Pressure/depth And Inflow Performance Relationship (ipr)

Chapter I: Pressure/Depth and Inflow Performance Relationship (IPR)

1

Pressure/Depth • Calculate Fluid Gradient given Density or pressure differential and the depth. • Calculate the Pressure given Depth and Gradient or Density • Calculate the equivalent Fluid column when given pressure and gradient or density • Estimate Fluid level or surface pressure when given pressure at depth and fluid gradient • Draw a simple pressure-depth plot. 2

Pressure and Depth

P=0

• Pressure and Depth is the FUNDAMENTAL relationship in the oil industry. depth

• Your understanding of the concept is critical to being a successful production engineer. P=?

3

Pressure Gradient • The easiest way to calculate pressure from depth is to use the pressure gradient of the given fluid. • Pressure gradients for incompressible fluids have units of pressure/depth. For example, psi/ft, bar/m.

• Pressure gradient seems difficult, but it is simply using the density of the fluid and converting units: • The density of pure water is 62.3-lb/ft³

4

Pressure Gradient • To convert to gradient: 62.3 lb/ft3 = 0.433 psi/ft

• This is the gradient for pure water (SG = 1) in Imperial units, remember it. • Specific Gravity is always relative to pure water • For Imperial Units, 1 lbf = 1lbm at standard gravity. For S.I. Units, be careful to correct for the accelerations gravity. 5

Calculating Pressure Gradient • Most of the time you will not be given a fluid gradient or an average specific gravity, you will need to calculate it. • First calculate your average specific gravity: SGAVG = SGH20*WC + SGOIL*(1-WC) • Then calculate your Gradient: GAVG = GH20*SGAVG • REMEMBER! GH2O equals 0.433 6

Calculating Pressure Gradient • If you are given the API gravity of oil instead of the SG, then use this formula:

SGOIL

141.5  131.5  API

7

Calculating Pressure Gradient Example • Water cut (WC) = 80% = 0.80 • Water SG = SGH20 = 1.04 • Oil Gravity = 25° API • Solution: – Oil Cut = (1-WC) = 1 - 0.80 = 0.20 – Oil Density: 141.5

SGOIL 

131.5  25

 0.904

– Average Density = SGAVG = SGH20*WC + SGOIL*(1-WC) – SGAVG = 1.04*0.80 + 0.904*0.20 = 1.013 – GAVG = SGAVG*GH20 = 1.013*(0.433-psi/ft) = 0.439-psi/ft 8

Pressure-Depth Plot

Pressure = 0 @ Surface

•

• • •

To find a pressure at a given depth, simply multiply the VERTICAL depth by the given fluid gradient. P = D*G For example, if my depth is 1200-ft and my gradient is 0.44psi/ft, then my pressure is 1200ft*0.44-psi/ft = 528-psi Assuming that the fluid is incompressible, this is a linear relationship. We can draw this on a graph that we call the pressure-depth plot. Obviously, denser fluids, and therefore higher fluid gradients, result in higher pressure.

Pressure ->

<- Vertical Depth

•

Increasing Density

Pressure-Depth Plot

Pressure  0 @ Surface Pressure ->

If the pressure at surface isn’t zero, then the whole line shifts over according to the surface pressure. <- Vertical Depth

•

Pressure-Depth Plot •

If the fluid doesn’t reach the surface, then there is some ‘fluid level’, or depth, where the pressure is zero and then the pressure increases according to the gradient. Fluid Level from Surface

<- Vertical Depth

Pressure ->

Similarly, if we know the pressure and the gradient, we can calculate the equivalent fluid column resulting from that pressure. H = P/G

•

Here the effect of increasing gradient is reversed, and a denser fluid results in a shorter fluid column for a given pressure.

Increasing Density

•

Fluid Height ->

Calculating the Fluid Height or Column

Pressure ->

Calculating Gradient

Pressure ->

Similarly, we can calculate gradient when given the pressure differential and the depth differential. G = P/D or G= P2-P1 / D2 – D1

• •

CAUTION! Unless we’re given both pressures and both depths, we don’t really know where this would be on the curve.

? <- Vertical Depth

•

?

