中南大学学报(英文版)

J. Cent. South Univ. (2016) 23: 1232-1242

DOI: 10.1007/s11771-016-0373-8

Interaction due to various sources in saturated porous media with incompressible fluid

Rajneesh Kumar¹, Ibrahim A. Abbas^{2, 3, 4}

1. Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India;

2. Department of Mathematics, Faculty of Science and Arts-Khulais, University of Jeddah, Saudi Arabia;

3. Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics,King Abdulaziz University, Jeddah, Saudi Arabia;

4. Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract:

The disturbance due to mechanical and thermal sources in saturated porous media with incompressible fluid for two-dimensional axi-symmetric problem is investigated. The Laplace and Hankel transforms techniques are used to investigate the problem. The concentrated source and source over circular region have been taken to show the utility of the approach. The transformed components of displacement, stress and pore pressure are obtained. Numerical inversion techniques are used to obtain the resulting quantities in the physical domain and the effect of porosity is shown on the resulting quantities. All the field quantities are found to be sensitive towards the porosity parameters. It is observed that porosity parameters have both increasing and decreasing effect on the numerical values of the physical quantities. Also the values of the physical quantities are affected by the different boundaries. A special case of interest is also deduced.

Key words:

axi-symmetry; incompressible porous medium; pore pressure; Laplace transform; Hankel transform; concentrated source and source over circular region；

1 Introduction

The dynamic response due to various sources in a saturated porous media with incompressible fluid are of great interest in geophysics, acoustic, soil and rock mechanics and many earthquake engineering problems.

BIOT [1] derived the basic equations of poroelastisity on the basis of energy principles. PREVOST [2] rederived these equations by use of mixture theory. ZENKIEWICZ et al [3], ZENKIEWICZ and SHIOMI [4] derived the basic equations of poroelasticity by the use of principle of continuum mechanics. GATMIRI and KAMALIAN [5] adopted the later approach because it is more flexible and is based on a set of parameters with a clear physical interpretation to discuss different type of problems. GATMIRI and NGUYEN [6] investigated two-dimensional problem for saturated porous media with incompressible fluid.

GATMIRI and JABBARI [7-8] discussed time domain Green’s functions for unsaturated soil for two-dimensional and three-dimensional solutions. GATMIRI and ESLAM [9] discussed the scattering of harmonic waves by a circular cavity in a porous medium by using complex function theory approach. GATMIRI et al [10] also discussed the two-dimensional transient thermo-hydro-mechanical fundamental solution of multiphase porous media in frequency and time domains.

BAI and LI [11] obtained the solutions for cylindrical cavity in saturated thermoporoelastic medium. KAUSHAL et al [12] discussed the response of frequency domain in generalized thermoelasticity with two temperatures. KUMAR et al [13] also discussed elastodynamics of an axisymmetric problem in an anisotropic liquid-saturated porous medium. LIU et al [14] discussed the relaxation effects of a saturated porous media using the two-dimensional generalized thermoelastic theory. SUVOROV and SELVADURAI [15] derived the constitutive equations of thermo- poroelasticity by using eigenstrain-eigenstress approaches. LIU et al [16] presented a model of the equations of a generalized theory with relaxation times for a saturated porous medium and investigated a two- dimensional problem due to time-dependent thermal/ mechanical source.

OLIVEIRA et al [17] discussed boundary element formulation of axisymmetric problems for an elastic half space.The influence of the finite initial strains on the axisymmetric wave dispersion in a circular cylinder embedded in a compressible elastic medium was discussed by AKBAROV and GULIEV [18]. GORDELIY and DETOURNA [19] investigated the displacement discontinuity method for modelling axisymmetric cracks in an elastic half-space. JABBARI and DEHBANI [20-21] studied axisymmetric and spherical symmetric problems in porothermoelastic solids. ABBAS [22] studied the natural frequencies of a poroelastic hollow cylinder. For our contribution, several problems have been solved by finite element method and analytical method [23-31]. LIU and CHEN [32] studied the problem of a micromechanical analysis of the fracture properties of saturated porous media.

In the present work, the disturbance is due to concentrated source and source over circular region in the time domain and frequency domain in saturated porous media with incompressible fluid. The transformed components of displacement, stress and pore pressure are obtained and depict graphically to show the effect of porosity on the resulting quantities.

