Gl b l Fl R iGlobal Flow Regimes• At any given time in the producing life of a
reservoir, the fluid flow condition existing may be characterized as eithery
a) transient, b) pse dostead state orb) pseudosteady-state or c) steady-state.
• What do these terms mean?
R i FlReservoir Flow
• Initially, in a virgin reservoir, the pressure at any fixed depth is constant.As production begins the pressure near the• As production begins, the pressure near the wellbore drops significantly as near-wellbore fluids expand to satisfy the imposed production condition.
• Far away from the well, no measurable pressure drop can be observed at early timespressure drop can be observed at early times – locations far away from the well are not
“aware” that the reservoir is being produced.
TransientTransient• As time progresses, pressure drops can be
meas red f rther and f rther a a from themeasured further and further away from the well, an increasing volume of the reservoir fluids expand to contribute to the well's production.
• During this period, the reservoir is said to be “infinite acting” and the flow is transient;infinite acting and the flow is transient; pressure drop at outer reservoir boundary is negligible.
• The pressure versus time behavior at the producing wellbore contains information about the reservoir permeability.p y
P d t d St tPseudosteady State• After a long time pressure drops can be• After a long time, pressure drops can be
measured at all reservoir locations– the entire reservoir is contributing to the g
well's production. • At this time, the pressure changes at the
same rate at every location in the reservoirsame rate at every location in the reservoir, i.e., dp(x,t)/dt = constant; pseudosteady state flow.
• The pressure versus time behavior at the wellbore reflects the volume of fluid (or the reservoir pore volume) contributing toreservoir pore volume) contributing to production.
Steady Statey• To see steady-state flow in a reservoir, we must
replace reservoir fluids at the same rate that we thremove them.
• This situation may occur if we have a recharge toThis situation may occur if we have a recharge to the system (an assocaited water aquifer) or may also occur in secondary and enhanced oil recovery operations e g waterflooding gasrecovery operations - e.g., waterflooding, gas injection, etc.
• During steady-state single-phase flow, nothing is changing in the reservoir, i.e., dp(x,t)/dt = 0. Note there is a pressure drop in the reservoir, andthere is a pressure drop in the reservoir, and pressure data contains information about recharging system parameters.
Comparisons of Flow Regimes
PumpAir
t = 1
q q
ViscousLiquid
ViscousLiquid
t 1
t = 100
Initial, t = 0 Transient
constantdDdt
=qq 0dD
dt=
ViscousLiquid
qD
Pseudosteady-state Steady-state
D ’ LDarcy’s Law
• For flow through a horizontal sand pack, flow rate is– Directly proportional to the pressure drop
across the packDirectl proportional to the (gross) area– Directly proportional to the (gross) area open to flow
– Inversely proportional to the length of theInversely proportional to the length of the pack
– Inversely proportional to fluid viscosityy p p y• Constant of proportion is the “permeability”
D ’ LDarcy’s Law• For single-phase, radial flow, Darcy’s equation isFor single phase, radial flow, Darcy s equation is
given by
rpkxvr ∂∂
−= −
μ310127.1
RB/(ft2-D)
r∂μ
( )rprhrkqB∂∂
×= −
μ31008.7 RB/D
Simplified Reservoir ModelSimplified Reservoir ModelReservoirReservoir
radius Damageradius
Wellradius rs re, pe
h
Cylindrical horizontal Reservoir; constant rate at every radius;Reservoir permeability, k; damaged zone permeability, ks;No gravity effects; constant viscosity formation volume factorNo gravity effects; constant viscosity, formation volume factor.
D ’ LDarcy’s Law
• Rate at any radius:( ) phrkB ∂−310087
• Pressure distribution
( )rprqB∂
×=μ
31008.7
• Pressure distribution– Integrate Darcy’s law over radius
O t ( d d )– Outer (undamaged zone)
∫∫ =× −ee rp drdpkh310087
( )∫∫ =×rrp r
dpqBμ
1008.7
Pressure Distribution – Outer Zone
• Perform integration
( ) ⎞⎛μ rqB2141
• Note: Even if there is no damaged zone
( ) ⎟⎠⎞
⎜⎝⎛μ
=−rr
khqBrpp eo
e ln2.141
• Note: Even if there is no damaged zone, pressure drop is greatest close to the wellbore radiuswellbore radius.– Why is this so?
