PET-332 Production Engineering I Reservoir Inflow Performance Introduction Any oil or gas production well is drilled
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PET-332 Production Engineering I
Reservoir Inflow Performance
Introduction Any oil or gas production well is drilled and completed to move the oil or gas from the reservoir to the stock tank or sales line. Transport of these fluids requires energy to overcome friction losses in the production system. The fluids must travel through the reservoir and the piping system and ultimately flow into a separator for gas-liquid separation. A production system can be relatively simple or can include many components in which energy or pressure losses occur. The selection and sizing of the individual components of the production system is very important since all the components are interrelated. 2
Production system
The production system is a composite term describing the entire production process. Starting from the reservoir and ending at the stock tank or even beyond.
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Production is expansion of fluids :: 1 Isothermal compressibility for a fluid is defined as
1 ∂V C=− V ∂P T Expansion of fluid may be written as
dV = C V dP 4
Production is expansion of fluids :: 2 For each fluid type in the reservoir :
∆Vtotal ≡ Production = Co Vo ∆P + C g Vg ∆P + Cw Vw ∆P
or
Production = ∆P ( Co Vo + C g Vg + Cw Vw
)
! ! ! Above formulation ignores the contribution of rock expansion. 5
Production is expansion of fluids :: 3 Typical compressibility values are:
1 Co = 15 10 psi −6 1 Cw = 8 10 psi −6 1 C g = 500 10 psi −6
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Flow into wellbore One of the most important concepts in the production system is the reservoir. Unless accurate predictions can be made about flow into the wellbore from the reservoir, the performance of the system can not be analyzed. The flow into the well depends on the drawdown or pressure drop in the reservoir :
q α
( pR − pwf )
The relationship between flow rate and pressure drop occurring in the porous medium can be very complex and depends on parameters such as rock properties, fluid properties, flow regime, fluid saturations, compressibility of the fluids and rock, formation damage or stimulation and drive mechanisms. 7
Inflow performance Development of the bottomhole pressure gauge in the late 1920s led to the practice of testing wells by simultaneous measurement of surface production rate and bottomhole pressure. The obvious reason to test a well is to determine what the production rate will be if a certain backpressure is exerted at the wellhead. The flow from the reservoir into the well has been called inflow performance. A plot of producing rate versus bottomhole flowing is called inflow performance relationship or IPR. IPR Since the early days of testing wells, most efforts have concentrated on the formulation of simple questions expressing the relation between the surface rate and bottomhole flowing pressure. The basic equation which relates the pressure drop and rate in a porous media is Darcy’s law. 8
Henry Darcy In 1856, while performing experiments for the design of sand filter beds for water purification, Henry Darcy proposed an equation relating apparent fluid velocity to pressure drop across the filter bed.
Henry Philibert Gaspard Darcy
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Darcy’s experimental apparatus
h1-h2
Q A L
h4 h2
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Darcy’s conclusion
Darcy concluded that rate goes through a sand pack are functions of cross-sectional area, length, pressure difference and a coefficient K that is a property of the sand pack and the fluid flows through it :
K A ( h1− h2 ) q= L
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Elements of coefficient K
Darcy’s K coefficient was determined to be a combination of
k
permeability of the sand pack and
µ viscosity of the fluid that flows through the sand pack That is
K=
k
µ
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Equation in differential form Under the following assumptions Darcy’s equation may be written in differential form :
kA dP q = vA = − µ dx Linear and horizontal system Constant cross section Incompressible liquid Laminar flow Fully saturating nonreactive liquid Single-phase system Constant temperature Fluid properties are constant with changing pressures
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Linear flow q p1
q
A
p2 L
For linear flow (constant area flow), the equation may be integrated to give the pressure drop occurring over distance L
∫
k dP
q L = − ∫ dx A 0 µ
P2
P1
If it is assumed that k, µ and q are independent of pressure, or that they can be evaluated at the average pressure in the system, the equation becomes
∫
P2
P1
qµ L dP = − dx ∫ 0 kA
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Linear flow (continued) Integration gives
θ k A( P1 − P2 ) q= µL where θ is a unit conversion factor. In Darcy units value of the conversion factor is 1.0 and in the field units is 1.127 10-3. The following table can be used as a guideline for the units : Variable
Symbol
Darcy
Field
Flow rate
q
cc/sec
Bbl/day
Permeability
k
darcys
md
Area
A
cm2
ft2
Pressure
P
atm
psia
Viscosity
µ
cp
cp
Length
L
cm
ft 15
Linear flow (continued) If P vs L is plotted on Cartesian coordinates, the equation produces a straight line of constant slope, (-qµ/kA). This tells us that the variation of pressure with distance is linear. If the flowing fluid is compressible, the in-situ flow rate is a function of pressure. Using the fact that the mass flow rate ρq, must be constant the following derivation will reveal the form of Darcy equation for compressible (real gas) flow :
PM ρ= zRT ρ q = ρ sc qsc
ρ q PTsc qsc = = q ρ sc PscT z
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Linear flow (continued)
qsc PscT Tsc qsc PscT Tsc
∫
L
∫
L
0
0
dL = −kA∫
P2
P1
kA dL = − µz
∫
P dP µz P2
P1
P dP
8.93 µ z TL P −P = qsc kA 2 1
2 2
Use field units with the above equation. T is in °R and µ and z are evaluated at the average reservoir pressure.
