*1.0) + Oil Component(0.23*2000*1.20) =
To understand the Welge solution after breakthrough, review the calculations in Part 5, as well as the solution.
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Learning Objectives
You are now able to:
Recognize immiscible fluid displacement and sources of dataused for calculations
Describe the assumptions and limitations of the Buckley-Leverett theory
Describe the derivation of the dispersed fractional flow equation
Recognize the impact of mobility, gravity, and capillary pressureon the dispersed fractional flow equation
Identify how the Welge solution can be applied before, during,and after breakthrough
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25
Aquifers
Reservoir Fluid Displacement Core
Learning Objectives
By the end of this lesson, you will be able to:
Determine the rock, fluid, operational, and reservoir geometricfactors affecting water influx (We) into your reservoir
Calculate We using different models
Understand which descriptive parameters you should change soyou can match the calculated We to the observed reservoirpressureCOPYR
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In This Section
Aquifer Models
Van Everdingen and Hurst
Pot Aquifers
Fetkovich Finite Aquifers
Carter-Tracy Aquifers
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27
Artesian Aquifers
Continuity from point of recharge
The fact that they are recharged Often, aquifers under the
sea are recharged better
Connection they have at point of contact with hydrocarbon reservoir
Head it can exert on hydrocarbon reservoir
Critical aspects of an aquifer:
Water Influx from Aquifers
Account for additional drive energy
Match historical reservoir pressure (OOIP)
Predict future pressure-production performance (rate vs. time)
Estimate oil-water contact movement
Why is it important to account for aquifer water encroachment?
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Water Influx from Aquifers
Aquifer influx depends more on aquifer propertiesthan hydrocarbon reservoir properties
• Location of wells relative to aquifer and watermovement
• Aquifer size
• Aquifer porosity and permeability
Why is it difficult to account for aquifer water encroachment?
Initial until transient reaches boundary Declines with continuing water influx Pressure may be maintained if aquifer outcrops
Water Influx Mechanism
1 Pressure drop in the reservoir
2 Water flows from aquifer to reservoir
3 Pressure gradient grows in the reservoir andaquifer pressure drops from depletion
4 Rock and water expansion in the aquifer balancesaquifer voidage
5 Aquifer pressure at outer boundary
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29
Aquifer Expansion
How long water encroachment takes is a function of:
Maximum water encroachment, We,aq
aqiaqaqwfaq,e ppPVccW
Aquifer size (PVaq = aquifer pore volume)
Aquifer permeability
Connection of reservoir and aquifer
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Water Influx Models
Pot Aquifer Small well
connected aquifer,limited use
Fetkovich Semi-Steady State
Good for lateperiods andlimited aquifers,but doesn’tdefine thetransient period
Van Everdingen and Hurst Most complete
and useful
Schilthuis Steady-State Almost never
applicable
Carter-Tracy Analytical model
good forsimulation
Van Everdingen and Hurst
Transient and bounded aquifer flow
Solution to hydraulic diffusivity equation
Most comprehensive approach to water influx
Requires superposition
Defines aquifer characteristics
Implemented in material balance programs,including MBAL
Characteristics
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31
Behavior of a Large Aquifer
RESERVOIR
AQUIFER
RESERVOIR
1 2 3 4
1 2 3 4
Transient moves out until boundaries reached,then PSS (pseudo-steady state) behavior
Reservoir/Aquifer Boundaries
ra
ro
Aquifer
ReservoirCOPYR
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Aquifer Pressure Distribution
t=0
t=1
t=10t=100
t=1000
pi
p
Radius
Pressure
raro
t=infinite
van Everdingen Hurst Solution (Field Units)
doee QpfrchW )119.1( 2
Where:We = cumulative water influx, reservoir bbls
= porosity, fraction
h = effective aquifer thickness, ft
ce = total compressibility of the aquifer, cw + cf, psi -1
ro = radius of the hydrocarbon reservoir, ft
f = fraction of the reservoir boundary exposed to the aquifer (0 < f < 1)
p = pressure drop across the original reservoir/aquifer boundary, psi
Qd = dimensionless cumulative influx term
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33
Dimensionless Influx, Qd
o
ad r
rr 2
* 0.00633
oewd rc
tkt
Where:ra = radius of the aquifer, ft or m
ro = radius of the original reservoir/aquifer interface, ft or m
Where:
k = aquifer permeability, mdt = time, days
Aquifer influx depends on the aquifer size, aquifer rock properties, water compressibility, the pressure drop the aquifer sees at the reservoir/aquifer interface, and how long that pressure drop has occurred.
