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DIFFUSIVITY EQUATIONDIFFUSIVITY EQUATION
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MODEL
MODEL : simplified version of the real system, by which thebehavior of the real system can be approximately but yet
representatively simulated.
MASS BALANCE EQUATIONS:MASS BALANCE EQUATIONS:for the considered extensive quantities
FLOW EQUATIONS:FLOW EQUATIONS: relate theextensive quantities to the significant statevariables of the problem
STATE EQUATIONS:STATE EQUATIONS: define thebehavior of the components of the system
INITIAL AND BOUNDARYINITIAL AND BOUNDARYCONDITIONS:CONDITIONS: must be defined after thedomain geometry has been established
MATHEMATICAL MODELMATHEMATICAL MODEL
THEORETICAL MODELTHEORETICAL MODELSet of simplifyingSet of simplifyingassumptionsassumptions
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MODEL SOLUTIONS
REALREALSYSTEMSYSTEM
MODEL COEFFICIENTSMODEL COEFFICIENTS:the transport coefficients of theconsidered extensive quantities
MATHEMATICALMATHEMATICAL
MODELMODEL
NUMERICAL METHODNUMERICAL METHODNeeded in the case of: non linearity of the equations
which constitute the model complexity of the boundary
conditions, etc.
ANALYTICAL METHODANALYTICAL METHODPreferable for the ease ofapplicability of the solutions
SOLUTIONS OFSOLUTIONS OF
MATHEMATICAL MODELMATHEMATICAL MODEL
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FLOW PROBLEM
Number of phasesNumber of phasessingle phase (kass)multiphase (krel)
Nature of fluidsNature of fluidscompressible (liquid)very compressible (gas)
GeometryGeometrymonodimensionalbidimensional (radial flow)tridimentional
Hydraulic regimeHydraulic regime
steady state flowpseudo-steady state flowtransient flow
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BASIC EQUATIONS
t
)()v(
=
CONTINUITY EQUATIONCONTINUITY EQUATION
STATESTATE EQUATIONSEQUATIONS
zRT
Mp=
)pp(c
00e = Liquid
Real Gas
FLOW EQUATIONSFLOW EQUATIONS(gravity effects are neglected)
Turbulent flow (Turbulent flow (ForchheimerForchheimer))
pkv
=
Laminar flow (Darcy)Laminar flow (Darcy)
2vvk
p +=
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DIFFUSIVITY EQUATION
MONOPHASIC FLOW OF A SLIGHTLY COMPRESSIBLE FLUID (LIQUID)
THROUGH A HOMOGENEOUS AND ISOTROPIC POROUS MEDIUM.
PRESSURE GRADIENTS ARE SMALL AND DARCYS LAW APPLIES:
t
p1
t
p
k
cp t2
=
=
=tc
k DIFFUSIVITY CONSTANTDIFFUSIVITY CONSTANT
MONOPHASIC FLOWMONOPHASIC FLOW: in the case of oil flow, water saturation is
equal to the irreducible value Swi, and pressure is always higher thanthe bubble point pressure; in the case of water flow the oil saturation is
equal to the residual value Sor
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PRESSURE PROFILE
9 Each time the well production is modified (i.e. rate change) a pressuredisturbance starts to propagate in the reservoir.
9 The DIFFUSIVITY EQUATIONDIFFUSIVITY EQUATION describes how this pressure disturbance
evolves within the reservoir.
Q
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AVERAGE PRESSURE
A VERAGE PRESSUREA VERAGE PRESSURE: the representative reservoir pressure atwhich the pressure-dependent parameters in the material
balance equations should be evaluated.
= V pdVV1
p
VOLUME AVERAGEDVOLUME AVERAGED
RESERVOIR PRESSURERESERVOIR PRESSURE
MEASUREMENT OF AVERAGE PRESSUREMEASUREMENT OF AVERAGE PRESSURE Theoretically:Theoretically: the average pressure could be measured in the wellbore
under static conditions if the well (or the field) had been shut in for an
infinitely long time so as to allow the reservoir pressure to reach
equilibrium.
In practice:In practice: the average pressure can be determined from buildup tests.
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Infinite Acting Radial Flow (I.A.R.F.)
HYPOTESIS: constant thickness of the producing layer, and
wellbore open to production across the entire layer thickness.
Therefore, the fluid flow is horizontal.
