Keywords

1 Introduction

Currently, mathematical modeling is one of the main tools to analyze and optimize oil and gas fields development, to solve the problems of seismic construction and bioengineering [1,2,3,4,5,6,7]. The model of poroelastic medium, allowing to describe fluid filtration in pores in together with a full-scale mechanical model of the stress-strain state of medium is usually used to describe the “solid”–”fluid” system.

Modern forms of these models were introduced by Biot [8]. Biot’s model correctly describes processes of elastic porous medium deformation and fluid flow in that medium. It is assumed that the space containing poroelastic medium is filled with a two-phase material, and one phase corresponds to the elastic skeleton, and the another one to the fluid in pores. Both phases are present at each point of the physical space, and the phase distribution in space is described by macroscopic quantities such as porosity.

Biot’s model allows us to solve a number of particular problems, among which the problems of wave propagation in homogeneous and layered poroelastic half-spaces are of particular interest [9,10,11,12,13,14,15,16,17]. However, the increasing complexity of computational schemes for boundary value problems requires the involvement of advanced methods such as boundary elements method (BEM). Possessing high accuracy and rigor of the approach, BEM is the most suitable method for considering nonstationary processes in semi-infinite bodies, since it ensures automatic fulfillment of the conditions for solution behavior at infinity. Despite the noted advantages of BEM, the possibility of modeling the poroelastic dynamics is mainly determined by the presence of the corresponding boundary integral equations (BIE) and fundamental solutions. Fundamental solutions and BIE of the dynamic theory of poroelasticity were obtained in [18,19,20,21]; in [22,23,24,25] different variants of boundary element schemes for the solution of problems in porodynamics are presented and results of numerical experiments are provided. Results of boundary element modeling of dynamics of poroelastic halfspaces are presented in [26,27,28,29,30].

The principal difference of poroelastic formulation from elastic and viscoelastic is that it allows to take into account the influence exerted by the fluid moving in pores on the behavior of the medium as a whole. Historically, on the basis of Biot’s theory, the existence of two longitudinal waves—fast and slow in porous medium was predicted. The slow longitudinal wave is caused by the movement of fluid particles of the pores relative to the porous skeleton and is peculiar only to porous media. In addition, the frictional interaction of viscous fluid and a skeleton leads to a significant dissipation of energy in the medium, which demonstrates viscoelastic behavior [31, 32]. The viscoelastic behavior of poroelastic medium can also be due to the viscoelastic properties of the skeleton [33,34,35]. Some results of simulation of wave processes in poroelastic solids with the use of BIE, BEM and various models of viscoelastic behavior of the skeleton are presented in [36,37,38].

The paper presents the modeling of wave propagation in homogeneous and inhomogeneous poroviscoelastic in semi-infinite bodies using time-step BEM scheme. A poroviscoelastic layer on a semi-elastic halfspace is considered as an implementation of the inhomogeneity model.

2 Problem Formulation

Basic poroelastic material is a two-phase material consisting of an elastic skeleton and compressible fluid or gas filler. Porous material of a volume V can be constructed as follows:

$$ V = V^{f} + V^{s} $$
(1)

where V is the total volume, \( V^{f} \) is the summary pore volume and \( V^{s} \) is the volume of the skeleton. It is assumed that filler can openly seep through the pores and all closed pores are assumed as a part of the skeleton. Then a correspondence principle is applied to the skeleton, so we extend poroelastic formulation to poroviscoelasticity.

