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๐ฝ =2๐๐๐โ
๐๐๐ต๐ ln๐๐๐๐ค
+ ๐ โ12 โ
1 โ ๐4
Productivity Index
๐ฝ =2๐๐๐โ
๐๐๐ต๐12
ln4๐ด
1.781๐ถ๐ด๐๐ค2+ ๐
๐ฝ =๐
เดค๐ โ ๐๐ค๐[=]
๐ฃ๐๐๐ข๐๐
๐๐ ๐ โ ๐๐๐ฆ
J is constant after transient flow period
Stages of Flow
๐๐ค๐(๐ก) = ๐๐ โ๐๐ ๐๐๐ต
2๐๐โ
1
2ln
4๐ด
1.781๐ถ๐ด๐๐ค2+
2๐๐๐ก
๐๐๐๐๐ด+ ๐
- When pressure disturbance first approaches boundary
- Short time period and difficult to model mathematically
๐1 โ ๐2 =๐๐
2๐๐โln
๐1๐2
โ1 โ ๐
2
๐1๐๐
2
โ๐2๐๐
2
๐ธ๐(โ๐ฅ) = ln(๐ฅ) + ๐พ โ
๐=1
โ
โ1 ๐๐ฅ๐
)๐(๐!
๐(๐, ๐ก) = ๐1 +๐๐
4๐๐โ๐ธ๐ โ
๐๐๐๐๐2
4๐๐ก
1. Infinite Acting/Transient Flow
๐๐๐ ๐ =)2๐๐โ( เดค๐ โ ๐๐ค
๐๐๐ต๐ ln๐๐๐๐ค
โ12 โ
1 โ ๐4 + ๐
2. Transitional/Late Transient Flow
3. Stabilized Flow
Pseudo/Semi SS
Closed drainage boundary
๐ = 00 < ๐ < 1
Actual Performance
Steady State (SS)
Open drainage boundary
๐ = 1
4. Depletion Flow
- Pwf limit reached; wellโฒs BHP held constant
- Wellโs flow rate begins to decline
- Exponential decline can be derived (equations shown to the left)
- Hyperbolic b-factor is empirical (refer to Decline Curve Analysis)
PSS:
**For gas flow, replace pressure with pseudopressure:
๐(๐) = เถฑ๐๐๐๐
๐ 2๐
๐๐ง๐๐
๐ก๐ =๐๐ก๐๐๐๐
๐๐ โ ๐lim โ๐๐ก๐๐
๐ฝ
๐๐ = ๐๐๐ก๐ +๐ฝ
๐เดค๐๐ โ ๐lim 1 โ ๐
โ๐ ๐กโ๐ก๐
เดค๐ ๐ก = ๐๐ โ๐๐
๐๐ก๐๐๐ก
เดค๐ ๐ก = ๐lim + เดค๐๐ โ ๐lim ๐โ๐ ๐กโ๐ก๐
๐ = ๐ฝ เดค๐ โ ๐๐ค๐
๐ = ๐ฝ เดค๐๐ โ ๐lim ๐โ๐ ๐กโ๐ก๐
๐๐ = โ๐๐๐๐๐ เดค๐
๐๐ก
๐
๐ =๐ฝ
๐๐๐๐ก[=]๐กโ1
)๐ธ๐(โ๐ฅ) โ ln(๐๐พ๐ฅ
Valid if ๐ฅ < 0.01
๐๐ค๐ ๐ก โ ๐๐ โ162.6๐๐ ๐๐ต๐
๐โlog ๐ก + log
๐
๐๐๐๐ก๐๐ค2โ 3.23
Aquifer Models
๐บ๐๐๐๐ธ๐ + ๐๐๐๐๐ธ๐ + ๐๐ธ๐ค + ๐๐๐๐ธ๐ + เธ๐๐ = ๐บ๐ โ ๐บ๐ผ๐ต๐ โ ๐ต๐๐ ๐ฃ1 โ ๐ ๐ฃ๐ ๐
+ ๐๐๐ต๐ โ ๐ต๐๐ ๐ 1 โ ๐ ๐ฃ๐ ๐
+ ๐๐ โ ๐๐ผ ๐ต๐คFree-gas
expansionFree-oil
expansion
Free-water expansion
Rock expansion Net gas withdrawal Net oil withdrawal
Net water withdrawalWater influx
๐บ๐ = Cumulative Gas Produced = ๐ ๐๐๐๐๐๐ = Original free oil = ๐๐๐ต๐บ๐๐๐ = Original free gas = ๐ ๐๐
Can combine free-water expansion and rock expansion into composite expansivity
๐๐ =๐๐ + ๐๐ค๐๐ค๐
1 โ ๐๐ค๐
๐ธ๐ = ๐ต๐ก๐ โ ๐ต๐ก๐๐๐ธ๐ = ๐ต๐ก๐ โ ๐ต๐ก๐๐
๐ =๐๐๐๐๐ค๐
๐ต๐ค๐๐ธ๐ =
๐๐๐ โ ๐๐
๐๐๐โ ๐๐๐ฅ๐ ๐๐ โ ๐๐ =
1
๐
๐๐
๐๐๐
๐ธ๐ค = ๐ต๐ค๐๐๐ค๐ฅ๐
๐ธ๐๐ค๐ = ๐ธ๐ + ๐ต๐๐๐๐๐ฅ๐ ๐ธ๐๐ค๐ โ ๐ธ๐
