- Email: [email protected]
Poroelastic Solid Flow with Material Point Method
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 175 (2017) 316 – 323
1st International Conference on the Material Point Method, MPM 2017
Poroelastic solid flow with material point method Bruno Zuada Coelhoa,∗, Alexander Rohea , Kenichi Sogab a Department b Department
of Geo-Engineering, Deltares, Boussinesqweg 1, 2629HV Delft, The Netherlands of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, United Kingdom
Abstract This paper presents the numerical modelling of one and two dimensional poroelastic solid flows, using the Material Point Method with double point formulation. The double point formulation offers the convenience of allowing for transitions in the flow conditions of the liquid, between free surface flow and groundwater flow. The numerical model is validated by comparing the solid flow velocity with the analytical solution. The influence of the Young’s modulus on the solid flow velocity is discussed for both one and two dimensional analysis cases. It is shown that the solid stiffness has an effect on the poroelastic flow velocity, due to swelling and bending for the one and two dimensional cases, respectively. c 2017 © Published by Elsevier Ltd. This by Elsevier B.V.is an open access article under the CC BY-NC-ND license 2016The TheAuthors. Authors. Published (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 1 st International Conference on the Material Point Method. Peer-review under responsibility of the organizing committee of the 1st International Conference on the Material Point Method Keywords: material point method; geocontainers; double point formulation; large deformations.
1. Introduction The modelling of large deformations is of great importance for engineering problems. In order to allow the modelling of problems involving the interaction between solid and fluid, as often is the case for offshore applications, a double point formulation has previously been developed [1–3]. This formulation extends the classical two-phase approach to model saturated solids [4], which cannot capture the transition in state for both fluid and solid phases. The key aspect of this formulation is that it considers two sets of Lagrangian material points to represent soil and water. Use of Lagrangian particles conserves mass and allows history dependent material models to be used. Also, the discrete equations for the momentum balances are obtained on the background grid similar to the finite element method with an updated Lagrangian formulation. Using the double point formulation, problems involving the fluidisation and sedimentation of solid can be modelled, as well as problems where free liquid flows through a porous solid, and there occurs a transition in the fluid state from free surface flow to groundwater. The modelling of geocontainers is one of the offshore applications that illustrates the necessity of the double point formulation. Geocontainers are sand filled containers encapsulated by a geotextile membrane, often used for coastal applications, such as revetments and breakwaters, mainly due to its resistance to erosion. Although the single ∗
Corresponding author. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 1st International Conference on the Material Point Method
doi:10.1016/j.proeng.2017.01.035
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
Fig. 1: Geometry of the poroelastic solid and water column.
point formulation is not the most appropriate strategy to model the installation of geocontainers, it has been the most widely used methodology [5–7]. The need for the double point formulation is related to the need to model both large deformations, and the interaction between the solid and liquid materials, and the change in flow conditions of the liquid (between free surface and groundwater flow), as the liquid flows around and through the solid. In the present paper, two problems where the transition of the fluid conditions occurs are analysed: one and two dimensional poroelastic flow through a porous solid immersed inside a Newtonian fluid. The latter resembles a geocontainer. The influence of the Young’s modulus on the flow velocity of the falling solid by gravity will be discussed. 2. One dimensional poroelastic solid flow 2.1. Analysis description The numerical analyses have been performed with the Material Point Method (MPM), following a double point formulation to simulate the interaction between solid and fluid [2,3]. The validation of the MPM has been performed by simulating an one dimensional poroelastic solid falling through a Newtonian liquid by gravity. This problem was chosen as it offers the convenience of a closed form analytical solution for the steady state velocity of the poroelastic solid flow. The geometry of the problem is shown in Figure 1. The dimensions H and L were assumed as 0.1m and 1m, respectively. The system of equations is solved explicitly in the time domain. The domain was discretised using low-order tetrahedral elements (in total 780 elements), with ten solid and ten liquid material points for the saturated poroelastic solid elements and ten liquid material points for the liquid elements. The nodes at the bottom of the column were fixed in the vertical direction. Both solid and liquid materials were assumed to be elastic. Table 1 presents the material properties. Table 1: Material properties for the poroelastic solid flow. Parameter Density solid [kg/m3 ] Solid Young modulus [kPa] Solid Poisson ratio [−] Solid initial porosity [−] Solid grain size diameter [m] Liquid density [kg/m3 ] Liquid bulk modulus [kPa] Liquid viscosity [kPa · s]