G hi l P Di t ib tiGraphical Pressure Distribution
Pressure versus radius
199020002010
psia
196019701980
ress
ure,
p
19401950
0 200 400 600 800 1000 1200 1400
P
Radius, ft
N tNote
• Most of the pressure drop occurs within the first few inches of the wellbore.
– As fluids approach the wellbore, the area available to flow is decreasing g(2πrh)
– Pressure losses increase as fluids approach wellbore.
R i d dReservoir drawdown
• Integrate Darcy’s law over entire reservoir
( )∫∫ =× −ee r
r
p
p rrkdrdp
qBhμ
31008.7 ( )wwf rpq μ
∫∫∫ +=× −ese rrp
kdr
kdrdp
Bh31008.7 ∫∫∫
swwf rr sp krrkp
qBμ
⎥⎤
⎢⎡
+ es rrqB l1l12.141 μ⎥⎦
⎢⎣
+=−e
e
w
s
swfe rkrkh
qpp lnlnμ
Si lifSimplify
⎤⎡⎥⎦
⎤⎢⎣
⎡++−=−
e
e
w
s
w
s
w
s
swfe r
rkr
rkr
rkr
rkh
qBpp ln1ln1ln1ln12.141 μ
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+=− se
wfe rr
kk
rr
khqBpp ln1ln2.141 μ
• For undamaged reservoir (k = ks)⎦⎣ ⎠⎝ wsw rkrkh
⎥⎦
⎤⎢⎣
⎡=− e
wfe rr
khqBpp ln2.141 μ
⎦⎣ wrkh
Ski F tSkin Factor
• Define skin factor, s, as
srk l1⎟⎞
⎜⎛
• Skin factor isw
s
s rks ln1⎟⎟
⎠⎜⎜⎝
−=
– A dimensionless number– = 0 if there is no damage– > 0 if there is damage– < 0 if the near-wellbore region is
ti l t dstimulated
P d ti it I dProductivity Index
• For steady state flow
( ) ==khqJ ( )
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛μ
=−
=
srrB
ppJ
w
ewfeo
ln2.141
– A positive skin factor will reduce the well’s productivityA ti ki ill i it– A negative skin will increase it
– Are there any other factors that influence a well’s productivity?well s productivity?
Average Reservoir PressureAverage Reservoir Pressure• Productivity Index is usually expressedProductivity Index is usually expressed
in terms of average reservoir pressure rather than external pressure, pep , pe
– Average pressure is also used in material balance calculationsmaterial balance calculations
– Obtainable from analysis of well test data ( ) ( )
rr
ddhee
φ ∫∫data
( )( ) ( )
( )22
,2,2r
rr
rr
drtrrpdrtrprhtp w
e
w
−=
φπ
=∫
∫
∫( )
2 we
r
rrdrrh
w
φπ∫
P d ti it I dProductivity Index
• Steady-State Flow
⎤⎡ ⎞⎛⎥⎦
⎤⎢⎣
⎡+−⎟⎟
⎠
⎞⎜⎜⎝
⎛μ=− s
rr
khqBpp
w
eowf 2
1ln2.141
• Productivity Index expression
kh( )
⎟⎟⎞
⎜⎜⎛
+−⎟⎟⎞
⎜⎜⎛
μ
=−
=
srB
khpp
qJewf
o1ln2.141 ⎟
⎠⎜⎝
+⎟⎟⎠
⎜⎜⎝
μ sr
Bw 2
ln2.141
N tNotes
• Productivity Index for steady state flow is constant– Pressure at each point in the
reservoir does not change with timeg• PI strongly influenced by the skin factor
– We would like to identify wells whereWe would like to identify wells where skin factor is large; we can increase production by a stimulation p oduct o by a st u at oworkover.
SkiSkin
• In practice, skin may be due to a variety of factors– Damage to formation due to invasion of
mud filtrate and mud solidsPartial penetration– Partial penetration
– Migration of finesAsphaltines– Asphaltines
• Treatment of skin will depend on the specific cause.cause.
