H2M Liakopoulos benchmark

This page is based on a Jupyter notebook.

#1-modules
from IPython.display import Image
from scipy.special import exp1
from vtk.util.numpy_support import vtk_to_numpy
import os
import pyvista as pv
pv.set_jupyter_backend("static")
import numpy as np
import matplotlib.pyplot as plt
#import vtk
import matplotlib.tri as tri
import vtuIO

#2-settings (file handling, title, figures)
fig_dir = "./figures/"

out_dir = os.environ.get('OGS_TESTRUNNER_OUT_DIR', '_out')
if not os.path.exists(out_dir):
    os.makedirs(out_dir)

prj_file_test = "liakopoulos_TH2M.prj"
pvd_file_test = f"{out_dir}/result_liakopoulos.pvd"
vtu_mesh_file = "domain.vtu"

Image(filename = fig_dir + "ogs-jupyter-lab.png", width=150, height=100)

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Image(filename = fig_dir + "h2m-tet.png", width=150, height=100)

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#3-description (markdown)

H2M process: Liakopoulos benchmark

Problem description

The Liakopoulos experiment was dealing with a sand column which was filled with water first and then drained under gravity. A sketch of the related model set-up including initial and boundary conditions is shown in the above figure. A detailed description of the underlying OGS model is given in Grunwald et al. (2022). Two hydraulic models have been compared; two-phase flow with a mobile and a Richards flow with an immobile gas phase coupled to mechanical processes. Due to the absence of analytical solutions various numerical solutions have been compared in the past (see Grunwald et al., 2022).

The model parameters are given in the below table.

Parameter	Value	Unit
Permeability	\(k^0_\textrm{S}\) = 4.5 \(\times\) 10\(^{-13}\)	m\(^2\)
Porosity	\(\phi\) = 0.2975	-
Young’s modulus	\(E\) = 1.3	MPa
Poisson ratio	\(\nu\) = 0.4	-
Dynamic viscosity of gas phase	\(\mu_\textrm{GR}\) = 1.8 \(\times\) 10\(^{-5}\)	Pa s
Dynamic viscosity of liquid phase	\(\mu_\textrm{LR}\) = 1.0 \(\times\) 10\(^{-3}\)	Pa s
Density of liquid phase	\(\rho_\textrm{LR}\) = 1.0\(\times\) 10\(^3\)	kg m\(^{-3}\)
Density of solid phase	\(\rho_\textrm{SR}\) = 2.0\(\times\) 10\(^3\)	kg m\(^{-3}\)

Numerical solution

mesh = pv.read(vtu_mesh_file)
print("inspecting vtu_mesh_file")
mesh

inspecting vtu_mesh_file

Header

Data Arrays

UnstructuredGrid	Information
N Cells	100
N Points	202
X Bounds	0.000e+00, 1.000e-01
Y Bounds	0.000e+00, 1.000e+00
Z Bounds	0.000e+00, 0.000e+00
N Arrays	2

Name	Field	Type	N Comp	Min	Max
bulk_node_ids	Points	uint64	1	0.000e+00	2.010e+02
bulk_element_ids	Cells	uint64	1	0.000e+00	9.900e+01

plotter = pv.Plotter()
plotter.add_mesh(mesh, show_edges=True)
plotter.view_xy()
plotter.add_axes()
plotter.show_bounds(mesh, xlabel="x", ylabel="y")
plotter.show()

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Running OGS

#run ogs
import time
t0 = time.time()
print(f"ogs -o {out_dir} {prj_file_test} > {out_dir}/log.txt")
! ogs -o {out_dir} {prj_file_test} > {out_dir}/log.txt
tf = time.time()
print("computation time: ", round(tf - t0, 2), " s.")

ogs -o /var/lib/gitlab-runner/builds/vZ6vnZiU/1/ogs/build/release-all/Tests/Data/TH2M/H2M/Liakopoulos/ogs-jupyter-lab-h2m-2d-liakopoulos liakopoulos_TH2M.prj > /var/lib/gitlab-runner/builds/vZ6vnZiU/1/ogs/build/release-all/Tests/Data/TH2M/H2M/Liakopoulos/ogs-jupyter-lab-h2m-2d-liakopoulos/log.txt
computation time:  14.65  s.