Symbol ρs Es ν n0 Ds ρl Kl μl
Value 2700 10 × 103 0.3 0.4 2 × 10−3 1000 20 × 203 1 × 10−6
317
318
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
2.2. Analytical solution The steady state velocity of a rigid poroelastic solid falling through a liquid by gravity offers the conviniency of an analytical solution [8]. The velocity v¯ of the solid follows:
v¯ = − μ κ
γ , + ρl √Fκ |v s |
(1)
where γ represents the submerged volumetric weight, κ the intrinsic permeability, and F the coefficient from Ergun’s drag force [9]: F= √
B An3/2
,
(2)
with parameters A and B being, respectively, 150 and 1.75. The intrinsic permeability κ is defined as [10]: κ=
D2s n3 , A (1 − n)2
(3)
where D s is the grain size diameter. 2.3. Results 2.3.1. Validation of the numerical model Figure 2 shows the comparison between the analytical and numerical results for the one dimensional poroelastic solid flow. It follows that the numerical results obtained with MPM are in agreement with the analytical solution. Due to the high permeability (solid grain diameter D s ) the transient behaviour of the flow is not noticeable in Figure 2. The numerical velocity computed with MPM oscillates around the analytical value (Figure 2b), initially (up to normalised time 0.1) with a sinusoidal shape, followed by a non periodic oscilations for higher times. The sinusoidal oscillation is related to the fact that the liquid is modelled as a compressive material, causing the oscillation of the poroelastic solid due to inertial force developed on the liquid. The non periodic oscillation is caused by element crossing, due to the discontinuity in the liquid concentration ratio, at the boundary between the liquid and the solid.
(a)
(b)
Fig. 2: Comparison between the analytical and numerical results: (a) displacement and (b) velocity of the poroelastic solid flow.
319
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
2.3.2. Solid Young modulus The effect of the solid Young’s modulus on the one dimensional poroelastic flow was investigated for a range of values ranging from 50 kPa to 50 MPa. The grain size diameter was 5 × 10−4 m, to which corresponds an intrinsic permeability of κ = 2.96 × 10−10 m2 (Darcy permeability: K = 2.96 × 10−3 ms−1 ). The particle diameter (porosity) was reduced in relation to the previous analysis, in order to capture the transient behaviour of the poroelastic solid. Figure 3a presents the results for the several Young’s moduli. Transient behaviour of the poroelastic solid is clearly identified for the analysis with a Young modulus of 50 kPa. Up to a normalised time of 0.004 the solid is accelerating, after which it reaches the steady state velocity. This transient behaviour is related to the consolidation that develops during the flow. The flow occurs because there is a pressure gradient between the top and bottom of the solid, which causes an excess of pore water pressure at the bottom. This results in the occurrence of consolidation (dissipation of the excess of pore water pressure) and swelling of the poroelastic solid (extension). For the remaining Young’s modulus cases, the transient response is less noticeable. This is related to the higher Young’s modulus that causes a faster consolidation and a smaller extension of the solid. This is illustrated in Figure 3b, which shows the solid extension for the different Young’s modulus cases. It is clear that the smaller the Young’s modulus is, the larger the solid extension is. For Young’s modulus of 5000 and 50000 kPa, the extension is very close to 0 %. The extension of the poroelastic solid can be expressed as a function of the consolidation time T , defined as:
T=
cv t , d2
(4)
where cv corresponds to the consolidation coefficient and d is the drainage path length. Figure 4 shows that the numerical results for the normalised extension of the poroelastic solid (defined as the ratio between the extension and the maximum extension) follow the trend of the analytical solution for the one dimensional consolidation problem (oedometer consolidation) [11]. The differences between the two results are related to the fact that the poroelastic flow does not have the same boundary conditions as the oedometer test, as the solid is moving, combined with liquid pressure fluctuations at the interface between the solid and liquid materials due to numerical issues. The solid velocity oscillates significantly around the average value for the softer solids (E = 50 and E = 500 kPa). These oscillations are likely related to the fact that these solids experience more extension, hence more liquid pressure oscillations within the solid.
