Numerical study of partially drained penetration and pore pressure dissipation in piezocone test

Abstract

The piezocone penetration test (CPTU) is commonly used as a fast and economical tool to identify soil profile and to estimate relevant material properties in soils ranging from fine to coarse-grained. Moreover, in the case of fine-grained soils (clays and silts), the consolidation coefficient and the permeability can be estimated through the dissipation test. Undrained conditions are commonly assumed for the interpretation of CPTU in fine-grained soils, but in soils such as silts, penetration may occur in partially drained conditions. This aspect is often neglected in data interpretation thus leading to an inaccurate estimate of soil properties. This paper investigates numerically the effect of partial drainage during penetration on the measured tip resistance and the subsequent pore pressure dissipation response contributing to a more accurate interpretation of field data. A realistic simulation of the cone penetration is achieved with the two-phase Material Point Method, modelling the soil response with the modified Cam-Clay model. The approach takes into account large soil deformations induced by the advancing cone, soil–water, and soil–structure interactions, as well as nonlinear soil behavior.

Appendices

Appendix 1: The two-phase MPM

The equations that describe the two-phase physics are: conservation of mass, conservation of momentum, and the constitutive relation. A detailed derivation of the equations presented in the following can be found in [53].

The momentum equation of the water phase is:

$$\rho_{\text{w}} \dot{\varvec{w}} + \frac{{n\gamma_{\text{w}} }}{k}\left( {\varvec{v} - \varvec{w}} \right) = \nabla p_{\text{w}} + \rho_{\text{w}} \varvec{g}$$

(13)

where v and w are the solid velocity and fluid velocity, respectively, ρ _w and γ _w are the density and the unit weight of the water, k is the Darcy’s permeability, n is the porosity, p _w is the pore water pressure and g is the gravity vector. The second term on the left-hand side represents the interaction between solid and fluid.

The momentum equation for the mixture is:

$$\left( {1 - n} \right)\rho_{\text{s}} \dot{\varvec{v}} + n\rho_{\text{w}} \dot{\varvec{w}} = \nabla \cdot \left( {\varvec{\sigma}^{\prime } +\varvec{\delta}p_{\text{w}} } \right) + \rho_{\text{sat}} \varvec{g}$$

(14)

where ρ _s is the density of the solid grains, ρ _sat = n ρ _w + (1 − n) ρ _s is the saturated density, σ′ is the effective stress, and $\varvec{\delta}= \left[ {1 1 1 0 0 0} \right]^{\text{T}}$.

The mass balance equation for the water phase gives the excess pore pressure increment:

$$\dot{p}_{w} = \frac{{K_{\text{w}} }}{n}\left[ {\left( {1 - n} \right)\nabla \cdot \varvec{v} + n\nabla \cdot \varvec{w}} \right]$$

(15)

where K _w is the bulk modulus of the water. The effective stress increment is calculated with the soil constitutive model.

The boundary $\partial \varOmega$ of the domain $\varOmega$ occupied by the continuum must satisfy the following conditions:

$$\partial \varOmega = \partial \varOmega_{u} \cup \partial \varOmega_{\tau } = \partial \varOmega_{w} \cup \partial \varOmega_{p}$$

$$\partial \varOmega_{u} \cap \partial \varOmega_{\tau } = \emptyset \quad {\text{and}}\quad \partial \varOmega_{w} \cap \partial \varOmega_{p} = \emptyset$$

where $\partial \varOmega_{w}$ and $\partial \varOmega_{p}$ are the prescribed velocity and prescribed pressure boundaries of the water phase, respectively, whereas $\partial \varOmega_{u}$ is the prescribed displacement (velocity) boundary of the solid phase and $\partial \varOmega_{\tau }$ is the prescribed total stress boundary.

The space discretization of the governing equations is derived from the weak form of Eqs. (13) and (14) by means of the shape functions N. The same shape functions are used to approximate the velocity of the solid and fluid phase.

