1 Introduction

Internal erosion is a phenomenon of progressive degradation of soils induced by hydrodynamic forces due to water flow. It can lead to serious hazards such as damage of dams and other geotechnical structures, and in extreme cases, might trigger off failure with serious and adverse consequences. The process of erosion can conceptually be divided into two phases. In the first phase, fine particles are carried away by water flow, leading to increased porosity, permeability and loss of mechanical properties. In the second phase, this process might eventually lead to the formation of macroscopically observable channels (piping) in which water flow takes place. While internal erosion resulting in piping is ultimately a discontinuous process, it can be simulated by a continuous approach in the first phase, which is the approach undertaken in the present paper. This approach is consistent with that of Vardoulakis et al. (1996, 2001) and Papamichos and Vardoulakis (2005), in modeling the erosion of sandstone during hydrocarbon production and is based on the continuum theory of mixtures. In their work, the rock medium is considered as a three phase porous material: solid grain, fluid and fluidized particles. The governing equations were derived from mass-balance considerations, an erosion law and the flow in the host rock described by Brinkman’s extension of Darcy’s law which was adopted to account for a smooth transition between channel and Darcian flows. The coupling has been introduced via a source term in the mass balance equations to describe the process of detachment and exchange of sand particles through erosion. The erosion constitutive law, related to the rate of eroded mass produced, was proposed according to the physical mechanisms of sand production observed in experiments. Numerical models were then developed in finite element/finite difference framework to solve the governing equations to deal with sand production processes due to radial flow as well as axial flow in the well bore of petroleum industry. It is also noted that these models have been mainly developed and applied for permeable soils.

The approach described above is different from that of Bonelli et al. (2006), Lachouette et al. (2008), and Golay et al. (2010, 2011), whose work deals with singular (or discontinuous) rather than diffused fluid/soil interfaces. In the latter approach, balance equations with jump conditions are used. The work of these authors is aimed at dealing with the progression of piping erosion process in which a continuous pipe generated in a cohesive soil is enlarged by a tangential flow of water. None of the structure of the erosion constitutive laws has been examined from perspective of irreversible thermodynamics, and this fact has been acknowledged by Bonelli et al. (2006).

In contrast to previous communications, the present work is based on rigorous thermodynamic principles (Wong et al. 2011), and highlights the fact that not only the hydrodynamic forces, but also the state of the soil skeleton, affect the erosion kinetics. The construction of the internal erosion law proceeds on the basis of the shear stress developed at the solid–fluid interface, inspired by the previous work of Reddi et al. (2000). It is then applied to analyse ground water seepage beneath a dam foundation to simulate the progressive appearance of a highly eroded zone, which precedes the formation of macroscopically visible flow channels.

2 Soil Components and Volume Fractions

The model is formulated directly at the macroscopic scale of observation, based on the mixture theory. In this approach, the soil is considered as a superposition of three continua: the solid skeleton ‘s’, the pore water ‘w’ and the mobilized solid particles in water ‘c’. Each of them is assumed to occupy the same space at the same time. Since we are more concerned with the behavior of the solid skeleton, the kinematics of the latter is privileged such that a Lagrangian description is used for the solid phase and an Eulerian description is used for the two others (Uzuoka and Borja 2011). In the sequel, index a will be used to denote a generic species whereas index β will be used to denote a species in the fluid phase. In other words:

$$ a \in \{ s,c,w\} ;\quad \beta \in \{ c,w\} $$
(1)

In order to treat correctly the momentum balance, we need to introduce the notion of volume fractions. Consider an infinitesimal soil element of initial volume dΩ0 and current volume dΩ t . The latter is the sum of individual volumes occupied by each of the three species: dΩ t  = dΩ s t  + dΩ w t  + dΩ c t . Figure 1 shows the relative volume fractions of a typical soil element at an arbitrary time t, as well as at the initial instant t = 0. The Jacobian defined by J = dΩ t /dΩ0. The current pore volume saturated by the pore fluid dΩ f t is expressed as ϕ dΩ0 or ndΩ t where ϕ = dΩ f t /dΩ0 (resp. n = dΩ f t /dΩ t ) denotes the Lagrangian (resp. Eulerian) porosity, with the obvious relation ϕ = nJ. In consequence, the current volume of the solid phase dΩ s t can either be written as (J − ϕ)dΩ0 or (1 − n)dΩ t . Due to erosion, a part of the initial volume of solid skeleton is transformed into mobilized soil particles in the fluid phase. ϕ er dΩ0 denotes the actual volume of these solid particles.

Fig. 1
figure 1

Definition of initial and current volume fractions

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To simplify the model, all species are assumed incompressible, with constant density. In particular, both the solid matrix and the fluidized particles possess the same intrinsic density: ρ s  = ρ c . The macroscopic description necessitates the introduction of the concept of mass contents m a as well as to distinguish apparent densities ρ a from intrinsic densities ρ a . In all the following, m a dΩ0 = ρ a dΩ t denotes the actual mass of component ‘a’ inside the current volume dΩ t , hence m a  = Jρ a. Moreover, c = dΩ c t /(dΩ w t  + dΩ c t ) and (1 − c) = dΩ w t /(dΩ w t  + dΩ c t ) denote respectively the volume fraction of mobilized solid particles and pore-water in the fluid phase. Using the above notation, it can easily be deduced that:

$$ \rho^{s} = (1 - n)\rho_{s} ;\quad \rho^{w} = n(1 - c)\rho_{w} ;\quad \rho^{c} = nc\rho_{s} $$
(2)
$$ m_{a} = J\rho^{a} ;\quad m_{s} = (J - \phi )\rho_{s} ;\quad m_{c} = \phi c\rho_{s} ;\quad m_{w} = \phi (1 - c)\rho_{w} $$
(3)

