Method estimates arctic well thermal insulation efficiency

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

For wells in an arctic environment, a simple method was developed for estimating the effect of thermal insulation on the maximum radius of thawing of permafrost and then determining thermal insulation efficiency.

Knowledge of the thawed radius is critical for predicting platform stability and well bore integrity.

Permafrost

Oil and gas flowing through the wells in permafrost areas can be at a high temperature, such as 190° F., that makes thawing of permafrost unavoidable during long-term production. If the thawed soil cannot withstand the load of the upper soil layers, consolidation will take place, and the corresponding settling can significantly shift the surface.

Estimates show that the magnitude of settlement (the center displacement of the thawed soil ring) and the axial compressive stress are proportional to the squared values of the thawed radius.¹

For injection wells, thawing of the permafrost greatly increases the heat loss from the well bore. Thus, knowledge of the thawed radius is critical for predicting platform stability and well bore integrity, as well as for estimating heat loss from wells.

In many cases an insulation layer for the producing strings can significantly reduce permafrost thawing.

Calculations

• Equations amd nomenclature [51,037 bytes]

• Equations amd nomenclature (cont.) [63,352 bytes]

A computer program was used to obtain a numerical solution for differential equations for the Stefan equation and the heat conductivity in the insulation and frozen and thawed zones. The numerical solutions show that Equation 1 (see equation box) can approximate the heat flow rate from the well bore.

The effective thermal conductivity of insulation (Kef) can be determined from Equation 2.

Usually for producing wells, the value of the dimensionless formation temperature is small so that to include a safety factor, one can neglect the heat flow to the frozen zone.

With Equation 1 at q = 0 and assuming a steady-state temperature distribution in the thawed zone, one obtains the solution of the Stefan equation (Equation 3).

The maximum value of the thawed radius (Hmax) is when J = 0. With J = 0, Equation 3 becomes Equation 4.

Equation 5 is a good approximation of Equation 4.

The ratio of the thickness of the thawing layers (around the well bore) with and without insulation can be expressed by the thermal insulation efficiency coefficient Y.

Equation 7 is obtained by combining Equations 3, 4, and 6.

Table 1 [87,602 bytes] shows the values calculated from Equation 7.

Example calculation

As an example, an oil company is considering installing insulated 3.5-in. tubing in a 17.5-in. well bore. The estimated effective thermal conductivity of insulation is K _ef = 0.040 BTU/hr-ft-°F. The problem is to obtain the values of the thermal insulation efficiency coefficient C after 2-10 years of production. The following data are known: r₁ = 1.496 in., r_w = 8.75 in., a_t = 0.030 sq ft/hr, K_t = 1.0 BTU/(hr-ft-°F.), q_f = 3,000 BTU/cu ft, T_o = 180° F., t₁ = 2 years = 17,520 hr, and t₂ = 10 years = 87,600 hr.

In Step 1, compute the dimensionless parameters as follows: J = (1.0/0.040) ln(8.75/1.496) = 44, I_f = (0.03 3 3,000)/(180 3 1.0) = 0.500, t_D1 = (0.030 3 17,520)/ (17.5/24)² = 990, t_D2 = (0.030 3 87,600)/(17.5/24)² = 4,940, t_D1/I_f = 990/0.500 = 1,980, and t_D2/I_f = 4,940/0.500 = 9,940.

In Step 2, determine the values of H_max for t₁ = 2 years and t₂ = 10 years from Equation 4 as follows: H_max,1 = 1 + 1.2582 3 1,980^0.4376 = 35.86, and H_max,2 = 1 + 1.1608 3 9,940^0.4477 = 72.51.

Next, in Step 3, find the values of function Y as follows: Y₁ = Y (44, 35.86) = 0.2395 and Y₂ = Y (44, 72.51) = 0.2754.

In the last step, Step 4, Equation 5 finds H₁ and H₂ as follows: H₁ = 1 + (35.86 - 1) 3 0.2395 = 9.35, and H₂ = 1 + (72.51 - 1) 3 0.2754 = 20.69.

Thus, insulation on the producing strings significantly reduced the permafrost thawing around the well bore. It is interesting to note the drastic reduction of the volume (mass) of the thawed formations.

Indeed, introducing the ratio of thawed volumes, M = [(H_max - 1)/(H - 1)]², we estimate that M1 = (34.86/8.35)² = 17.4 and M₂ = (71.51/19.69)² = 13.2.

Reference

1. Palmer, A., "Thawing and the Differential Settlement of the Ground Around Oil Wells in Permafrost, the U.S.S.R.," Second International Conference, National Academy of Science, Washington, D.C., July 13-28, 1978.

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