I.M. KutasovMultiSpectrum Technologies Santa Monica, Calif.

To determine formation (true) skin and the rate-dependent skin, a semitheoretical equation is proposed for relating the critical value of flow rate (qc) to formation permeability, formation porosity, and gas/oil dynamic viscosity.

An accurate evaluation of skin is important for designing remedial treatments or evaluating gas well productivity.1 2

Three examples illustrate the proposed equation. In all cases, the actual gas/oil flow rates are compared with the calculated critical flow rate.

### NON-DARCY FLOW

Pressure and flow tests in many high-capacity gas wells have shown that pressure drop near the well bore cannot be estimated from the Darcy equation in which the pressure gradient is proportional to the flow velocity (laminar flow).

To describe the additional pressure drop at the laminar-turbulent flow regime transition, Forchheimer proposed an equation in which the non-Darcy term is proportional to the squared flow velocity. Most researchers assume that because of non-Darcy flow, the apparent skin estimated from build-up pressure is rate dependent.

The following equation relates flow rate (q), apparent skin factor (s), and true formation skin (sf):

s = sf + Dq (1)

To estimate the rate factor for non-Darcy flow (D), one can use empirical correlations,2 3 but usually the parameters sf and D are determined by two pressure buildup tests or by multirate flow tests. However, some studies show that rate dependent skin in an oil well is due to increased gas saturation near the well bore.1

As mentioned by Meehan and Schell, an analogous situation in a gas well would be an increased condensate saturation near the well bore.1 With increased flow rates (and corresponding increased pressure gradients near the well bore) one can expect an increased gas (condensate) saturation or water saturation near the well bore of oil and gas wells.

Thus, it is reasonable to assume that in some cases true formation skin (sf) cannot be considered as a constant in Equation 1. Moreover, Equation 1 can be used only when flow rate (q) exceeds some critical value of flow rate qc. Here we assume that non-Darcy flow regime exists only at q qc.

It is known that the dimensionless Reynolds number (Re) is used as a criterion to distinguish between laminar flow and turbulent flow. For fluid/gas flow through porous media:

[SEE FORMULA (2)]

where:

d = Length of the porous matrix

v = Velocity

p = Fluid or gas density

m = Dynamic viscosity In many cases, the mean grain diameter is taken as the length. Collins suggests that the representative length in the Reynolds number is d = (K/f)1/2, where K is permeability and f is the porosity.4

For complex media, Darcy flow occurs for Reynolds numbers Re 2.3, and Forchheimer flow occurs for 5

Thus, the critical flow rate can be estimated from the following:

[SEE FORMULA (3)]

Radial flow velocity can be obtained from:

[SEE FORMULA (4)]

where:

rw = Well bore radius

h = Net pay thickness

It is assumed that the length of the porous matrix5 is given by:

[SEE FORMULA (5)]

where:

A = 182

Ko = Formation permeability at a water saturation of 0

Alpha is defined by:

[SEE FORMULA (6)]

From Equations 3-6 and assuming that skin does exist near the well bore, one can obtain the generalized equation for critical flow:

[SEE FORMULA (7)]

where:

Ks = Effective permeability of the skin zone

fs = Effective porosity of the skin zone

The following semiempirical equation has been found to accurately represent many experimental data points:5

[SEE FORMULA (8)]

where:

so = Specific surface of the particles (surface area per unit volume)

k = Experimentally determined Kozeny-Carman constant

If one assumes kso2 is a constant for the formation and skin zone, then from Equation 8 we obtain:

[SEE FORMULA (9)]

Combining Equations 7 and 9 and introducing oil field units we obtain Equations 10 and 11.

