The Depth of the Zone of Thermal Influence of Highways

Galkin Aleksandr ORCID: 0000-0002-5924-876X Doctor of Technical Science Professor, Chief Researcher of the Melnikov Permafrost Institute SB RAS 677010, Russia, Yakutsk, ul. Merzlotnaya, 36, IMZ SO RAN. Laboratoriya geotermii kriolitozony afgalkin@yandex.ru Other publications by this author

Introduction.The thermal regime of road coverings and foundations is an important factor determining their reliable and safe operation in the cryolithozone ^[1,2,3,4,5].

Especially, the negative influence of the thermal factor affects the construction and operation of roads, when large ice inclusions are present in the soils of the road base in the zone of thermal influence of the road ^[6,7,8,9]. Under normal natural conditions, the thermal regime of the heliothermozone (zones of annual heat turnover) is quite stable and, practically, has not changed for centuries ^[10,11,12]. Under anthropogenic influence, which includes the construction of highways, the thermal and humidity regime of the annual heat turnover zone changes significantly ^{[13,14,15,16]}. This leads to the development of negative cryogenic processes, such as soil heaving, frost-breaking cracking of road clothes and foundations, cavern formation, etc. ^{[17,18,19,20]}. Even at negative temperatures, ice inclusions and ice-saturated dispersed rocks change their strength characteristics, which affects the reliability and safety of road operation in the cryolithozone ^{[21,22,23,24]}. In this regard, the forecast of the thermal regime of road foundations is a mandatory and important element of the justification of design decisions for the construction and reconstruction of highways in the zones of distribution of solid and island permafrost ^{[25,26,27,28,29]}.  One of the design parameters that determine the choice of technical solutions and the technology of road construction is the depth of the zone of seasonal thermal influence of the road. The depth of the zone of thermal influence depends on many factors, the main of which is the thermal conductivity of the soils of the road base. At the same time, as a rule, in the zone of thermal influence, soils are heterogeneous, both due to the natural layered texture and thermophysical properties, and due to seasonal changes in their iciness (humidity) within the active layer. The purpose of these studies was to quantify the possibility of replacing the layered environment of the road's soil base with an equivalent homogeneous soil with an average coefficient of thermal conductivity in thermal calculations.

Calculation method.The depth of the thermal influence of the highway can be determined using the well-known solution of the one-dimensional unsteady Fourier equation under boundary conditions of the first kind, obtained by the integral method for a homogeneous medium, according to the formula ^[30]:

                                                                                                    (1)                                                                    

Where: H is the depth of the zone of thermal influence of the road, m; a is the thermal conductivity of the soil, m2/s; ? is the time, s.

Such a formula entry is not very convenient in engineering calculations, where the time of the studied thermal processes is calculated in weeks, months or years. Therefore, using the dependence given in ^[7], we transform formula (1) to the form:

                                                                                      (2)                                                                     

Where: N is time, months.

Writing in this form is very convenient, since the coefficient of thermal conductivity of most soils has the order "P · 10-6".

The coefficient of thermal conductivity of layered soil can be defined as:

 a) arithmetic mean

                                                                                                          (3)                                                               

      b) weighted average value

                                                                                         (4)                                               

It is known that the arithmetic mean value of the parameter is a special case of the weighted average, and it is true, in this case, if the thicknesses of the individual layers of the dirt base of the road are equal. Indeed, if we assume that = , then expression (4) is converted to the form                       

                =  =                     (5)                           

As you can see, formulas (3) and (5) for determining the equivalent coefficient of thermal conductivity of the soil coincide.

To assess the degree of influence of the method of averaging the coefficient of thermal conductivity of the soil on the final result (the depth of the zone of thermal influence of the road), we determine the calculated error by the formula

  =  (1                                         

Where H ₁ and H ₂ are the depth of the thermal influence of the road, determined by formula (1) when calculating the coefficient of thermal conductivity of the soil according to formulas (3) and (4), respectively.

In engineering calculations, an acceptable relative error is considered to be an error not exceeding 10%. Using this assumption, it is possible to determine the permissible ratio of the thermal conductivity coefficients calculated by formulas (3) and (4).

                                                             ? ? (a _v/a _a) ? 0.81 (6)                                                                       

Where a _b and a _a are the coefficients of thermal conductivity of the soil, determined by formulas (4) and (3), respectively.

For example, consider a simple two-layer dirt road base. In this case, the root relation included in expression (5) can be written as follows

? = (a _b/a _a) =2(a ₁ ? ₁+a ₂ ? ₂)/(a ₁+a ₂) (? ₁+? ₂) (7)

We introduce the parameter "k", which characterizes the degree of inequality of the thermal conductivity coefficients of materials of individual layers, i.e. _a2 = k · _a1. Similarly, the ratio of the individual thicknesses of the layers is assumed to be equal to some parameter "m", i.e. ?2 = m_·?1.