Spatial Profiles

#alternative way
pv.set_plot_theme("document")
pv.set_jupyter_backend("static")
pt1 = (0,0,0)
pt2 = (0,1,0)
yaxis = pv.Line(pt1,pt2,resolution=2)
#print(yaxis)
line_mesh = mesh.slice_along_line(yaxis)
y_num = line_mesh.points[:,1]
reader = pv.get_reader(pvd_file_test)

reader.set_active_time_value(0.)
mesh = reader.read()[0]
line_mesh = mesh.slice_along_line(yaxis)
p_gas0 = line_mesh.point_data["gas_pressure"]
s_wat0 = line_mesh.point_data["saturation"]
p_cap0 = line_mesh.point_data["capillary_pressure"]
u_y0 = line_mesh.point_data["displacement"].T[1]

reader.set_active_time_value(120.)
mesh = reader.read()[0]
line_mesh = mesh.slice_along_line(yaxis)
p_gas120 = line_mesh.point_data["gas_pressure"]
s_wat120 = line_mesh.point_data["saturation"]
p_cap120 = line_mesh.point_data["capillary_pressure"]
u_y120 = line_mesh.point_data["displacement"].T[1]

reader.set_active_time_value(300.)
mesh = reader.read()[0]
line_mesh = mesh.slice_along_line(yaxis)
p_gas300 = line_mesh.point_data["gas_pressure"]
s_wat300 = line_mesh.point_data["saturation"]
p_cap300 = line_mesh.point_data["capillary_pressure"]
u_y300 = line_mesh.point_data["displacement"].T[1]

reader.set_active_time_value(4800.)
mesh = reader.read()[0]
line_mesh = mesh.slice_along_line(yaxis)
p_gas4800 = line_mesh.point_data["gas_pressure"]
s_wat4800 = line_mesh.point_data["saturation"]
p_cap4800 = line_mesh.point_data["capillary_pressure"]
u_y4800 = line_mesh.point_data["displacement"].T[1]

reader.set_active_time_value(7200.)
mesh = reader.read()[0]
line_mesh = mesh.slice_along_line(yaxis)
p_gas7200 = line_mesh.point_data["gas_pressure"]
s_wat7200 = line_mesh.point_data["saturation"]
p_cap7200 = line_mesh.point_data["capillary_pressure"]
u_y7200 = line_mesh.point_data["displacement"].T[1]