? ?

Set 1.01

14

Well Productivity • Upon completion of this section, you should be able to: • Explain "Productivity Index" as a summary of reservoir performance.  Use Productivity Index equation derived form Darcy's Law to calculate performance for a given well (QMAX when given PR and PI, or calculate PWF or Q given the other).  Calculate a Productivity Index from well test data.

• Explain the Vogel relationship for a well producing oil, water, and gas Compare Vogel to "PI" for a given case  Use Vogel’s relationship to calculate well performance when given a test point.  Use a Vogel relationship combined with PI when the reservoir static pressure is greater than the bubble point  Use Vogel combined with PI on an arithmetic basis when the water cut is high but the well pressure is less than bubble point

• Correct IPR for depth to its proper location 15

Well Productivity - Inflow

• Simply enough, Inflow is the relationship between pressure and flow rate at the sand face. • Inflow is THE MOST IMPORTANT issue for well performance and evaluation.

16

Well Productivity For the remainder of the program, we are going to make the assumption that fluid always flows from high pressure toward low pressure. Some of you may recognize that this is not exactly true. The exactly true expression is fluid always flows from high potential toward low potential.

17

Well Productivity The difference between "pressure" and "potential" is the elevation (or height) and the elevation potential can be calculated from the equation *gc * h.

14.7 psi

We have already seen how pressure increases with the depth in a column of fluid.

6"

14.9 psi

18

Inflow – Darcy’s Experiments • The relationship between pressure and Flow rate was first studied extensively by the scientist Henry Darcy (1803-1858). • He created pressure differentials across a porous media and measured the resulting flow rates that resulted from those pressures. • His experiments resulted in what is now known as ‘Darcy’s Law’ (1856) and are the benchmark for permeability. In fact, the unit of permeability is called the ‘Darcy’ (D). Permeable Medium: Fluid Properties: Viscosity, Volume Factor P1

Area, Length, Permeability

Direction of Flow

P0

19

Darcy’s Law • For general flow through porous Media:

k * A *( P0  P1 ) Q  *L

• But we’re working with oil reservoirs, not general porous media…

20

Well Productivity Pr is the average reservoir Shut-in pressure Reservoir

Perforations

Pr

Pr 21

Darcy's Law for radial flow into a wellbore: Pr

Pr

Reservoir outer "drainage" boundary

Fluid Flow

Q=? Pwf

Fluid Flow

Pr 22

Darcy's Law for radial flow into a wellbore: For the system just described, Darcy's Law looks like:

qo 

7.08 x 10

 o Bo

-3

ln

k o h ( Pr

re rw

P wf ) S

qo = flow rate ko = effective permeability h = effective feet of pay o = average viscosity Pr = reservoir pressure Pwf = wellbore pressure re = drainage radius rw = wellbore radius Bo = formation volume factor Note: (Pr - Pwf) is the drawdown pressure

23

Darcy's Law for radial flow into a wellbore: All of the data necessary for this equation is usually not available. But if we make the assumption that ko, h, re, rw, Bo and o are constant for a particular well (this is a pretty good assumption), the equation becomes:

k 1 k2 k 3

qo  k 4k5

ln

( Pr

k6

k7

P wf ) k8

24

Darcy's Law for radial flow into a wellbore: Simplifying...

qo  K (Pr  Pwf )

25

Darcy's Law for radial flow into a wellbore: Re-arranging terms, we obtain...

1 Pwf   qo + Pr K

26

Darcy's Law for radial flow into a wellbore: This is an equation of the form" y=mx+b" which is a straight line. Furthermore, the line has a slope of "m" and a Y-intercept of "b".

The constant, K, has a special name: Productivity Index or "PI" for short.

1 Pwf   qo + Pr K 27

Darcy's Law for radial flow into a wellbore: Pressure - PSI

Intercept = Pr Slope = -1/K

Pwf

0 Q - Flow Rate (BPD)

0

28

Darcy's Law for radial flow into a wellbore: The Productivity Index (PI) is equal to the flow rate divided by the "drawdown":

PI 

qo

(P  P ) r

wf

29

Example Darcy's Law for radial flow into a wellbore: Consider the following example: Pr = 2,300 psi, and Pwf = 1,200 psi @ qo = 1,150 bpd What is the Productivity Index (PI) of the well?