We will denote by m t the total mass content, in other words m t  = m s  + m c  + m w . From Fig. 1, it is readily seen that:

$$ \phi = \phi_{0} + \varepsilon_{v} + \phi_{er} $$
(3a)

3 Fundamental Conservation Equations

3.1 Mass Conservation Equations

The mass balance for a generic phase a in the current state can be written as:

$$ \frac{{d^{a} \rho^{a} }}{dt} + \rho^{a} div\varvec{v}_{a} = \hat{m}_{a} $$
(4)

where v a is the velocity of phase ‘a’ and \( \hat{m}_{a} d\Upomega_{0} dt \) denotes the rate of mass increase of species ‘a’ in volume element dΩ t within time interval dt resulting from phase change due to erosion. Recall that the material derivative of a quantity A, following the motion of a particle of species ‘a’ is defined by (Coussy 2004):

$$ \frac{{d^{a} A}}{dt} = \frac{\partial A}{\partial t} + {\mathbf{grad}}\varvec{A} \cdot \varvec{v}^{a} $$
(5)

It is however more useful to express the material derivatives from the point of view of the solid skeleton. Using the previously derived quantities and relations, it can be shown that the above can be recast into the following form (Pereira 2005):

$$ \dot{m}_{a} + Jdiv\varvec{w}_{a} = \hat{m} \quad {\text{with}} \quad \dot{A} = \frac{{d^{s} A}}{dt} $$
(6)
$$ \varvec{w}_{a} = \rho^{a} \varvec{v}_{ar} ;\quad \varvec{v}_{ar} = (\varvec{v}_{a} - \varvec{v}_{s} ) $$
(7)

The notation \( \dot{A} \) is more convenient and has the advantage of flexibility as will become evident later. w a and v ar are respectively the mass flux and the velocity of species ‘a’ relative to the solid skeleton ‘s’. Since erosion only involves mass exchange between species ‘s’ and ‘c’, it is easy to see that we have:

$$ \hat{m}_{w} = 0;\quad \varvec{w}_{s} = \varvec{v}_{sr} = 0;\quad \hat{m}_{s} + \hat{m}_{c} = 0 $$
(8)

Note that the \( \hat{m}_{s} \) is by definition the rate of mass loss per unit initial volume. It can easily be deduced from Fig. 1:

$$ \hat{m}_{s} = - \hat{m}_{c} = - \rho_{s} \dot{\phi }_{er} $$
(8a)

3.2 Momentum balance equations

We consider a given volume Ω t in the current configuration bounded by an external surface \( \partial \Upomega_{t} \). The balance of linear momentum writes:

$$ \sum\limits_{a} {\frac{{d^{a} }}{dt}} \int\limits_{{\Upomega_{t} }} {\rho^{a} \varvec{v}_{a} d\Upomega_{t} } = \sum\limits_{a} {\int\limits_{{\Upomega_{t} }} {\rho^{a} \varvec{g}d\Upomega_{t} } + } \sum\limits_{a} {\int\limits_{{\partial \Upomega_{t} }} {{\varvec{\sigma}} \cdot \varvec{n}dS_{t} } } $$
(9)

The left hand side of the above is the total rate of change of linear momentum of the entire mass while the right hand side represents the total instantaneous external applied forces, σ being the total stress tensor and n the external unit normal vector. Applying the classic Gauss theorem and that of material derivatives of volume integrals (Coussy 2004), and using the mass conservation equation previously derived, we get:

$$ div{\varvec{\sigma}} + \sum\limits_{a} {\left( {\rho^{a} ({\mathbf{g}} - {\varvec{\gamma}}_{a} ) - \frac{1}{J}\hat{m}_{a} \varvec{v}_{a} } \right)} = 0 $$
(10)

We shall assume that the total stress can be decomposed as the sum of partial stresses (Pereira 2005):

$$ {\varvec{\sigma}} = {\varvec{\sigma}}^{s} + {\varvec{\sigma}}^{w} + {\varvec{\sigma}}^{c} $$
(11)

To simplify, we will assume that the stresses of the liquid species are hydrostatic:

$$ {\varvec{\sigma}}^{\beta } = - n_{\beta } P_{\beta } {\mathbf{I}};\quad \beta \in \{ c,w\} $$
(12)

where n β and P β are the current volume fraction and pressure of ‘β’ species, expressed by:

$$ n_{c} = nc;\quad n_{w} = n(1 - c) $$
(13)

Since the fluidized particles are completely surrounded by pore water, it is reasonable to suppose that they share the same hydrostatic stress. To simplify further, we will suppose that they also share the same velocity field. Summarizing:

$$ P_{c} = P_{w} ;\quad \varvec{v}_{c} = \varvec{v}_{w} $$
(14)

In order to apply a consistent notation, we will denote these quantities by P f and v f .

4 Thermodynamic Principles and the Fundamental Inequality

To derive the fundamental dissipation inequality, which is the starting point of our modeling, we consider again a generic volume Ω t in the current configuration (Fig. 2).

