EQUATION CORRECTS KILL-WATER DENSITY VALUE IN DEEP, HOT WELLS

D. Ram Babu
Oil India Ltd.
Duliajan, India

A pressure-depth relationship is derived for a water column in deep and hot wells. In these wells, calculations based on normal water gradients may lead to well control problems.

Often, plain water is the kill fluid for well servicing operations. Apart from its ready availability, another major advantage of using water is that its density is higher than that of the reservoir fluids. This allows a column of water to exert sufficient static pressure to arrest the inflow of reservoir fluids into the well bore from normally pressured reservoirs.

The usual practice is to calculate hydrostatic pressures by assuming a constant pressure gradient of 0.433 psi/ft. This constant gradient assumes a standard water density value of 8.338 ppg at 59 F. and atmospheric pressure.

In wells, the water column temperature increases with depth. In high-temperature deep wells, the temperature may be as high as 400 F. Water density at a pressure of 5,787 psig and 392 F. is 7.435 ppg1 or 0.387 psi/ft, significantly lower than the standard water density.

Using the published data on water density at different pressures and temperatures, 2 3 Equation 1 (see equation and nomenclature box) relates the density of water p (ppg) at different temperatures, T (F.), and pressure P (psia).1

In this equation, po, a, , t are empirical constants. Po and To represent the standard conditions of 14.7 psia and 59 F.

The objective of the present study is to derive a pressure-depth relationship using Equation 1 for temperature increasing linearly with depth of the well.

ANALYSIS

Recently, an empirical model was proposed for the liquid fraction in water muds.4

Following the same procedure, the static pressure gradient in a water column can be given by Equation 2, in which Z is depth in feet.

Equation 3 assumes that temperature varies linearly with depth. The b is the temperature gradient.

Equation 4 is obtained after substituting Equation 1 in Equation 2 and solving the resulting differential equation with the help of Equation 3 and with boundary conditions at Z = 0, P = Po, T = To.

The error function, F(Z), and its arguments are given by Equations 5-8.

DISCUSSION

Equation 1 was derived from 103 data points of water density in the pressure range of 1,436-26,002 psig and in the temperature range of 68-392 F.

Linear regression analysis was used to obtain the empirical constants in Equation 1. With the constants, estimated values of water density were observed to be close to the measured values with an average error of 0.172%. The constants are given in Equations 9-12.

After substituting these constants in Equations 5-8, the pressures at different depths are calculated from Equation 4 and plotted in Fig. 1 as equivalent hydrostatic gradients in psi/ft for different temperature gradients.

The dotted portion of the gradients is an extrapolation beyond the temperature range in which the constants given by Equations 9-12 are valid.

Fig. 1 shows that the hydrostatic gradients at different temperature gradients are lower than the normally assumed value of 0.433 psi/ft. Furthermore, the hydrostatic gradients at higher temperature gradients are lower than those at lower temperature gradients.

EXAMPLE

Consider a well that is draining a reservoir at 23,000 ft with a geothermal gradient of 0.018 F./ft and a reservoir pressure of 10,500 psig (0.420 psi/ft). If the well is killed with water, the calculation with a hydrostatic gradient of 0.433 psi/ft estimates the bottom hole pressure of 10,825 psig or a 325 psi overbalance.

But after standing for several hours, the water column can reach near geothermal conditions. The hydrostatic gradient then is estimated from Equation 4 or Fig. 1 to be about 0.407 psi/ft or 10,175 psig. The well is therefore underbalanced by 325 psi.

In this case, the potential for flow from the formation exists in the well and hence water should not be used to kill the well.

ACKNOWLEDGMENT

The author expresses his gratitude to the management of Oil India Ltd. for its permission to publish this article.

REFERENCES

Kutasov, I.M., "Water FV factors at higher pressures and temperatures," OGJ, Mar. 20, 1989, pp. 102-104.
Dodson, C.R., and Standing, M.B., Pressure-volume-temperature and solubility relations for natural-gas-water mixtures, API Drilling and Production Practices, 1944, p. 176.
Burnham, C.V., Halloway, J.M., and Davis, N.F., "Thermodynamic properties of water up to 1,000 C. and 10,000 bars," Special Paper No. 132, the Geological Society of America, 1969.
Babu, D.R., "Effect of P-p-T behavior of water muds on static pressures during deep well drilling," Journal of Petroleum Science and Engineering, to be published in 1993.