Dynamics of a Drilling Column with a Nonlinear Law of its Interaction with Well Walls

International Journal of Applied Engineering Research and Development (IJAERD) ISSN(P): 2250–1584; ISSN(E): 2250–9383 Vol. 10, Issue 1, Jun 2020, 31–42 © TJPRC Pvt. Ltd.

DYNAMICS OF A DRILLING COLUMN WITH A NONLINEAR LAW OF ITS INTERACTION WITH WELL WALLS MARDONOV B. M1, DZHANZAKOV I. I2 & RAKHMATOV R. R3 1,3

Tashkent University of Information Technologies named after Muhammad al-Khwarizmi., Tashkent, Uzbekistan 2

Atyrau Institute of Gas Oil, Kazakhstan, Atyrau, Kazakhstan

ABSTRACT In the process of dealing with various types of complications, in particular with sticking and, as oilfield experiments show, occurs at large values of displacement of the pipe surface, and this fact indicates the necessity of using a nonlinear model of drilling mechanics. The stress state of the drill string in contact with the borehole wall according to the nonlinear (cubic) law under the action of vertical dynamic force is investigated, and the effect of geometric non-linearity on the stress state of the string is studied.

KEYWORDS: Drill String, Model, Nonlinear Law, Contact Interaction, Bore & Sticking

1. INTRODUCTION Many problems of drilling mechanics, mathematically formulated using linear differential equations, strictly speaking, are nonlinear. In fact, they should be described by nonlinear differential equations, where the influence of many factors can be taken into account, which, in some circumstances, play a significant role in the drilling process.

Original Article

Received: Dec 20, 2019; Accepted: Jan 10, 2020; Published: Mar 03, 2020; Paper Id.: IJAERDJUN20204

The linearization of the problems of drilling mechanics is carried out entirely in order to avoid mathematical difficulties associated with solving nonlinear differential equations, and to use for a clear interpretation of the results of the theory of linear equations.

The widespread use of mechanics in the field of technology, in particular in drilling, led to the need to increase the accuracy of the analysis of the results of the studied processes, as well as to the description of effects that could not be taken into account by linear theories, despite the fact that some of them, as noted above, have already become play an important role in solving technical problems. In recent years, they began to deal with mathematically incomparably more difficult, but also more attractive non-linearized problems, and therefore a large number of separate studies have accumulated at present [1-7]. When using the fundamentals of the theory of elasticity in drilling mechanics, two linearization processes are realized. The first of them, concerning the geometry of a deformed body, is based on the assumption that the dimensionless quantities determining the deformation are very small. Therefore, it is possible to accurately describe the deformed state, discarding the higher degrees of deformation as small quantities in comparison with the first degrees, and this linearization process is called geometric. The second linearization process relates to the physical properties of the material and the contact force of the interaction of the pipe tubes with the external environment (with a rock mass or with the walls of the well). This process is based on the assumption that stresses and strains are connected by a linear dependence (Hooke's law), and the interaction force of the pipe sections with the external medium obeys a linear law (physical linearization).

Drill strings in wells are hollow, heavy rods with stepwise varying physical properties (for example, rigidity), and

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Mardonov B. M., Dzhanzakov I. I. & Rakhmatov R. R.

each link in the string can be deformed according to geometrically nonlinear laws. They interact with the walls of the well, both hydrodynamic and contact, and are always curved, both due to the curvature of the well itself, and under the influence of its own weight and contact forces, as well as centrifugal forces in the case of rotation of the column. In this case, the curvature of the axis of the column can be significantly affected by the geometric nonlinearity of the deformation of the pipes of the column. Depending on whether the behavior of the pipe string is studied in a static or dynamic setting, ordinary non-linear differential equations or non-linear partial differential equations have to be solved, in the latter case, having an approved numerical scheme, for example, by the finite element method, the problem can be reduced to ordinary differential equations. First of all, it is necessary to calculate the static stability according to Euler.

When studying the vibrations and stability of geometrically nonlinear drill strings under the action of dynamic loads, new effects appear that are related to the nonlinearity of the deformation of the pipe tubes and the nature of their loading. Such tasks, in particular, include the task of quickly loading the column with an axial compressive load. If the rate of increase of the compressive force is sufficiently high, then the sections of the drill string do not have time to move in the direction normal to the axis of the string.

In connection with this, there are various possibilities of constructing a nonlinear theory of drilling mechanics.