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Radial flow
pw
re
pe
rw h
Darcy’s law can be used to calculate the flow into a well where the fluid is converging radially into a relatively small hole. In this case area open to flow (A=2πrh) is not constant and must be included in the integration : 18
Radial flow (continued) 2π r h k dP q= µ dr In the case of small compressibility fluids, change in the liquid volume may handled with formation volume factor correction :
2π r h ko dP qo Bo = µo dr Pe re dr ko 2π h ∫ ( ) dP = qo ∫ Pwf µ B rw r o o When integrating the previous equation, it is usually assumed that the pressure function f (p)= ko/µoBo is independent of pressure or it can be evaluated at average pressure in the well’s drainage volume. 19
Radial flow (continued) This is necessary because no simple analytical equation for this term as a function of pressure can be formulated. Under the previous assumptions the integration leads to the following equation
2π ko h ( Pe − Pwf ) qo = re µo Bo ln( ) rw For field units
0.00708 ko h ( Pe − Pwf ) qo = re µo Bo ln( ) rw where r and h are in ft and Bo is in Bbl/STB. 20
Radial flow (continued) The equations we derived so far applies for steady-state (Pe=constant), laminar flow and a well in the center of a circular drainage area. It is more useful if we rewrite the equations in terms of directly measurable quantities (such as average reservoir pressure) and stabilized flow (pseudo-steady state) :
0.00708 ko h ( PR − Pwf ) qo = re µo Bo ln(0.472 ) rw It must be noted that in radial systems with a centered well in the middle, the average reservoir pressure happens at 0.472 re under stabilized flow conditions. 21
Radial flow (continued) For the gas flow
∫
Pe
Pwf
PdP =
qsc µ gTPsc Z 2πhk gTsc
dr ∫rw r re
r qsc µ gTPsc Z ln( e ) rw 2 2 Pe − Pwf = π hk gTsc In terms of average reservoir pressure
703 10 −6 k g h( PR2 − Pwf2 ) qsc = re µ g Z T ln(0.472 ) rw 22
Flow types-1 Transient flow is defined as a flow condition which radius of pressure wave propagation from wellbore has not reached the boundaries of the reservoir. During the transient flow a small portion of the reservoir contributes to production. Therefore the well behaves as if it is producing from an infinitely large reservoir. This condition is only applicable for a relatively short period after some pressure disturbance has been created in the reservoir. In this case, both the pressure and time derivative of pressure are themselves functions of both position and time:
P = f (r , t ) ∂P = f (r , t ) ∂t 23
Flow types-2 (transient flow) Assuming single-phase oil flow in porous media, constant rate solution of the diffusivity equation gives
162.6qo µo Bo ko log t + log Pwf = Pi − − 3.23 + 0.87 S 2 φµo ct rw ko h Pwf Pi qo µo t ko h ø ct rw
flowing bottom-hole pressure, psia initial reservoir pressure, psia oil production rate, STB/day oil viscosity, cp flow time, hours effective horizontal permeability to oil, md reservoir thickness, ft porosity total compressibility, 1/psi wellbore radius, ft
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Flow types-3 After an initial production period with transient well pressure and rate, the outer boundary starts affecting production at the wellbore and flow stabilizes. When stabilization is reached, the constant pressure boundary at the limits of the reservoir causes steady-state flow. flow Wells producing under steady-state conditions do not experience depletion, since average reservoir pressure remains constant. In such reservoirs the volumetric average reservoir pressure is approximately located at 0.61re. Stabilized flow from wells with no-flow boundaries is usually referred to as pseudosteady-state flow. flow This type of production results from depletion, and a major consequence is that average reservoir pressure declines. In such reservoirs the volumetric average reservoir pressure is approximately located at 0.472re.