The momentum equation of the fluid can be written as:

$$\int\limits_{\varOmega } {\varvec{N}^{T} \rho_{\text{w}} \varvec{N}\dot{\varvec{w}}{\text{d}}\varOmega } = \int\limits_{{\partial \varOmega_{p} }} {\varvec{N}^{T} \bar{\varvec{p}}{\text{d}}S} - \mathop \int \limits_{\varOmega }^{{}} \varvec{B}^{T}\varvec{\delta}p_{\text{w}} {\text{d}}\varOmega + \mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} \rho_{\text{w}} \varvec{g}{\text{d}}\varOmega - \mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} \frac{{n\rho_{\text{w}} g}}{k}\varvec{N}\left( {\varvec{w} - \varvec{v}} \right){\text{d}}\varOmega$$

(16)

where now w and v are the nodal fluid velocity and nodal solid velocity respectively, $\bar{\varvec{p}}$ is the applied pressure at the boundary, $\varvec{B}^{T}$ is the matrix of the derivatives of the shape functions. Equation (1) is the matrix form of Eq. (16) where

$$\varvec{M}_{{\mathbf{w}}} = \mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} \rho_{\text{w}} \varvec{N}{\text{d}}\varOmega$$

(17)

$$\varvec{F}_{{\mathbf{w}}}^{{{\mathbf{ext}}}} = \mathop \int \limits_{{\partial \varOmega_{p} }}^{{}} \varvec{N}^{T} \bar{\varvec{p}}{\text{d}}S + \mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} \rho_{\text{w}} \varvec{g}{\text{d}}\varOmega$$

(18)

$$\varvec{F}_{{\mathbf{w}}}^{{{\mathbf{int}}}} = \mathop \int \limits_{\varOmega }^{{}} \varvec{B}^{T}\varvec{\delta}p{\text{d}}\varOmega$$

(19)

$$\varvec{F}_{{\mathbf{w}}}^{{{\mathbf{drag}}}} = \mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} \frac{{n\rho_{\text{w}} g}}{k}\varvec{N}\left( {\varvec{w} - \varvec{v}} \right){\text{d}}\varOmega$$

(20)

The momentum equation of the mixture can be written as:

$$\mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} \left( {1 - n} \right)\rho_{\text{s}} \varvec{N}\dot{\varvec{v}}{\text{d}}\varOmega = - \mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} n\rho_{\text{w}} \varvec{N}\dot{\varvec{w}}{\text{d}}\varOmega + \mathop \int \limits_{{\partial \varOmega_{\tau } }}^{{}} \varvec{N}^{T} \bar{\varvec{\tau }}{\text{d}}S + \mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} \rho_{\text{sat}} \varvec{g}{\text{d}}\varOmega - \mathop \int \limits_{\varOmega }^{{}} \varvec{B}^{T}\varvec{\sigma}{\text{d}}\varOmega$$

(21)

where $\bar{\varvec{\tau }}$ is the vector of applied total stress. Equation (2) is the matrix form of Eq. (21), where

$$\varvec{M}_{{\mathbf{s}}} = \mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} \left( {1 - n} \right)\rho_{\text{s}} \varvec{N}{\text{d}}\varOmega$$

(22)

$$\bar{\varvec{M}}_{{\mathbf{w}}} = \mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} n\rho_{\text{w}} \varvec{N}{\text{d}}\varOmega$$

(23)

$$\varvec{F}_{{}}^{{{\mathbf{ext}}}} = \mathop \int \limits_{{\partial \varOmega_{\tau } }}^{{}} \varvec{N}^{T} \bar{\varvec{\tau }}{\text{d}}S + \mathop \int \limits_{\varOmega }^{{}} \varvec{N}^{T} \rho_{{\text{sat}}} \varvec{g}{\text{d}}\varOmega$$

(24)

$$\varvec{F}_{{}}^{{{\mathbf{int}}}} = \mathop \int \limits_{\varOmega }^{{}} \varvec{B}^{T}\varvec{\sigma}{\text{d}}\varOmega$$

(25)

For convenience in numerical implementation, the lumped mass matrices are used [23].

The Euler–Cromer time integration scheme is adopted. The solution sequence of a single time step can be summarized in the following steps:

• 1. The momentum equations for the fluid (Eq. 1) and the mixture (Eq. 2) are initialized by mapping the quantities from the MP to the mesh nodes by means of interpolation functions.