First, geometric linearization can be abandoned, but physical linearization can be maintained, therefore, it can be assumed that Hooke’s law and the linear relationship between the contact force and the displacement of the column sections of the column (Winkler’s law) are valid for large deformations and relative mixing. Secondly, one can proceed from the assumption that under deformations that admit geometric linearization without fear, there is a noticeable deviation from Hooke's law. This assumption, confirmed by the actual behavior of many materials, as well as by the dependences between the contact force and the displacements of the cross sections of the column pipe, forces even at small deformations and displacements to introduce additional terms in the Hooke laws and the dependences of the contact force on displacement that are nonlinear with respect to deformations and displacements. In this paper, we use a model of the nonlinear theory of drilling mechanics, where the linear Hooke law for the pipe material holds, and the relationship between the contact on the pipe surface and the displacements of the pipe sections (Winkler's law) obeys the nonlinear law. Such a model of drilling mechanics to solve a number of contact problems associated with the elimination of various types of complications when drilling oil and gas wells [8-10]. The case of subordination of the contact friction force to the Coulomb dry law was considered in [5]. Analysis of the results of experimental data on the determination of the contact force of interaction [9, 10] indicates a more complex dependence of this force on the displacement of the pipe section relative to the soil and borehole walls.

2. STATEMENT OF THE PROBLEM The process of eliminating sticks, as shown by oilfield experiments, occurs at large values of displacement of the pipe surface, and this circumstance indicates the need to use the above non-linear model of drilling mechanics to solve problems associated with assessing the stress state of pipe pipes in the process of eliminating sticks. Consider the problem of the action of axial force on a drill string with a total length

Ox

l

in the presence of a contact zone

l1 < x < l 2 (where the axis

is directed along the axis of the string from top to bottom) with the borehole walls.

The equation for the movement of the cross sections of the column in the presence of a nonlinear contact force on

the pipe surface is written as Impact Factor (JCC): 4.9648

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Dynamics of a Drilling Column with a Nonlinear Law of its Interaction with Well Walls

ES

∂ 2 u1 ∂ 2 u1 = E ρ + P0 (t )δ ( x) at 0 < x < l1 , ∂x 2 ∂t 2

(1)

∂ 2u2 ∂ 2u2 ES = Eρ 2 + F (u 2 ) at l1 < x < l 2 , ∂x 2 ∂t

(2)

∂ 2u3 ∂ 2u3 ES = Eρ 2 at l 2 < x < l , ∂x 2 ∂t

(3)

where

u 2 ( x, t ) −

the external medium, materials, respectively,

P0 (t ) −

l2

33

is the longitudinal movement of an arbitrary section in the zone of contact of the column with

u1 ( x, t ) and u3 ( x, t ) − are outside this zone, E and

ρ−

are Young's moduli, density of

S = π ( D 2 − d 2 ) / 4 , D and d − are the external and internal diameters of the pipes, t − time,

is the axial force applied in the cross section of the column with

x = 0, δ ( x) − is the Dirac function, l1

and

are the coordinates of the beginning and end of the zone of contact of the column with the walls of the well,

respectively,

F (u 2 ) −

is the contact force acting on the surface of the column pipe. In the future, we assume

F = k1u 2 ( x, t ) + k 2 u 23 ( x, t ) (where k1 and k2 are constants determined on the basis of experiments). Equations (1) (3) are written in the form

∂ 2 u1 ∂ 2 u1 ∂ 2u3 ∂ 2u3 = = + P ( τ ) δ ( ξ ) at 0 < ξ < ξ , at ξ 2 < ξ < 1 , 0 1 ∂ξ 2 ∂τ 2 ∂ξ 2 ∂τ 2

(4)

∂ 2u2 ∂ 2u2 = + αu 2 (1 + β u 22 ) at ξ1 < ξ < ξ 2 2 2 ∂ξ ∂τ

(5)

where

ξ = x / l , τ = ct / l , u i = u / l , ξ1 = l1 / l , ξ 2 = l 2 / l , c = E / ρ , P0 = P0 / ES , δ = lδ (x) ,

α = k1l 2 / ES , β = k 2 l 2 / k1 .