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Flow types-4
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Reservoir pressure profile The pressure distribution in a radial drainage reservoir can be analyzed by plotting pressure versus radius as predicted by radial flow equation (stabilized flow) :
141.2qo µo Bo 141.2qo µo Bo P = PR − ln(0.472 re ) + ln( r ) ko h ko h For steady state flow
141.2qo µo Bo 141.2 qo µo Bo P = PR − ln(0.61 re ) + ln( r ) ko h ko h
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Reservoir pressure profile (continued) A plot pressure versus radius for typical well conditions shows the large increase in pressure gradient as the fluid increases in velocity near the wellbore. Approximately one-half of the total pressure drawdown occurs within a 15 ft radius from the well. For gas flow, the pressure drop around the wellbore is even more severe.
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Reservoir pressure profile (continued) Examination of the previous equation reveals that a plot of P versus ln(r) will result in a straight line of constant slope m,
141.2 qo µo Bo m= ko h For gas flow, a plot of P2 versus ln(r) results in straight line of slope m,
m=
1422 qsc µ g z T kgh 29
Reservoir pressure profile (continued) P - ln(r) plot will have a constant slope if all of the terms on the righthand side of the equation remain constant. A different slope and a different Pwf would be obtained for each flow rate qo.
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Productivity index Earlier in the chapter it is stated that the flow into the wellbore depends on the drawdown or pressure drop in the reservoir such a way that :
qo α
( pR − pwf )
Perhaps the simplest way to relate rate and pressure drop is to use a straight-line IPR, which states that the rate is directly proportional to pressure drawdown in the reservoir. The constant of proportionality is called productivity index “ J “. Then the above equation becomes
qo = J ( pR − pwf ) 31
Productivity index (continued) or
J=
qo PR − Pwf
From Darcy equation productivity index becomes
J=
0.00708 ko h re µo Bo ln(0.472 ) rw
Above equations are valid only if the flowing wellbore pressure is above the bubble point pressure.
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Specific productivity index The specific productivity index is defined to account for formation thickness :
qo Js = h ( PR − Pwf )
From Darcy equation specific productivity index becomes
Js =
0.00708 ko
re µo Bo ln(0.472 ) rw
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Total productivity index The total productivity index is defined to account for total barrels of liquid (oil + water) or total production from a well :
qo + qw JT = ( PR − Pwf ) From Darcy equation total productivity index becomes
0.00708 h k kw o JT = + r µ B µ B ln(0.472 e ) o o w w rw 34
Classifying productivity index Common oil field usage is to classify a well as having either low, intermediate, or high PI. The following limits may be used to classify the productive oil wells :
< 0.5
Low
Between 0.5 - 1.5
Intermediate
> 1.5
High
PI’s of 0.01 are not uncommon on the low side while PI’s of 50-100 are not uncommon on the high side. 35
Straight line IPR
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Example #1 A well producing from a reservoir having an average pressure of 2085 psig produced at a rate of 282 STB/day when bottomhole pressure was 1765 psig. Calculate : 1.
The productivity index J
2.
The production rate if Pwf is reduced to 1485 psig
3.
The bottomhole pressure necessary to obtain an inflow of 400 STB/day
4.
The inflow rate if Pwf is reduced to zero (called absolute open flow potential, AOF) 37
Solution #1
qo 282 1. J = = = 0.88 STB / day − psi PR − Pwf 2085 − 1765 2. q0 = J ( PR − Pwf ) = 0.88 * ( 2085 − 1485) = 528 STB / day 3. Pwf = PR −
qo 400 = 2085 − = 1630 psig J 0.88
4. qo ,max = J ( PR − 0) = 0.88 * ( 2085) = 1835 STB / day
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Dependency of pressure function f(P) The predictions made in the previous example are valid only if J remains constant. This implies that the pressure function f(p)=ko/µoBo remains constant, which is seldom the case. Viscosity increases with pressure, while oil formation volume factor decreases with pressure. The composite effect is that (1/µoBo) decreases almost linearly with pressure.
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Productivity index from Darcy’s equation The productivity index can also be expressed as following from the theoretical point of view :
0.00708 h PR ko qo = dP ∫ re Pwf µo Bo ln(0.472 ) rw PR qo ko 0.00708 h J≡ = dP ∫ P r ( PR − Pwf ) ( P − P ) ln(0.472 e ) wf µo Bo R wf rw
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