๐ธ๐๐ค๐ โ ๐ธ๐
*๐๐ ๐๐ค ๐๐๐ ๐๐ ๐๐๐ ๐๐๐๐๐๐๐ก๐๐
๐บ๐ผ = Gas Injected = ๐ ๐๐W๐[=]๐ ๐ต
๐ธ๐๐ค๐ = ๐ธ๐ + ๐ต๐๐๐๐๐ฅ๐
Original Free Oil EstimationP/z Plot โ OGIP Estimation
Plot applicable for undersaturated reservoirs with Volatile, Black, Undersaturated, or Dead Oils
เดคP
zi
Water influx
No water influx
เดฅ๐
๐ณ
เดคP
zab
๐๐ โ ๐๐
Gab = Gr
G = OGIP
เดค๐
๐ง=
เธ
เดค๐
๐ง๐
โเดค๐
๐ง๐
1
๐บ๐บ๐ โ ๐บ๐ผ
intercept slope
๐ ๐น =๐บ๐๐บ
Recovery factor
๐น = ๐๐๐๐๐ธ๐๐ค๐ ๐บ๐๐๐ = 0
Zero intercept
slope = ๐๐๐๐
Water influx
No water influx
๐
๐๐จ๐ฐ๐
Carter-Tracy (1960)
Drive Indices
- Constant terminal-rate solution- Incorporates transient and SSS flow- Preferred model for simulators
๐๐๐น
= ๐ผ๐๐ค๐
๐บ๐๐๐๐ธ๐
๐น= ๐ผgd
๐e = ๐โฒ(๐๐ + ๐๐ค)๐ฅ๐
- Assumes instantaneous response- No time dimension
- Assumes SSS relationship- Not applicable for transient response
- Constant terminal-pressure solution- Time based (superposition principle)- Applicable for transient and SSS flow
Fetkovitch (1971)
Hurst and Van Everdingen (1949)
๐๐๐๐๐ก
= ๐ฝ ๐๐ โ เดค๐
๐๐ = ๐ต ๐ฅ๐๐๐ท ๐ก๐ท
Pot Aquifer (Coats, 1970)
๐๐ธ๐ค๐น
+๐๐๐๐ธ๐
๐น= ๐ผ๐๐ค๐
Gas cap drive
Solution gas drive
Formation drive
Natural water drive
๐๐๐๐๐ธ๐
๐น= ๐ผ๐ ๐
**Different definition can be used to account for an artificial water drive
Drive Indices must add up to 1.0
๐บ๐๐๐๐ธ๐๐ค๐ + ๐๐๐๐๐ธ๐๐ค๐ + ๐๐ = ๐น = (๐บ๐ โ ๐บ๐ผ)๐ต๐ โ ๐ต๐๐ ๐ฃ
1 โ ๐ ๐ฃ๐ ๐ + ๐๐
๐ต๐ โ ๐ต๐๐ ๐
1 โ ๐ ๐ฃ๐ ๐ + (๐๐ โ ๐๐ผ)๐ต๐ค
๐๐ = Cumulative Oil Produced = ๐๐๐ต๐น = Reservoir Voidage[=]๐ ๐ต๐ธ๐๐ค๐[=] เต๐ ๐ต
๐ ๐๐ ๐ธ๐๐ค๐[=] เต๐ ๐ต
๐๐๐ต
๐๐
Compressibility Factor
Black Oil Phase Diagram Undersaturated Reservoir Key Properties
API Gravity
๐ด๐๐ผยฐ =141.5
๐พโ 131.5
Specific Gravity
๐พ๐ =๐๐
๐๐๐๐ ๐๐ถ=
๐๐
๐๐๐๐
๐พ๐ =๐๐๐๐ค ๐๐ถ
Relate ๐ต๐ to z-factor
๐ = ๐๐ ๐๐ ๐ = ๐
Pressure Dependent Fluid Properties
๐๐จ
BP
400 800 1200 1600
Pressure (psi)
๐๐ญ๐จ40
20
60
1
1.5
1.1
1.20.02
๐๐ = ๐ง๐๐ ๐ 0.04
Pressure (psi)
0.02
0.03
0.01
400 800 1200 16000.01
๐ฉ๐
1.1 โ Free Gas1.2 โ Volatilized Oil2.1 โ Solution Gas2.2 โ Free Oil1 โ Gaseous Phase (Gas + Volatilized Oil)2 โ Oleic Phase (Oil + Solution Gas)
Oil
Pis
ton
๐ต๐ก๐ =เตฏ๐ต๐(1 โ ๐ ๐ฃ๐๐ ๐ ) + ๐ต๐(๐ ๐ฃ๐ โ ๐ ๐ฃ
1 โ ๐ ๐ ๐ ๐ฃ