Productivity Index – Steady State oduct ty de Steady StateRadial Liquid Flow
• In terms of external reservoir pressure
( ) ⎞⎛ ⎞⎛==
khqJo
• In terms of average reservoir pressure
( )⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛μ
−s
rrB
pp
w
ewfeo
ln2.141
• In terms of average reservoir pressure
==khqJ ( )
⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛μ
=−
=
srrB
ppJ
w
ewfo
21ln2.141
⎠⎝ ⎠⎝
A l ti TAccumulation Term
• During pseudosteady state flow,
( )∂ trp
• Reservoir flow equation for pseudosteady
( ) constant,==
∂∂ A
ttrp
state
pk⎟⎞
⎜⎛ ∂∂006330 Ac
rpkr
rr tφ=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
μ∂∂00633.0
B d C ditiBoundary Conditions
• The previous differential equation is secondorder; we need two boundary conditions
• In addition we have an unknown constant, A– Outer boundary sealed:
I b d t t t t
0=∂∂
errp
– Inner boundary at constant rate:
( )31.127 10 2 k prh qBr
πμ
−⎡ ⎤∂× =⎢ ⎥∂⎣ ⎦
wrrμ ∂⎣ ⎦
D t i ti f C t t ADetermination of Constant A
• Integrate flow equation over reservoir
φ=⎟⎟⎞
⎜⎜⎛ ∂∂
∫∫006330rr
rdrAcdrpkree
⎟⎟⎞
⎜⎜⎛ −φ∂∂
φ=⎟⎟⎠
⎜⎜⎝ ∂μ∂ ∫∫00633.0
22wet
rt
r
rrAcpkrpkr
rdrAcdrr
rr
ww
• Inner boundary condition
⎟⎟⎠
⎜⎜⎝
=∂μ
−∂μ 200633.0
wet
rr rr
rr
we
( )31.127 10 2k p qBr
r hμ π−
⎡ ⎤∂=⎢ ⎥∂ ×⎣ ⎦ ( )1.127 10 2
wrr hμ π∂ ×⎣ ⎦
Rate of Change of Pressure with ate o C a ge o essu e ttime
• Solve for A2 2c A r r qBφ ⎛ ⎞−
( )30.00633 2 1.127 10 25 615
t e wc A r r qBh
p qB
φπ−
⎛ ⎞−=−⎜ ⎟ ×⎝ ⎠
∂
( )2 2
5.615
e w t
p qBAt h r r cπ φ∂
= =−∂ −
• During pseudo-steady state flow, pressure is a linear function of time; slope inversely proportional to pore volumeproportional to pore volume.
P d ti it I dProductivity Index
• Pseudosteady State Flow
( ) 1fkh p p r− ⎛ ⎞
• If skin were included
( ) 1ln141.2 2
e wf e
w
kh p p rqB rμ
⎛ ⎞= −⎜ ⎟
⎝ ⎠
( ) 1lne wf ekh p p r s
− ⎛ ⎞= − +⎜ ⎟ln
141.2 2w
sqB rμ
+⎜ ⎟⎝ ⎠
( )q khJ = =
⎛ ⎞⎛ ⎞( ) 1141.2 ln2
e wf e
w
p p rB sr
μ⎛ ⎞− ⎛ ⎞
− +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
N tNote• During Pseudosteady state flow, pressure is
h i ith ti h J ( PI) ichanging with time; however J (or PI) is a constant.
– External pressure (or average average) and wellbore pressure are changing at exactly the same rate, so difference between themthe same rate, so difference between them is constant.
• We can also derive a Productivity Index• We can also derive a Productivity Index equation in terms of average pressure.
( ) 3q khJ = =
⎛ ⎞⎛ ⎞( ) 3141.2 ln4
wf e
w
p p rB sr
μ⎛ ⎞− ⎛ ⎞
− +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
N i l R iNon-circular Reservoirs
• Productivity Index expressed in terms of reservoir area, A, and Dietz shape factor, CA
q kh( )
21 2.2458141.2 ln2
wf
w A
q khJp p AB s
r Cμ
= =⎛ ⎞− ⎛ ⎞
+⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠w A⎝ ⎠⎝ ⎠
Transient FlowTransient Flow• Transient Flow includes a set of transientTransient Flow includes a set of transient
flow regimes
– Wellbore storage dominated flow– Spherical flow
Radial Flow– Radial Flow– Linear Flow– etcetc
• These are all subsets of an overall transient flow regime
R ll Li S S l tiRecall Line-Source Solution
• Pressure Drop in an “Infinite” system– Well modeled by a “zero-radius” line
( )t
rEkhqB
trEi
khqBtrppi ⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−=−
ημ
ημ 2
1
2
46.70
46.70,
duekhqB
tkhtkhu
∫∞ −
=
⎟⎠
⎜⎝
⎟⎠
⎜⎝
μ
ηη
6.70
44
E1(x) =-Ei(-x)ukh
tr∫η4
2
• Solution valid at all radii at all times