res eng 3 - mat balance

8/22/2019 Res Eng 3 - Mat Balance

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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


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


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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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 =

+


33/58

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



34/58


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 =


35/58

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



36/58


MBMB equationequation linearizationlinearization METHODMETHOD

g

gigP

B

)BB(GG

=

xay =)BB(GGB gigPg =

PgGB

G

)BB( gig

Recovery factor


37/58

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


38/58

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


39/58

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


40/58

)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



41/58


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


42/58

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

+=


Initial pressure of reservoir

Pressure drop due to the production

Chemical composition of gas

Time (through We)

Drive Index equation


43/58

q

1BG

BWW

BG

)BB(G

gp

wpe

gp

gig =

+

WDIWDI

DDIDDI

1

Dr

iveIndex

0 t


44/58

b) Oil reservoirsb) Oil reservoirs

Undersaturated-Oil Reservoir withConstant Porous Volume


45/58

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 =


46/58

=

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 =


47/58

pNBNBNB1cooi

oio =

pBcBB oiooio =

p)cSc(S1

NBpBNcBN fwiw

wi

oioiooP +

+=

pS1

cSccNBBN

wi

fwiwooioP

++=

pS1

cSc)S1(cNBBN

wi

fwiwwiooioP

++=

c e,o


48/58

pcNBBN e,ooioP =

o

ie,ooiP

B

)pp(cNBN

=ppp i =

tcosRGOR s ==

psp NRG =

wo S1S =

Methods for oil reserves evaluation


49/58

Nxy =pcNBBN e,ooioP =

)pp(cB ie,ooi

oPBN

N

Recovery factor


50/58

)pp(cB

B

N

Nie,o

o

oiP =


Reservoir Initial pressure

Pressure drop due to production

Equivalent compressibility of oil( 10-5 psi-1)

Saturated-Oil Reservoirs with ConstantPorous Volume


51/58

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 +=+


52/58

)]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


53/58

General equation:

++

++=

g

wpe

g

oPoioiSP

gi

oiSiP

B

BWW

B

)BN(NmNBNB)RN(N

B

NBmNRG

SATURATIONS


54/58

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


55/58

AVERAGEPRESSURE/

INITIALPR

ESSURE(%

)

PRODUCED OIL / OOIP (%)

GOR


56/58

PRODUCED OIL/OOIP (%)

GOR

(Mcf

/bbl)

PRODUCTION DRIVE MECHANISM IDENTIFICATION


57/58

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 ++=


58/58

whenwhen m=0m=0

fwo EE

F

+ STRONG AQUIFER

MODERATE AQUIFER

VOLUMETRIC DEPLETION

pN

OOIP

N

res eng 3 - mat balance

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