(a)
(b)
Fig. 3: Effect of the solid Young’s modulus on: (a) poroelastic flow velocity and (b) swelling of the poroelastic solid due to consolidation.
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Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
Fig. 4: Normalised extension of the poroelastic solid due to consolidation (bold line represents moving average).
Fig. 5: Geometry of the two dimensional poroelastic flow (simplified geocontainer).
3. Submerged two dimensional poroelastic solid flow 3.1. Analysis description This section presents the modelling of a two dimensional poroelastic solid flow case, which resembles a simplified geocontainer installation, and shows the appropriateness of the MPM double point formulation for the analysis of this type of problem. Figure 5 shows the geometry of the problem (dimensions assumed as: H = 1, L = 2 and B = 4 m). The problem was assumed to be symmetric, with the nodes at the bottom of the model fixed in the vertical direction, and at the side in the horizontal direction. In total the model comprised 6300 elements with the following material point distribution: • 10 material points for solid and liquid phases in the poroelastic material; • 20 materials points for the liquid phase in the liquid material. The poroelastic solid and liquid solids were assumed as elastic materials. Table 2 presents the adopted properties for the solid and liquid materials.
321
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323 Table 2: Material properties for the two dimensional poroelastic solid flow. Parameter Density solid [kg/m3 ] Solid Young modulus [kPa] Solid Poisson ratio [−] Solid initial porosity [−] Solid grain diameter [m] Liquid density [kg/m3 ] Liquid bulk modulus [kPa] Liquid viscosity [kPa · s]
Symbol ρs Es ν n0 Ds ρl Kl μl
Value 2700 10 × 503 0.3 0.5 2 × 10−3 1000 100 × 103 1 × 10−6
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 6: Flow of the two dimensional poroelastic solid at times: (a) t = 1.25s, (b) t = 2.5s, (c) t = 3.75s, (d) t = 5s, (e) t = 6.25s, (f) t = 7.5s.
3.2. Overall response Figure 6 shows, at several distinct times, the movement of the poroelastic solid through the fluid. The poroelastic solid is found to move down through the liquid as a rigid body, i.e. without significant bending. From the figure it is possible to see that the water flows into the bottom of the solid, as the blue material points (initially liquid material points) enter the black material points (solid material points), and flows out of the top of the solid as the green
322
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
(a)
(b)
Fig. 7: Effect of the solid Young modulus on the: (a) vertical displacement and (b) vertical velocity of point A of the poroelastic solid.
Fig. 8: Deformation of the poroelastic solid for a Young modulus of 500 kPa at t = 5s.
material points (initially groundwater liquid material points) leave the black material points (solid material points). The movement of the solid is mainly due to the liquid that flows around the solid, rather than through it. 3.3. Solid Young modulus The effect of the solid Young’s modulus on the two dimensional poroelastic flow was investigated for three different values: 500, 5000 and 50000 kPa. All the remaining properties were the same as presented in Table 2. Figure 7 shows the effect of the Young’s modulus on the solid vertical displacement and velocity for point A (see Figure 5). The velocity is found to increase with the reduction of the solid Young modulus. This increase of solid velocity is related to the bending deformation of the poroelastic solid. As the stiffness decreases, the solid exhibits higher deformation due to bending. As the solid bends, the water flow around it is facilitated, and the solid velocity increases. This is illustrated in Figure 8, which shows the poroelastic solid displacement for a Young modulus of 500 kPa at t = 5s (Figure 6d shows the poroelastic solid displacement at the same time). 4. Conclusions This paper presented the validation and two numerical applications of the MPM double point formulation [2,3].