๐ต๐ก๐ =เตฏ๐ต๐(1 โ ๐ ๐ ๐๐ ๐ฃ) + ๐ต๐(๐ ๐ ๐ โ ๐ ๐
1 โ ๐ ๐ ๐ ๐ฃ
๐ถ๐ =๐๐๐ต๐
+๐๐
๐ต๐๐ ๐ฃ
๐ถ๐ =๐๐
๐ต๐+
๐๐๐ต๐
๐ ๐
Reservoir Conditions
๐ถ๐ =๐๐๐ต
๐ ๐ต
P > BP
Oleic
Gaseous
AqueousCapillary Pressure creates
transition zone between phases
Surface Conditions
P < BP
Isothermal
๐ต๐ =
๐ต๐ = ๐ ๐ =
๐ ๐ฃ =1
2
2
1
๐ต๐ก๐ = [=]๐ ๐ต
๐๐๐ต
+
+
๐ต๐ก๐ = [=]๐ ๐ต
๐ ๐๐+
+
Component ConcentrationsLegend
OIL
2
21
1
1.1 1.1 1.1
1.1
2.12.1
2.1
2.2 2.2 2.2 2.21.2
1.2
1.2
๐ต๐[=] เต๐ ๐ต
๐๐๐ต ๐ต๐[=] เต๐๐๐
๐ ๐๐๐ ๐ [=] เต๐ ๐๐
๐๐๐ต๐ ๐ฃ[=] เต
๐๐๐ต๐ ๐๐
Gas
Oil
Gas
Gas
Oil
๐ถ๐ค =๐๐๐ต๐
๐ต๐ =๐๐๐ถ
๐๐๐ถ๐ง๐
๐[=]
๐๐๐
๐ ๐๐
1 kPa = 0.1450 psi1 MPa = 10 bar1 atm = 14.696 psi1 atm = 1.013 bar
1 in = 2.54 cm1 ft = 0.3048 m1 mile = 5,280 ft
1 m3 = 6.2898 bbl1 bbl = 5.6146 ft3
1 bbl = 42 US gal1 ft3 = 7.4805 gal
R = 459.67 + ยฐFK = 273.15 + ยฐCยฐF = 1.8 ยฐC + 32
1 acre = 43,560 ft2
1 m2 = 10.764 ft2
1 Darcy = 9.8692 x 10-9 cm2
Euler (ฮณ) = 0.5772 = ln(1.781)
Gravitational Constant = 9.806 m/s2
Natural logarithm base = 2.71828
Gas Constant = 8.314 ฮคPa โ m3 mol K
1 g/cm3 = 62.428 lbm /ft3
1 dyne = 2.248 x 10-6 lbf
1 mD/cp = 6.33 x 10-3 ft2/psi-day
1 lbm = 453.592 g
1 cp = 1.0 mPa-s
1 ฮคkg l = 8.347 ฮคlbm gal
1000 ฮคkg m3 = 0.4335 ฮคpsi ft
1 Newton = 1 x 105 dynes
1 Darcy = 1.0623 x 10-11 ft2
Standard Temperature = 60ยฐF
Standard Pressure = 14.696 psia
Water densityโat SC = 62.37 ฮคlbm ft3
Molar Mass of Air = 28.966 g/mol
VMIG = 379.3 ฮคscf lbmol @ 14.696 psia
Univ. Gas = 10.732 ฮคpsia โ ft3 lbmol โ R
๐ =
๐=1
5
๐๐๐๐๐๐๐๐ต๐
Volumetrics
๐1 = โ๐ด0 + ๐ด10
2
Isopach Map
Each contour line represents a line of constant thickness in the reservoir
4030
50
2010
50
0
๐๐ = ๐ดโ๐
๐๐ = average oil saturation๐๐ = reservoir pore volume
๐2 = โ๐ด10 + ๐ด20
2
๐ = เต๐๐๐๐
๐ต๐[=]๐๐๐ต ๐บ = เต
๐๐๐๐๐ต๐
[=]๐ ๐๐
Pressure
Material Balance Equation
Formation Skin
Decline Curve Analysis
Cumulative Recovery ๐๐
Reservoir FluidsConversions and Constants
๐ผ๐ is usually 0.01 โ .02ยฐ๐น/๐๐ก
๐ = ๐๐ ๐ข๐๐๐๐๐ + ๐ผ๐๐ง
๐ = ๐๐ ๐ข๐๐๐๐๐ + ๐ผ๐๐ง
๐ง = depth ๐ผ๐ = ๐๐๐D
epth
Gas Kick
OWOC
OGOC
๐๐๐
๐๐ค๐๐๐บ๐
Normal Pressure Gradient
๐ผ๐ =
0.433๐๐ ๐
๐๐กifโfreshโH2O
0.465๐๐ ๐
๐๐กifโbrine
Well Testing
๐๐ก = ๐๐ + ๐๐๐๐ข๐๐