The system of equations (4) and (5) is integrated under the following initial and boundary conditions

ui = 0 ,

∂u i = 0 at τ = 0 ∂τ

∂u ∂u1 = 0 , at ξ = 0 , 3 = 0 at ξ = 1 ∂ξ ∂ξ

u1 = u 2 ,

∂u1 ∂u 2 ∂u 2 ∂u 3 at ξ = ξ1 , u 2 = u 3 , at ξ = ξ 2 = = ∂ξ ∂ξ ∂ξ ∂ξ

(6)

(7)

(8)

3. SOLVING METHOD The boundary value problem (6) - (8) for the system of equations (4) and (5) is nonlinear due to the presence of the term www.tjprc.org

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β u13

Mardonov B. M., Dzhanzakov I. I. & Rakhmatov R. R.

in equation (5), and therefore its exact analytical solution is absent. To find an approximate solution, we use the

method proposed in [10], the essence of which is to reduce the boundary-value problem to finding a solution to an infinite system of nonlinear ordinary equations. According to this method, its solution to the boundary value problem can be represented as ∞

u i = ∑ Tn (τ ) X ni (ξ )

(9)

n =1

X ni (ξ ) is the eigenfunctions of the corresponding linear boundary value problem ( β = 0 ), which are

where

chosen as follows

X n1 = cos µ nξ at 0 < ξ < ξ1 ,

X n 2 = X n1 (ξ1 ) cos λ n (ξ − ξ1 ) +

X n′1 (ξ1 )

X n 3 = X n 2 (ξ 2 ) cos µ n (ξ − ξ 2 ) +

µ n = λ2n + α , λ n

where

λn

sin λ n (ξ − ξ1 ) at ξ1 < ξ < ξ 2 ,

X n′ 2 (ξ 2 )

µn

sin µ n (ξ − ξ 2 ) at ξ 2 < ξ < 1 ,

- roots of equation

X n′ 2 (ξ 2 ) cos µ n (1 − ξ 2 ) − µ n X n 2 (ξ 2 ) sin µ n (1 − ξ 2 ) = 0 .

Moreover, the eigenfunctions satisfy the orthogonality condition ξ1

∫X

n1

ξ2

1

ξ1

ξ2

X k 1 dξ + ∫ X n 2 X k 2 dξ + ∫ X n 3 X k 3 dξ = 0 at n ≠ k

0

(10)

Substituting the expressions u i (ξ ,τ ) from (1) into equations (4) and (5), we obtain ∞

∑ (T ′′ + µ n =1

n

T ) X n1 = − P0 (τ )δ (ξ ) ,

2 n n

∞

∞

n =1

n =1

∑ (Tn′′ + µ n2Tn ) X n 2 + αβ ∑ ( X n 2Tn ) 3 = 0 , ∞

∑ (T ′′ + µ n =1

n

T ) X n3 = 0 ,

2 n n

We multiply equalities (11) - (13), respectively, by the functions intervals

(11)

(12)

(13)

X k 1 (ξ ) , X k 2 (ξ ) , X k 3 (ξ ) integrate in the

0 < ξ < ξ1 , ξ1 < ξ < ξ 2 and ξ 2 < ξ < 1 , add the result

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Dynamics of a Drilling Column with a Nonlinear Law of its Interaction with Well Walls

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ξ2 1 ∞ ξ2 ξ1  (Tn′′ + µ T )∫ X n1 X k1dξ + ∫ X n2 X k 2 dξ + ∫ X n3 X k 3dξ  + αβ ∑ ∫ ( X n2Tn )3 X k 2 dξ = −P0 X k1 (0) ∑ n=1 n=1 ξ1  0  ξ1 ξ2 ∞

2 n n

In view of the orthogonality condition (11), we obtain

Tn′′ + µ T + 2 n n

Where

Xn =

αβ Xn

∑T ∫ X ξ k =1

ξ2

∫X ξ 1

ξ2

∞

k

3 k2

X n 2 dξ = − P0

1

ξ2

2 n1

X n1 (0) .

Xn

(14)

1

dξ + ∫ X dξ + ∫ X n23 dξ - norm of eigenfunction.

2 n2

ξ1

ξ2

Expression (14) forms an infinite system of equations for the functions of

Tn (τ ) , which is solved by the

truncation method.