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
The one dimensional poroelastic flow was used to illustrate the accuracy of the numerical results. Additionally, it was shown that there is a transient response of the poroelastic flow velocity, where the solid swells due to consolidation. The poroelastic solid velocity was found to have a sinusoidal oscillation around the analytical flow velocity, due to the fluid compressibility, and element crossing. However the average value of the velocity is in agreement with the analytical solution. A possible mitigation for these oscillations can be the implementation of incompressible fluid, and a redifinition of the transition zone between solid and liquid to address the discontinuity in the liquid concentration ratio. The two dimensional poroelastic analysis resembled the installation of a geocontainer. It was found that the flow velocity of the falling poroelastic solid depends on the solid Young’s modulus. As the solid Young modulus reduces, the solid bending curvature is greater, which improves the flow around the solid, therefore the solid velocity is higher. Acknowledgements The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. PIAP-GA-2012-324522 ”MPM-DREDGE”. References [1] S. Bandara, K. Soga, Coupling of soil deformation and pore fluid flow using material point method, Computers and Geotechnics 63 (2015) 199–214. [2] M. Martinelli, A. Rohe, Modelling fluidisation and sedimentation using material point method, in: 1 st Pan-American Congress on Computational Mechanics, CIMNE, 2015, pp. 1–12. [3] M. Martinelli, A. Rohe, Modelling fluidisation and sedimentation with the material point method, in: Incontro Annuale dei Giovani Ingegneri Geotecnici, 2015, pp. 1–4. [4] O. Zienkiewicz, A. Chan, M. Pastor, B. Schrefler, T. Shiomi, Computational Geomechanics, John Wiley & Sons, 1999. [5] Z. Wie¸ckowski, Enhancement of the Material Point Method for Fluid-Structure Interaction and Erosion, Technical Report, Seventh Framework Programme, 2013. [6] J. Grabe, E. Heins, G. Qiu, K. Werth, Numerical investigation of loading on geotextile sand containers, in: 10th International Conference on Geosynthetics, International Geosynthetics Society, 2014, pp. 1–6. [7] F. Hamad, D. Stolle, P. Vermeer, Modelling of membranes in the material point method with applications, International Journal for Numerical and Analytical Methods in Geomechanics 39 (2015) 833–853. [8] M. Martinelli, 2 Layer Formulation. Joint MPM Software, Technical Report, Deltares, 2015. [9] S. Ergun, Fluid flow through packed column, Chemical Engineering Progress 2 (1952) 89–94. [10] J. Bear, Dynamics of Fluids in Porous Media, Elsevier, 1972. [11] A. Verruijt, Soil Mechanics Geomechanics, Delft University of Technology, 2006.
323
ScienceDirect Procedia Engineering 175 (2017) 316 – 323
1st International Conference on the Material Point Method, MPM 2017
Poroelastic solid flow with material point method Bruno Zuada Coelhoa,∗, Alexander Rohea , Kenichi Sogab a Department b Department
of Geo-Engineering, Deltares, Boussinesqweg 1, 2629HV Delft, The Netherlands of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, United Kingdom
Abstract This paper presents the numerical modelling of one and two dimensional poroelastic solid flows, using the Material Point Method with double point formulation. The double point formulation offers the convenience of allowing for transitions in the flow conditions of the liquid, between free surface flow and groundwater flow. The numerical model is validated by comparing the solid flow velocity with the analytical solution. The influence of the Young’s modulus on the solid flow velocity is discussed for both one and two dimensional analysis cases. It is shown that the solid stiffness has an effect on the poroelastic flow velocity, due to swelling and bending for the one and two dimensional cases, respectively. c 2017 © Published by Elsevier Ltd. This by Elsevier B.V.is an open access article under the CC BY-NC-ND license 2016The TheAuthors. Authors. Published (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 1 st International Conference on the Material Point Method. Peer-review under responsibility of the organizing committee of the 1st International Conference on the Material Point Method Keywords: material point method; geocontainers; double point formulation; large deformations.
1. Introduction The modelling of large deformations is of great importance for engineering problems. In order to allow the modelling of problems involving the interaction between solid and fluid, as often is the case for offshore applications, a double point formulation has previously been developed [1–3]. This formulation extends the classical two-phase approach to model saturated solids [4], which cannot capture the transition in state for both fluid and solid phases. The key aspect of this formulation is that it considers two sets of Lagrangian material points to represent soil and water. Use of Lagrangian particles conserves mass and allows history dependent material models to be used. Also, the discrete equations for the momentum balances are obtained on the background grid similar to the finite element method with an updated Lagrangian formulation. Using the double point formulation, problems involving the fluidisation and sedimentation of solid can be modelled, as well as problems where free liquid flows through a porous solid, and there occurs a transition in the fluid state from free surface flow to groundwater. The modelling of geocontainers is one of the offshore applications that illustrates the necessity of the double point formulation. Geocontainers are sand filled containers encapsulated by a geotextile membrane, often used for coastal applications, such as revetments and breakwaters, mainly due to its resistance to erosion. Although the single ∗
Corresponding author. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 1st International Conference on the Material Point Method
doi:10.1016/j.proeng.2017.01.035
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
Fig. 1: Geometry of the poroelastic solid and water column.
