Dynamic stress concentration in an infinitely long cylindrical cavity due to a point spherical source embedded within a fluid-saturated poroelastic formation of infinite extent

doi:10.1007/s10483-025-3203-6

Abstract

Abstract:

The effects of a harmonically exciting monopole source on an infinitely long cylindrical cavity embedded entirely within a fluid-saturated poroelastic formation of infinite extent are examined theoretically. It is assumed that the source is located outside the cavity at a specified distance from the borehole axis. The magnitudes of the hoop and radial stresses beside the pore pressures exerted on the interface and inside the porous medium surrounding the borehole are calculated and discussed. Biot's poroelastic modeling along with three types of boundary conditions for the cylindrical interface including the ideal fluid, empty borehole, and rigid inclusion with a hard boundary is employed for the analysis. Utilizing a proper translational addition theorem for expressing the incident spherical wave in terms of cylindrical wave expansions, the proposed boundary conditions at the interface are satisfied. Stresses are formulated by means of wave potential functions in a three-dimensional (3D) manner. The effects of the frequency and the radial distance between the source and borehole on the induced stresses are examined for the first cylindrical modes over frequency spectra. Two permeability conditions for the interface and three types of soils for the porous formation are considered throughout the analysis. To give an overall outline of the study, a numerical example is presented. The results clearly indicate that the distance is a key parameter and has considerable effects on the induced stress values. In addition, the interface permeability condition and soil characteristics play an important role in determining the dynamic response of the borehole. Finally, the obtained results are compared with the relevant analyses existing in the literature for some limit cases, and good agreement is achieved.

Key words: point source, cylindrical cavity, porous medium, wave-field transformation, stress concentration

2010 MSC Number:

O353.4

H. HOSSEINI, O. BALILASHAKI. Dynamic stress concentration in an infinitely long cylindrical cavity due to a point spherical source embedded within a fluid-saturated poroelastic formation of infinite extent. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 139-156.

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Figures/Tables 10

Fig. 1

Table 1

Biot's input parameter values"

Parameter	Ridgefield sandstone (SI)	Stiff soil (SI)	Soft soil (SI)
$ϕ 0$	0.37	0.33	0.44
$α$	1.58	2.015	1.63
$κ$ /m²	$2.77 × 10 − 11$	$5 × 10 − 12$	$1.1 × 10 − 10$
$μ$ /(N·m^-2)	$2.89 × 1010$	$(4.55 − 0.227 i) × 109$	$6 × 106$
$K fl$ /(N·m^-2)	$2.55 × 109$	$2.223 × 109$	$2.55 × 109$
$Λ$ /m	$1.94 × 10 − 5$	$9 × 10 − 8$	$5.7 × 10 − 5$
$ρ s$ /(kg·m^-3)	2 500	2 650	2 650
$K s$ /(N·m^-2)	$4.99 × 1010$	$3.6 × 1010$	$3.5 × 1010$
$K 0$ /(N·m^-2)	$5.24 × 109$	$(6.21 − 0.207 i) × 109$	$1.25 × 107$
$ρ fl$ /(kg·m^-3)	1 000	1 000	1 000
$η$ /(kg·m^-1·s^-1)	0.001	0.001	0.001

Table 1

Fig. 2

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Fig. 9

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