4. ANALYSIS OF THE RESULTS Figures 1 and 2 show the curves of the displacement of the upper section of the column (Figure 1 a, b) and the voltage in the section

x = x1 (Figure 2 a, b) of dimensionless time τ = ct / l for linear (a) nonlinear (b) interaction laws columns

with a borehole wall. In the calculations, α

= 3, k 2 / k1 = −0.1 , l = 1000 м , P0 = 1.3ρSgl ρ = 6400кг / м 3 ,

D = 0.114 м , d = 0.086 м , E = 2 ⋅ 10 4 МПа .

From the analysis of the curves it follows that the presence of nonlinearity with a soft characteristic ( β

<0)

leads to an increase in the displacement of the cross section of the column and a decrease in voltage in the cross section of the beginning of the formation of the interaction zone.

(a) (b) Figure 1: Dependences of the Displacement of the Upper Section of the Column (m) on the Dimensionless Time

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τ = tau = ct / l

U 1 = u1 (0,τ )

for Linear (a) and Nonlinear (b) Contact Force.

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Mardonov B. M., Dzhanzakov I. I. & Rakhmatov R. R.

(a) (b) Figure 2: Dependences of the Voltage in the Column Section on the Dimensionless Time

τ = tau = ct / l

x = x1 sigma = σ ( x1 ,τ ) (MPa)

for Linear (a) and Nonlinear (b) Contact Force.

Now we consider the case of oscillation of a geometrically nonlinear weighted drill string during torsion.

Torsion of a long-stemmed rod can lead to its curvature as a result of loss of stability of a straight shape. This is explained by the fact that the torque acting in a plane perpendicular to the axis of the rod, when the latter deviates from a rectilinear shape, leads to the appearance of bending moments. At a critical value of the torque, the components of the bending moments reach such values that the elastic forces of the rod are not able to return it to its original position. In the general case, the problem of the stability of the rods under the action of torques is non-conservative, and they must be solved taking into account the complex deformed state of the rod caused by the action of axial, bending, and torques. However, in some cases, Greenhill's formula can be used to determine critical moments [11]

P π2  Mk  =   + EJ l 2  2 EJ  2

where

M−

(1.38)

is the torque,

P−

is the compressive force. As can be seen from expression (10), the compressive

force leads to a decrease in the critical moment. If the compressive force is known, then from (1.38) it is possible to determine the critical value of the torque

M k = 2 EJ

π2 l

If we denote by

2

−

P EJ

ϕ ( x, t )

the angle of rotation of the cross section of the rod, then the expression of the torque,

taking into account the nonlinear deformation of the shear

∂ϕ , can be expressed by the formula ∂x

 J p 2  ∂ϕ  2  ∂ϕ M = GJ p 0 1 + γ 2    J p 0  ∂x   ∂x 

where

G−

is the shear modulus,

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J po

Dн4 − Dв4 = 32

is the polar moment of inertia of the rod sections,

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Dynamics of a Drilling Column with a Nonlinear Law of its Interaction with Well Walls

J p2 =

2 Dн6 − Dв6 , 9 64

37

γ 2 − is the coefficient characterizing the nonlinearity of the shear. The full work of deformation is

calculated by the formula 2 2 Dн4 − Dв4 l  ∂ϕ   2γ 2 Dн2 − Dв2  ∂ϕ   U= G   dx ∫0  ∂x  1 + 3 4 16 4  ∂x   

π

The kinetic energy of the rod will be equal to

T= Here

πm Dн4 − Dв4 l  ∂ϕ  4

m−

16

2

∫  ∂t  dx 0

is the linear mass of the core material. In sections

x = 0 and x = l , a variable moment ± M 0 (t ) is

applied. We represent the angle of rotation of the rod in the form

ϕ=

M 0l πx πx sin + q (t ) cos , GJ p 0π l l

For potential and kinetic energy, we obtain the expressions

π3

Dн4 − Dв4 U= G [ M 02 + q 2 + c( M 04 + 2 M 02 q 2 + q 4 )] , 128 l T=

πm( Dн2 − Dв2 )l 128

qɺ 2

M 0l 2 2 2 2 , c = π γ 2 ( Dн − Dв ) / 8l GJ p 0π

where,

M0 =

Taking

q (t ) for the generalized coordinate, we compose the Lagrange equation of the second kind

ɺɺ , qɺɺ + b1 q + b2 q 3 = − M 0 where

(1.39)

b1 = (1 + 2cM 02 )ω 2 , b2 = 2cω 2 , ω =

The value of

G ( Dн2 + Dв2 )π 2 ml 2

c in equation (1.39) characterizes the nonlinearity of the rod during torsion. The equation is

integrated under the following initial conditions

q (0) = 0 , q (0) = 0 .