๐ต =๐๐๐ถ๐๐ ๐ถ
Diffusivity EquationMass Balance leads to Continuity Equation
โ1
๐
)๐(๐๐๐ข๐๐๐
=)๐(๐๐
๐๐กโ
)๐(๐๐ข๐ฅ๐๐ฅ
=)๐(๐๐
๐๐ก
Introduce Darcyโs Law
Introduce Formation Volume Factor
Introduce Compressibility
Introduce Diffusivity Constant ๐ผ =๐
๐๐๐๐ก
๐๐
๐๐ก=
๐ผ
๐
๐
๐๐๐
๐๐
๐๐
1D Diffusivity Equation
๐๐๐๐ข๐๐ = ๐ต๐
๐๐
1
๐ต๐๐ =
1
๐
๐๐
๐๐๐
๐๐
๐๐ก= ๐ผ
๐2๐
๐๐ฅ2
๐ข๐ = โ๐
๐๐ป๐ = โ
๐
๐
๐๐
๐๐
โ๐ป(๐๐ข) =)๐(๐๐
๐๐ก
๐๐
๐๐ก= ๐ผ ๐ป2๐
Reservoir Pressure and Temperature
๐๐ = เถฑ๐ก=0
๐ก
๐(๐ก)๐๐ก
๐ = ๐๐๐โ๐ท๐๐ก
๐ =๐๐
1 + ๐๐ท๐๐กฮค1 ๐
๐๐ =๐๐
๐
)๐ท๐(1 โ ๐๐๐
1โ๐ โ ๐1โ๐ ๐ท = ๐ท๐๐
๐๐
๐
๐๐ =๐๐ โ ๐
๐ท๐๐ท = ๐ท๐
๐๐ =๐๐๐ท๐
ln๐๐๐
Rate ๐b-factor Cumulative Recovery ๐๐ Decline Rate ๐ท
Exponential
Hyperbolic
Harmonic
๐ = 0
๐ = 1
0 < ๐ < 1
๐ =๐๐
(1 + ๐ท๐๐ก)๐ท = ๐ท๐
๐
๐๐
0 0
ODE: โ1
๐
๐๐
๐๐ก= ๐ท๐
๐
๐๐
๐; ๐ = ๐๐ @๐ก = 0;
๐ท๐
๐๐๐ เถฑ
๐ก=0
๐ก
๐๐ก = โ เถฑ๐๐
๐ ๐๐
๐๐+1
Time Time
Pro
du
ctio
n R
ate
(q)
Pro
du
ctio
n R
ate
(q)
Economic Limit
Economic LimitAs b-factor increases, wellโs economic life increases
EL
๐ = 1๐ = 0 ๐ = 0.5
EL๐ = 1๐ = 0 ๐ = 0.5
๐ = 1๐ = 0 ๐ = 0.5
00
Rec
ov
ery
๐
๐
Reservoir Engineering โ Primary Recovery
Created by James Riddle with guidance from Dr. Matthew T. Balhoff and contributions from Jenny Ryu and Matt Mlynski
100
1000
10
1
100
1000
10
1
Contact [email protected] with comments/suggestions
(Arps, 1940)
Fluid Type ๐๐๐ ๐ข ๐๐๐จ๐ข ๐๐ฉ ๐๐ฉ ๐๐ฏ ๐๐ฌDry Gas >0 0 >0 0 0 --Wet Gas >0 0 >0 >0 Rvi 0
Condensate >0 0 >0 >0 >0 0Volatile Oil 0 >0 >0 >0 >0 >0Black Oil 0 >0 >0 >0 0 >0Undersaturated Oil 0 >0 >0 >0 0 RsiDead Oil 0 >0 0 >0 -- 0
Drawdown Test Analysis
๐๐ค๐ = ๐๐ โ162.6๐ต|๐๐ ๐|๐
๐โlog
๐ก๐ + ๐ฅ๐ก
๐ฅ๐ก
๐ = 162.6๐๐ ๐๐ต๐๐
โ๐๐๐ข
1 10 100 1000
Semi-log Plot (k & s)
P1
PSS Flow
Well Storage
Transient Flow
Pre
ssu
re
time (hours)
๐ = 1.151๐๐ค๐ ,1 โ ๐๐ค๐
|๐๐๐ข|โ log
๐๐๐๐๐๐ก๐๐ค
2๐+ 3.23
mbu = [psi/log cycle]
10000 1000 100 10
Pre
ssu
re
Semi-log Plot (k & s)
Horner time เตซ๐ก๐ + ๐ฅ๐ก ฮค) ๐ฅ ๐ก
๐๐ค๐ ,1
๐ฅ๐ก = 1
๐ด =๐๐ ๐๐ต๐
๐๐กโ๐๐๐ ๐ [=]consistentโunits
๐[=]๐๐
๐ = 1.151๐๐ โ ๐1
|๐๐|โ log
๐๐๐๐๐๐ก๐๐ค
2๐+ 3.23
๐[=]๐๐
๐๐ก[=]๐๐ ๐โ1
๐๐ ๐[=]STB/day
๐[=]๐๐ ๐
โ[=]๐๐ก
๐๐ค[=]๐๐ก
log๐ถ๐ด = log4๐ด
1.781๐๐ค2 + 0.87s +
๐0 โ ๐๐|๐๐|
Buildup Test Analysis