point formulation is not the most appropriate strategy to model the installation of geocontainers, it has been the most widely used methodology [5–7]. The need for the double point formulation is related to the need to model both large deformations, and the interaction between the solid and liquid materials, and the change in flow conditions of the liquid (between free surface and groundwater flow), as the liquid flows around and through the solid. In the present paper, two problems where the transition of the fluid conditions occurs are analysed: one and two dimensional poroelastic flow through a porous solid immersed inside a Newtonian fluid. The latter resembles a geocontainer. The influence of the Young’s modulus on the flow velocity of the falling solid by gravity will be discussed. 2. One dimensional poroelastic solid flow 2.1. Analysis description The numerical analyses have been performed with the Material Point Method (MPM), following a double point formulation to simulate the interaction between solid and fluid [2,3]. The validation of the MPM has been performed by simulating an one dimensional poroelastic solid falling through a Newtonian liquid by gravity. This problem was chosen as it offers the convenience of a closed form analytical solution for the steady state velocity of the poroelastic solid flow. The geometry of the problem is shown in Figure 1. The dimensions H and L were assumed as 0.1m and 1m, respectively. The system of equations is solved explicitly in the time domain. The domain was discretised using low-order tetrahedral elements (in total 780 elements), with ten solid and ten liquid material points for the saturated poroelastic solid elements and ten liquid material points for the liquid elements. The nodes at the bottom of the column were fixed in the vertical direction. Both solid and liquid materials were assumed to be elastic. Table 1 presents the material properties. Table 1: Material properties for the poroelastic solid flow. Parameter Density solid [kg/m3 ] Solid Young modulus [kPa] Solid Poisson ratio [−] Solid initial porosity [−] Solid grain size diameter [m] Liquid density [kg/m3 ] Liquid bulk modulus [kPa] Liquid viscosity [kPa · s]
Symbol ρs Es ν n0 Ds ρl Kl μl
Value 2700 10 × 103 0.3 0.4 2 × 10−3 1000 20 × 203 1 × 10−6
317
318
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
2.2. Analytical solution The steady state velocity of a rigid poroelastic solid falling through a liquid by gravity offers the conviniency of an analytical solution [8]. The velocity v¯ of the solid follows:
v¯ = − μ κ
γ , + ρl √Fκ |v s |
(1)
where γ represents the submerged volumetric weight, κ the intrinsic permeability, and F the coefficient from Ergun’s drag force [9]: F= √
B An3/2
,
(2)
with parameters A and B being, respectively, 150 and 1.75. The intrinsic permeability κ is defined as [10]: κ=
D2s n3 , A (1 − n)2
(3)
where D s is the grain size diameter. 2.3. Results 2.3.1. Validation of the numerical model Figure 2 shows the comparison between the analytical and numerical results for the one dimensional poroelastic solid flow. It follows that the numerical results obtained with MPM are in agreement with the analytical solution. Due to the high permeability (solid grain diameter D s ) the transient behaviour of the flow is not noticeable in Figure 2. The numerical velocity computed with MPM oscillates around the analytical value (Figure 2b), initially (up to normalised time 0.1) with a sinusoidal shape, followed by a non periodic oscilations for higher times. The sinusoidal oscillation is related to the fact that the liquid is modelled as a compressive material, causing the oscillation of the poroelastic solid due to inertial force developed on the liquid. The non periodic oscillation is caused by element crossing, due to the discontinuity in the liquid concentration ratio, at the boundary between the liquid and the solid.
(a)
(b)
Fig. 2: Comparison between the analytical and numerical results: (a) displacement and (b) velocity of the poroelastic solid flow.