Considering a boundary-value problem for Biot’s model of fully saturated poroelastic continuum in Laplace domain in terms of four unknowns (displacements \( \bar{u}_{i} \) and pore pressure \( \bar{p} \)) the set of differential equations take the following form [30]:

$$ \begin{aligned} G\bar{u}_{i,jj} & + \left( {K + \frac{G}{3}} \right)\bar{u}_{j,ij} - (\psi - \beta )\bar{p}_{,i} - s^{2} (\rho - \beta \rho_{f} )\bar{u}_{i} = - \bar{F}_{i} , \\ \frac{\beta }{{s\rho_{f} }}\bar{p}_{,ii} & - \frac{{\phi^{2} s}}{R}\bar{p} - (\psi - \beta )s\bar{u}_{i,i} = - \bar{a},\quad x \in\Omega, \\ \end{aligned} $$
(2)

Boundary conditions:

$$ \begin{aligned} \bar{u}(x,s) & = f(x,s),\quad x \in\Gamma ^{u} ,\quad \bar{u} = \left( {\bar{u}_{1} ,\bar{u}_{2} ,\bar{u}_{3} ,\bar{p}} \right), \\ \bar{t}(x,s) & = g(x,s),\quad x \in\Gamma ^{\sigma } ,\quad \bar{t} = \left( {\bar{t}_{1} ,\bar{t}_{2} ,\bar{t}_{3} ,\bar{q}} \right), \\ \end{aligned} $$

where \( \Gamma ^{u} \) and \( \Gamma ^{\sigma } \) denotes boundaries for boundary conditions of 1st and 2nd kind respectively, \( G,\;K \) are elastic moduli, \( \phi = V^{f} /V \) is porosity, \( \bar{F}_{i} ,\;\bar{a} \) are bulk body forces.

$$ \beta = \frac{{\kappa \rho_{f} \phi^{2} s}}{{\phi^{2} + s\kappa (\rho_{a} + \phi \rho_{f} )}},\psi = 1 - \frac{K}{{K_{s} }}\text{and}\,R = \frac{{\phi^{2} K_{f} K_{s}^{2} }}{{K_{f} (K_{s} - K) + \phi K_{s} (K_{s} - K_{f} )}} $$

are constants reflecting interaction between skeleton and filler, \( \kappa \) is permeability. Further, \( \rho = \rho_{s} (1 - \phi ) + \phi \rho_{f} \) is a bulk density, \( \rho_{s} ,\;\rho_{a} ,\;\rho_{f} \) are solid, apparent mass density and filler density respectively, \( K_{s} ,K_{f} \) are elastic bulk moduli of the skeleton and filler respectively. Apparent mass density \( \rho_{a} = C\phi \rho_{f} \) was introduced by Biot to describe dynamic interaction between fluid and skeleton. \( C \) is a factor depending on the pores geometry and excitation frequency.

The governing equation system (2) in matrix form can be written as follows:

$$ B\bar{u} = F,\quad \bar{u}^{\rm T} = (\bar{u}_{i} ,p),i = 1,3 $$
$$ {\rm B} = \left[ {\begin{array}{*{20}c} {G\nabla^{2} + \left( {K + \frac{1}{3}G} \right)\partial_{i} \partial_{j} - s^{2} (\rho - \beta \rho_{f} )} & { - (\psi - \beta )\partial_{i} } \\ { - s(\psi - \beta )\partial_{j} } & {\frac{\beta }{{s\rho_{f} }}\nabla^{2} - \frac{{\phi^{2} s}}{R}} \\ \end{array} } \right]. $$

Boundary conditions are:

$$ u(x,s) = f(x,s)\,\text{on}\,\Gamma ^{u} ,t_{n} (x,s) = g(x,s)\,\text{on}\,\Gamma ^{\sigma } . $$

