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Petro physics

Capillary Pressure Hysteresis&Capillary End effect

Name: Nalluri ViswanathCAPILLARY PRESSURE HYSTERESISCapillary pressure curves show a marked hysteresis depending on whether the curve is determined under a drainage process or an imbibition process.After completing measurements of capillary pressure for primary drainage, the direction of saturation change can be reversed, and another capillary pressure relationship can be measuredit is usually called an imbibition relationship. The primary drainage and imbibition relationships generally differ significantly for a gas/water system. This difference is called capillary pressure hysteresis.

At any wetting phase saturation, the drainage capillary pressure is higher than the imbibition capillary pressure.

At a capillary pressure of zero, the spontaneous imbibition curve terminates at a wetting phase saturation that may or may not correspond to the true residual non-wetting phase saturation depending on the wettability of the rock.

Cycles of capillary pressure measurements.

Capillary pressure hysteresis can be explained in a variety of ways.

Energy considerationsContact AngleDisplacementPore structure

Energy considerationsIt was shown from energy considerations that more work is required for a non-wetting phase to displace a wetting phase than for a wetting phase to displace a non-wetting phase. This means that at any level of saturation, more work is required during the drainage capillary pressure measurement than during the imbibition measurement. So the capillary pressure on the drainage cycle will be greater than on the imbibition cycle to displace the same volume of fluid.Contact AngleDuring drainage, the wetting phase recedes from the porous medium and the contact angle is the receding contact angle, R. During imbibition, the wetting phase advances into the porous medium and the contact angle is the advancing contact angle, A. Since R is less than A, 2cosR /rm, the drainage capillary pressure, is larger than 2cosA /rm, the imbibition capillary pressure at the same saturation state.DisplacementWhen the capillary pressure experiment is reversed to measure the spontaneous imbibition curve, the pressure in the non-wetting phase is reduced to allow the wetting phase to be imbibed. As the wetting phase is imbibed into the rock, some non-wetting phase will be trapped in certain pores. This trapping causes the wetting phase saturation on the imbibition curve to be less than on the drainage curve at the same capillary pressure.Pore StructureDuring drainage, the pore is initially full of the wetting fluid at a capillary pressure. Next, the capillary pressure is increased to a higher value to drain some of the wetting fluid.Next, we consider the imbibition process. At c, the capillary pressure is high at a wetting phase saturation of nearly zero. After the wetting fluid has been imbibed to the equilibrium level, the imbibition capillary pressure will be approximately the same as the drainage capillary pressure of Drainage capillary pressure, because the mean curvature of the interfaces at c and d are about the same. However, the wetting phase saturation at d is considerably lower than at b. Thus, at the same capillary pressure, the wetting phase saturation for imbibition is less than for drainage. This is hysteresis.

Pore Structure

Drainage and imbibition capillary pressure curves showing the type of fluid producedCapillary ImbibitionConsider a reservoir consisting of two layers with different permeabilities and capillary pressure curves as shown in figure.Initially, both layers are in capillary equilibrium at their respective irreducible water saturations.Water flooding the two layers.

Capillary ImbibitionCapillary End EffectDuring steady-state, immiscible displacement in the bulk of the core plug there is a constant saturation and the capillary pressure corresponds to it.At the outflow face, however, the capillary pressure is zero and hence the wetting phase saturation is one. Therefore, whatever the wetting phase saturation is in the bulk core, at the outflow end face it approaches 1. we observe more wetting phase coming out than it would be according to the bulk saturation condition. This is called "end capillary effect".

Mathematical Analysis of Capillary End EffectDarcy's law for the wetting and non-wetting phases is given by

(7.42) (7.43)

Let us define the relative permeability's of the wetting and non-wetting phases as

(7.44) (7.45)

Eqs.(7.42) and (7.43) can be written in terms of the relative permeabilities as

(7.46) (7.47)

Capillary equilibrium gives (7.48)

Assuming incompressible fluids, then (7.49)

the saturation constraint gives (7.51)

Subtracting Eq.(7.46) from (7.47) and rearranging gives (7.52)

Substituting Eqs.(7.48) and (7.49) into (7.52) gives upon rearrangement

(7.53)

Let the true fractional flow of the wetting phase be defined as (7.54)

Let an approximate fractional flow of the wetting phase be defined as

(7.55)

Both f w and F w are functions of saturation. Substituting Eqs.(7.54) and (7.55) into (5.53) gives the true fractional flow of the wetting phase as (7.57) (7.58)

Let the spontaneous imbibition capillary pressure curve be given in terms of its Leverett J-function as (7.59)

Substituting Eqs.(7.58) and (7.59) into (7.57) gives the true fractional flow of the wetting phase as

(7.60)

The term in the inner bracket on the right side of Eq.(7.60) is a dimensionless number (7.61)

Substituting Eq.(7.61) into (7.60) gives (7.62)

Let the dimensionless time be defined as

(7.63)

Substituting Eq.(7.63) into (7.50) gives the continuity equation for the wetting phase as (7.64)

Substituting Eq.(7.62) into (7.64) gives (7.66)

Let us examine in detail the fractional flow of the wetting phase at the outlet end of the core. Applied to the outlet end of the core, Eq.(7.62) can be written as

(7.67)

J+ is the J-function inside the porous medium, J is the J-function outside the porous medium and xD is a small distance in the neighborhood of the outlet end of the porous medium (7.69)

Depending on the values of Ncap , krnw , and J+ , it is possible for the following inequality to prevail during the displacement:

(7.70)

IF (7.71)

Then (7.72)

at the outlet end of the core. Because the fractional flow of the wetting phase is zero at the outlet end of the core, the wetting phase cannot flow out of the core but instead will accumulate there raising the wetting phase saturation to an abnormal level. This is the capillary end effect phenomenon at work.

How can capillary end effect be eliminated from the experiment? The condition for eliminating the capillary end effect is obtained from Eq.(7.69) as (7.75) or (7.76)Thus, Ncap should be as small as possible in the experiment to eliminate capillary end effect.The only means to control Ncap in the experiment is through the injection rate, q. Examination of Eq.(7.61) shows that Ncap can be made small by the use of a high injection rate in the experiment. Substituting Eq.(7.61) into (7.76) gives the condition for the injection rate to eliminate capillary end effect as

(7.78)

Experimental Evidence of Capillary End EffectPerkins (1957) has presented experimental data that show capillary end effect at work. He conducted waterfloods in laboratory cores at two rates, one below the critical rate for capillary end effect and one above the critical rate. The core was 12 inches in length and 1.25 inches in diameter. The oil and water viscosities were 1.8 and 0.9 cp.The low injection rate was 2.4 ft/day whereas the high injection rate was 36 ft/day. The injected water was 0.1 normal sodium chloride solution.The core was instrumented with two current electrodes and nineteen potential electrodes distributed along its length.Wetting phase saturation profiles at low injection rate

Wetting phase saturation profiles at high injection rate

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