t๐[=]โ๐
๐[=]๐๐
๐[=]๐๐ ๐
๐[=]๐๐
โ[=]๐๐ก
๐๐ค[=]๐๐ก
๐๐ ๐[=]STB/day
Linear Plot (A & ๐ถ๐ด)
PSS Flow
Transient Flow
P0
1:2
๐ถ๐ด = 31.62
๐ถ๐ด = 30.88 ๐ถ๐ด = 22.6
๐ถ๐ด = 5.381:4
โโmT from semi-log plot
๐๐ก[=]๐๐ ๐โ1
๐ด, ๐๐ค2 = ft2
Pre
ssu
re
time (hours)0 200 400 600
mT =[psi/log cycle]
๐ก๐ =๐๐
๐๐ ๐
๐๐ ๐ =[psi/hr]
๐ = 162.6๐๐ ๐๐ต๐๐
โ๐๐
Field Units
(โ)๐๐๐๐ ๐ โ
๐๐ ๐ก๐๐๐๐๐๐๐
=๐ฝ๐ ๐ก๐๐๐ฝ๐๐๐๐
=ln ฮค๐๐ ๐๐ค + ๐ ๐๐๐๐
ln ฮค๐๐ ๐๐ค + ๐ ๐ ๐ก๐๐
(+)๐๐๐๐ ๐ โ
Skin is unitless and specific to each well
๐ฅ๐๐ ๐๐๐ = 0.869|๐|๐
๐ฅ๐๐ ๐๐๐ =๐๐ ๐๐ต๐
2๐๐โ๐
๐ = transient slope
Darcyโs Law for Multiple Phases
1D: ๐ข๐ = โ๐๐๐๐
๐๐
๐๐๐
๐๐ฅ+ ๐๐๐sin๐ผ
๐๐ค =1 +
๐๐๐๐๐ข๐๐
๐๐๐๐๐ฅ
โ ๐ฅ๐๐sin๐ผ
1 +๐๐๐๐๐ค๐๐๐ค๐๐
1 โ ๐๐sin๐ผ
1 +๐๐๐๐๐ค๐๐๐ค๐๐
1
1 +๐๐๐/๐๐๐๐๐ค/๐๐ค
1D flow of oil and water๐ = phase
๐๐๐๐
๐๐ก+
๐๐ข๐
๐๐ฅ= 0
๐ข๐ = โ๐๐๐
๐๐เทจ๐ ๐ป๐๐ + ๐๐๐๐ปโ
Mass Balance for Multiphase
if constant ๐, ๐๐
๐๐ =๐ข๐
๐ข๐ข๐ =
๐๐
๐ด๐ข = ๐ข๐
๐๐๐๐
๐๐ก+ ๐ข
๐๐๐
๐๐ฅ= 0
๐๐๐๐๐ฅ
=๐๐๐๐๐ฅ
โ๐๐๐ค๐๐ฅ
๐๐ = 1
๐ข๐ + ๐ข๐ค๐ข
= 1
Combine Darcyโs Law, definition of fractional flow, and capillary pressure
Introduce capillary pressure
Fractional Flow Derivation
๐๐ =๐๐๐๐๐ฅ๐๐
๐ข๐๐
๐ผ = rsvr dipโangle
if
๐๐ = 0
Multiphase Flow
if
๐ผ = 0
๐ผ = dispersivity
Miscible DisplacementMechanisms of Mixing
1. Molecular Diffusion ๐ท๐
๐ท๐ =๐ท๐๐
=๐ท๐๐น๐
2. Convective Mixing (Dispersion)
3. Small Scale Permeability Variations
๐พ๐ฟ >> ๐พ๐
๐น =๐
๐๐
๐ = tortuosity
๐พ๐ = ๐ท๐ + ๐ผ๐๐ฃ๐ฝ๐
๐พ๐ฟ = ๐ท๐ + ๐ผ๐ฟ๐ฃ๐ฝ๐ฟLongitudinal
Transverse
๐ถ๐ท =1
2๐๐๐
)๐ฅ โ ๐ฃ(๐ก โ ๐ก๐
)4๐พ๐ฟ(๐ก โ ๐ก๐ โ ๐๐๐
๐ฅ โ ๐ฃ๐ก
4๐พ๐ฟ๐ก
)๐๐๐๐(๐) = 1 โ ๐๐๐(๐
๐ถ๐ท =๐ โ ๐๐
๐๐๐๐ โ ๐๐
๐ถ๐ = ๐ถ๐ท|๐๐=1
Convection - Dispersion
+๐๐ฅ๐๐๐๐
2๐๐๐๐
๐ฅ๐ + ๐ก๐
2 ฮค๐ก๐ ๐๐๐
๐ถ๐ท(๐ฅ๐,๐ก๐) =1
2๐๐๐๐
๐ฅ๐ โ ๐ก๐
2 ฮค๐ก๐ ๐๐๐
2nd term negligible
๐๐ถ๐ท๐๐ก๐
+๐๐ถ๐ท๐๐ฅ๐
โ1
๐๐๐
๐2๐ถ๐ท
๐๐ฅ๐2 = 0
๐ถ๐ท = 0.5โโwhenโโ๐ก๐ = 1
Pulse/Slug Injection (from time 0 to ๐ก๐ )
๐ฅ๐ฅ๐ = 3.625๐ก๐
๐๐๐
Co
nce
ntr
atio
n ๐ถ
๐ท
Dimensionless Distance ๐ฅ๐
Effect of Peclet Number
๐๐๐ = 1000
๐๐๐ = 10๐๐๐ = 100
๐๐๐ =๐ข๐ฟ
๐๐ซ๐ฟโก
๐ฟ
๐ผ๐ฟ
๐ก๐ = 0.5
๐ฅ๐ =๐ฅ
๐ฟ๐ก๐ =
๐ข๐ก
๐๐ฟ
๐ฅ๐|๐ถ๐ท=0.1 โ ๐ฅ๐|๐ถ๐ท=0.9 = ๐ฅ๐ฅ๐
๐ฅ๐ฅ๐ = widthโofโmixingโzone๐ถ๐ท = 0.1
๐ถ๐ท = 0.9
๐ท๐ = effective diffusivity
๐ฃ =๐ข
๐