2 Governing equations

Following GATMIRI and NGUYEN [6], the field equations are as follows.

Equation of motion:

                                                 (1)

Generalized Darcy’s law:

                                               (2)

Constitutive relation:

                               (3)

Flow conservation for the fluid phase:

                                                 (4)

where u_i is the displacement of the solid skeleton; p denotes the fluid pressure; W_i represents the average displacement of the fluid relative to the solid. The elastic constants λ and μ are drained Lame’s constants. ρ_f is the fluid density; ρ_s is the solid density; ρ=(1-n)ρ_s+nρ_f is the density of solid-fluid mixture and m=ρ_f/n is the mass parameter where n is the porosity; k is the permeability coefficient. α and M are material parameters which describe the relative compressibility of the constituents. fi and γ denote the body force and the rate of fluid injection into the media, respectively.

Equations (1) and (2) with the aid of Eqs. (3) and (4) in the absence of body force and the rate of fluid injection into the media can be reduced to

                (5)

                   (6)

where

3 Formulation of problem

We consider a saturated porous media with incompressible fluid whose boundaries are parallel to the plane z=0 in the cylindrical polar coordinate system (r, θ, z). We consider a two dimensional axi-symmetric problem with symmetry about z-axis, so that all the quantities are maintained independent of θ and The complete geometry of the problem is shown in Figs. 1(a) and (b). We assume the components of displacement vector as

                  (7)

Fig. 1 Concentrated source acting on saturated porous media with incompressible fluidhalf space (a) and source over circular region acting on saturated porous media with incompressible fluid half space (b)

We define the non-dimensional quantities:

                   (8)

where ω is the constant having the dimensions of frequency.

The displacement components u_r and u_z are related to the potential functions and as

                   (9)

We define the Laplace and Hankel transforms as follows:

                                  (10)

                       (11)

where J_n( ) is the Bessel function of the first kind of index n.

Making use of the dimensionless quantities defined by Eq. (8) on Eqs. (5) and (6) and with the aid of Eqs. (7) and (9) (after suppressing the prime) and applying the Laplace and Hankel transforms defined by Eqs. (10) and (11) on the resulting quantities after simplification, we obtain

                           (12)

                             (13)

where

and

Using the solution of Eqs. (13) and (15) and satisfying the radial conditions that and as z→∞, we obtain the values of and as follows:

                                 (14)

                                         (15)

                              (16)

where m₁ and m₂ are given by when n=1, 2, are the roots of Eq. (15); where and the coupling constants are given by i=1, 2.

4 Boundary conditions and solution of problem

The boundary conditions at z=0 are

                                (17)

where P₁ and P₃ are the magnitudes of the forces; P₂ is the constant pressure applied on the boundary; F(r) and δ( ) are the known functions defined later in the manuscript.

Applying Laplace and Hankel transforms, we have

                                 (18)

The stress components are obtained with the aid of Eqs. (3), (7), (10), (11), (16) and (17) as

       (19)

 (20)

    (21)

   (22)

                  (23)

where

and

① For normal force P₂=P₃=0;

② For tangential force P₁=P₂=0;

③ For pressure source P₁=P₃=0.

5 Case study

5.1 Case 1: Concentrated source

The solution due to concentrated source is obtained by substituting

                                  (24)

Applying Laplace and Hankel transforms on Eq. (24), we obtain

5.2 Case 2: Source over circular region

The solution due to source over the circular region of non-dimensional radius a is obtained by setting F(r)=Applying Laplace and Hankel transforms on these quantities, we obtain

5.3 Frequency domain

In this case we assume the time harmonic behaviour as

and the boundary conditions Eq. (19) takes the form:

The expressions for displacement, stress and pore pressure in frequency domain can be obtained by replacing s by iω in the expressions of time domain Eqs. (2)-(23).

5.4 Special case

In the absence of porous incompressible fluid, the boundary conditions can be reduced toand the corresponding expressions for stress components in elastic half space, we obtain

                         (25)

                             (26)

where

Taking P₃=0 and P₁=0 in Eqs.(25)-(26), we obtain respectively the stress components for the normal and tangential forces.

