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Petr 3520 Homework 21: Capillary Pressure, J Function

Capillary Pressure Equations
This equation was derived for the capillary rise of a fluid in a capillary tube. The pressure difference across the meniscus = difference in gh in the two fluids from the meniscus down to the free surface. pc = po – pw = difference in phase pressures pc = gh = (w – o)gh = (w – o)h from fluid rise in a capillary tube  = mass density;  = g = specific weight of fluid [lb/ft3]	(Equation pC 1)
This equation was derived from a force balance on the fluid meniscus in a capillary tube. The upward force is the vertical component of the fluid surface tensile force, and the downward force is the weight of the fluid in the tube. wo = oil-water surface tension [dyne/cm]  = wetting fluid contact angle with capillary tube r = radius of capillary tube	(Equation pC 2)
Use this equation to convert a pc vs. Sw curve to height above the water-oil-contact (100% water).	(Equation pC 3) (pC 1 rearranged)
This the dimensionless Leverett J-Function. It was devised to be a dimensionless function to correlate capillary pressure curves from different regions in the same reservoir. Units: pc [psi], wo [dyne/cm], k [md],  [fraction], J [unitless]

1. Look up “Capillary Action” on www.wikipedia.org and read the entry. List three things you learned about capillarity.

2. In 1940, Leverett placed 10 ft long sand-packed tubes into water and measured saturations along the length of the tube. (This was water-air capillarity, different from the water-oil case in a reservoir.) Explain his two tests (a) dry sand in tube in water, and (b) water-saturated tube in water. Draw a picture of these two curves (on the same h vs. Sw axes). Label the imbibition and drainage curves. At a given height, h, which curve has the larger water saturation, Sw?

3. Explain the mercury injection method for measuring capillary pressure. A capillary pressure vs. saturation curve can indicate the pore size distribution in the core. How? Draw two examples and explain them.

4. What is the “threshold pressure” (also called “pore entry pressure”). Give an example of a field case when we need to be concerned about pore entry pressure.

5. For this equation: Briefly explain how it is derived. How is it used? What are po and pw? What is ? What is ?

6. For this equation: Briefly explain how it is derived (in your own words). What is ? What is ? What is r? How do these affect capillary pressure?

7. A capillary pressure curve shows pc vs. Sw. This suggests that pc is a function of Sw. But is it? What is (are) the most foundational scientific principle(s) that cause(s) capillary rise?

8. Color in the capillary rise in various sizes of capillary tubes and draw the resulting h vs. Sw curve from the average Sw at a particular height from the “bundle of capillary tubes” model.

Capillary Pressure Bundle of Tubes 1

9. What other ways do you, as a reservoir engineer, have to determine the height of the transition zone (saturations vs. height)?

10. How does the existence of a water-oil transition zone affect:

(a) Where you complete wells? Why?

(b) Calculation of the volume of oil in place?

11. A special core analysis provided the capillary pressure curve given below for a reservoir whose o = 54 lb/ft3, and w = 66 lb/ft3. Please do the following:

(a) Convert the pc vs. Sw curve to a height vs. Sw curve. [Answer: At pc = 0.8 psi, h = 9.6 ft, etc.]

(b) Sketch the water-oil transition zone on the figure below by carefully drawing construction lines (it’s best to use a ruler) from the curve to the reservoir. Label the heights h and the saturations at those heights.

Capillary Pressure 2009c.png

12. Convert the capillary pressure curve given above to a J function curve, plus this data: wo = 48 dyne/cm, k = 24 md,  = 16%,  = 30. [Answer: At pc = 0.8 psi, J = 0.0608 (unitless)]

Capillary Pressure vs

http://www.digitalformation.com/Documents/CPRP.pdf

Notes and Terms	Equations
Capillary pressure (pc) equation (developed for a capillary tube (liquid-air)) as a function of (in consistent units): Capillary pressure (pc) in [dyne/cm2] Liquid-air interfacial tension () in [dyne/cm] Liquid-solid tube material contact angle () in [degrees] Capillary tube radius (r) in [cm]
For pressure in [psi] and r in [microns]: Capillary pressure (pc) in [psi] Liquid-air interfacial tension () in [dyne/cm] Liquid-solid tube material contact angle () in [degrees] Capillary tube radius (r) in [micron = 10-6 m = m] Note: 1 psi = 68,947.6 dyne/cm2 and 1 cm = 10-2 m = 104 m
To find pore throat diameter (r) vs. pc from mercury injection test: Capillary pressure (pc) in [psi] Mercury-air interfacial tension ( = 480 dyne/cm] Mercury-rock contact angle ( = 140°) Capillary tube radius (r) in [micron = 10-6 m = m]	with  = 480 and  = 140°,

The use of Hg injection to determine pore throat size distribution is well established. However, the capillary pressure vs. saturation curve obtained by Hg injection cannot directly be used for a reservoir. The Hg-air pc data should be converted to oil-water, gas-oil, gas-water, or whatever the actual system is by using the following equation:

where,

Term	Typical Value(s)
pc(o-w) = oil-water capillary pressure	n/a
pc(air-Hg) = air-mercury capillary pressure	n/a
0-w = oil-water interfacial tension	0-w = 48 [dyne/cm]
air-Hg = air-mercury surface tension	air-Hg = 480 [dyne/cm]
0-w = oil-water contact angle	0-w = 30°
air-Hg = air-mercury contact angle	air-Hg = 140°
g-w = gas-water interfacial tension	g-w = 72 [dyne/cm]
g-o = gas-oil interfacial tension	g-o = 24 [dyne/cm]
g-w = gas-water, g-o = gas-oil contact angle	g-w = g-o = 0°

1. Explain why all oil and/or gas reservoirs have a minimal water saturation (Swc (“connate”) or Swi (“interstitial”), usually ranging from 10-30%) throughout the reservoir?

2. Explain the mercury injection method for measuring capillary pressure. A capillary pressure vs. saturation curve can indicate the pore size distribution in the core. How? What equation applies to Hg injection?

3. What are the advantages/disadvantages to the Hg injection method for determining a capillary pressure curve?

4. Once a capillary pressure curve is determined from Hg injection data, can this be used directly for an oil reservoir? Why or why not? If not, how can it be converted to a suitable capillary pressure curve?

5. What is the “threshold pressure” (also called “pore entry pressure”). Give an example of a field case when we need to be concerned about pore entry pressure.

6. For each capillary tube bundle size distribution given below, draw the corresponding capillary pressure curve shape.

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