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RADIAL FLOW EQUATIONS
Transient flowTransient flow
tp
kc
rp
r1
rp t2
2
=+
CONSTANT TERMINAL RATE :Rate of water influx=const for t
calculation of pressure drop
CONSTANT TERMINAL PRESSURE:boundary pressure drop=const for t
calculation of water influx rate
Van Everdingen-Hurst
SOLUTIONSSOLUTIONS
to diffusivity equation
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WATER ENCROACHMENT
rw rw
VanVan EverdingenEverdingen--Hurst ModelHurst Model CarterCarter--TracyTracyModelModel
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VAN EVERDINGEN-HURST SOLUTION
t
p
k
c
r
p
r
1
r
p t2
2
=
+
=
==
=
===
>
==
w
e
w
eie
e
wiww
i
r
r0
r
p
r
rp)t,r(p
rr
tcospp)t,r(prr
0t
rpp0t
Initial and boundary conditions:
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VAN EVERDINGEN-HURST AQUIFER
Dimensionless water influxDimensionless water influx
w
eD
r
r,tQ
Dimensionless timeDimensionless time 2wtw
Drc
kt
t =
Dimensionless radiusDimensionless radius
=
w
eD
r
rr
Cumulative Water encroachment:Cumulative Water encroachment:
=
w
eDwe
rr,tQpBW
where B: water influx constantB: water influx constant hrcB wt2
2 =hfrcB wt
22 =
360
=f
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FUNCTION Q(re/rw,tD)
Dimensionless Time, tD
Wa
terIn
flux,
Q
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CARTER-TRACY AQUIFER
The CarterThe Carter--Tracy solution is not an exact solution to theTracy solution is not an exact solution to thediffusivity equation, it is an approximationdiffusivity equation, it is an approximation
CONSTANT WATER INFLUX RATE
over each finite time interval.
Condition:Condition:
( ) ( ) ( ) ( )[ ]
( ) ( )
( ) ( ) ( )
+=
nD1nDnD
nD1nen
1nDnD1nene 'ptp
'pWpBttWW
Where : B=Van Everdingen-Hurst water influx constant
n=current time step
n-1=previous time step
pn=total pressure drop, pi-pn
pD=dimensionless pressure
pD=dimensionless pressure derivative
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PRODUCTION DRIVEPRODUCTION DRIVE
MECHANISMSMECHANISMS
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OIL RECOVERY
Traditionally, oil recovery was subdivided into three stages which
described the production from a reservoir in a chronological
sense:
P r im a r y r e co v e r y P r i m a r y r e co v e r y production due to energy naturallyexisting in a reservoir
Se co n d a r y r e co v e r y Se co n d a r y r e co v e r y water flooding (and gas injection) for
pressure maintenance
Te r t i a r y r e c o v e r y Te r t i a r y r e c o v e r y processes that use miscible gas,
chemicals, and/or thermal energy to displace additional oil
after secondary recovery processes become uneconomical
However, many reservoir production operations are not
conducted in the specified order. Thus, the designation of
En h a n c e d O i l Re c o v e r y became more accepted instead of theterm tertiary recovery in the petroleum engineering literature.
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DRIVE MECHANISMS
DEPLET I ON DR I VE ( GAS ) : DEP LET I ON DR I VE ( GAS ) : reservoir with constant porous volume
gas recovery: 80-90% GOIP
DEPLET I ON DR I V E ( O I L ) : D EPLET I ON DR I V E ( O I L ) :
undersaturated oil reservoir with constant porous volume
source of energy: expansion of solution gas
oil recovery: 2-5% OOIP
D I SSOLVED GAS DR I VE: D I SSOLVED GAS DR I VE:
saturated oil reservoir
oil recovery: 15-20% OOIP
Decreasing ofpressure:
p< pb
Gas liberation andexpansion.