5.5 Numerical results and discussion

For numerical computation, the following values of the various physical parameters are taken from GATMIRI and NGUYEN [6]:

λ=12.5 MPa,μ=8.33 MPa,K_s=10⁵ MPa,

K_f=0.22×10⁴ MPa,ρ_s=2600 kg/m³,ρ_f=1000 kg/m³,

k=0.001 m/s, α=1, n=0.3, ω=1.

The values of normal stress σ_zz, tangential stress σ_zs and pore pressure p for fluid saturated incompressible porous medium (FSPM) and empty porous medium (EPM) are shown due to concentrated source and source applied over the circular region. The computation is carried out for two values of dimensionless time t=0.50 and t=0.75 at z=1 in the range 0≤r≤10.

The solid lines either without central symbols or with central symbols represent the variations for t=0.1, whereas the dashed lines with or without central symbols represent the variations for t=0.50. Curves without central symbols correspond to the case of FSPM whereas those with central symbols correspond to the case of EPM.

5.6 Time domain

Figure 2 shows the variation of normal stress component σ_zz w.r.t distance r for both FSPM and EPM due to concentrated normal force. The value of σ_zz first increases in the range 0≤r≤7 and then starts decreasing for FSPM as r increases and in the case of EPM, its value oscillates as r increases for both values of time.

Figure 3 shows the variation of normal stress component σ_zz w.r.t distance r for both FSPM and EPM due to concentrated tangential force. The value of σ_zz oscillates for FSPM as r increases for both values of time whereas in the case of EPM, the value of σ_zz starts with sharp decrease and then oscillates for both values of time as r increases.

Fig. 2 Variation of normal stress σ_zz with distance r due to concentrated normal force

Fig. 3 Variation of normal stress σ_zz with distance r due to concentrated tangential force

Figure 4 shows the variation of normal stress component σ_zz w.r.t distance r for FSPM due to concentrated pressure source. The value of σ_zz is more in the range 3≤r≤6 and less in the range 6≤r≤9 for FSPM as r increases for time t=0.50 whereas for the time t=0.75 its value converges near the boundary surface.

Figure 5 shows the variation of pore pressure p w.r.t distance r for FSPM due to concentrated normal force. The value of p decreases in the range 0≤r≤3.5 and then oscillates for FSPM as r increases for time t=0.50 whereas for the time t=0.75 the value of p decreases and then oscillates as r increases.

Figure 6 shows the variation of pore pressure p w.r.t distance r for FSPM due to concentrated tangential force. The value of p decreases sharply for FSPM and then oscillates as r increases for both values of time.

Figure 7 shows the variation of pore pressure p w.r.t distance r for FSPM due to concentrated pressure source. The value of p converges near the boundary surface for FSPM as r increases for time t=0.50 whereas for the time t=0.75 the value of p increases sharply and then oscillates as r increases.

Fig. 4 Variation of normal stress σ_zz with distance r due to concentrated pressure source

Fig. 5 Variation of pore pressure p with distance r due to concentrated normal force

Fig. 6 Variation of pore pressure p with distance r due to concentrated tangential force

Figure 8 shows the variation of tangential stress σ_zr w.r.t distance r for both FSPM and EPM due to concentrated normal force. The value of σ_zr first decreases in the range 0≤r≤3 and then oscillates for FSPM as r increases for both value of time whereas the value of σ_zr decreases sharply and then oscillates for EPM as r increases for both value of time.

Fig. 7 Variation of pore pressure p with distance r due to concentrated pressure source

Fig. 8 Variation of tangential stress σ_zr with distance r due to concentrated normal force

Figure 9 shows the variation of tangential stress σ_zr w.r.t distance r for both FSPM and EPM due to concentrated tangential force. The value of σ_zr first increases monotonically and then converges near the boundary surface for FSPM and for EPM its value almost closes zero as r increases for both values of time.

Figure 10 shows the variation of tangential stress σ_zr w.r.t distance r for FSPM due to concentrated pressure source. The value of σ_zr first increases and then oscillates for FSPM as r increases for time t=0.50 whereas for the time t=0.75 the value of σ_zr starts with initial increase and then converges near the boundary surface.