• 2. Eq. (1) is solved for the fluid acceleration at time t

  $$\dot{\varvec{w}}^{t} = \varvec{M}_{{\mathbf{w}}}^{{\varvec{t} - 1}} \left[ {\varvec{F}_{{\mathbf{w}}}^{{{\mathbf{ext}},\varvec{t}}} - \varvec{F}_{{\mathbf{w}}}^{{{\mathbf{int}},\varvec{t}}} - \varvec{F}_{{\mathbf{w}}}^{{{\mathbf{drag}},\varvec{t}}} } \right]$$

  (26)

• 3. Eq. (2) is solved for the solid acceleration at time t

  $$\dot{\varvec{v}}^{t} = \varvec{M}_{{\mathbf{s}}}^{{\varvec{t} - 1}} \left[ { - \bar{\varvec{M}}_{{\mathbf{w}}}^{\varvec{t}} \dot{\varvec{w}}^{t} + \varvec{F}_{{}}^{{{\mathbf{ext}},\varvec{t}}} - \varvec{F}_{{}}^{{{\mathbf{int}},\varvec{t}}} } \right]$$

  (27)

• 4. The velocities of the MP are updated using nodal accelerations and interpolation functions:

  $$\varvec{w}_{p}^{{t +\Delta t}} = \varvec{w}_{p}^{t} + \mathop \sum \limits_{i = 1}^{{n_{e} }}\Delta tN_{i,p} \dot{\varvec{w}}^{t}$$

  (28)
  $$\varvec{v}_{p}^{{t +\Delta t}} = \varvec{v}_{\varvec{p}}^{\varvec{t}} + \mathop \sum \limits_{i = 1}^{{n_{e} }}\Delta tN_{i,p} \dot{\varvec{v}}^{\varvec{t}}$$

  (29)

  where n _e is the number of nodes per element, $N_{i,p}$ is the interpolation function of node i evaluated at the position of material point p, and Δt is the time increment

• 5. The nodal velocities w ^t+Δt and v ^t+Δt are then calculated from the updated MP momentum

• 6. Nodal velocities are integrated to get nodal incremental displacements: $\Delta \varvec{u}^{{\varvec{t} +\Delta \varvec{t}}} =\Delta t\varvec{v}^{{\varvec{t} +\Delta \varvec{t}}}$

• 7. Fluid and solid strains at MP are calculated as

  $$\Delta\varvec{\varepsilon}_{\text{w}}^{{t +\Delta t}} = \varvec{B}_{\varvec{p}}^{\varvec{t}}\Delta t\varvec{w}^{{\varvec{t} +\Delta \varvec{t}}} \quad {\text{and}}\quad\Delta\varvec{\varepsilon}_{\text{s}}^{{t +\Delta t}} = \varvec{B}_{\varvec{p}}^{\varvec{t}}\Delta t\varvec{v}^{{\varvec{t} +\Delta \varvec{t}}}$$

  (30)

  where $\varvec{B}_{\varvec{p}}^{\varvec{t}}$ is the matrix of the derivative of the shape functions at the location of material point p.

• 8. Stresses are updated according to the constitutive relation

• 9. Water pressure at MP p is updated as:

  $$p_{w,p}^{{t +\Delta t}} = p_{w,p}^{t} +\Delta t\frac{{K_{w} }}{n}\left[ {\left( {1 - n} \right)\Delta\varvec{\varepsilon}_{{{\mathbf{vol}},\varvec{s}}}^{{\varvec{t} +\Delta \varvec{t}}} + n\Delta\varvec{\varepsilon}_{{{\mathbf{vol}},\varvec{w}}}^{{\varvec{t} +\Delta \varvec{t}}} } \right]$$

  (31)

  where $\Delta\varvec{\varepsilon}_{{{\mathbf{vol}},\varvec{s}}}^{{\varvec{t} +\Delta \varvec{t}}}$ and $\Delta\varvec{\varepsilon}_{{{\mathbf{vol}},\varvec{w}}}^{{\varvec{t} +\Delta \varvec{t}}}$ are the volumetric strains of solid and fluid, respectively.

• 10. The positions of MP are updated using the displacements of the solid phase

• 11. The book-keeping is updated using the new position of MP

Appendix 2: The contact algorithm

This appendix briefly summarizes the contact algorithm presented by Bardenhagen et al. [7] and its extension to the two-phase problem of cone penetration.

Let us consider two bodies, labelled A and B, in contact. The nodal velocities of the single body (v _A , v _B ) and the nodal velocity of the system of bodies (v _S ) are predicted by solving the respective momentum equations. A node is identified as a contact node if the velocity of the single bodies differs from the one of the combined system. If so, the algorithm checks if the bodies are approaching or separating. If they are separating, there is no need for correction and the velocities correspond to the single-body velocities. Otherwise, the occurrence of sliding is checked.