Fig. 2
figure 2

Current configuration considered to derive the fundamental dissipation inequality. Ω t is a generic part of a body in movement, with distributed body force f, surface traction T = σ·n, heat flux q, and distributed heat source R

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The first law of thermodynamics stipulates that energy must be globally conserved. In the present context, we can write:

$$ \frac{D}{Dt}(K + E) = P_{ex} + Q $$
(15)

The left hand side is the rate of increase of total energy of the entire mass, with K and E being respectively the kinetic and internal energies. The right hand side is simply the external supply of mechanical power P ex and thermal power Q to matters inside the current volume Ω t . The four quantities K, E, P ex , and Q can be expressed in terms of local variables:

$$ \begin{gathered} K = \int\limits_{{\Upomega_{t} }} {\sum\limits_{a} {\frac{1}{2}\rho^{a} \varvec{v}_{\varvec{a}}^{2} d} \Upomega_{t} } ;\quad E = \int\limits_{{\Upomega_{t} }} {\sum\limits_{a} {\frac{1}{2}\rho^{a} e_{m}^{a} d\Upomega_{t} } } ; \hfill \\ P_{ex} = \sum\limits_{a} {\left( {\int\limits_{{\Upomega_{t} }} {\rho^{a} \varvec{v}_{a} \cdot {\mathbf{g}}d\Upomega_{t} } + \int\limits_{{\partial \Upomega_{t} }} {\varvec{n} \cdot {\varvec{\sigma}}^{a} \cdot \varvec{v}_{a} dS_{t} } } \right)} ;\quad Q = \int\limits_{{\Upomega_{t} }} {Rd\Upomega_{t} } - \int\limits_{{\partial \Upomega_{t} }} {\varvec{q} \cdot \varvec{n}dS_{t} } \hfill \\ \end{gathered} $$
(16)

where e a m is the specific internal energy per unit mass of species ‘a’, R the volumetric heat source and q the heat flux. Again, simplification using classic theorems of divergence and material derivatives lead to the partial differential equation:

$$ \begin{aligned} & {\varvec{\sigma}} :\varvec{d} + \sum\limits_{\beta } {\sigma^{\beta } :{\mathbf{grad}}(\varvec{v}_{\beta r} )} + \sum\limits_{a} {(div{\varvec{\sigma}}^{\beta } + \rho^{\beta } ({\mathbf{g}} - \gamma^{\beta } ))} \cdot \varvec{v}_{\beta r} \\ & - \left( {\frac{\partial e}{\partial t} + div(e\varvec{v}_{s} ) + \sum\limits_{\beta } {div\left( {e_{m}^{\beta } \varvec{w}_{\beta } } \right)} } \right) - \frac{1}{2J}\hat{m}_{c} \varvec{v}_{cr}^{2} + R - div\varvec{q} = 0 \\ \end{aligned} $$
(17)

where d is the strain rate tensor, e is the total internal energy per unit current volume: \( e = \sum\nolimits_{a} {\rho^{a} e_{m}^{a} }. \) The second law of thermodynamics states that the entropy increase of an isolated system cannot be larger than the external heat supply divided by the absolute temperature:

$$ \frac{D}{Dt}\int\limits_{{\Upomega_{t} }} {sd\Upomega_{t} } \ge \int\limits_{{\Upomega_{t} }} {\frac{R}{T}d\Upomega_{t} } - \int\limits_{{\partial \Upomega_{t} }} {\frac{1}{T}\varvec{q} \cdot \varvec{n}dS_{t} } $$
(18)

where \( s = \sum\nolimits_{a} {\rho^{a} s_{m}^{a} } \) is the entropy per unit overall volume and s a m the entropy per unit mass of species ‘a’. Using the material derivative as in Eq. (5) and Gauss’ Theorem, we obtain the local form of the second law:

$$ T\left( {\frac{\partial s}{\partial t} + div(s\varvec{v}_{s} ) + div\left( {s_{m}^{a} \varvec{w}_{a} } \right)} \right) - R + div\varvec{q} - \frac{1}{T}\varvec{q} \cdot {\mathbf{grad}}T \ge 0 $$
(19)

Introducing the specific free energy:

$$ \psi = e - Ts $$
(20)

Then combining (17), (19) and (20) leads after simplification to:

$$ \begin{aligned} & {\varvec{\sigma}} :\varvec{d} - s\frac{{d^{s} T}}{dt} - \frac{{d^{s} \psi }}{dt} - \psi div\varvec{v}_{s} + \sum\limits_{\beta } {{\varvec{\sigma}}^{\beta } } :{\mathbf{grad}}\varvec{v}_{\beta r} + \sum\limits_{\beta } {(div{\varvec{\sigma}}^{\beta } + \rho^{\beta } ({\mathbf{g}} - {\varvec{\gamma}}^{\beta } )) \cdot \varvec{v}_{\beta r} } \\ & \quad + T\sum\limits_{\beta } {div\left( {s_{m}^{\beta } \varvec{w}_{\beta } } \right) - div\left( {e_{m}^{\beta } \varvec{w}_{\beta } } \right) - } \frac{1}{2J}\hat{m}_{c} \varvec{v}_{cr}^{2} - \frac{1}{T}\varvec{q} \cdot {\mathbf{grad}}T \ge 0 \\ \end{aligned} $$
(21)