๐พ = dispersioncoefficient
๐พ๐ฟ
)ln(๐ฃ
Diffusion and Dispersion
Diffusion controls
Dispersion controls
๐ฝ๐ โ 1.2
๐๐ > ๐๐,100 ๐ธ๐ด = 1
๐๐,100 = ๐๐,๐ต๐๐1โ๐ธ๐ด,๐ต๐
0.274
๐๐ค|๐๐ค2 =๐๐๐
๐๐๐ + 1
๐๐โ = ๐๐,100
โ +๐๐ โ ๐๐,100
๐๐
Buckley-Leverett Extension to 2-D๐๐ < ๐๐,๐ต๐ ๐๐,๐ต๐ < ๐๐ < ๐๐,100Definitions
๐๐โ =
๐๐๐๐๐ธ๐ด
=1
๐๐คโฒ |๐๐ค2
๐๐ค2 = ๐๐ค5 = ๐๐ค๐ = ๐๐ค|๐ฅ๐=1
๐๐,๐ต๐ = ๐๐๐ธ๐ด,๐ต๐ าง๐๐ค๐ โ ๐๐ค๐ = ๐๐,๐ต๐
๐ธ๐ด,๐ต๐ = 0.546 +.0317
๐ าง๐ +
0.30223
๐๐เดค๐ โ .005097๐ าง๐
๐๐,๐ต๐โ =
๐๐,๐ต๐๐๐๐ธ๐ด
= าง๐๐ค๐ โ ๐๐ค๐
๐ธ๐ด = ๐ธ๐ด,๐ต๐ + 0.274ln ฮค๐๐ ๐๐,๐ต๐
๐๐โ = effective ๐ก๐
๐1 = 3.65๐ธ๐ด,๐ต๐ ๐2 = ๐1 + ln ฮค๐๐ ๐๐,๐ต๐
๐๐ = าง๐๐ค2 โ ๐๐ค๐ ๐๐
๐๐ = าง๐๐ค2 โ ๐๐ค๐ ๐ธ๐ด๐๐; าง๐๐ค2 = ๐๐ค2 + ๐๐โ 1 โ ๐๐ค2
๐๐โ = ๐๐,๐ต๐
โ 1 + ๐1๐โ๐1 ๐ธ๐ ๐2 โ ๐ธ๐ ๐1
าง๐๐ค2 = าง๐๐ค5 = าง๐๐ค าง๐๐ค๐ = แ๐๐ค
๐ าง๐ =๐๐๐ค + ๐๐๐ | าง๐๐ค๐๐๐๐ค + ๐๐๐ |๐๐ค๐
๐๐๐๐ข๐๐๐
=1
๐๐
0.274๐๐,๐ต๐ ๐๐ค๐ โ ๐๐ค๐
๐ธ๐ด,๐ต๐ าง๐๐ค๐ โ ๐๐ค๐
- For low viscosity oil or condensate (NI/NP) > 1
- For similar fluid viscosities (NI/NP) = 1
Well Patterns
5-spot (NI/NP) = 1
- For viscous oil (NI/NP) < 1
Rules of Thumb
Injector
Producer
๐๐๐ =๐๐ค|๐๐ค๐ 1 โ เต
๐๐๐๐ข๐๐๐
1 โ ๐๐ค|๐๐ค๐ 1 โ เต๐๐๐๐ข
๐๐๐+ เต
๐๐๐๐ข๐๐๐
Reservoir Simulation
Improved Oil Recovery
Created by James Riddle with guidance from Dr. Matthew T. Balhoff and contributions from Jenny Ryu and Matt Mlynski Contact [email protected] with comments/suggestions
Reservoir Engineering โ Improved Recovery
๐ ๐ + ๐ ๐ + ๐๐, ๐ท, ๐๐ค , ๐ต
๐ด, ๐, ๐๐ = 1
๐๐๐๐ต
๐+12
=
๐๐ ๐๐ค,๐+1๐ ๐๐+1 ๐ต ๐๐+1
if ๐ท๐+1 > ๐ท๐
๐๐ ๐๐ค,๐
๐ ๐๐ ๐ต ๐๐if ๐ท๐ > ๐ท๐+1
เตฏ๐๐(๐๐ค,๐เตฏ๐๐(๐๐ค,๐+1
Upwinding Harmonic Mean
๐๐ด
๐ฅ๐ฅ๐+
12
=2๐๐๐ด๐๐๐+1๐ด๐+1
๐๐๐ด๐๐ฅ๐ฅ๐+1 + ๐๐+1๐ด๐+1๐ฅ๐ฅ๐
Heterogeneities
๐๐+
12
=๐๐๐๐ต
๐+12
๐๐ด
๐ฅ๐ฅ๐+
12
๐ =๐๐ด
๐๐ต๐ค๐ฅ๐ฅ
Finite Differences Approximation
๐(๐ฅ + ๐ฅ๐ฅ) = ๐(๐ฅ) + ๐โฒ(๐ฅ)๐ฅ๐ฅ +1
2!๐โณ(๐ฅ)๐ฅ๐ฅ2 +
1
3!๐"โฒ(๐ฅ)๐ฅ๐ฅ3+. . .Taylor Series:
ErrorDeriv.
)๐(๐ฅ๐ก
)๐(๐ฅ๐ก2
)๐(๐ฅ๐ฅ2
)๐(๐ฅ๐ก
FD ApproximationType
)๐(๐ก + ๐ฅ๐ก) โ ๐(๐ก
๐ฅ๐ก
)๐(๐ก) โ ๐(๐ก โ ๐ฅ๐ก
๐ฅ๐ก
)๐(๐ก + ๐ฅ๐ก) โ ๐(๐ก โ ๐ฅ๐ก
2๐ฅ๐ก
)๐(๐ฅ + ๐ฅ๐ฅ) โ 2๐(๐ฅ) + ๐(๐ฅ โ ๐ฅ๐ฅ
๐ฅ๐ฅ2
๐๐
๐๐ก
๐2๐
๐๐ฅ2
๐๐
๐๐ก
๐๐
๐๐ก
Forward
Backward
Centered
Centered
Neumann ฮค๐๐ ๐๐ฅ = 0
ึ๐ท๐+1
= ๐ป +1
๐ฅ๐ก๐ฉ
โ11
๐ฅ๐ก๐ฉ ึ๐ท
๐+ ึ๐ธ
ึ๐ท๐+1