The sliding condition is based on the contact forces. If the contact force is lower than the maximum value allowed by the contact law, the bodies stick to each other and the velocity is equal to the system velocity. If the bodies are sliding one respect to the other, then the single-body velocity is corrected in such a way that no interpenetration occurs and the contact force respects the Coulomb friction law.

The corrected nodal velocity at the contact node k, is calculated as:

$$\tilde{\varvec{v}}_{{\varvec{k},\varvec{A}}} = \varvec{v}_{{\varvec{k},\varvec{A}}} - \left[ {\left( {\varvec{v}_{{\varvec{k},\varvec{A}}} - \varvec{v}_{{\varvec{k},\varvec{S}}} } \right) \cdot \varvec{n}_{{\varvec{k},\varvec{A}}} } \right]\varvec{n}_{{\varvec{k},\varvec{A}}} + \left[ {\left( {\varvec{v}_{{\varvec{k},\varvec{A}}} - \varvec{v}_{{\varvec{k},\varvec{S}}} } \right) \cdot \varvec{n}_{{\varvec{k},\varvec{A}}} } \right]\mu \varvec{t}_{\varvec{k}}$$

(32)

where $\varvec{v}_{{\varvec{k},\varvec{A}}}$ and $\varvec{v}_{{\varvec{k},\varvec{S}}}$ are predicted velocity for the single body and the coupled system, respectively, $\varvec{n}_{{\varvec{k},\varvec{A}}}$ is the outward normal vector to body A at the node k, $\varvec{t}_{\varvec{k}}$ is the tangential vector, μ is the friction coefficient. The second term on the right-hand side represents the correction for the normal component in order to ensure no interpenetration. The third term represents the correction for the tangential component in order to respect the contact law. In the cone penetration problem, the system velocity coincides with the cone penetration velocity (prescribed velocity).

Once the corrected velocity has been computed, the corrected nodal acceleration is calculated:

$$\dot{\varvec{v}} = \frac{{\tilde{\varvec{v}}_{{\varvec{k},\varvec{A}}}^{{\varvec{t} + {\varvec{\Delta}}\varvec{t}}} - \tilde{\varvec{v}}_{{\varvec{k},\varvec{A}}}^{\varvec{t}} }}{{\Delta t}}$$

(33)

where the superscript t + Δt indicates the velocity computed by Eq. (32), and t indicates the velocity at the previous time step. The corrected nodal values are used to update the MP quantities (velocity, position, strain, stress and so on).

In the two-phase analyses, in addition to the solid velocity, also nodal water velocities and accelerations must be corrected in order to consider the interaction between the water phase and the impermeable structure.

The contact algorithm for the fluid phase is similar to the one presented above for the solid phase. In order to prevent the inflow of water into the cone, the normal component of the fluid velocity must be equal to the normal component of the cone velocity. The corrected velocity for the water at the contact node k takes the form:

$$\tilde{\varvec{w}}_{{\varvec{k},\varvec{A}}} = \varvec{w}_{{\varvec{k},\varvec{A}}} - \left[ {\left( {\varvec{w}_{{\varvec{k},\varvec{A}}} - \varvec{v}_{{\varvec{k},{\mathbf{cone}}}} } \right) \cdot \varvec{n}_{{\varvec{k},\varvec{A}}} } \right]\varvec{n}_{{\varvec{k},\varvec{A}}}$$

(34)

After recalculating the velocity of the contact node k at time t + Δt, the corrected acceleration vector at the node must be recalculated as:

$$\dot{\varvec{w}} = \frac{{\tilde{\varvec{w}}_{\varvec{k},\varvec{A}}^{{\varvec{t}} + {\varvec{\Delta}}\varvec{t}} - \tilde{\varvec{w}}_{{\varvec{k},\varvec{A}}}^{\varvec{t}} }}{{\Delta t}}$$

(35)

where $\tilde{\varvec{w}}_{{\varvec{k},\varvec{A}}}^{{\varvec{t} + {\varvec{\Delta}}\varvec{t}}}$ is the velocity computed by Eq. (34) and $\tilde{\varvec{w}}_{{\varvec{k},\varvec{A}}}^{\varvec{t}}$ is the water velocity at the previous time step. The corrected water acceleration is used in the momentum equation for the mixture (Eq. 2), which is solved to obtain the acceleration of the solid phase.