[SEE FORMULA (10)]

[SEE FORMULA (11)]

where:

cg = 1.407 x 109

Co = 1.722 x 106

Bg = Gas formation volume factor, res bbl/Mscf

Bo = Oil formation volume factor, res bbl/st-tk bbl

h = Net pay thickness, ft

Ko = Formation permeability at water saturation sw = 0, md

qcg = Critical gas production rate, Mscfd

qco = Critical oil production rate, st-tk b/d

rw = Well bore radius, ft

tg = Specific gravity of gas, Dimensionless (air = 1)

to = Specific gravity of oil, dimensionless (water = 1)

f = Porosity, dimensionless

m = Dynamic viscosity, cp

### EXAMPLES

An oil well, Reference 6, pp. 142-43, has the following properties: h = 8 ft, K = 96 md, sw = 0.35, f = 0.1, rw 0.33, to = 1.00 (assumed), s = -5.0, m = 1.0 cp, qo = 800 st-tk b/d, and Bo = 1.25 res bbl/st-tk bbl.

A permeability of Ko = 377 md was estimated from the following semitheoretical generalized permeability equation:7

[SEE FORMULA (12)]

The K/Ko f(sw, f = 0.10) function is shown in Fig. 1 (41205 bytes) and the critical flow rate was calculated from Equation 10 as qco = 6,580 st-tk b/d.

Comparing this value with the actual oil flow rate q,) = 800 st-tk b/d, we can see that the apparent skin factor (s) is equal to the true formation skin (sf).

In another example, a gas well shown in Reference, 6 p. 138, has the following reservoir parameters: h = 84 ft, Ko = 6.92 md, d) = 0.16, rw = 0.292 ft, tg = 0.8, s = 21.12, m = 0.0201 cp, qg = 3,010 Mscfd, and Bg = 1.00 res bbl/Mscf.

For this gas well, the critical flow rate from Equation 11 is qcg = 25.1 bcfd. In this case qcg is more than 8,000 times larger than the actual flow rate qg of 3.01 MMcfd. It is obvious That the Reynold's number is very small (Re

In a third example, a gas well (Reference 8, p. 45) has the following parameters: h 28 ft, K = 9.77 md, sw = 0.3, f = 0.18, rw = 0.3 ft, tg = 0.7 (assumed), s = 4.27, m = 0.028 cp, qg = 5,256 Mscfd, and Bg = 0.944 res bbl/Mscf.

A permeability of Ko = 32.4 md was calculated from Equation 12. The critical flow rate for this gas well, from Equation 11, is qcg = 8.20 bcfd. Because the actual flow rate of qg = 5.256 MMcfd is much lower, we can conclude that only the laminar flow regime is possible near the well bore and the apparent skin factor is equal to the true formation skin.

### REFERENCES

1. Meehan, D.N., and Schell, E.J., "An Analysis of Rate-Sensitive Skin in Gas Wells," SPE Paper No. 12176, 58th Annual Technical Conference and Exhibition, San Francisco, Oct. 5-8, 1983.

2. McLeod, H.O. Jr., "The Effect of Perforating Conditions on Well Performance," JPT, January 1983, pp. 31-38.

3. Lee, W.J., "Pressure-Transient Test Design in Gas Formations," JPT, October 1987, pp. 1185-95.

4. Bear, J., Dynamics of Fluids in Porous Media, American Elsevier, 1972, pp 123-27.

5. Fand, R.M., Kim, B.Y.K., Lam, A.C.C., and Phan, R.T., "Resistance to the Flow of Fluids Through Simple and Complex Porous Media Whose Matrices are Composed of Randomly Packed Spheres," journal of Fluids Engineering, September 1987, pp. 268-74.

6. Matthews, C.S., and Russel, D.G., Pressure Buildup and Flow Tests in Wells, SPE Monograph Series, 1967, pp. 138-143.

7. Kutasov, I.M., "Equation predicts effective gas permeability OGJ, Aug. 19, 1991, pp. 63-64.

8. Lee, J., Well Testing, SPE Monograph Series, 1982, P. 45.

*Copyright 1995 Oil & Gas Journal. All Rights Reserved.*