In present paper we consider piecewise homogeneous solid \( \Omega \) in Euclidian space \( R^{3} \) with coordinate system \( Ox_{1} x_{2} x_{3} \). Solid \( \Omega \) is enclosed with boundary denotes as \( \Gamma \), boundaries of \( \Omega _{k} \, (k = 1, \ldots ,K) \) are denoted as \( \Gamma _{k} \). Each part \( \Omega _{k} \) is assumed to be isotropic. Material parameters of each \( \Omega _{k} \) are denotes by upper index « k » So, governing equations for each part \( \Omega _{k} \) in matrix form take a following form:

$$ B^{k} \bar{u}^{k} = 0,\quad \bar{u}^{k} = (\bar{u}_{{^{i} }}^{k} ,p^{k} ),\,i = 1,3 $$
$$ \left[ {\begin{array}{*{20}c} {G^{k} \nabla^{2} + \left( {K^{k} + \frac{1}{3}G^{k} } \right)\partial_{i} \partial_{j} - s^{2} (\rho^{k} - \beta^{k} \rho_{f}^{k} )} & { - (\psi^{k} - \beta^{k} )\partial_{i} } \\ { - s(\psi^{k} - \beta^{k} )\partial_{j} } & {\frac{{\beta^{k} }}{{s\rho_{f}^{k} }}\nabla^{2} - \frac{{\phi^{k2} s}}{{R^{2} }}} \\ \end{array} } \right] $$

where \( \bar{u}^{k} (x,s) \)—generalized displacements vector at point \( x = (x_{1} ,x_{2} ,x_{3} ) \). Assumed that \( u^{k} (x,t) \) fulfill zero initial condition:

$$ u^{k} (x,\,0) = u^{k} (x,\,0) = 0 $$

Following boundary conditions are employed for each \( \Omega _{k} \):

$$ \begin{array}{*{20}c} {u_{l}^{k} (x,t) = f_{l}^{k} (x,t), \, x \in\Gamma ^{u} \cap\Gamma _{k} { ,}\quad l = \overline{ 1 , 3} ;} \\ {t_{l}^{k} (x,t) = g_{l}^{k} (x,t),\quad x \in\Gamma ^{\sigma } \cap\Gamma _{k} ;} \\ {u_{l}^{k} (x,t) = u_{l}^{m} (x,t), \, t_{l}^{k} (x,t) = - t_{l}^{m} (x,t), \, x \in {\varGamma^{\prime}}_{km} \, .} \\ \end{array} $$

Here, \( \Gamma ^{u} \) and \( \Gamma ^{\sigma } \) are parts of boundary \( \Gamma \) of body \( \Omega \), along which displacements and surface tractions, respectively, are assigned; \( {\varGamma^{\prime}}_{ks} \) is the contact boundary of parts \( \Omega _{k} \) and \( \Omega _{s} \). Functions \( f_{l}^{k} (x,t) \) and \( g_{l}^{k} (x,t) \) are assigned functions of the coordinates and time.

Poroviscoelastic solution is obtained from poroelastic solution by means of the elastic-viscoelastic correspondence principle, applied to skeleton’s moduli K and G in Laplace domain. Forms of functions \( \bar{K}(s) \) and \( \bar{G}(s) \) depend on chosen viscoelastic model.

In present paper, standard linear solid model is employed:

$$ \begin{aligned} \bar{K}(s) & = K^{\infty } \cdot \left[ {(\chi - 1)\frac{s}{s + \eta } + 1} \right], \\ \bar{G}(s) & = G^{\infty } \cdot \left[ {(\chi - 1)\frac{s}{s + \eta } + 1} \right] \\ \end{aligned} $$

The equilibrium and instantaneous values of the relaxation function associated with material modules are connected as follows:

$$ \chi = K^{0} /K^{\infty } = G^{0} /G^{\infty } $$

Equilibrium and instantaneous values are denoted by «\( \infty \)» and «0» respectively.