PI 

1150

(2300 - 1200)

= 1.046 bbl/day/psi

30

Darcy's Law for radial flow into a wellbore: What is the maximum flow rate the well will produce? The maximum flow rate occurs at the maximum drawdown (Pwf = 0).

PI 

qmax

(P  0 )

or

qmax  Pr x PI

r

qmax  2300 x 1.046 = 2406 BPD 31

Example – Calculating PWF • Now that we know our PI, we can calculate the PWF for a given Q. • Two-Step Method: – – – –

Calculate P, then calculate PWF. Example: Q = 2000-bpd. P = Q/PI = 2000-bpd/1.046-bpd/psi P = 1912-psi PWF = PR - P = 2300-psi – 1912-psi = 388-psi

• One-Step Method: – PWF = PR – Q/PI = 2300-psi – 2000-bpd/1.046-bpd/psi = 388-psi

32

Example – Calculating Q • Similarly, we may want to calculate Q when we know PWF. • Two-Step Method – – – –

Calculate P, then calculate Q Example: PWF = 500-psi P = PR – PWF = 2300-psi – 500-psi = 1800-psi Q = P*PI = (1800-psi)*(1.046-bpd/psi) = 1883-bpd

• One-Step Method – Q = (PR – PWF)*PI = (2300-psi – 500-psi)*(1.046-bpd/psi) = 1883-psi 33

1.02

34

Darcy's Law for radial flow into a wellbore: The straight-line PI works great for single phase fluid (i.e. water, oil, or water/oil*) flowing into a wellbore, but what happens if gas comes "out of solution" in the reservoir?

* Even though water and oil are two separate phases, they are considered single phase since they are both liquid.

35

Darcy's Law for radial flow into a wellbore: What happens when the gas comes out of solution? Darcy's law works just as well for a single phase gas as it does for a single phase oil. Let's look qualitatively at what will happen to the flow rate of gas. -3

qg 

7.08 x 10 kg h Pr

 g Bg

ln

re rw

P wf 0.75 38

Darcy's Law for radial flow into a wellbore: First of all, the permeability, k, will be much higher for gas. What will this do to the flow rate?

-3

qg 

7.08 x 10 kg h Pr

 g Bg

ln

re rw

P wf

0.75

39

Darcy's Law for radial flow into a wellbore: First of all, the permeability, k, will be much higher for gas. What will this do to the flow rate? Secondly, the viscosity of gas, g is typically about 50 times lower than that of oil. What will this do to the flow rate? -3

qg 

7.08 x 10 kg h Pr

 g Bg

ln

re rw

P wf 0.75 40

Darcy's Law for radial flow into a wellbore: These two factors will give the gas a much higher flow rate than liquid inside the reservoir. QG>>QL In addition, the gas also takes up more space than it did when it was dissolved in the oil. This will cause the rock pores to be filled up eliminating any place for the oil to go. This causes the gas migration situation to accelerate.

41

Darcy's Law for radial flow into a wellbore: What will this do to the flow of oil and water?

The liquid flow we get as the pressure is lowered will be less than we would predict using a straight-line PI.

42

Pressure drops as we move toward the wellbore Pb Gas will begin to form here

Pr

Pr 43

Darcy's Law for radial flow into a wellbore: Graphically it would look like this:

Pressure - PSI

Pr < Pb

Darcy's law predicted Qmax

Pwf Actual Qmax

0 0

Q - Flow Rate (BPD) 44

Inflow Performance Relationship - IPR: We use instead Vogel's IPR curve. The equation is:

Q Q(max)

= 1 - 0.2

Pwf Pr

- 0.8

Pwf

2

Pr

where qo(max) is the maximum flow rate the well can produce.