Fig. 9 Variation of tangential stress σ_zr with distance r due to concentrated tangential force

Fig. 10 Variation of tangential stress σ_zr with distance r due to concentrated pressure source

Figure 11 shows the variation of normal stress σ_zz w.r.t distance r for both FSPM and EPM due to normal force over circular region. The value of σ_zz first increases in the range 0≤r≤7 and then starts decreasing for FSPM as r increases and in the case of EPM, its value oscillates as r increases for both values of time.

Figure 12 shows the variation of normal stress σ_zz w.r.t distance r for both FSPM and EPM due to tangential force over circular region. The value of σ_zz starts oscillating for FSPM as r increases for both values of time and the value of σ_zz for EPM first decreases and then starts oscillating as r increases for both values of time.

Figure 13 shows the variation of normal stress σ_zz w.r.t distance r for FSPM due to pressure source over circular region. The value of σ_zz first increases in the range 0≤r≤6 then decreases for FSPM as r increases for both values of time.

Fig. 11 Variation of normal stress σ_zz with distance r due to normal force over circular region

Fig. 12 Variation of normal stress σ_zz with distance r due to tangential force over circular region

Fig. 13 Variation of normal stress σ_zz with distance r due to normal force over circular region

Figure 14 shows the variation of pore pressure p w.r.t distance r for FSPM due to normal force over circular region. The value of p decreases sharply in the range 0≤r≤3.5 and then oscillates as r increases for both values of time.

Fig. 14 Variation of pore pressure p with distance r due to normal force over circular region

Figure 15 shows the variation of pore pressure p w.r.t distance r for FSPM due to tangential force over circular region. The value of premain almost close to zero for FSPM as r increases for time t=0.5 whereas for the time t=0.75 the value of p decreases sharply and then oscillates as r increases.

Fig. 15 Variation of pore pressure p with distance r due to tangential force over circular region

Figure 16 shows the variation of pore pressure p w.r.t distance r for FSPM due to pressure source over circular region. The value of p starts with initial increase and then oscillates for FSPM as r increases for time t=0.50 whereas for the time t=0.75 the value of p increases sharply in the range 0≤r≤5 and then starts decreasing as r increases.

Fig. 16 Variation of pore pressure p with distance r due to pressure source over circular region

Figure 17 shows the variation of tangential stress σ_zr w.r.t distance r for both FSPM and EPM due to normal force over circular region. The value of σ_zr first decreases and then oscillates for FSPM as r increases for both value of time whereas the value of σ_zr decreases gradually and then oscillates for EPM as r increases for both value of time.

Fig. 17 Variation of tangential stress σ_zr with distance r due to normal force over circular region

Figure 18 shows the variation of tangential stress σ_zr w.r.t distance r for both FSPM and EPM due to tangential force over circular region. The value of σ_zr first increases monotonically and then remains almost close to zero for FSPM for both values of time and first increases monotonically and then converges near the boundary surface for the time t=0.75 for EPM as r increases.

Figure 19 shows the variation of tangential stress σ_zr w.r.t distance r for FSPM due to pressure source over circular region. The value of σ_zr increases sharply and then oscillates as r increases for FSPM for both value of time.

Fig. 18 Variation of tangential stress σ_zr with distance r due to normal force over circular region

Fig. 19 Variation of tangential stress σ_zr with distance r due to pressure source over circular region

5.7 Frequency domain

Figure 20 shows the variation of normal stress component σ_zz w.r.t distance r for both FSPM and EPM due to concentrated normal force. The value of σ_zz first increases monotonically in the range 0≤r≤2 and then start oscillating for FSPM as r increases for both values of time where its value oscillates as r increases for t=0.50 and converges near the boundary surface for t=0.75 for EPM.

Figure 21 shows the variation of normal stress component σ_zz w.r.t distance r for both FSPM and EPM due to concentrated tangential force. The value of σ_zz first decreases sharply and then starts oscillating for FSPM as r increases for both values of time where its value oscillates as r increases for both values of time for EPM.

Figure 22 shows the variation of normal stress component σ_zz w.r.t distance r for FSPM due to con-centrated pressure source. The value of σ_zz first decreases and then start oscillating for FSPM as r increases for both values of time.