SgScg :gas is movable
Gas expansionforce oil out ofthe pore space
Decreasing of oil production:
decreasing So
ko o (qoko/o )
increasing GOR
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T = constantO
ppb
dp
kg
gBgqg A=
DISSOLVED GAS DRIVE: MULTI-PHASE FLOW
OilOil
GasGas
ko
oBoqo A= ( )dr
dp( )dr
GAS SATURATION
R
ELATIVEPERM
EABILITY
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DRIVE MECHANISMS
GAS CAP D R I VE: GAS CAP D R I VE:
oil reservoir with initial gas cap
pressure must fall slowly in order to favor gas cap drive compare todepletion drive as greater recovery is obtained
source of energy: expansion of the gas cap, and expansion of solution gasas it is liberated
oil recovery: 25-30% OOIP
W ATER DR I V E: W ATER DR I V E:
oil reservoir with active aquifer
water drive supports totally or partially the reservoir pressure that tendsto decrease due to production
water moves into the pore spaces originally occupied by oil, replacing theoil and displacing it to the producing wells.
source of energy: expansion of the aquifer
oil recovery: 40% OOIP
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MATERIAL BALANCEMATERIAL BALANCE
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Reserves Evaluation: Material Balance
Material Balance Evaluation (MBE) is based on production history data
NNremaining = NNinitial- Nremoved
MBEMBEINPUTINPUT OUTPUTOUTPUTpressure and production
histories, PVT
reserve evaluation anddrive mechanism
identification
MBE can be written in terms of volumes but, because volumes vary withpressure, they must be referred at the same conditions (i.e. standard orstock tank conditions).
MBE (or Schilthuis equation) expresses the concept that in the reservoirthe algebraic sum of volume variations of oil, gas and water must be null.
The reservoir is considered as a system described by overall parameters,i.e. by total volumes of oil, gas and water and by the values of the
average pressure (pav) and average saturation at each moment. This
assumption is equivalent to consider the reservoir at equilibrium.
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Material Balance
Because of production, the reservoir pressure decreases from theinitial value pi until the average pressure value pav
PRODUCEDPRODUCEDFLUIDSFLUIDS
The removed fluids brought back inthe reservoir must occupy the
volume invaded by the remaining
fluids due to the effect ofp
avi ppp =
ORIGINAL RESERVOIRORIGINAL RESERVOIR
SYSTEM AT INITIALSYSTEM AT INITIAL
PRESSUREPRESSURE ppii
EXPANSION
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Material Balance vs Numerical Simulation
PRESSURE DATA
PRODUCTION DATA
PVT DATA
NONO GEOLOGICAL MODEL ISGEOLOGICAL MODEL ISNEDEEDNEDEED
PETROPHYSICAL CHARACTERISTICS
PRODUCTION DATA
PVT DATA
THE GEOLOGICAL MODEL ISTHE GEOLOGICAL MODEL ISNEDEEDNEDEED
H y d r o c a r b o n s o r i g i n a l l y i n p l a ce
D r i v e m e ch a n i sm
MATERIALBALANCE
SIMULATION
P r e ss u r e v a l u e s
Sa t u r a t i o n v a lu e s
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a) Gas reservoirsa) Gas reservoirs
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Gas Reservoir with Constant Porous Volume
RP GGG =
gi
wip
B
)S1(VG
=
g
wipR
B
)S1(VG
=
giwip GB)S1(V =
g
gi
g
wip
P B
GB
GB
)S1(V
GG =
=
ggigP
B
)BB(GG
=
h d f l i
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Methods for gas reserves evaluation
p/z METHODp/z METHOD
g
gigP
B
)BB(GG
=
=
=i
i
g
giP
p
p
z
z1G
B
B1GG
=
zp
zp
pzGG
i
i
i
iP
TT
p
p
z
)S1(V
B
)S1(V
G
sc
sc
i
i
wip
gi
wip
=
=
= z
p
z
p
TT
p
)S1(V
Gii
sc
sc
wipP
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Pwipsc
sc
i
i G)S1(V
T
T
p
z
p
z
p
= bxay
zp
PG
ii
z
p
G PGG
z
p
Possibleuncertainty
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1. Water1. Water--drive Gasdrive Gas ReservoirsReservoirs
+
=
T
1
p
TW
z
p
p
p
z
z1GG
sc
sce
i
iP
z
p
PG
i
i
z
p
wrongG
GWC
original GWC
Sw = 1
SwiSg = 1-Swi
Sw = 1-Sgr
Sgr
Gas is trapped behind the foreheadwater that is moving forward
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In the presence of water drive, the production of gas must beaccelerated to maximize recovery so as to evacuate the gas
before the less mobile water can catch-up and trap significant
quantities of gas behind the advancing flood front.