46

Inflow Performance Relationship - IPR:

Using this curve, if we know Qmax and Pr, we can calculate the wellbore pressure for any flowrate Vogel Inflow 1.2 1

Pwf/PR

0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

Q/Qmax

47

Inflow Performance Relationship - IPR: •

Consider our previous example…

•

Pr = 2,300 psi Pwf = 1,200 psi @ qo = 1,150 bpd

•

48

Inflow Performance Relationship - IPR: First we need to calculate Q/Qmax:

•

Q

Q(max) Q Q(max) •

Then…

= 1 - 0.2

= 1 - 0.2 Q(max) =

Pwf

- 0.8

Pr 1200 2300

Pwf

2

Pr 2

- 0.8

1200

1150-bpd 0.678

2300

= 0.678

= 1696 bpd 49

Inflow Performance Relationship - IPR: •

Compare this to the Qmax we got from Darcy's equation of 2406 bpd. The well has lost 710 bpd (~-30%) in capability due to gas interference. More importantly, if we wanted to produce 2000 bpd and sized a pump for this based on Darcy's PI, we would be pretty disappointed. Vogel vs. PI for given test point 2500

Pwf (psi)

2000 1500 1000 500 0 0

500

1000

1500

2000

2500

3000

Q (bpd) 50

Inflow Performance Relationship - IPR: •

Now, calculate Pwf at a flow rate of 1000 bpd. The first thing to do is to calculate Q/Qmax: Q/Qmax = 1000/1696 = 0.5896

Now we can use the curve to get Pwf/Pr

51

Inflow Performance Relationship - IPR: Read Pwf/Pr on the Y axis - about 0.6 Vogel Inflow 1.2 1

Pwf/PR

0.8

~0.6 0.6 0.4 0.2

0.5896 0 0

0.2

0.4

0.6

0.8

1

1.2

Q/Qmax

52

Inflow Performance Relationship - IPR: •

Pwf/Pr = 0.60 so Pwf = Pr * 0.60. 2300*0.60 is 1380 psi. This is the wellbore pressure for a flow rate of 1000 bpd.

53

Solving for PWF/PR using Vogel Equation • Alternatively, we can solve for PWF/PR directly using the Vogel equation. • The problem is that the equation isn’t easy to solve. • It’s a quadratic equation, which means that we can solve it using the quadratic formula:

b  b  4ac x 2a 2

• But we need to convert the Vogel equation first…

• Where…

2

ax  bx  c  0 2

 PWF   PWF 0.8    0.2   PR   PR

  Q   1  0    QMAX 

54

Solving for PWF/PR using Vogel Equation • In our example…  PWF   PR

 0.2  0.22  4  0.8  (0.5896  1)  0.602  2  0.8 

• And now that we know PWF/PR, we can calculate PWF: PWF = (PWF/PR)*PR = 0.602*(2300-psi) = 1385-psi

55

Inflow Performance Relationship - IPR • If you don’t like that formula with all the small coefficients, we can multiply through by 5 to get something somewhat nicer: 2

 PWF   PWF   Q  4  1  0     5  PR   PR   QMAX  • Then we can substitute into the Quadratic equation and simplify to get the following formula:

 PWF   PR

  Q   0.125  1  81  80  QMAX  

   56

Inflow Performance Relationship - IPR: • If you want to get REALLY simple, you can even multiply through to have a single equation for PWF:

 Q PWF  PR  0.125  1  81  80  Q MAX 

  

• For our previous example…

Pwf = 2300 X 0.125 -1 +

81-80(1000/1696)

P wf = 1385-psi 57

1.03

58

Combined IPR We saw that we could use Darcy's law when gas was not a problem (Pwf > Pb). We also saw how to use Vogel's IPR for cases where Pwf < Pb. What about a case where PR is above Pb?

59

Combined IPR All we have to do in this case is use Darcy's law for Pr > Pwf > Pb and Vogel's IPR for the portion where Pb > Pwf > 0. Let's say, for our problem, we have a Pb of 1800 psi. Graphically it would look like:

60

Combined IPR:

Pr=2300 2500

We use a straight line PI above Pb

2000 Pb=1800

We use VOGEL below Pb

1500 Pressure - psi 1000 500

Qv = PI x Pb / 1.8 Qv

Qb

0 0

500

Qb = PI x (Pr-Pb)

1000 Flow Rate - BPD

1500

2000

Qtot-max = Qb + Qv

Pwf =0 .125x Pb {-1+[81-80(q-qb)/(qtmx-qb)]^.5} 61

1.04

62

Composite Vogel IPR: Vogel's relationship works reasonably well for water cuts below 50%. For higher water cuts, a method has been developed which takes an arithmetic average of the PI and IPR equations to yield a "composite IPR“. For a given PWF, therefore, Composite predicts more flow than Vogel but less flow than straight-line PI.