= ึ๐ท๐
+ ๐ฅ๐ก๐ฉโ1 ึ๐ธ โ ๐ป ึ๐ท๐
Crank-Nicholson ๐ = 0.5
Explicit:
Implicit:
๐๐ โ ๐๐+1๐ฅ๐ฅ
= 0; ๐๐ = ๐๐+1
Dirichlet ๐(0, ๐ก) = ๐๐ต1
N N+1
CFL ConditionStability
**Necessary for stability of IMPES
Explicit Method
(1D)
= time stepP ( , )P x t
x
P Pin
= grid block
Analytical (Continuous) Solution Numerical (Discrete) Solution
1 โ ๐ ๐ป +1
๐ฅ๐ก๐ฉ ึ๐ท
๐+1=
1
๐ฅ๐ก๐ฉ โ ๐๐ป ึ๐ท
๐+ ึ๐ธ
0 1
Boundary Condition
๐๐ต =๐0 + ๐1
2; ๐0 = 2๐๐ต โ ๐1
๐ฅ๐ฅ
req
๐ฅ๐ฅ
๐ฅ๐ฅ
req
Pwf
Pl
P2
q2rw
rwr๐๐ = 0.28
ฮค๐๐ฆ ๐๐ฅเต1 2๐ฅ๐ฅ2 + ฮค๐๐ฅ ๐๐ฆ
เต1 2๐ฅ๐ฆ2เต1 2
ฮค๐๐ฆ ๐๐ฅเต1 4 + ฮค๐๐ฅ ๐๐ฆ
เต1 4
เตฏ๐๐ค = โ๐ฝ๐๐ค(๐๐ โ ๐๐ค๐
๐๐๐ = ๐ฅ๐ฅ๐โ
๐2 โ 0.2๐ฅ๐ฅ
๐2 = ๐๐ โ๐๐ค๐๐ต๐ค2๐๐โ
๐๐๐ฅ๐ฅ
๐๐๐
Radial Solution
๐ฝ๐๐ค =
2๐โ ๐๐ฅ๐๐ฆ
๐๐ต๐ค ln ฮค๐๐๐ ๐๐ค + ๐
Saturation (solve explicitly)Pressure (solve implicitly)
๐ + ๐ +๐
๐ฅ๐ก๐๐+1 =
๐
๐ฅ๐ก๐๐ + ๐ ๐๐ค
๐+1 = ๐๐ค๐ + ๐12
โ1 โ๐๐ค๐๐+1 + ๐๐ค โ ๐๐ก๐ค ๐
๐+1 โ ๐๐
๐ = ๐๐ค +๐ต๐๐ต๐ค
๐๐ ๐ = ๐๐ค +๐ต๐๐ต๐ค
๐๐ ๐๐ =๐๐๐๐๐๐ก,๐
๐ต๐ค
๐ 12,๐ =๐๐๐๐
๐ต๐ค,๐๐ฅ๐ก๐๐ก = ๐๐ค๐๐ค + ๐๐๐๐ + ๐๐ ๐๐ก๐ค,๐ = ๐๐ค,๐
๐ ๐๐ค,๐ + ๐๐,๐
IMPES (Implicit Pressure and Explicit Saturation)
๐ = ๐๐ค +๐ต๐๐ต๐ค
๐๐
๐ โก๐ผ๐ฅ๐ก
๐ฅ๐ฅ 2โค 0.5
๐ข๐ฅ๐ก
๐ฅ๐ฅ< 1
Variables change from
block to block
IOR: Improved Oil Recovery
Hydraulic Fracturing
Pressure Maintenance
Other- Microbial- Acoustic
Thermal
Tertiary Recovery (EOR)
Gas Injection
Artificial Lift
Secondary Recovery
Natural Flow
Primary Recovery
Waterflooding
Chemical- Steam- Hot Water- Combustion
- Carbon Dioxide- Hydrocarbon- Nitrogen Flue
- Alkali- Surfactant- Polymer
IOR
Adapted from SPE-SPE-8490B, 87864
Stages of Recovery
เตฏ๐๐๐ = ๐ธ๐ด๐ธ๐ท๐ธ๐ผ(๐๐๐ โ ๐๐๐
3D Cumulative Oil Recovery
๐ธ๐ท = Displacement Efficiency
๐ธ๐ผ = Vertical Sweep Efficiency
๐ธ๐ด = Areal Sweep Efficiency
EA =Areaโcontactedโbyโfluid
Totalโarea
EI =Crossโsectionalโareaโcontactedโbyโfluid
Totalโcrossโsectionalโarea
ED =Amountโofโoilโdisplaced
Totalโmoveableโoilโ
๐ธ๐ท =๐๐๐ โ าง๐๐
1 โ ๐๐ค๐ โ ๐๐๐๐ธ๐ด is a ๐ ๐ าง๐ ,โ๐ก๐ , pattern
**Oil Saturation constant during Primary Recovery
Effect of Trapped Gas on Residual Oil
Capillary Desaturation Curve
๐๐ฃ๐ =๐ข๐
๐[=] ฮค(๐ฃ๐๐ ๐๐๐ข๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐๐ฆ ๐๐๐๐๐๐ )
(Lake, 2014)
Prod.