01.0
02.0
9.04.0
2.0
k
k
M
g
rg
w
rw
=
=
THE GAS CAN MOVE 100 TIMES
FASTER THAN WATER BY WHICHIT IS BEING DISPLACED
RESIDUAL GAS SATURATION BEHIND THE WATER FRONT
Sgr = 0.3 - 0.4
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dp
dz
z
1
p
1cg =
gwi
fwwwigge c
S1
cSc)S1(cc
++=
2.2. Anomalous initial pressureAnomalous initial pressure ReservoirsReservoirs
gge cc
com
pressibility
p
cg
cfcw
only when pressure decreases
PG
i
iz
p
z
p
Overpressured f(ce)
Gas compressibility
predominates f(cg)
G
Methods fo gas ese es e al ation
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Methods for gas reserves evaluation
RAMAGOSTRAMAGOST--FARSHAD METHODFARSHAD METHODHp: 1) constant effective formation compressibility
2) limited water drive
g
gigP
B)BB(GG =
=z
p
z
p
p
zGG
i
i
i
iP
Accounting for the effective or equivalent formation compressibility:
wi
fwwie
S1ccSc
+=
( ) GzGp
z
pppS1
cSc1z
p
i
pi
i
iiwi
fwiw =
+
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p /z
o r
p /z(1-
p
c e
)
Assuming:
( )
+
= ppS1
cSc1
z
py i
wi
fwiw
pGx =
Gz
pslope
i
i=
i
i
zperceptint =
p/zp/z methodmethod
RamagostRamagost--FarshadFarshad methodmethod
GpGOIP Gwrong
Methods for gas reserves evaluation
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Methods for gas reserves evaluation
American GasAmerican GasAssociationAssociation (AGA)(AGA) METHODMETHOD
g
gigP
B
)BB(GG
=
p
i
igig
gpG
p
z
p
z
p
z
BB
BGG
=
=
p
i
ii
i
i
ii
i
p
i
iG
z
p
z
p z
p
zz
ppzz
pp
G
p
z
p
z p
z
G
=
=
G
G
z
p
z
p
z
p p
i
i
i
i =
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pi
i
i
i GlogG
z
p
logz
p
z
plog +=
xay +=
G
z
p
log i
i
z
p
z
plog
i
i
pGlog
45
GG
Methods for gas reserves evaluation
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Methods for gas reserves evaluation
MBMB equationequation linearizationlinearization METHODMETHOD
g
gigP
B
)BB(GG
=
xay =)BB(GGB gigPg =
PgGB
G
)BB( gig
Recovery factor
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Recovery factor
i
i
g
giP
p
p
z
z1
B
B1
G
G==
RECOVERY IS A FUNCTION OF:
Initial pressure of reservoir
Pressure drop due to the production
Chemical composition of gas
RECOVERY IS INDEPENDENT FROM TIME
Water-Drive Gas Reservoirs
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Water Drive Gas Reservoirs
RP GGG =
g
wpegiP
B
)BWW(GBGG
= We IS THE WATER ENCROACHMENT
VanVan EverdingenEverdingen--HurstHurst equationequation::
= we
Dwew,
2
we r
r
,tQpchr2W
We = f (B, tc, pw, aquifer geometry)
2wew, rhc2B =B = AQUIFER STRENGTH (CONSTANT)
k
rct
2wt
c =tc = CHARACTERISTIC TIME
pw pressure difference at inner radius of the aquifer
Havlena-Odeh Method
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Havlena Odeh Method
g
wpegiP
B
)BWW(GBGG =
wpegiggP BWWGBGBBG +=
egigwpgP W)BB(GBWBG +=+
or F = GEg+We
)BB(
WG
)BB(
BWBG
gig
e
gig
wpgP
+=
+
g
e
g E
WG
E
F+=or
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)BB(BWBG
gig
wpgP
+
G
STRONG AQUIFER
MODERATE AQUIFER
VOLUMETRIC DEPLETION
gE
F
pG
GOIP
Havlena-Odeh method allows identification of the driving mechanism:
if We=0: volumetric depletion; estimate of G
if the (F/Eg) plot is a concave downward shaped arc: water drive
if the production rate is constant, the aquifer strength can be qualitativelyassessed
Methods for gas reserves evaluation
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Methods for gas reserves evaluation