319
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
2.3.2. Solid Young modulus The effect of the solid Young’s modulus on the one dimensional poroelastic flow was investigated for a range of values ranging from 50 kPa to 50 MPa. The grain size diameter was 5 × 10−4 m, to which corresponds an intrinsic permeability of κ = 2.96 × 10−10 m2 (Darcy permeability: K = 2.96 × 10−3 ms−1 ). The particle diameter (porosity) was reduced in relation to the previous analysis, in order to capture the transient behaviour of the poroelastic solid. Figure 3a presents the results for the several Young’s moduli. Transient behaviour of the poroelastic solid is clearly identified for the analysis with a Young modulus of 50 kPa. Up to a normalised time of 0.004 the solid is accelerating, after which it reaches the steady state velocity. This transient behaviour is related to the consolidation that develops during the flow. The flow occurs because there is a pressure gradient between the top and bottom of the solid, which causes an excess of pore water pressure at the bottom. This results in the occurrence of consolidation (dissipation of the excess of pore water pressure) and swelling of the poroelastic solid (extension). For the remaining Young’s modulus cases, the transient response is less noticeable. This is related to the higher Young’s modulus that causes a faster consolidation and a smaller extension of the solid. This is illustrated in Figure 3b, which shows the solid extension for the different Young’s modulus cases. It is clear that the smaller the Young’s modulus is, the larger the solid extension is. For Young’s modulus of 5000 and 50000 kPa, the extension is very close to 0 %. The extension of the poroelastic solid can be expressed as a function of the consolidation time T , defined as:
T=
cv t , d2
(4)
where cv corresponds to the consolidation coefficient and d is the drainage path length. Figure 4 shows that the numerical results for the normalised extension of the poroelastic solid (defined as the ratio between the extension and the maximum extension) follow the trend of the analytical solution for the one dimensional consolidation problem (oedometer consolidation) [11]. The differences between the two results are related to the fact that the poroelastic flow does not have the same boundary conditions as the oedometer test, as the solid is moving, combined with liquid pressure fluctuations at the interface between the solid and liquid materials due to numerical issues. The solid velocity oscillates significantly around the average value for the softer solids (E = 50 and E = 500 kPa). These oscillations are likely related to the fact that these solids experience more extension, hence more liquid pressure oscillations within the solid.
(a)
(b)
Fig. 3: Effect of the solid Young’s modulus on: (a) poroelastic flow velocity and (b) swelling of the poroelastic solid due to consolidation.
320
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
Fig. 4: Normalised extension of the poroelastic solid due to consolidation (bold line represents moving average).
Fig. 5: Geometry of the two dimensional poroelastic flow (simplified geocontainer).
3. Submerged two dimensional poroelastic solid flow 3.1. Analysis description This section presents the modelling of a two dimensional poroelastic solid flow case, which resembles a simplified geocontainer installation, and shows the appropriateness of the MPM double point formulation for the analysis of this type of problem. Figure 5 shows the geometry of the problem (dimensions assumed as: H = 1, L = 2 and B = 4 m). The problem was assumed to be symmetric, with the nodes at the bottom of the model fixed in the vertical direction, and at the side in the horizontal direction. In total the model comprised 6300 elements with the following material point distribution: • 10 material points for solid and liquid phases in the poroelastic material; • 20 materials points for the liquid phase in the liquid material. The poroelastic solid and liquid solids were assumed as elastic materials. Table 2 presents the adopted properties for the solid and liquid materials.
321
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323 Table 2: Material properties for the two dimensional poroelastic solid flow. Parameter Density solid [kg/m3 ] Solid Young modulus [kPa] Solid Poisson ratio [−] Solid initial porosity [−] Solid grain diameter [m] Liquid density [kg/m3 ] Liquid bulk modulus [kPa] Liquid viscosity [kPa · s]
Symbol ρs Es ν n0 Ds ρl Kl μl
Value 2700 10 × 503 0.3 0.5 2 × 10−3 1000 100 × 103 1 × 10−6
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 6: Flow of the two dimensional poroelastic solid at times: (a) t = 1.25s, (b) t = 2.5s, (c) t = 3.75s, (d) t = 5s, (e) t = 6.25s, (f) t = 7.5s.