63

Composite Vogel IPR • Consider the plot of Vogel, PI, and Composite on the same flow/pressure coordinates. This shows 60% water correction. Com posite, Vogel, and PI for given test point 2500

Pwf (psi)

2000 1500 1000 500 0 0

500

1000

1500

2000

2500

3000

Q (bpd)

64

Composite Vogel IPR • Calculating Composite Vogel IPR • The trick for calculating composite Vogel IPR is to consider a PWF and to find the appropriate flow rate at that given pressure. – Start with your test point and calculate the pure Vogel and the pure PI inflow curves. – For a given PWF, average the pure-Vogel and pure-PI flow rates based on the water cut to find the composite flow rate: – QCOMP(PWF) = QPI(PWF)*WC + QVOGEL(PWF)*(1-WC) – Example: using our previous data, we calculated the QMAX from PI as 2405-bpd and the QMAX from Vogel as 1696-bpd. If we assume a 60% water cut, then: – QCOMP(PWF=0) = (2405-bpd)*(0.6) + (1696-bpd)*(0.4) = 2121-bpd

65

Composite Vogel IPR • You can summarize this with a table of Q vs. Pwf Q, Vogel Q, PI Q, comp PWF: psi bpd bpd bpd 2300 2070 1840 1610 1380 1150 920 690 460 230 0

0 292 556 794 1004 1188 1344 1473 1574 1649 1696

0 240 481 721 962 1202 1443 1683 1924 2164 2405

0 261 511 750 979 1196 1403 1599 1784 1958 2121 66

Composite Vogel IPR • Calculating Composite Vogel IPR • Another method is to divide the flow rate into water and oil and treat them separately. – Water inflow is straight-line PI – Oil in flow is pure Vogel – Then re-combine the individual flow rates into a total flow rate. Composite, Vogel, and PI for given test point

2500

Pwf (psi)

2000

+

1500

=

1000 500 0 0

500

1000

1500

Q (bpd)

2000

2500

67

Composite and Combined IPR:  Finally, we can consider both combined (straight-line plus curve) and composite on the same IPR.  Graphically it would look like this, where qt is the composite flow: Pressure - psi

Oil IPR

Composite IPR

Water PI

qb

Flow Rate - BPD

qo(max) qt(max)

qw(max) 68

1.05

71

The “Skin” effect (van Everdingen & Hurst) Skin is a wellbore phenomenon, that causes an additional pressure drop in the near-wellbore region:

141.2 q o Bo q o ( p) skin  S S 2 ko h ko h

72

Inflow Performance Curve Summarizing: • If we do not have a fluid level, we must use some type of inflow relation to calculate a wellbore pressure. We can then convert this pressure to feet using a specific gravity. • Depending on the reservoir fluid, we may use a straight line PI, an IPR, or a combination of the two.

74

Inflow, Summary Basic Tool: Pressure-Flow Plot Factors that Affect Inflow: – – – – – – – –

Permeability Viscosity of wellbore fluid Reservoir size (effective radius and height) Well bore radius Static pressure Skin fluid phases (gas and liquid, water and oil) Oil properties (Bubble point, volume factor) 75

1.06

77

Darcy's Law for radial flow into a wellbore: In some cases, the PI can also be improved slightly by acidizing or fracturing. Acidizing cleans up "skin" on the perforations and can improve porosity in limestone reservoirs by making larger holes for oil flow. Before Skin Damage

After

Acid

78

Darcy's Law for radial flow into a wellbore: Fracturing can also improve permeability by making large cracks near the wellbore. Before

After

79

Darcy's Law for radial flow into a wellbore: The area of the reservoir around the wellbore is the most critical in terms of flow restrictions.

Why is this the case?

80