To proceed further, introduce the chemical potential (which for simplicity will be identified with Gibb’s specific enthalpy) for the liquid phase species:

$$ \mu_{\beta } = e_{m}^{\beta } + \frac{{p_{\beta } }}{{\rho_{\beta } }} - Ts_{m}^{\beta } $$
(22)

Substitution of Eq. (22) into (21) and use of (6) lead after simplification to:

$$ \begin{aligned} & {\varvec{\sigma}} :\varvec{d} - s\frac{{d^{s} T}}{dt} + \sum\limits_{\beta } {\frac{1}{J}\mu_{\beta } \frac{{d^{s} m_{\beta } }}{dt}} - \frac{{d^{s} \psi }}{dt} - \psi div\varvec{v}_{s} - \frac{{\hat{m}_{c} }}{J}\left( {\frac{{\varvec{v}_{cr}^{2} }}{2} + \mu_{c} } \right) \\ & \quad - \sum\limits_{\beta } {\varvec{w}_{\beta } \left( {s_{m}^{\beta } {\mathbf{grad}}T + {\mathbf{grad}}\mu_{\beta } - ({\mathbf{g}} - \gamma^{\beta } )} \right) - } \frac{1}{T}\varvec{q} \cdot {\mathbf{grad}}T \ge 0 \\ \end{aligned} $$
(23)

Note that the term involving the square of the relative velocity of fluidized particles can be neglected for realistic levels of groundwater seepage. At this point we introduce the isothermal condition) (dT = 0), neglect the acceleration terms (γ a = 0) and using the following definitions and relations:

$$ \Uppsi d\Upomega_{0} = \psi d\Upomega_{t} ;\quad \frac{{d^{s} \Uppsi }}{dt} = J\left( {\frac{{d^{s} \psi }}{dt} + \psi div\varvec{v}_{s} } \right) $$
(24)

Furthermore, we’ll limit ourselves to the case of small transformation, so that:

$$ J \approx 1;\quad \phi \approx n;\quad \frac{{d^{s} }}{dt} \approx \frac{\partial }{dt};\quad \Uppsi \approx \psi ;\quad \varvec{d} = \dot{{\varvec{\varepsilon}} } $$
(25)

In consequence, notation \( \dot{A} \) from now on will mean \( \frac{\partial A}{dt}, \) which is much more convenient to write. Under these conditions, Eq. (23) can be simplified to:

$$ {\varvec{\sigma}} :\dot{\varvec{\varepsilon}} + \sum\limits_{\beta } {\mu_{\beta } \dot{m}_{\beta } - \dot{\psi } - \hat{m}_{c} \mu_{c} + \sum\limits_{\beta } {( - {\mathbf{grad}}\mu_{\beta } + {\mathbf{g}})} } \cdot \varvec{w}_{\beta } \ge 0 $$
(26)

To proceed further we decompose the total free energy into the sum of the free energies of the skeleton, the fluidized solid particles and the pore water:

$$ \psi = \psi_{s} + m_{c} \psi_{m}^{c} + m_{w} \psi_{m}^{w} $$
(27)

We will also make use of the following classic thermodynamic relations for the fluid phases (keeping in mind that dT = 0 is assumed):

$$ de_{m}^{\beta } = - P_{\beta } d\left( {\frac{1}{{\rho_{\beta } }}} \right) + Tde_{m}^{\beta } $$
(28)
$$ d\psi_{m}^{\beta } = - P_{\beta } d\left( {\frac{1}{{\rho_{\beta } }}} \right) - s_{m}^{\beta } dT $$
(29)
$$ d\mu_{\beta } = \left( {\frac{1}{{\rho_{\beta } }}} \right)dP_{\beta } - s_{m}^{\beta } dT $$
(30)

With Eqs. (28), (29) and (30), Eq. (26) can be simplified to:

$$ \Phi = \Phi_{s, \to } + \Phi_{t} = ({\varvec{\sigma}} :\dot{{\varvec{\varepsilon}}} + P\dot{\phi } - \hat{m}_{c} \mu_{c} - \dot{\psi }_{s} ) + \sum\limits_{\beta } {( - {\mathbf{grad}}\mu_{\beta } + {\mathbf{g}})} \cdot \varvec{w}_{\beta } \ge 0 $$
(31)

where Φ s,→ represents the dissipation due to the skeleton irreversible behavior plus that due to the erosion process, and Φ t is the dissipation due to the relative motion of liquid phase across the solid skeleton. In view of the form of the dissipation inequality, we postulate that:

$$ \psi_{s} = \psi_{s} ({\varvec{\varepsilon}} ,\phi_{er} ) $$
(32)

where ϕ er is the irreversible porosity change due to erosion. This allows the decomposition of Φ s,→ to:

$$ \Phi_{s, \to } = \Phi_{s} + \Phi_{ \to } = \left( {{\varvec{\sigma}} + P_{f} {\mathbf{I}} - \frac{{\partial \psi_{s} }}{\partial {\varvec{\varepsilon}} }} \right):{\varvec{\varepsilon}} + \left( {P_{f} \dot{\phi }_{er} - \hat{m}_{c} \mu_{c} - \frac{{\partial \psi_{s} }}{{\partial \phi_{er} }}\dot{\phi }_{er} } \right) \ge 0 $$
(33)

On account of the different natures of the dissipative mechanisms, it is reasonable to assume that each contribution to the total dissipation should be independently non negative: Φ s  ≥ 0; Φ →≥0; Φ t  ≥ 0. We now develop constitutive equations which allow each of the three dissipations Φ s , Φ → and Φ t to be rendered non negative.