In this case, formula (7) is converted to the following form

? =                                        (8)      

It is easy to check that in the case when "k" or "m" are equal to one, the ratio "ab/aa" is also equal to one. This means that the two methods of averaging give the same result. Otherwise, we can say that the weighted average and arithmetic mean values of the soil thermal conductivity coefficient are equal to each other. Using expressions (6) and (8), after simple transformations, it is possible to obtain an equation of the functional relationship between the parameters "k" and "m", which will provide an error less than acceptable

                                                  (9)

Results and discussion.

To achieve the goal, variant calculations were carried out according to the above formulas, the results of which are presented in the form of 2D and 3D graphs in Figures 1-4.  Figure 1 shows the dependence of the ratio of the averaged thermal conductivity coefficients of a two-layer soil base "?" (formula 8) on the degree of deviation of the thickness and thermophysical properties of individual layers from each other, which are characterized by the parameters "k" and "m".

Fig.1. The ratio of the average coefficients of thermal conductivity of a two-layer soil base depending on the degree of deviation of the thickness and thermophysical properties of individual layers from each other

As can be seen from the graphs in the figure, this dependence is clearly nonlinear. Moreover, at the point of equality of parameters (m = k = 1), the nature of the curves changes qualitatively. If earlier, when the parameter "m" increased, the parameter "?" increased, then after the equality point it decreases. This is typical for all values of the parameter "k". But, in quantitative terms, the rate of change of the parameter "?" decreases with an increase in the parameter "k" throughout the considered area of change of the parameter "m".

 Fig.2 shows graphically the results of calculations of the error in determining the depth of the zone of thermal influence when replacing the weighted average coefficient of thermal conductivity with the arithmetic mean.

Fig.2. Percentage error in determining the depth of the zone of thermal influence at different ratios of the parameters "m" and "k", characterizing the degree of deviation of the properties and thicknesses of individual layers from each other

The analysis of the graphs in the figure shows that with certain ratios of the parameters "m" and "k" characterizing the degree of deviation of the properties and thicknesses of individual layers from each other, the calculation error may exceed the permissible in engineering practice. At the same time, the margin of error (e ? 10%) is sufficiently large, and covers a wide range of changes in the parameters "m" and "k". For clarity, Figure 3 shows a 3D graph of the change in the percentage error in determining the depth of the zone of thermal influence at different ratios of the parameters "m" and "k".

Fig.3. Dependence of the absolute value of the averaging error

 from the values of the parameters "k" and "m",

It can be seen from the figure that the area of the ratio of the parameters "k" and "m" for an error value less than 10% is significantly larger than the area of exceeding the error values is greater than the permissible one. That is, in practice, the probability of falling into the permissible error range when determining the depth of the thermal influence zone of the road with a two-layer soil foundation is much higher.

Figure 4 shows graphs obtained as a result of variant calculations using formula (9), which allow us to determine the ratio of the parameters "k" and "m", providing an error less than the permissible value.

Fig.4. The ratio of the parameters "k" and "m", providing an error

less than the permissible value.

As can be seen from the figure, the curves are a mirror image of each other. That is, it was possible to simply change the designations on the axes without resorting to calculations. Actually, this follows from a simple analysis of formula (9 ) expressing the functional dependence between the parameters "m" and "k". If we express the parameter "m" as a function of "k" (change the designation of the function and argument), then the type of functional dependence will not change.

Conclusion.  Simple engineering dependences are obtained to determine the error that occurs when calculating the depth of the thermal influence zone of the road by replacing the layered structure of the road base with an equivalent single-layer with an average coefficient of thermal conductivity. For clarity, a simple example of calculating the thermal resistance of a two-layer structure of a soil base is used. For a comprehensive assessment, intermediate parameters are introduced that characterize the degree of deviation of the thermophysical properties and thicknesses of individual layers from each other. A concrete example of calculating the depth of the zone of thermal influence in the two-layer soil of the road base is considered. An equation of the functional relationship between the parameters characterizing the degree of deviation of the thickness and thermophysical properties of individual layers from each other is obtained, which provides an error in calculating the depth of the zone of thermal influence less than the permissible value. The results of numerical calculations are presented in the form of 2D and 2D graphs, which allow us to visually assess the influence of the range of changes in the values of the thermal conductivity coefficients of individual soil layers on the legality of using various methods of constructing an equivalent single-layer road foundation structure. The article is primarily of methodological importance and allows for a concrete example to trace in detail the ways of qualitative and quantitative analysis of the effect of averaging the initial parameters on the final result during engineering calculations. The article can be useful for both design engineers and practitioners of the road industry, as well as researchers in the field of engineering geocryology. In methodological terms, the article may be of interest to graduate students studying in various specialties of the direction 1.6. "Earth Sciences", as well as students studying in the specialty 08.02.05 "Construction and operation of highways and airfields".  Further research should be aimed at assessing the effect of averaging the initial parameters on the accuracy of calculating such important values in the design of roads in the cryolithozone as, for example, the depth of thawing or freezing of the road base.