Figure 3 shows graphs of the dependence of the angle of rotation of the cross sections of the rod action of the moment

M 0 according to the law M 0 = 1 − cos ω 0 t for various values of the parameters с and ω 0 . It

can be seen that the nonlinearity, depending on the frequency of the different ways. For small values of www.tjprc.org

x = 0 under the

ω0 ,

affects the torsional vibrations of the rod in

ω 0 , the nonlinearity in the case under consideration decreases the vibration amplitudes. editor@tjprc.org

38

Mardonov B. M., Dzhanzakov I. I. & Rakhmatov R. R.

ω0 = 1 c=0

c =1

ω 0 = 10 c=0

c =1

Figure 3: Torsional Vibrations of the Rod at Various Values of Parameters Let the torsional motion of the rod occur under the action of the compression force performs transverse vibrations in the

yOz

c and ω 0 .

P , we believe that the rod

plane in two directions. We accept that the transverse motion of the rod

sections does not affect its torsional vibrations. The equations of motion of the rod in the case of geometric linearity are written in the form [1]

m

∂ 2v ∂ 4v ∂ 2ϕ + EJ + Pz =0 0 ∂t 2 ∂x 4 ∂x 2

(1.40)

m

∂2w ∂ 4w ∂ 2ϕ + EJ − Pz =0 0 ∂t 2 ∂x 4 ∂x 2

(1.41)

where v( x, t ) and

w( x, t ) are the displacements of the rod sections along the axes Oy and Oz , and

eccentricity. Equations (1.40) and (1.41) introducing dimensionless variables and quantities by the formulas τ

ξ = x / l , z0 = z0 / l , α =

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π 3GJ p 0 2 EJ

,

P=

=

z0 − t l

is

EJ , ml

Plz 0 taking into account (14) we bring to the form πGJ p 0

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Dynamics of a Drilling Column with a Nonlinear Law of its Interaction with Well Walls

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∂ 2v ∂ 4v + − αP [ M 0 (τ ) sin πξ + q (τ ) cos πξ ] = 0 ∂τ 2 ∂ξ 4

(1.42)

∂2w ∂4w + + αP [ M 0 (τ ) sin πξ + q (τ ) cos πξ ] = 0 ∂τ 2 ∂ξ 4

(1.43)

Equations (1.42) and (1.43) are integrated under the conditions

v = 0, w = 0 ,

v = 0, w = 0 ,

∂v ∂w = 0, = 0 at τ = 0 , ∂τ ∂τ

∂ 2v ∂2w = 0 , = 0 at ξ = 0 and ξ = 1 , which solution is obtained by the Fourier method ∂ξ 2 ∂ξ 2

∞

∞

n =1

n =1

v = ∑ Tn sin nπξ , w = −∑ Tn sin nπξ

where

Tn =

α τ P sin[n 2π 2 (τ − ς )][M 0 (ς ) + q(ς )]dς 2nπ ∫0

.

ω 0 = 0. 5 c=0

c =1

ω0 = 1 c=0

c =1

ω 0 = 10 c=0

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c =1

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Mardonov B. M., Dzhanzakov I. I. & Rakhmatov R. R.

Figure 4: Transverse Vibrations of the Rod at Various Values of the Parameters Figure 4. the dependences of the midpoint of the elastic line on the dimensionless time moment

τ

c and ω 0 . under the action of the

M 0 according to the law M 0 = 1 − cos ω 0 t are presented for different values of the parameters с and ω 0 . It

is assumed in the calculations that

α = 0.01 , P = 0.005 . It is seen that taking into account the nonlinearity of the rod

during torsion can lead to a decrease in the deflection of the rod at small parameter

ω 0 . With the growth of this parameter,

the rod performs a high-frequency oscillation with large amplitudes.

5. CONCLUSIONS •

The presence of nonlinearity with a soft characteristic leads to an increase in the displacement of the column cross section and a decrease in voltage in the initial interaction zones.

•

In the case of the implementation of the cubic law, a significant change in the frequency composition of the column vibrations occurs, and with soft characteristics, stresses of different signs may appear in the pipe sections of the column.