5 Constitutive Model of Solid Skeleton

For non dissipative solid skeleton behavior, the intrinsic dissipation is null Φ s  = 0. We thus deduce that:

$$ {\varvec{\sigma}}^{\prime } = {\varvec{\sigma}} + P_{f} {\mathbf{I}} = \frac{{\partial \psi_{s} }}{\partial {\varvec{\varepsilon}} } $$
(34)

where we recognize the classic Terzaghi’s effective stress. Under isothermal conditions, the free energy is equivalent to the elastic energy. Taking inspiration from damage mechanics, we suppose that erosion and the subsequent enlargement of pores can be assimilated to a kind of damage (Lemaitre 1991). Assuming the behavior to be isotropic and in the spirit of a simplified approach, we postulate that:

$$ \psi_{s} = \frac{1}{2}{\varvec{\varepsilon}} :C(\phi ):{\varvec{\varepsilon}} ;\quad C(\phi ) = \frac{1 - \phi }{{1 - \phi_{0} }}C^{0} $$
(35)

where C 0 and C(ϕ) are the elastic stiffness tensors at the initial (when ϕ = ϕ 0) and current state. For isotropic behavior, we have:

$$ C^{0} = \frac{{vE_{0} }}{(1 + v)(1 - 2v)}{\mathbf{I}} \otimes {\mathbf{I}} + \frac{{E_{0} }}{(1 + v)}{\text{II}} $$
(36)

In the above, I is the second order identity tensor with components δ ij , the components of the tensor product are \( ({\mathbf{I}} \otimes {\mathbf{I}})_{ijkl} = \delta_{ij} \delta_{kl} \) whereas II is the fourth order identity tensor, with components \( II_{ijkl} = \frac{1}{2}(\delta_{ik} \delta_{jl} + \delta_{il} \delta_{kj} ). \) Using the notation \( \varepsilon_{v} = {\mathbf{I}}:\varepsilon \) for the volume strain, these lead to:

$$ {\varvec{\sigma}} = C(\phi ):{\varvec{\varepsilon}} - P_{f} {\mathbf{I}} = \frac{{vE(\phi){\varvec{\varepsilon}}_{v} }}{(1 + v)(1 - 2v)}{\mathbf{I}} + \frac{E(\phi )}{(1 + v)}{\varvec{\varepsilon}} - P_{f} {\mathbf{I}}; \quad E(\phi ) = \frac{1 - \phi }{{1 - \phi_{0} }}E_{0} $$
(37)

6 Constitutive Model of Erosion

Note that the erosion model must satisfy Φ →≥0. From classical thermodynamics, we have:

$$ \mu_{c} = \frac{{P_{f} }}{{\rho_{s} }} + \frac{RT}{{M^{s} }}Ln(x_{c} ) \approx \frac{{P_{f} }}{{\rho_{s} }} $$
(38)

In the above, R is the universal gas constant, M s and x c respectively the molar mass and molar fraction within the fluid phase of fluidized solid particles. Neglecting the term involving molar concentration x c and on account of (8a) and (38), we get:

$$ \Phi_{ \to } = - \frac{{\partial \psi_{s} }}{{\partial \phi_{er} }}\dot{\phi }_{er} \ge 0 $$
(39)

This inequality shows that the erosion kinetics also depends on the state of solid skeleton, via the free energy ψ s . Using (35) and accounting for the relation ϕ = ϕ 0 + ε v  + ϕ er , we obtain:

$$ \frac{{\partial \psi_{s} }}{{\partial \phi_{er} }} = - \frac{1}{2}{\varvec{\varepsilon}} :C^{0} :{\varvec{\varepsilon}} < 0 $$
(40)

The inequality Φ →≥0 is therefore reduced to:

$$ \dot{\phi }_{er} \ge 0 $$
(41)

This condition is however not very restrictive. In order to construct an erosion model, we have to go back to fundamental physical principles. At the local scale, internal erosion is related to the shear stress τ exerted by the pore fluid on the solid skeleton. Assuming that no erosion takes place below a certain threshold stress τ c , a possible simplified model is (Brivois et al. 2007):

$$ \dot{\phi }_{er} = \alpha \langle \tau - \tau_{c} \rangle $$
(42)

In the above, α is a positive material constant depending on the sensitivity of the soil to hydrodynamic forces, while the Maclaurin brackets are defined by \( \langle x\rangle = x \) if x ≥ 0 and \( \langle x\rangle = 0 \) if x < 0. Hence (41) is automatically satisfied by construction. Reddi et al. (2000), inspired by the classic Poiseuille flow in a cylindrical tube (“Appendix”), postulated that:

$$ \tau = \rho {\mathbf{g}}\left\| \varvec{i} \right\|\frac{{D_{r} }}{4};\quad D_{r} = 4\sqrt {2\kappa (\phi )/\phi } $$
(43)

in which D r is some kind of representative pore diameter, i the normalized piezometric gradient and κ(ϕ) the intrinsic permeability of the porous solid, which depends on the current porosity ϕ. This leads to the proposal of Fuijisawa et al. (2010):

$$ \dot{\phi }_{er} = \alpha \left\langle {\sqrt {\frac{2\kappa (\phi )}{\phi }} \left\| {{\mathbf{grad}}[P_{f} + \rho_{f} {\mathbf{g}}z]} \right\| - \tau_{c} } \right\rangle $$
(44)

Note that these authors only considered non-deformable porous solids, whereas solid skeleton deformations are accounted for in the present model. To complete the construction, we also need to include the equation governing pore fluid flow.