HAVLENAHAVLENA--ODEH METHODODEH METHOD
g
wpegiP
B
)BWW(GBGG
=
)BB(WG
)BB(BWBG
gig
e
gig
wpgP
+=
+
45
AQUIFER TOO STRONG
AQUIFER TOO WEAK CORRECT MATCH
)BB(BWBG
gig
wpgP
+
)BB(
W
gig
e
G
Recovery factor
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Recovery factor
)BWW(GB1
BB1
GG wpe
gg
gip +=
T
1
p
T
z
p
)BWW(G
1
p
p
z
z
1G
G
sc
sc
wpei
ip
+=
RECOVERY IS A FUNCTION OF:
Initial pressure of reservoir
Pressure drop due to the production
Chemical composition of gas
Time (through We)
Drive Index equation
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q
1BG
BWW
BG
)BB(G
gp
wpe
gp
gig =
+
WDIWDI
DDIDDI
1
Dr
iveIndex
0 t
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b) Oil reservoirsb) Oil reservoirs
Undersaturated-Oil Reservoir withConstant Porous Volume
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Constant Porous Volume
RP NNN =
oi
wip
B
)S1(VN
= oiwip NB)S1(V =
=
o
f
o
w
o
wipP
B
V
B
V
B
)S1(VNN
=
o
fwoiP
B
VVNBNN
pSVcV wpww =
p
V
V
1c
= pcVV =
pVcV pff =
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=
o
pfwipwoiP
BpVcpSVcNBNN
]p)cSc(VNB[NBBNfwiwpoiooP
+=
p)cSc(S1
NBNBNBBN fwiw
wi
oioiooP +
+=
p)cSc(S1
NB)BB(NBN fwiw
wi
oioiooP +
+=
p
V
V
1c
=
oiwip NB)S1(V =
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pNBNBNB1cooi
oio =
pBcBB oiooio =
p)cSc(S1
NBpBNcBN fwiw
wi
oioiooP +
+=
pS1
cSccNBBN
wi
fwiwooioP
++=
pS1
cSc)S1(cNBBN
wi
fwiwwiooioP
++=
c e,o
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pcNBBN e,ooioP =
o
ie,ooiP
B
)pp(cNBN
=ppp i =
tcosRGOR s ==
psp NRG =
wo S1S =
Methods for oil reserves evaluation
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Nxy =pcNBBN e,ooioP =
)pp(cB ie,ooi
oPBN
N
Recovery factor
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)pp(cB
B
N
Nie,o
o
oiP =
RECOVERY IS A FUNCTION OF:
Reservoir Initial pressure
Pressure drop due to production
Equivalent compressibility of oil( 10-5 psi-1)
Saturated-Oil Reservoirs with ConstantPorous Volume
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RP GGG =
+=
g
oPoiSPSiP
B
B)NN(NBR)NN(NRG
GAS DISSOLVEDIN OIL
FREE GAS
CUMULATIVE PRODUCTION RATIO
P
PP
N
GR =
oPoigSPgSigPP B)NN(NBBR)NN(BNRBRN +=
)BBBRBR(N)BBRBR(N ooigSgSiogSgPP +=+
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)]RR(BBB[N)]RR(BB[N SSigoioSPgoP +=+
Nxy =
N
)]RR(BB[N SPgoP +
)RR(BBB SSigoio +
Combined Gas-Cap Drive and Water DriveSaturated-Oil Reservoirs
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General equation:
++
++=
g
wpe
g
oPoioiSP
gi
oiSiP
B
BWW
B
)BN(NmNBNB)RN(N
B
NBmNRG
SATURATIONS
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Sor,gSg =1- Sor,g-Swi
Swi
Sg=1-Swi
GAS
OIL
WATER
1
2
3
4
5Swi1
So
Swi
2
3
So=0
Sg =1- So-Swi
Sor,w
Sg 04
Sw =1- Sor,w
Sw =15 So=0
Sg =0
SSor,gor,g SSor,wor,w
Reservoir Pressure Decline
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AVERAGEPRESSURE/
INITIALPR
ESSURE(%
)
PRODUCED OIL / OOIP (%)
GOR
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PRODUCED OIL/OOIP (%)
GOR
(Mcf
/bbl)
PRODUCTION DRIVE MECHANISM IDENTIFICATION
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efwgo W)EmEN(EF +++=
wPSPgoP BW)]R(RB[BNF ++=
)R(RBBBE SSigoioo +=
)B(BB
BE gig
gi
oig =
p)
cS(c)S(1
B
m)(1E fwiwwi
oi
fw ++=
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whenwhen m=0m=0
fwo EE
F
+ STRONG AQUIFER
MODERATE AQUIFER
VOLUMETRIC DEPLETION
pN
OOIP
N