3 Boundary-Element Approach

Boundary-value problem can be reduced to the BIE system as follows [26,27,28,29,30]:

$$ \frac{{1 - \alpha_{\Omega } }}{2}\upsilon_{i} ({\mathbf{x}},s) + \int\limits_{\Gamma } {\left( {T_{ij} ({\mathbf{x}},{\mathbf{y}},s)\upsilon_{i} ({\mathbf{y}},s) - T_{ik}^{0} ({\mathbf{x}},{\mathbf{y}},s)\upsilon_{i} ({\mathbf{x}},s) - U_{ij} ({\mathbf{x}},{\mathbf{y}},s)t_{i} ({\mathbf{y}},s)} \right)d\Gamma = 0} , $$

where \( {\mathbf{x}},{\mathbf{y}} \in\Gamma \) \( U_{ij}^{{}} ,\;T_{ij}^{{}} \) are fundamental and singular solutions, \( T_{ij}^{0} \) contains the isolated singularities, \( {\mathbf{x}} \in\Gamma \) is an arbitrary point. Coefficient \( \alpha_{\Omega } \) equals to 1 in case of finite domain and −1 in case of infinite domain.

Boundary surface of our homogeneous solid is discretized by quadrangular and triangular elements and triangular elements are assumed as singular quadrangular elements. The Cartesian coordinates of an arbitrary point of the element are expressed through the coordinates of the nodal points of this element, using shape functions of the local coordinates. Shape functions are quadratic polynomials of interpolation. We use reference elements: square \( \xi = (\xi_{1} ,\xi_{2} ) \in \left[ { - 1, \, 1} \right]^{2} \) and triangle \( 0 \le \xi_{1} + \xi_{2} \le 1, \) \( \xi_{1} \ge 0,\xi_{2} \ge 0 \), and each boundary element is mapped to a reference one by the following formula:

$$ y_{i} (\xi ) = \sum\limits_{l = 1}^{8} {N^{l} (\xi )y_{i}^{\beta (k,l)} } ,\quad \quad i = 1,\,2,\,3, $$

where l is local node number in element k, \( \beta (k,l) \) is global node number, \( N^{l} (\xi ) \) are shape functions. Goldshteyn’s displacement-stress mixed model is performed. To discretize the boundary surface eight-node biquadratic quadrilateral elements are used, generalized displacements and tractions are approximated by linear and constant shape functions, respectively.

Subsequent application of collocation method leads to the system of linear equations. As with the collocation nodes we take the approximation nodes of boundary functions. Gaussian quadrature are used to calculate integrals on regular elements. However, if an element contains a singularity, algorithm of singularity avoiding or order reducing is applied. When singularity is excluded we use an adaptive integration algorithm. An appropriate order of Gaussian quadrature is chosen from primarily known necessary precision, if it is impossible, the element is subdivided to smaller elements recursively.

Solving the system of linear equations leads to the solution of the initial boundary-value problem in Laplace domain.

4 Laplace Transform Inversion

Consider a method based on the theorem of the integration of the original—the stepped method of numerical inversion of Laplace transform. Consider the following integral:

$$ y(t) = \int\limits_{0}^{t} {f(\tau )d\tau } . $$
(3)

Integral (3) gives rise to Cauchy problem for an ordinary differential equation:

$$ \frac{d}{dt}x(t) = sx(t) + C,x(0) = 0. $$

Integral (3) is substituted for by a quadrature sum, weighting factors of which are determined using Laplace representation \( \overline{f} \) and the linear multi-step method [39]. Further derivation is based on the results of those works. The traditional stepped method of integrating the original consists in that integral (3) is calculated using the following relation:

$$ \begin{array}{*{20}c} {y(0) = 0,\quad y(n\Delta t) = \sum\limits_{k = 1}^{n} {\omega_{k} (\Delta t),} \quad n = 1, \ldots N,} \\ {\omega_{n} (\Delta t) = \frac{{R^{ - n} }}{2\pi }\int\limits_{0}^{2\pi } {\bar{f}\left( {\frac{{\gamma \left( {Re^{i\,\varphi } } \right)}}{{\Delta t}}} \right)e^{ - in\,\varphi } d\varphi } } \\ \end{array} $$

where \( \Delta t \) is time step; \( \gamma (z) = {3 \mathord{\left/ {\vphantom {3 {2 - 2z + {{z^{2} } \mathord{\left/ {\vphantom {{z^{2} } {2.}}} \right. \kern-0pt} {2.}}}}} \right. \kern-0pt} {2 - 2z + {{z^{2} } \mathord{\left/ {\vphantom {{z^{2} } {2.}}} \right. \kern-0pt} {2.}}}} \); п is number of a time step, \( n = 0,N \); R is parameter of the method.