7 Darcy’s Law

Recalling that all fluid species share the same velocity v f and pressure P f , the dissipation due to transport can be simplified to:

$$ \Phi_{t} = ( - {\mathbf{grad}}P_{f} + \rho_{f} {\mathbf{g}}) \cdot {\varvec{V}}^{D} \ge 0 $$
(45)

With ρ f  = (1 − c)ρ w  + cρ s the over-all fluid density and V D the classic Darcy velocity:

$$ \varvec{V}^{D} = \phi (\varvec{v}_{f} - \varvec{v}_{s} ) $$
(46)

A generalized Darcy’s law which verifies the above inequality writes:

$$ \varvec{V}^{D} = k(\phi ,c)( - {\mathbf{grad}}P_{f} + \rho_{f} {\mathbf{g}}) $$
(47)

where k is a positive scalar (or symmetric definite positive tensor for anisotropic behaviors) which represents the permeability. It does not only depend on the porosity but also on the concentration of fluidized particles c since the latter would affect the viscosity of the pore fluid. At this point note that the intrinsic and the dimensional hydraulic conductivities, k(ϕ) and k(ϕ, c) are related to each other via the particle concentration-dependent viscosity η(c) of the pore fluid:

$$ k(\phi ,c) = \frac{\kappa (\phi )}{\eta (c)} $$
(47a)

The dependency of the intrinsic conductivity on porosity can be expressed by the classic Kozeny-Carmen equation:

$$ \kappa (\phi ) = \frac{{\phi^{3} }}{{(1 - \phi )^{2} }}\left( {\frac{{\phi_{0}^{3} }}{{(1 - \phi_{0} )^{2} }}} \right)^{ - 1} \kappa_{0} $$
(47b)

where κ 0 is the value of κ(ϕ) at the initial state, when ϕ = ϕ 0. Regarding the dependency η(c), the classic equation of Einstein is used:

$$ \eta (c) = \eta_{w} (1 + 2.5c) $$
(47c)

With η w the viscosity of water at zero concentration of fluidized particles. We thus arrive at:

$$ k(\phi ,c) = \frac{{\kappa_{0} }}{{\eta_{w} (1 + 2.5c)}}\frac{{\phi^{3} }}{{(1 - \phi )^{2} }}\left( {\frac{{\phi_{0}^{3} }}{{(1 - \phi_{0} )^{2} }}} \right)^{ - 1} $$
(47d)

8 Governing Equations

We begin by considering the mass conservation of fluid species. Combining relations (3), (6), (7) for the fluid species {c, w} followed by their addition, we get, after accounting for the identity (3a):

$$ \dot{{\varvec{\varepsilon}}}_{v} + div(\varvec{V}^{D} ) = 0 $$
(48)

In other words, the rate of macroscopic volumetric expansion is equal to the fluid influx, which is consistent with the assumption of solid phase incompressibility even during its transformation into fluid phase due to erosion. Invoking Darcy’s law and the definition of volumetric strain ε v  = div u, we get the first governing equation of the problem:

$$ div(\dot{\varvec{u}}) + div(k(\phi ,c)( - {\mathbf{grad}}P_{f} + \rho_{f} {\mathbf{g}})) = 0 $$
(49)

Applying the mass conservation Eq. (6) only to fluidized particles, and on account of the relations (3), (7), (8a) and (48), we derive the second governing equation:

$$ \phi \dot{c} - k(\phi ,c){\mathbf{grad}}P_{f} \cdot {\mathbf{grad}}c + k(\phi ,c)\rho_{f} {\mathbf{g}} \cdot {\mathbf{grad}}c = (1 - c)\dot{\phi }_{er} $$
(50)

We now turn our attention to the mechanical equilibrium of the system. Neglecting acceleration, the overall equilibrium of a soil element writes:

$$ div(\sigma ) + \bar{\rho }{\mathbf{g}} = 0;\quad \bar{\rho } = (1 - \phi )\rho_{s} + \phi (1 - c)\rho_{w} + \phi c\rho_{s} $$
(51)

\( \bar{\rho } \) being the overall density. Combining the above with the constitutive Eq. (37), we get the third governing equation:

$$ div\left( {\frac{1 - \phi }{{1 - \phi_{0} }}C^{0} :{\varvec{\varepsilon}} } \right) - {\mathbf{grad}}P_{f} + \bar{\rho }{\mathbf{g}} = 0 $$
(52)

The fourth, which is also the last governing equation is simply the erosion law (44). Note that Eq. (3a) implies that ϕ er can be expressed in terms of ϕ. We have thus formulated a coupled problem on the four fields (u, P f , ϕ, c), defined by the four partial differential Eqs. (49), (50), (52) and (44).

To summarize, for an erodible material, its behavior is completely described by 6 material constants, namely: α, τ c , κ 0, E 0, v, ϕ 0. The last parameter ϕ 0 is needed to monitor the evolution of both the hydraulic conductivity and the stiffness. For simplicity, in what follows, we will suppose that the density of the solid matrix is the same for all of soils in the numerical example of Sect. 10, with ρ s  = 26 kN m−3. Recall that g = 9.81 m s−2, ρ w  = 1,000 kg m−3 and the viscosity of water w  = 0.001 Pa s are commonly used values.

