EQUATION CALCULATES THAWING TIME AROUND WELLS COMPLETED IN PERMAFROST

I.M. Kutasov
MultiSpectrum Technologies
Santa Monica, Calif.

For oil and gas wells completed in permafrost, a semitheoretical equation has been developed to relate the production time to the radius of thawing.

Wells in Siberia, Alaska, and the Canadian Arctic have the problem of mechanical and thermal interaction between the permafrost and borehole.

The temperature of oil/gas flowing through the permafrost area can be as high as 190 F. and, therefore, permafrost thawing cannot be avoided in long-term production.

An insulation layer (with reasonable thickness) around the producing strings can only delay permafrost thawing.

If the thawing soil cannot withstand the load of overlying formations, consolidation and the corresponding soil settlement can significantly shift the surface. Settlement in the vertical direction is more serious when the shear stress acts downward on the casing, causing compressive stresses that can deform casing.'

Estimates show that the magnitude of the settlement (the center displacement of the thawing soil ring) and the axial compressive stress are proportional to the squared values of the radius of thawing. Thus, knowing the radius of thawing is critical for predicting platform stability and well bore integrity.

During drilling the radius of thawing increases as mud is circulated. If the well is shut-in, from physical considerations, it is clear that the radius of thawing will continue to increase because of the heat energy stored in the thawed zone.

With restoration of the well's thermal regime, the radius of thawing decreases to its initial value (rw, well radius).

During production, the thawing radius again increases with time. For long-term production, the thermal disturbance of the formation from drilling can be neglected.

A computer program was used to obtain a numerical solution of a system of differential equations of heat conductivity (for insulation, frozen, and thawed zones) and the Stefan equation.

These solutions were combined with the results of hydrodynamic modeling 2 to develop an empirical expression for the ratio of heat flows at the thawed zone/frozen zone interface.

Introducing this relationship into the Stefan equation, and assuming a steady-state temperature distribution in the insulation and in the thawed zone, 2-3 we obtained the solution of the Stefan equation, Equation 1 below.

EXAMPLE

As an example, assume that to avoid significant surface shifts, the radius of thawing should not exceed 70 in. The allowable production time is then calculated with the following steps:

Initial data for the well configuration in Fig. 1 is:

[SEE FORMULA]

The subscripts for the thermal conductivities K are: c-cement, s-steel, d-diesel, and f-foam.

Thermal properties of the formation, the values of undisturbed formation temperature, and temperature of the producing oil are:

[SEE FORMULA]

For Step 1 of the calculations, the effective thermal conductivity of insulation, Kef, can be determined from Equation S. The denominator of the right-hand side of Equation 8 is equal to:

[SEE FORMULA]

In the last step, Step 5, Equation 1 is used to find It and the allowable production time is Determined from Equation 6:

[SEE FORMULA]

REFERENCES

Palmer, A., "Thawing and the Differential Settlement of the Ground Around Oil Wells in Permafrost, The U.S.S.R. Contribution to the Second International Conference, National Academy of Sciences, Washington D. C., July 13-28, 1978, pp. 619-24.
Kutasov, I.M., "Thermal Parameters of Wells Drilled in Permafrost Regions," Nedra, Moscow, 1976.
Bondarev, E. A., and Krasovitskiy, B.A., "Thermal of Oil and Gas Boreholes" Nauka, Novosibirsk, 1974.