The traditional method uses a constant-step trapezoid method for integrating. Consider the following formula of constructing \( \omega_{n} \) based on a variable step:

$$ \omega_{n} (\Delta t) = \frac{{R^{ - n} }}{2\pi }\sum\limits_{k = 0}^{L - 1} {\left[ {\bar{f}\left( {\frac{{\gamma (Re^{{in\varphi_{k} }} )}}{\Delta t}} \right)e^{{ - in\varphi_{k} }} + \bar{f}\left( {\frac{{\gamma (Re^{{in\varphi_{k + 1} }} )}}{\Delta t}} \right)e^{{ - in\varphi_{k} + 1}} } \right]\frac{{\left( {\varphi_{k + 1} - \varphi_{k} } \right)}}{2}.} $$

5 Numerical Example

The problem of the Heaviside-type load \( H\left( t \right) \) acting on the surface of a poroelastic layer located on a poroelastic halfspace is considered (Fig. 1). Two variants of geometry problem are considered—with a layer thickness of 5 and 10 m. A vertical load \( t_{3} = - 1000\;{\text{N/m}}^{ 2} \cdot \,H\left( t \right) \) is specified on a surface area of 1 m2, the rest of the surface is traction-free and permeable. At the boundary between the layer and halfspace, the flow, the force, displacements and pore pressure are assumed to be unknown functions. The parameters of the poroelastic soil and the rock are given in Table 1. Moduli characterizing the elastic behavior of the porous material in accordance with the drained and undrained models are also given in Table 1. Dynamic responses of vertical displacements \( u_{3} \) at the point P, located at a distance of 10 m from the area of load application are shown in Figs. 2, 3, 4, 5, 6, 7, 8, 9 and 10.

Fig. 1
figure 1

Layered half space under vertical load

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Table 1 Poroelastic constants for various materials
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Fig. 2
figure 2

Displacement at point P versus time: different depth of the layer

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Fig. 3
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Vertical displacement at point P versus time. Comparison of poroelastic and elastic solutions of the soil and rock for layer of 5 m depth

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Fig. 4
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Vertical displacement at point P versus time. Comparison poroelastic and elastic solutions of the soil and rock for layer of 10 m depth

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Fig. 5
figure 5

Vertical displacement at point P versus time. Poroviscoelastic dynamic analysis of the soil and rock for layer of 10 m depth

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Fig. 6
figure 6

Vertical displacement at point P versus time. Poroviscoelastic dynamic analysis of the soil and rock for layer of 5 m depth

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Fig. 7
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Vertical displacement at point P versus time. Poroviscoelastic dynamic analysis of the rock for layer of 10 m depth

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Fig. 8
figure 8

Vertical displacement at point P versus time. Poroviscoelastic dynamic analysis of the rock for layer of 5 m depth

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Fig. 9
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Vertical displacement at point P versus time. Poroviscoelastic dynamic analysis of the soil for layer of 10 m depth

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Fig. 10
figure 10

Vertical displacement at point P versus time. Poroviscoelastic dynamic analysis of the soil for layer of 5 m depth

