A model of propagation of thermal fields in permafrost from various engineering objects operating in Arctic regions is considered. The proposed model includes the most significant technical and climatic parameters affecting the formation of thermal fields in the surface layer of the soil. The main objective of the study is a long-term forecasting of changes in the dynamics of permafrost boundaries during operation of cluster sites of northern oil and gas fields. Such a forecast is obtained by simulation of complex system consisting of heat or cold sources and frozen soil, thawing of which can lead to the loss of the bearing capacity and possible technogenic and environmental accidents. For example, the sources of heat can be production wells, and the sources of cold can be seasonal cooling devices that are used to stabilize the soil. To minimize the impact of heat sources on permafrost, various options for thermal insulation are used, and to preserve the original temperature regime of the top layer of soil, riprap materials consisting of sand, concrete, foam concrete, or other heat insulating material are used. The developed set of programs was used in the design of 12 northern oil and gas fields. To solve the described problem in a complex three-dimensional area, substantial computational resources are required. The computing time of one variant can often exceed 10-20 hours of machine time on a supercomputer. To speed up the numerical calculations, multicore processors are used. Numerical calculations illustrate the possibility of a developed set of programs for making long-term forecasts for determining changes in the boundaries of the permafrost zones, and show that on multicore processors it is possible to achieve acceleration close to the theoretical one.
The term “permafrost” was introduced into the English literature by S.W. Muller [
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
the zone of seasonal thawing of the soil, which is determined by geographical
coordinates, the intensity of solar radiation, and other climatic factors. The climate changes
related with, for example, the warming may lead to the essential permafrost
degradation especially in high-latitudes regions, which may influence on the environment of
the globe [
The development of the permafrost regions is especially important for Russia,
since about 93% of Russian natural gas and 80% of oil are produced in these areas. At
the same time, the development of these regions leads to negative consequences
associated with the thawing of permafrost, which leads to the formation of dangerous
geological phenomena called thermokarst. The average thickness of the permafrost
varies from 10 to 800 meters, and the components of the permafrost soils have various
physicochemical properties that can vary in all directions. In summer, as a result of
positive temperatures and solar radiation, seasonal thawing of the upper soil layer
occurs; in winter, the reverse process is observed. Computer simulation of such
seasonal processes is described in [
To solve the described problems, mathematical models, numerical methods of
solution, and codes are developed for modeling non-stationary thermal fields in the
near-surface soil layer in a complex three-dimensional area. The computational time
of the problems solution, for which a large number of variants with different
parameters are required, could exceed tens of hours of computer time on a supercomputer.
The parallel approaches are considered [
Problem Statement and Mathematical Model Let = ( , , , ) be the soil temperature at the time moment , where ( , , ) is a point of the computational area Ω = {( , , ): 0 ≤ ≤ , 0 ≤ ≤ , − ≤ ≤ 0}. The area is a 3D “box” (see Fig. 1), in which the cylinder-shaped well Ω1 and the cold sources (SDCs) Ω2 are excluded. The axes x and y are parallel to the soil surface, and the axe z is directed vertically down into Ω. In Fig. 1 there presented two variants of the unit combinations: a single well and a well surrounded by SCDs. (1) (2) y Ω solar radiation z x insulation Ω1 e g n a h c x e l a m r e h t Ω2 y t i v i s s i m e
T (0, x, y, z) T0 (x, y, z), where ρ=ρ(x,y,z) is density [kg/m3], T*=T*(x,y,z) is temperature of phase transition, с1(x, y, z), for T T * сν(T) = с2 (x, y, z), for T T *
is specific heat [J/(kg K)],
1(x, y, z), for T T *
λ(T)= 2 (x, y, z), for T T * is thermal conductivity [W/(m K)], δ is Dirac
δfunction, k=k(x,y,z) is specific heat of phase transition. The justification of the
applicability of this equation for solving problems of the Stefan type is presented in
[
To use the numerical methods it is necessary to set the boundary conditions at the lateral boundaries of Ω. Let suppose
T i
Ti (t), i 1, 2. T x xLx
T 0, y yLy 0, T z zLz 0.
In this case, the computational domain should be chosen large enough to avoid the influence of the boundary conditions (5) on the thermal fields in the computational domain Ω created by the objects Ωi. 3
Numerical Calculations and Performance Study
To apply a numerical method of solution of the problem (1)–(5) it is necessary to
evaluate the parameters included into boundary condition (3). In [
The described algorithm is implemented in the “Wellfrost” certified software package and in the various modifications, which was tested at 12 northern oil and gas fields. Also the experimental data and the results of numerical calculations was compared in view to determining the boundary of the location of the zero isotherm, which determines the boundary of the area of soil thawing around the producing well. In 2012 the accuracy of numerical calculations was verified for the Russkoye oil field (Yamalo-Nenets Autonomous Okrug, Russia), for which the obtained numerical results are in accordance with the experimental results, and the accuracy reached 5% after 3 years from the start of the field operation.
In Figures 2 and 3 the calculated thermal fields are shown for 5 years of operation around the well without heat-insulating shells (a) and with a combined heat-insulating shell (b), as well as with a system of 8 SCDs.
The permafrost soil temperature is -0.7°С, the temperature of the fluid in the well is 50°С. Calculations allow to evaluate the influence of heat or cold sources, the zone of seasonal thawing and freezing of the upper soil layer. To assess the long-term impact, calculations are carried out for a time period of up to 50 years, the time step does not exceed 24 hours. The calculation time depends both on the size of the computational grid and on the complexity of the objects introduced into the computational domain.
Table 1 shows the considered variants of the calculations. Fig. 4 shows a graph of the change of calculation time for each of the model variants. The calculations were carried out on a grid with 91x91x51 nodes. 6 FSrCeDezsi.ng and thawing of the soil around the well without thermal insulation and 8 7 SFrCeDezsi.ng and thawing of the soil around the well with a simple insulation and 8 8 FSrCeDezsi.ng and thawing of the soil around the well with a complex insulation and 8 The computing time increases nonlinearly when the number of the included object grows, as well as, the complexity of the objects, even if the grid size remains the same. T1, s T6, s 1 2 3 4 5 6 7
8
The model variants For studying the parallel algorithm running on cores, the speedup Sn T1 Tn and efficiency En Sn n coefficients were used. The resulted values are consistent with the theoretical ones, obtained by the Amdahl’s law Sn 11 2.2 , where n 0.35 is the serial code proportion.
Conclusion The developed models and algorithms make it possible to simulate the propagation of unsteady thermal fields in frozen ground from producing wells at well pads, taking into account the possible placement of cooling devices around the wells. The numerical experiments show the possibility of obtaining a long-term forecast in the dynamics of permafrost boundaries for various options for well operation using additional technical systems and heat-insulating materials. The complications and additional constructions inserted into the model by taking into account various additional sources of cold, for example the SCDs, and the use of thermal insulation increase the computational time. Parallel computations approaches significantly reduced the computational time in solving such problems. A parallel software package for multicore processors using OpenMP technology has been developed.
Acknowledgments This work was supported by the RFBR (project No. 19-07-00435).