•

The geometric nonlinearity of the deformation practically does not affect the law of movement of the lower end of the column during the descent operation. The presence of nonlinearity leads to an increase in the descent speed and the appearance of high-frequency oscillations with low amplitudes near its average value.

•

Taking into account the nonlinearity of the rod during torsion can lead to a decrease in the deflection of the rod for small values of the frequency of the external action, while a significant increase in the amplitude

REFERENCES 1.

Volmir A. S. Stability of elastic systems. Moscow: Fizmatgiz, 1963, 880 p.

2.

Kauderer G. Non-linear mechanics. M. Izd. in. lit. 1961.

3.

Novozhilov V. V. Fundamentals of the nonlinear theory of elasticity M.-L. OGIZ, 1948, 211 p.

4.

Hayasi. Nonlinear vibrations in physical systems. M. " world ", 1968, 204 p.

5.

Tyurekhodjaev A. N. Some actual problems of nonlinear mechanics. Materials of the III international scientific conference “Actual problems of mechanics and mechanical engineering” vol. I. Almaty, June 17-10, 2009, pp. 27-54.

6.

Khadzhieva L. A. Models of nonlinear deformable systems. Practical use. Almaty: Cossack University, 2004, 64 p.

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Dynamics of a Drilling Column with a Nonlinear Law of its Interaction with Well Walls 7.

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Hassan, Q. M., Bakr, H., Sultan, H. A., Hassan, R. M., & Emshary, C. A. (2017). Evolution of far-field diffraction patterns and nonlinear optical properties of SAE 70 oil. International Journal of Applied and Natural Sciences, 6, 181-188.

8.

Mardonov B. M., Khadzhieva L. A. The study of the dynamic stability of drill rods (columns) with geometric nonlinearity of their deformation by the finite element method. “non-newton systems in the oil and gas industry” of the International Scientific Conference dedicated to the 85th anniversary of academician Azad Khalil oglu Mirzadzhanzade. November 21 - 22, 2013, Baku. p.166-168.

9.

Mardonov B. M., Khadzhieva L. A. Dynamic stability of a geometrically nonlinear drill string under variable loads.

International scientific and practical conference "Ecology and the Oil and Gas Complex" dedicated to the 80th anniversary of the academician National Academy of Sciences of the Republic of Kazakhstan, Doctor of Geological and Mineralogical Sciences, Professor Muftakh Diarovich Diarov, Atyrau, October 10-11, 2013, pp. 63-68.

10. Cister E. G., Mikheev V. L. About resistance to movement of a drilling tool in a well. - "Oil industry", 1987, No. 9, pp. 3-8.

11. Janzakov I. I. The mechanics of drill strings under sticking conditions. Atyrau, 2006.

12. Musa, N. W. Investigation on Torsional Vibration of Drill String in Cylindrical Cavities of Vertical Bore-Hole with Liquid Medium.

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AUTHOR’S PROFILE

Mardonov Botir Mardonovich, Professor, Doctor of Physical Mathematical Sciences. Faculty of Software engineering, Tashkent University Information and Technologies, Tashkent, Uzbekistan. He was born in 1942. He graduated from the Moscow State University named after M. Lomonosov with a specialty “Mechanics and Mathematics”. Academician of the Academy of Natural Sciences and National Academy of Kazakhstan. Honored Worker of Science of Uzbekistan. His scientific interests: continuum mechanics, aerodynamics, mathematical modeling of technological processes. The author of more than 200 scientific articles, 20 monographs and textbooks.

Islam Isagalievich Dzhanzakov, Vice-Rector of Science, Atyrau Institute of Oil and Gas of the Republic of Kazakhstan.

Doctor of Technical Sciences, Professor. Specialist in the field of drilling and well development and oil field development.

The author of more than 100 scientific articles, 5 monographs and textbooks.

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42

Mardonov B. M., Dzhanzakov I. I. & Rakhmatov R. R.

Rakhmatov Rabbim Rakhmatovich, Doctor of Physical and Mathematical Sciences. Faculty of software engineering, Tashkent University Information and Technologies named after Muhammad al Khwarizmi, Uzbekistan. He was born in 1955. He graduated from the Institute of Mechanics and seismic resistance of constructions in the specialty “Solid mechanics”. He is head of the department of “Higher mathematics”. His scientific interests: continuum mechanics, aerodynamics, mathematical modeling of technological processes. He is an author of more than 50 scientific articles, 1 monographs.

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