wW
w OO W
OGG
O
Water-Wet
WF
O
GO W
WGO
WF
W
Oil-Wet
WF
WF
OW
๐๐ค =1 โ ๐๐
๐ 1 โ ๐ ๐sin๐ผ
1 +1 โ ๐ ๐
๐๐๐๐
1
1 +1 โ ๐ ๐
๐๐๐๐
๐๐๐ = าง๐๐ค โ ๐๐ค๐
๐ =๐๐๐ค๐๐๐
=ฮค๐๐๐ค ๐๐คฮค๐๐๐ ๐๐
At Breakthrough
๐ก๐,๐ต๐ =1
๐๐ค๐โฒ |๐๐ค๐
๐๐๐,๐ต๐ = ๐ก๐,๐ต๐๐๐๐ ๐๐๐ = 1 โ ๐๐ค|๐๐ค๐
Wat
er S
atu
rati
on
(๐ ๐ค
)
Fractional Flow โ 1D Water Flood
๐๐๐ค = ๐๐๐ค๐ ๐๐
Mobility Ratio
๐๐๐ = relativeโmobility
๐๐ =ฮค๐๐๐ค
๐ ๐๐คฮค๐๐๐
๐ ๐๐
Corey Equations
๐๐๐ = ๐๐๐๐ (1 โ S)๐
๐ =๐๐ค โ ๐๐ค๐
1 โ ๐๐ค๐ โ ๐๐๐
Fractional Flow
๐๐๐ =
๐๐๐๐๐ ๐ฅ๐๐
๐ข๐๐
๐๐คโฒ =
๐๐๐ค๐๐๐ค
=๐๐๐ค๐๐
๐๐
๐๐๐ค
Important Definitions
๐ฅ๐ =๐ฅ
๐ฟt๐ =
๐ข๐ก
๐๐ฟ[=]PVI
๐๐ =๐๐๐๐๐
๐ต๐[=]STB
After Breakthrough
๐ก๐ =1
๐๐คโฒ |๐๐ค๐
(๐๐ค๐ usually 1)
าง๐๐ค = ๐๐ค๐ +๐๐ค๐ โ ๐๐ค|๐๐ค๐
๐๐คโฒ |๐๐ค๐
แ๐๐ค = ๐๐ค๐ +๐๐ค๐ โ ๐๐ค|๐๐ค๐
๐๐คโฒ |๐๐ค๐
Decreasing ๐๐
Breakthrough
1 โ ๐๐๐
๐๐ค๐ ๐๐ค๐
แ๐๐ค = าง๐๐ค,๐ต๐
Draw BT tangent line
from initial ๐๐คF
ract
ion
al F
low
(๐ ๐ค
)
Water Saturation (๐๐ค)
าง๐๐ค = avg. ๐๐ค inโsweptโarea
Dimensionless Distance (๐ฅ๐)
๐๐ค๐
๐ฅ๐๐
๐ฅ๐ = ๐ก๐๐๐คโฒ ๐๐ค
๐๐ค๐
Dimensionless Distance (๐ฅ๐)
๐๐ค๐
Shock Front
๐ฅ๐๐
๐๐ค๐๐๐ค๐
๐๐ค๐๐ฅ๐๐
แ๐๐ค
แ๐๐คaverage ๐๐ค
behind shock
Effect of Mobility Ratio
Mo=0.1
Mo=100Mo=10Mo=1
Decreasing ๐๐
Fra
ctio
nal
Flo
w (
๐ ๐ค)
Pore Volumes Injected (t๐)
Cu
mu
lati
ve O
il (
๐๐
๐)
Effect of ๐๐ on Oil Recovery
Effect of Gravity
*Gas flood would be inverse effect
ฮฑ = โ 15โ
ฮฑ = 15โฮฑ = 0โ
Fra
ctio
nal
Flo
w (
๐ ๐ค)
Fra
ctio
nal
Flo
w (
๐ ๐ค) Effect of Wettability
WWOW
As ๐๐ค curve shifts right,
breakthrough occurs later and produce oil quicker
๐ ๐ค๐
๐๐๐ =๐ต๐๐ต๐ค
๐๐ค|๐๐ค๐
1 โ ๐๐ค|๐๐ค๐
Pore Volumes Injected
Before Breakthrough
Increasing water
wettability
Increasing reservoir dip angle
if
๐ผ = 0
(๐๐ค๐ usually 1)
๐๐๐๐๐
*Oil-Wet Rock is inverse
3-Phase Permeability
๐๐๐ค isโaโfunctionโof ๐๐ค
Mixed-Wet Rock
Water Wet Rock
Oil Wet Rock
Petrophysics Review
Permeability Variations
เดค๐ =ฯ ๐๐โ๐ฯ โ๐
Vertical Variation
เดค๐ =ฯ ๐ฟ๐
๐ฟ๐๐๐
Horizontal Variation
เดค๐ =ln ฮค๐๐ ๐๐ค
ln ฮค๐๐ ๐๐โ1
๐๐
Radial Variation
เดค๐ = ๐๐๐๐+1. . . ๐๐ฮค1 ๐ Geometric
Mean
Capillary Pressure
๐๐(๐ง) = ๐ฅ๐๐๐๐ง
๐๐ =2๐cos๐
๐
๐๐1๐๐2
=ฮค๐2 ๐2
ฮค๐1 ๐1
๐ฝ(๐๐ค) =๐๐๐
๐
๐
๐๐(๐๐ค) = ๐๐๐ค โ ๐๐ค
๐๐ = ๐๐ฟ 1 +๐
๐
โ๐๐
๐๐ฅ=
๐ ๐ข
๐+ ๐ฝ๐๐ข2
Water Wet Rock
๐๐๐ค๐ < ๐๐๐
๐
๐๐ค > 0.5 when ๐๐๐ค curveintersects ๐๐๐ curve
Relative Permeability
๐๐ = ๐๐๐๐
Darcyโs Law Variations
๐ท = ๐ + ๐๐โ๐ข = โ๐
๐
๐๐ท
๐๐ฅ
WW Rock
Forcheimer (valid at high flow rates, Re>1)
๐ฝ material constant; depends on pore structure
Klinkenberg Effect (valid for gas flow at low pressure)
๐๐ฟ = liquid perm
๐๐ = gas perm
b factor important when k < 10md๐๐ฟ[=]๐โ
๐๐๐ ๐๐ , ๐๐ค ๐๐๐ ๐๐
๐๐๐ค ๐๐ค, ๐๐
๐๐๐ ๐๐ ๐๐๐ ๐๐
๐๐๐ค ๐๐ค, ๐๐
๐๐๐ ๐๐ , ๐๐ค
๐๐๐ ๐๐
Rel
ativ
e P
erm
eab
ility
WaterSaturation
๐๐๐๐ค๐
๐๐๐ค๐
1-๐๐๐,๐ค๐๐ค๐๐๐ค๐๐๐ 1-๐๐๐๐๐,๐ค
LiquidSaturation
๐๐๐๐ค
๐๐๐ค
๐๐๐๐
๐๐๐๐๐
๐๐๐๐
๐๐๐
1-๐๐๐๐๐
1-๐๐๐๐๐๐,๐
๐๐๐๐๐,๐
๐๐๐,๐
๐๐ค๐๐๐
Rel
ativ
e P
erm
eab
ility
๐๐ค = non-wetting phase