Fig. 20 Variation of normal stress σ_zz with distance r due to normal force (frequency domain)

Fig. 21 Variation of normal stress σ_zz with distance r due to tangential source (frequency domain)

Fig. 22 Variation of normal stress σ_zz with distance r due to pressure source (frequency domain)

Figures 23 and 25 show the variation of pore pressure p w.r.t distance r for FSPM due to concentrated normal and pressure sources. The value of p first increases sharply in the range 0≤r≤2 and then oscillates for FSPM as r increases for both values of time.

Fig. 23 Variation of pore pressure p with distance r due to normal force (frequency domain)

Figure 24 shows the variation of pore pressure p w.r.t distance r for FSPM due to concentrated tangential source. The value of p decreases and then oscillates for FSPM as r increases for both values of time.

Fig. 24 Variation of pore pressure p with distance r due to tangential force (frequency domain)

Figure 26 shows the variation of tangential stress σ_zr w.r.t distance r for both FSPM and EPM due to concentrated normal force. The value of σ_zr first increases sharply and then starts oscillating for time t=0.50 whereas for the time t=0.75, the value of σ_zr first increases and then converges near the boundary surface as r increases for FSPM. And for EPM it value first increases monotonically and then starts oscillating for time t=0.50 whereas for the time t=0.75 the value of σ_zr converges near the boundary surface as r increases.

Fig. 25 Variation of pore pressure p with distance r due to pressure source (frequency domain)

Fig. 26 Variation of tangential stress σ_zr with distance r due to normal force (frequency domain)

Figure 27 shows the variation of tangential stress σ_zr w.r.t distance r for both FSPM and EPM due to concentrated tangential source. The value of σ_zr decreases gradually for time t=0.50 whereas for the time t=0.75 the value of σ_zr first increases and then converges near the boundary surface as r increases for FSPM and for EPM it value first decreases monotonically and then starts to oscillate for time t=0.50 whereas for the time t=0.75 the value of σ_zr converges near the boundary surface as r increases.

Figure 28 shows the variation of tangential stress σ_zr w.r.t distance r for FSPM due to concentrated pressure source. The value of σ_zr decreases gradually and then starts oscillating for time t=0.50 whereas for the time t=0.75 the value of σ_zr first decreases and then converges near the boundary surface for FSPM as r increases.

Fig. 27 Variation of tangential stress σ_zr with distance r due to tangential force (frequency domain)

Fig. 28 Variation of tangential stress σ_zr with distance r due to pressure source (frequency domain)

6 Conclusions

1) In the present work, we obtain the components of displacement, stress and pore pressure due to concentrated source and source over circular region in the time domain and frequency domain in saturated porous media with incompressible fluid.

2) Near the application of the source, the porosity effect decreases the values of σ_zz for normal force, tangential force and pressure source where it increases the values of σ_zr fornormal force and tangential force but decreases the values for pressure source, due to concentrated source in the time domain.

3) In frequency domain, porosity effect increases the values of σ_zz, σ_zr and p for normal force, tangential force and pressure source for source over circular region. Also away from the source the porosity effect decreases the values of σ_zz for normal force and increases the values for tangential force and pressure source whereas it decreases the value of σ_zr for normal force and monotonically increases for tangential force and pressure source.

4) Thus, the problem analyzed is a significant problem of continuum mechanics. The results obtained as a consequence of this research work should be beneficial for researchers working on saturated porous media with incompressible fluid. By introducing various sources, the assumed model presents a more realistic model for future study.

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(Edited by DENG Lü-xiang)

Received date: 2015-01-27; Accepted date: 2015-05-20

Corresponding author: Rajneesh Kumar, PhD; E-mail: Rajneesh_kuk@rediffmail.com

Abstract: The disturbance due to mechanical and thermal sources in saturated porous media with incompressible fluid for two-dimensional axi-symmetric problem is investigated. The Laplace and Hankel transforms techniques are used to investigate the problem. The concentrated source and source over circular region have been taken to show the utility of the approach. The transformed components of displacement, stress and pore pressure are obtained. Numerical inversion techniques are used to obtain the resulting quantities in the physical domain and the effect of porosity is shown on the resulting quantities. All the field quantities are found to be sensitive towards the porosity parameters. It is observed that porosity parameters have both increasing and decreasing effect on the numerical values of the physical quantities. Also the values of the physical quantities are affected by the different boundaries. A special case of interest is also deduced.