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Figure 2 presents the comparison of vertical displacements, calculated for the layer with thicknesses of 5 m и 10 m and labeled \( u_{3}^{{5\;{\text{m}}}} \) and \( u_{ 3}^{{ 1 0\,{\text{m}}}} \), respectively. Figure 2 also represents solutions for displacements \( \left( {u_{3}^{hs} } \right) \), calculated for the same values of the halfspace material parameters, which describe layer material. It can be seen that until the appearance of the fast longitudinal wave \( \left( {t \approx 0.01\,{\text{s}}} \right) \) all three curves are almost graphically indistinguishable, however, quantitative differences are observed further. In the moment of Rayleigh wave \( \left( {t \approx 0.047\,{\text{s}}} \right) \) appearance and up to the moment \( t \approx 0.057\,{\text{s}} \) the displacement behavior is described by the relation \( u_{3}^{{5\;{\text{m}}}} \le u_{3}^{hs} \le u_{3}^{{10\;{\text{m}}}} \). The time moment \( t \approx 0.057\,{\text{s}} \) is marked by an increase in displacements \( u_{3}^{{5\;{\text{m}}}} \) and \( u_{3}^{{10\;{\text{m}}}} \), and the displacements \( u_{3}^{{5\;{\text{m}}}} \) increase much more rapidly and exceed \( u_{3}^{{10\;{\text{m}}}} \) and \( u_{3}^{hs} \) by amplitude. The observed effect is explained by the influence of longitudinal waves reflected from the halfspace and is less noticeable in the case of a larger layer thickness due to the considerable wave dispersion in porous medium.

Figures 3 and 4 show the comparison of displacements \( u_{3}^{{5\;{\text{m}}}} \) and \( u_{3}^{{10\;{\text{m}}}} \) ith the displacements calculated using elastic models, respectively. In both cases, the solution obtained from the undrained material model is the best approximation to the poroelastic solution, but the differences are also observed here due to the influence of the waves reflected from the halfspace. Figure 4 shows that the amplitude of the Rayleigh wave of the poroelastic solution is smaller than the corresponding amplitudes of the elastic solutions.

In order to obtain poroviscoelastic solutions standard linear solid model is employed. Parameter \( \eta \) characterize viscosity, and parameter \( \chi = K^{0} /K^{\infty } = G^{0} /G^{\infty } \) characterize dependence between equilibrium and instantaneous values of material modules. In further computations parameter \( \chi = 4 \).

Figures 5, 6, 7, 8, 9 and 10 demonstrate the influence of viscoelastic properties of the skeleton on the dynamic response of vertical displacements. Figures 5 and 6 represent the vertical displacements on the surface of a viscoelastic layer located on a poroelastic halfspace. With increase of the parameter \( \eta \), it is seen that viscoelastic solution approaches poroelastic solution, but decrease of \( \eta \) leads to reduce of displacement amplitude. The influence of dissipative effects on the surface displacements of the five-meter layer is especially noticeably at \( \eta = 10^{3} \) when the amplitude of the Rayleigh wave displacement does not exceed the amplitude of the longitudinal wave displacements (Fig. 6).

Figures 7 and 8 represent vertical displacements on the surface of poroelastic layer located on a porous viscoelastic halfspace. It can be observed that the dissipative effects associated with viscoelastic behavior of the halfspace have minor effect on the dynamic responses of displacements on the surface of both the 5 and 10 m layers.

Figures 9 and 10 show vertical displacements on the surface of a viscoelastic layer located on the poroelastic halfspace. In this case, the displacement curves almost coincide with the displacement curves presented in Figs. 5 and 6, and the corresponding comments can be repeated. Taking into account the comments made regarding the results in Figs. 7 and 8, we can say that the dynamic responses of the surface displacements of the layer are mainly affected by the viscoelastic properties of the layer.

6 Conclusion

The Biot’s mathematical model of poroelastic material is given in the present paper. The systems of equations of the theory of dynamic poroelasticity and the formulation of boundary value problems in Laplace transforms are formulated. The technique of numerical inversion of the Laplace transform based on step method is presented. The boundary-element solutions for a problem involving a vertical load acting on the surface of a poroelastic layer located on a poroelastic halfspace are presented. The effect of the viscoelastic properties of the skeleton of porous material on vertical displacements on the surface layer is studied. It is noted, that the form of wave patterns is mainly influenced by dissipative effects caused by the viscoelastic behavior of the layer.