The system of partial differential equations involves 6 scalar fields in 3D problems and 5 scalar fields in 2D problems. They are highly non linear and not easy to solve. One important limiting case can be obtained by neglecting the deformations of the solid skeleton by setting u = ε = 0. Equation (49) is then reduced to:

$$ div(k(\phi ,c)( - {\mathbf{grad}}P_{f} + \rho_{f} {\mathbf{g}})) = 0 $$
(53)

Together with Eqs. (50) and (44), they define a 3-field problem (P f , ϕ, c) which is much faster to solve. This has been used to check the consistency of the construction before treating the more general 4-field problem.

9 Numerical Example and Comments

In order to illustrate the applicability of the model developed, we propose to use it to analyze a typical dam-on-foundation problem, where erosion takes place underneath the dam due to high local hydraulic gradients. The computations are aimed at showing how the internal erosion may weaken the foundation material of the dam, which will eventually lead to piping. However, since formulation is based on continuum mechanics approach, the passage from a continuous medium to a discontinuous medium cannot be accounted for, and the model ceases to apply when macroscopic discontinuities appear.

9.1 The Computer Code COMSOL

COMSOLMultiphysics® is a commercial finite element simulation package which provides a software environment for all steps in the modeling process, namely: defining the geometry, meshing, specifying the physics, solving and visualizing the results, particularly of coupled phenomena. This package has been used to set up, solve and visualize the results of the numerical example below in accordance with the formulation developed in the present paper.

9.2 Geometric Configuration

The geometry and the dimensions of this numerical example are schematically shown in Fig. 3. (h1 = 50 m, h2 = 50 m, AB = 15 m, BC = 8 m, CD = 13 m, KE = 20 m, EF = 151.6 m, FI = 34.8 m, IJ = 124.6 m, JL = 19 m, GH = 10.5 m, MN = 350 m). The dam comprises a core wall BCIHGF (Mat 1) with a low permeability clay surrounded by a more permeable backfill soil ABEF and CDJI (Mat 2). It sits on an underlying soil mass (Mat 3), below the interface KEFGHIJL. The dam retains a reservoir of water on the left upstream side AE. In reality, a free water surface will appear in the region CDJI. To simplify, we will assume that this free water surface will follow the boundary ABCDJ. In other words, water flow will take place everywhere inside the dam. Numerical computations show that this assumption has no significant influence on the final results, because the bulk of the flow takes place underneath the core wall and the water pressure inside the block CDJI remains largely atmospheric. Moreover, the upstream water surface is supposed to be aligned with the top of the dam ABCD. The problem is analyzed under the assumption of plane strain conditions. The finite element mesh of the physical domain is shown in Fig. 4.

Fig. 3
figure 3

Schematic configuration of the dam on foundation example

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Fig. 4
figure 4

Finite element mesh of the physical domain, including core wall, back fill and foundation soil

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9.3 Material Parameters

As mentioned in Sect. 8, each erodible material is described by 6 material parameters: α, τ c , κ 0, E 0, v, ϕ 0. The numerical values adopted in this example problem are summarized in the following Table 1.

Table 1 Material parameters used in the numerical example
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9.4 Initial Conditions

Initially, the displacements and strains are identically null everywhere (reference state). The initial stresses are non null. However in the present case, the behavior is supposed to be elastic and the initial stress state does not play any role and need not to be specified. Hydraulically, pore pressures at the initial instant are supposed to be hydrostatic everywhere. The initial porosity and particle concentration are supposed to be homogeneous, with ϕ 0 = 0.31 and c 0 = 0.

9.5 Boundary Conditions (BC)

On the upstream surface AEK, a hydrostatic pore pressure P w is imposed. Its value is computed from the position of the point considered relative to the free water surface. Mechanically, the applied surface force density is given by −P w n, where n is the outward unit normal. Along the surface ABCD the pore water pressure is zero and the surface surcharge is 90 kPa (to account for the part of the dam above the water level). Along the downstream surface DJL, both the pore water pressure and the surface force are null. At the boundary KM the pore water pressure and surface pressure are respectively (h1 + h2 − y)γ w , and \( \left( {h2 - y} \right) \times \left( {\gamma_{sat} - \gamma_{w} } \right)\frac{v}{1 - v} \) (v is the Poisson ratio), while along the boundary LN the pore water pressure and surface pressure are respectively (h2 − y)γ w , and \( \left( {h2 - y} \right) \times \left( {\gamma_{sat} - \gamma_{w} } \right)\frac{v}{1 - v}. \) MN is specified as a Neumann boundary and is assumed to be impervious, with null displacement.

9.6 The numerical results

Figure 5a shows the evolution of the porosity inside the dam and the underlying foundation soil with time. As internal erosion takes place, the porosity which was spatially uniform everywhere initially (Fig. 5a (a)) begins to increase with elapsed time. The porosity increase is particularly apparent in the foundation soil beneath the dam. The two points experiencing the highest increase in porosity, that is suffering the most internal erosion, are located directly beneath the clay core and at the downstream toe of the dam. Here the increased porosity which appears in Fig. 5a (b), continues to increase in Figs. 5a (c) and 8a (d), and leads to a highly eroded zone which precedes the formation of a piping channel linking the two localities. Beyond this, internal erosion will lead to the creation of tubes or pipes of fluid flow with extremely large porosities and virtually discontinuous soil mass where the piping has occurred. At the stage of piping, the applicability of the current model and formulation would have been surpassed, and further simulation must revert to an approach taking into consideration the discontinuous soil/fluid interface, as mentioned above.