Function of Two Dimensionless Terms
Dimensionless Influx, Qd 2 ≤ rd≤ 4
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Dimensionless Influx, Qd 5 ≤ rd ≤ 10
Calculation Procedure
1
2
3
4
Estimate a dimensionless radius rd from regional geology
Calculate a dimensionless time td based uponaquifer properties and real time
Determine dimensionless water influx, Qd as a function of rd and td
Using the observed reservoir p, calculate water influx for the time period
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Application of Superposition
If you want to calculate the We that has occurred after 24 months, you need to take into account everything that has occurred before then
At 24 months:p1 has been occurring since time = 0 mo., or for 24 months,
+ p2 has been occurring since time = 6 mo., or for 18 months,
+ p3 has been occurring since time = 12 mo., or for 12 months,
+ p4 has been occurring since time = 18 mo., or for 6 months
So the calculated We at any time must take into account all of the
p’s that have happened before then, and for how long they have happened
Application of Superposition
Etrc
ktt
oewd
2
00633.0
We = (1.119hcero2f)Σ(pQd )=C Σ(pQd )
rd = ra/ro
Water Influx Calculations Let:
(E-constant)
(C-constant)
Time, yrs p, psi p p, psi
0 4500 -- --
1 4250 4375 125
2 3900 4075 300
3 3500 3700 375
For example: rd = 10, E = 0.01 days-1, C = 150 rb/psi
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Application of Superposition Example
After 1 Year:
After 2 Years:
After 3 Years:
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1
Predicting Aquifer Pressure Support
We We We
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Fetkovich Finite Aquifer
Used for finite aquifers• Neglects transient effects
Does not require superposition
Closely approximates van Everdingen and Hurstunder pseudo-steady state conditions
Based upon productivity index and pressuredepletion concepts
Overview
Fetkovich Finite Aquifer
Where:qw = water influx rate from aquifer; rbbl/d
J = productivity index for the aquifer, rbbl/d/psi
pa = average aquifer pressure, psi
pr = pressure at the aquifer-reservoir boundary, psi
Where:pi = initial aquifer pressure, psi
Wei = encroachable water-in-place at pi, rbbls
We = cumulative water influx, rbbls
Based on two equations:
)( rawe ppJq
dt
dW ie
ei
ia pW
W
pp
Productivity index for the aquifer Aquifer material balance
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Fetkovich Finite Aquifer
ei
ninrna
i
eine W
tJppp
p
WW exp1)( ,1,,
Solved over n repeated time increments:
Where:
ei
eina W
Wpp 11,
21,,
,
nrnrnr
ppp nee WW ,
Fetkovich Method
Where:k = absolute permeability, md
w = width, ft
L= length of linear aquifer, ft
f = fraction of reservoir boundary exposed to aquifer (0<f<1)
ra = radius of aquifer, ft
ro = radius of the original reservoir/aquifer interface, ft
h = aquifer thickness, ft
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Aquifers in Numerical Simulation
Analytical Carter-Tracy (numerical approximation to
van Everdingen and Hurst), Fetkovich, etc.
Coupled to edge or bottom of grid
Numerical Aquifer represented with discrete grid cells
Each cell can have its own uniqueproperties
Aquifers Summary
Constant terminal pressure solution
Most comprehensive approach
Time-based (superposition principle)
Applicable to transient, late transient and PSS flow
Tedious calculation procedure
Assumes PSS relationship
Not applicable for transient response
Avoids superposition calculations
Assumes instantaneous response (no time dimension)
Quick and simple
Calculates maximum influx
Van Everdingen and Hurst
Fetkovich
Pot Aquifer
Carter-Tracy
de QpCW
)( rae ppJ
dt
dW
pPVcW aqte
Constant terminal rate solution
Applicable to transient, late transient and PSS flow
Avoids extensive superposition calculations
Often preferred to analytical
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This description can then be used to project future performance,but the reliability of this forecast is limited if the aquifer has notyet reached pseudo-steady state
Key Learnings
1 Aquifer performance is predominantly determined by the aquifer properties: size, degree/manner of connection to reservoir and permeability
2 The aquifer goes through the same transient flow into pseudo-steady state behavior that we see in oil or gas wells
3 Aquifer behavior can be modeled by various means, with the van Everdingenand Hurst and Carter-Tracy models being the most technically correct
4Historical reservoir production and pressure data is used to allow the engineer to “identify” an aquifer description that makes the material balance calculations converge on a reservoir and aquifer description that honors field performance
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Learning Objectives
You are now able to:
Determine the rock, fluid, operational, and reservoir geometricfactors affecting water influx (We) into your reservoir
Calculate We using different models
Understand which descriptive parameters you should change soyou can match the calculated We to the observed reservoirpressure