3.2. Overall response Figure 6 shows, at several distinct times, the movement of the poroelastic solid through the fluid. The poroelastic solid is found to move down through the liquid as a rigid body, i.e. without significant bending. From the figure it is possible to see that the water flows into the bottom of the solid, as the blue material points (initially liquid material points) enter the black material points (solid material points), and flows out of the top of the solid as the green
322
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
(a)
(b)
Fig. 7: Effect of the solid Young modulus on the: (a) vertical displacement and (b) vertical velocity of point A of the poroelastic solid.
Fig. 8: Deformation of the poroelastic solid for a Young modulus of 500 kPa at t = 5s.
material points (initially groundwater liquid material points) leave the black material points (solid material points). The movement of the solid is mainly due to the liquid that flows around the solid, rather than through it. 3.3. Solid Young modulus The effect of the solid Young’s modulus on the two dimensional poroelastic flow was investigated for three different values: 500, 5000 and 50000 kPa. All the remaining properties were the same as presented in Table 2. Figure 7 shows the effect of the Young’s modulus on the solid vertical displacement and velocity for point A (see Figure 5). The velocity is found to increase with the reduction of the solid Young modulus. This increase of solid velocity is related to the bending deformation of the poroelastic solid. As the stiffness decreases, the solid exhibits higher deformation due to bending. As the solid bends, the water flow around it is facilitated, and the solid velocity increases. This is illustrated in Figure 8, which shows the poroelastic solid displacement for a Young modulus of 500 kPa at t = 5s (Figure 6d shows the poroelastic solid displacement at the same time). 4. Conclusions This paper presented the validation and two numerical applications of the MPM double point formulation [2,3].
Bruno Zuada Coelho et al. / Procedia Engineering 175 (2017) 316 – 323
The one dimensional poroelastic flow was used to illustrate the accuracy of the numerical results. Additionally, it was shown that there is a transient response of the poroelastic flow velocity, where the solid swells due to consolidation. The poroelastic solid velocity was found to have a sinusoidal oscillation around the analytical flow velocity, due to the fluid compressibility, and element crossing. However the average value of the velocity is in agreement with the analytical solution. A possible mitigation for these oscillations can be the implementation of incompressible fluid, and a redifinition of the transition zone between solid and liquid to address the discontinuity in the liquid concentration ratio. The two dimensional poroelastic analysis resembled the installation of a geocontainer. It was found that the flow velocity of the falling poroelastic solid depends on the solid Young’s modulus. As the solid Young modulus reduces, the solid bending curvature is greater, which improves the flow around the solid, therefore the solid velocity is higher. Acknowledgements The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. PIAP-GA-2012-324522 ”MPM-DREDGE”. References [1] S. Bandara, K. Soga, Coupling of soil deformation and pore fluid flow using material point method, Computers and Geotechnics 63 (2015) 199–214. [2] M. Martinelli, A. Rohe, Modelling fluidisation and sedimentation using material point method, in: 1 st Pan-American Congress on Computational Mechanics, CIMNE, 2015, pp. 1–12. [3] M. Martinelli, A. Rohe, Modelling fluidisation and sedimentation with the material point method, in: Incontro Annuale dei Giovani Ingegneri Geotecnici, 2015, pp. 1–4. [4] O. Zienkiewicz, A. Chan, M. Pastor, B. Schrefler, T. Shiomi, Computational Geomechanics, John Wiley & Sons, 1999. [5] Z. Wie¸ckowski, Enhancement of the Material Point Method for Fluid-Structure Interaction and Erosion, Technical Report, Seventh Framework Programme, 2013. [6] J. Grabe, E. Heins, G. Qiu, K. Werth, Numerical investigation of loading on geotextile sand containers, in: 10th International Conference on Geosynthetics, International Geosynthetics Society, 2014, pp. 1–6. [7] F. Hamad, D. Stolle, P. Vermeer, Modelling of membranes in the material point method with applications, International Journal for Numerical and Analytical Methods in Geomechanics 39 (2015) 833–853. [8] M. Martinelli, 2 Layer Formulation. Joint MPM Software, Technical Report, Deltares, 2015. [9] S. Ergun, Fluid flow through packed column, Chemical Engineering Progress 2 (1952) 89–94. [10] J. Bear, Dynamics of Fluids in Porous Media, Elsevier, 1972. [11] A. Verruijt, Soil Mechanics Geomechanics, Delft University of Technology, 2006.
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