Fig. 5
figure 5

a Spatial distribution of porosity at different times. b Variation of porosity with time for points E, F, G, H, I, J

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The variation of the porosity at points E, F, G, H, I, J along the dam-foundation soil interface is shown in Fig 5b. The plots reaffirm previous statements that the highest porosities are found directly below the clay core (points G, H) and at the downstream toe of the dam (point J). Although point J has a lower erosion rate initially than points G and H, by the end of the analysis period (1.6 × 109 s) it experiences a higher erosion rate and records a higher porosity value than the other points.

Figure 6 shows the seepage velocity (and pore pressure gradient) reaching the highest levels within the foundation soil at the points directly beneath the clay core and at the downstream toe of the dam, with elapsed time. With the low hydraulic conductivity (permeability) clay core acting as a flow barrier, the path of least resistance for seepage is via the more pervious foundation soil as anticipated. The evolution of the seepage velocity and pore pressure distribution shown in Fig. 6 is a direct consequence of this. It is also worthwhile noting that the points with the highest increases in porosity in Fig. 5 correlate to the points with the highest seepage and pressure gradients shown in Fig. 6. The reason for this may be inferred from the constitutive equation for erosion, (44), which is a function of the hydraulic gradient, hydraulic conductivity and porosity of the soil, the same parameters that govern seepage. Thus in this case, the parameters which cause an increase in the seepage velocity will also generally lead to an increase in the erosion rate. Figure 6 shows that the pore pressure in upper zone of the dam to the right of the clay core is virtually atmospheric. This justifies the simplifying assumption made earlier to specify the free water surface along the boundary ABCDJ.

Fig. 6
figure 6

Spatial distribution of pore pressure and seepage velocity at different times

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The concentration of the fluidized particles is shown increasing with time and space in Fig. 7. It is observed that the points with the highest concentrations are in the vicinity of the locations which recorded the largest porosity increase (Fig. 5). This would indicate that the erosion rate is either maintained or still increasing for the period of analysis. As internal erosion increases and the fluidized particles are transported further downstream from where the erosion first occurs, the plume of fluidized particles spreads over an increasing area with elapsed time.

Fig. 7
figure 7

Spatial distribution of fluidized particles at different times

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The effects of erosion rate on the degradation of the dam are illustrated in Figs. 8 and 9 showing the variation of horizontal and vertical displacements at point A with time for various erosion rates. The erosion rate is varied in terms of the coefficient α, the scaling factor of Eq. (44). Here α is an intrinsic property reflecting the nature of the soil and construction method, such that all else being equal the higher its value, the higher will be the erosion rate. α is increased from the lowest value (9.67 × 10−11) to the highest value (3.48 × 10−10) in 3 successive steps by a factor of 2, 1.5 and 1.2. Figures 8 and 9 show that an increase in erosion rate will lead to an increase in the displacements at point A at a given time. The displacements increase nonlinearly approximately as a power function in time, the degree of which is directly proportional to the erosion rate. As a result, the differences in the displacements between successive erosion rates are accentuated at large times, as exemplified for instance by the vertical displacements computed for α = 2.90 × 10−10 and 3.48 × 10−10 in Fig. 9. Relative to t = 0, namely the no erosion case, the vertical displacement increases are 0.03 m and 0.034 m (resp. α = 2.90 × 10−10 and 3.48 × 10−10) at t = 0.8 × 109s, and 0.073 m and 0.087 m at t = 1.6 × 10−9s. These results show that the vertical displacement increase between α = 3.48 × 10−10 and α = 2.90 × 10−10 at t = 0.8 × 109 is 0.004 m, with this increasing to 0.014 m at t = 1.6 × 109 s. This is a consequence of the damage model (Eq. 35) which has yielded nonlinear power law type damage effects in time and with respect to the erosion rate due to internal erosion.

Fig. 8
figure 8

Variation of horizontal displacement at point A with time, for 4 different values of α, showing the effects of erosion on the mechanical degradation

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Fig. 9
figure 9

Variation of vertical displacement at point A with time, for 4 different values of α, showing the effects of erosion on the mechanical degradation

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10 Conclusions

In this paper, a numerical solution on the internal erosion of porous soil materials within a geotechnical setting is presented. The solution is developed within the framework of the conservation laws of mass and momentum, and consideration of the principles of thermodynamics, the latter revealing important insights facilitating the formulation of thermodynamically consistent constitutive equations of the solid skeleton, fluid flow and internal erosion. The numerical solution is formulated within continuum poro-mechanics and finite element setting. A numerical example of a dam-on-foundation has been used to demonstrate the applicability of the numerical solution to simulate the evolution of internal erosion, in this case, especially of the foundation soil beneath the dam. The effects of internal erosion leading to increase in porosity and fluidized particle concentration, and the mechanical degradation of the dam, in time and space were simulated by the numerical solution. The results of the erosion kinetics have been shown to accord well with intuitive reasonings.