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Segregated Flow
Reservoir Fluid Displacement Core
Learning Objectives
By the end of this lesson, you will be able to:
Identify the theoretical differences between segregated anddispersed flow
Describe how segregated flow is characterized
Identify what causes segregated flow instability in oildisplacement by water
Identify the critical rate calculation for unstable displacement
Identify the pore level displacement efficiency calculation forsegregated flow
Explain why coning occurs and how it can be avoided
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In This Section
Gravity
MobilityRatio
Contact Tilt
Stability of Displacement Processes
Water Under-Running
Maximum Pore Level DisplacementEfficiency
Special Segregated Flow Case - Coning
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Gravity
Very thin transition zone
Water-oil-contact
Saturations are verticallysegregated based on fluiddensity
Residual oil saturationbehind the front
Low rates of flooding
Large fluid densitydifference
Good vertical permeability
Small thickness with smallcapillary transition zone
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Mobility Ratio
wSro
oSrw
wi
orw
k
kM
@
@
Water
Sw=1 - Sor
So=Sor
Oil
Sw = Swi
So = Soi
Thin transition zone
Mobility ratio based on endpoint saturations
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Contact Tilt
Water-oil Contact
Contact tilt occurs for dipping reservoirs as a function of reservoirprocessing rate (qt )
wt
Srwa
w
o
ro
rw
q
AkkG
k
kM
G
TanMGTan
orw
sin000488.0
1
@
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Stability of Displacement Processes
M <= 1
Unconditionally Stable
UnstableViscous forces are strongerthan gravity (density) forces
G < M-1M > 1
Oil Zone(light oil)
Oil Zone(viscous oil)
Water Zone
Water Zone
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Maximum Pore Level Displacement Efficiency
• This equation is referring to only the lab-derived fraction of pore levelrecoverable oil from kr curves of total oil in pore space of thereservoir, since behind the front in segregated flow, the oil saturationis at the end point, Sor
• Two phases are present, but only one phase is flowing
oi
rooiD S
SSE
• Dispersed flow ED atbreakthrough depends on theaverage saturation behindthe front at that time
• Segregated flow ED atbreakthrough depends on theresidual saturation of oilbehind the front at that timeResidual Oil Saturation
Sw@ which Kro 0) (1
Sor
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Wow! Nice Oil Well!
Oil
Kh
Kv
Qc Here is a well that is capable ofproducing 1347 bopd
It has no geologic barriers through theinterval and it has a relatively highmacro kv/kh
Qo = 0.00708 * K * Kro * H * (Preservoir – Pbhf)
(μo * Bo * (ln(re/rw) + s))
Where: K (md) = 15Kro = 0.9H (ft) = 25Preservoir (psia) = 3500Pbhf = 500
Oil Viscosity = 0.5Bo (RB/STB) = 1.4Skin = 0Drainage Radius (ft) = 500Wellbore Radius (ft) = 0.25
Qo (STBOPD) = 1347 per well
Whoa, Not So Fast!
∆P > 0.433 * ( γw - γo ) * hc
Qc
Oil
Water
Kh
KvD
hc
h
Water coning into a vertical well is due to exceeding a critical drawdown toovercome gravity (see cone of water in the figure below)
When the ∆P exceeds the valueshown below under the geometricconditions indicated, water willcone into a well of relativelyhomogeneous reservoir character
By staying below the critical rate(see animation on water_drive), it ispossible to avoid coning
However, this rate is, in the majorityof cases, sub-economical
Criteria for Coning
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Bottom Water Drive Vertical Well - Coning
Oil
Water
Kh
KvD
hc
h
Critical Rate qDC - maximumoil flow rate that will avoid acone breakthrough into thewell
qDC is a function of:• Oil zone thickness, h
• Perforation interval, D
• FWL distance away
For a vertical well, you cannow write the qDC
qDC = 0.0246 * k * (ρw - ρo) * (h2 - D2)
μo * Bo * [ln(re/rb)]
How Much Can You Produce to Avoid Water?
Where: K (md) = 15Kro = 0.9H (ft) = 25D (perfs) = 0.5Density Oil (psi/ft) = 0.35Density Water (psi/ft) 0.45Oil Viscosity = 0.5Bo (RB/STB) = 1.4Skin = 0Drainage Radius (ft) = 500Wellbore Radius (ft) = 0.25
Qo (STBOPD) = 4 per well
Qo = 0.0246 * Ka * Kro * H * (ρw - ρo) * (h2 - D2)
μo * Bo * (ln(re/rw) + S)
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If You Exceed the Critical Drawdown an Upside Down Cone Forms Perpendicular to the Iso-Potential Lines
Vertical Well
FWL
xe
With a vertical well the Xe is the distance you can perforate the well, and since you want to get away from water, you perforate as little as possible
xe = effective length
If you had a horizontal well, you could place the well parallel to the FWL, and the drawdown would be spread over the longer Xe
FWL
xe = length of horizontal
xe
Horizontal Well
If You Exceed the Critical Drawdown an Upside Down Cone Forms Perpendicular to the Iso-Potential Lines
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When is Vertical Well Coning Impact Greatest?
The oil is viscous or heavy (low API or high SG)
The effective distance to the water is low
Production rate is higher than the critical rate
Reservoir is oil wet
The macro kv/kh is high, i.e., there is good communication to water, and there are no barriers to perforations
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Learning Objectives
You are now able to:
Identify the theoretical differences between segregated anddispersed flow
Describe how segregated flow is characterized
Identify what causes segregated flow instability in oildisplacement by water
Identify the critical rate calculation for unstable displacement
Identify the pore level displacement efficiency calculation forsegregated flow
Explain why coning occurs and how it can be avoided
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Vertical Layering of Reservoir Units
Reservoir Fluid Displacement Core
Learning Objectives
By the end of this lesson, you will be able to:
Recognize the techniques available to separate a reservoir intovertical layers
Explain how Lorenz techniques can divide flow units by the porestructure that occursCOPYR
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In This Section
Dykstra-Parsons
CoefficientLorenz
Techniques
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k vs. % greater than
Permeability Variation (VDP Steps)
Dykstra Parsons Coefficient of Variability VDP
1 Arrange core samples in order of decreasing kvalue
2 Prepare cumulative frequency plot on log-normalpaper
3 Draw a straight line through data points puttingmore weight on central points
4 Calculate V from k50 , k84.1
Permeability Variation VDP
Percent Greater Than
K84.1
K50
K,
md
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Permeability Variation VDP
Where:V = coefficient of variation
= standard deviation
x = mean value
log-normal distribution: x = geometric mean 68.3% of values within mean +/-1 standard deviation
k84.1 = mean + 1 standard deviation
Permeability Variation VDP
Advantages• Ability to compare
across reservoirsworldwide for useas an analog
• Allows for easyelimination ofoutliers
Disadvantages• Eliminating outliers
may result in data1 – 2 standarddeviations awayfrom the mean
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Sample Lorenz Plot – SPE Monograph 3
Lorenz Coefficient
Coefficient values range from 0 to 1 in order of increasing heterogeneity
It is important to retain the hin both sides of the axis
• kh is in Darcy’s Law, indicating cumulative flow capacity at pores
Most often coupled with porethroat size
Alternatively, Φ multiplied byh in the X axis as in thevolumetric formula andindicates storage capacity, orpore body size
Therefore this is a plot of thevariation in pore throatgeometry
Sample Lorenz Plot – SPE Monograph 3
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Sample Lorenz Plot – SPE Monograph 3
Increasing Heterogeneity
Lorenz Coefficient
Coefficient values range from 0 to 1 in order of increasing heterogeneity
Lorenz Plot Honoring Stratigraphy
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Learning Objectives
You are now able to:
Recognize the techniques available to separate a reservoir intovertical layers
Explain how Lorenz techniques can divide flow units by the porestructure that occurs
While the fractional flow curve is usually calculated ignoring the effect of gravity, the gravity term (for non horizontal flow) can be utilized to calculate the potential benefit of processing the reservoir at a slower rate.
A mobility ratio of less than 1.0 is favorable and leads to pistonlike displacement and better performance in both technical and economic terms.
Key Learnings
A fractional flow curve is calculated from the relative permeability curve, coupled with the reservoir oil and water viscosities.
The shape of the fractional flow curve can be graphically analyzed to identify the efficiency of the displacement process before water breakthrough and after
The mobility ratio is the mobility of the displacing fluid (behind the front) / the mobility of the displaced fluid.
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PetroAcademyTM Applied Reservoir Engineering Skill Modules
This is Reservoir Engineering Core
Reservoir Rock Properties Core
Reservoir Rock PropertiesFundamentals
Reservoir Fluid Core
Reservoir Fluid Fundamentals
Reservoir Flow Properties Core
Reservoir Flow PropertiesFundamentals
Reservoir Fluid Displacement Core
Reservoir Fluid Displacement Fundamentals
Properties Analysis Management
Reservoir Material Balance Core
Reservoir Material Balance Fundamentals
Decline Curve Analysis and Empirical Approaches Core
Decline Curve Analysis and Empirical Approaches Fundamentals
Pressure Transient Analysis Core
Rate Transient Analysis Core
Enhanced Oil Recovery Core
Enhanced Oil Recovery Fundamentals
Reservoir Simulation Core
Reserves and Resources Core
Reservoir Surveillance Core
Reservoir Surveillance Fundamentals
Reservoir Management Core
Reservoir Management Fundamentals
Reservoir Fluid Displacement Core
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