plt.rcParams['figure.figsize'] = (10,6)
plt.rcParams['lines.color'] = 'red'
plt.rcParams['legend.fontsize'] = 7
fig1, (ax1, ax2) = plt.subplots(2,2)
ax1[0].set_ylabel(r"$p_g$ / Pa")
ax1[1].set_ylabel(r"$p_c$ / Pa")
ax1[1].yaxis.set_label_position("right")
ax1[1].yaxis.tick_right()
ax2[0].set_ylabel(r"$s_l$ / -")
ax2[1].set_ylabel(r"$u_y$ / m")
ax2[1].yaxis.set_label_position("right")
ax2[1].yaxis.tick_right()
ax2[0].set_xlabel(r"$y$ / m")
ax2[1].set_xlabel(r"$y$ / m")
#gas pressure
ax1[0].plot(y_num, p_gas0, label=r"$p_g$ t=0")
ax1[0].plot(y_num, p_gas120, label=r"$p_g$ t=120")
ax1[0].plot(y_num, p_gas300, label=r"$p_g$ t=300")
ax1[0].plot(y_num, p_gas4800, label=r"$p_g$ t=4800")
ax1[0].plot(y_num, p_gas7200, label=r"$p_g$ t=7200")
ax1[0].legend()
ax1[0].grid()
#capillary pressure
ax1[1].plot(y_num, p_cap0, label=r"$p_c$ t=0")
ax1[1].plot(y_num, p_cap120, label=r"$p_c$ t=120")
ax1[1].plot(y_num, p_cap300, label=r"$p_c$ t=300")
ax1[1].plot(y_num, p_cap4800, label=r"$p_c$ t=4800")
ax1[1].plot(y_num, p_cap7200, label=r"$p_c$ t=7200")
ax1[1].legend()
ax1[1].grid()
#liquid saturation
ax2[0].plot(y_num, s_wat0, label=r"$s_l$ t=0")
ax2[0].plot(y_num, s_wat120, label=r"$s_l$ t=120")
ax2[0].plot(y_num, s_wat300, label=r"$s_l$ t=300")
ax2[0].plot(y_num, s_wat4800, label=r"$s_l$ t=4800")
ax2[0].plot(y_num, s_wat7200, label=r"$s_l$ t=7200")
ax2[0].legend()
ax2[0].grid()
#vertical displacement
ax2[1].plot(y_num, u_y0, label=r"$u_y$ t=0")
ax2[1].plot(y_num, u_y120, label=r"$u_y$ t=120")
ax2[1].plot(y_num, u_y300, label=r"$u_y$ t=300")
ax2[1].plot(y_num, u_y4800, label=r"$u_y$ t=4800")
ax2[1].plot(y_num, u_y7200, label=r"$u_y$ t=7200")
ax2[1].legend()
ax2[1].grid()

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Contour plots

theme = "Vertical cross-section"
print(theme)
file_vtu = f"{out_dir}/result_liakopoulos_t_7200.vtu"
m_plot=vtuIO.VTUIO(file_vtu, dim=2)
triang=tri.Triangulation(m_plot.points[:,0],m_plot.points[:,1])
p_plot = m_plot.get_point_field("gas_pressure")
s_plot = m_plot.get_point_field("saturation")
u_plot = m_plot.get_point_field("displacement").T[1]
fig, ax = plt.subplots(ncols=3, figsize=(5,10))
plt.subplots_adjust(wspace=2)
#fig.tight_layout()
#plt.subplot_tool()
contour_left = ax[0].tricontourf(triang, p_plot)
contour_mid = ax[1].tricontourf(triang, s_plot)
contour_right = ax[2].tricontourf(triang, u_plot)
fig.colorbar(contour_left,ax=ax[0],label='$p$ / [MPa]')
fig.colorbar(contour_mid,ax=ax[1],label='$S$ / [-]')
fig.colorbar(contour_right,ax=ax[2],label='$u$ / [m]')
plt.show()

Vertical cross-section
WARNING: Default interpolation backend changed to VTK. This might result in
slight changes of interpolated values if defaults are/were used.

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OGS links

description: https://www.opengeosys.org/docs/benchmarks/
project file: https://gitlab.opengeosys.org/ogs/ogs/-/blob/master/Tests/Data/TH2M/H2M/Liakopoulos/liakopoulos_TH2M.prj

Credits

Norbert Grunwald for set-up the OGS benchmark, https://gitlab.opengeosys.org/ogs/ogs/-/blob/master/Tests/Data/TH2M/H2M/Liakopoulos/liakopoulos_TH2M.prj

References

Grunwald, N., Lehmann, C., Maßmann, J., Naumov, D., Kolditz, O., Nagel, T., (2022): Non-isothermal two-phase flow in deformable porous media: systematic open-source implementation and verification procedure. Geomech. Geophys. Geo-Energy Geo-Resour. 8 (3), art. 107 https://doi.org/10.1007/s40948-022-00394-2

Kolditz, O., Görke, U.-J., Shao, H., Wang, W., (eds., 2012): Thermo-hydro-mechanical-chemical processes in porous media: Benchmarks and examples. Lecture Notes in Computational Science and Engineering 86, Springer, Berlin, Heidelberg, 391 pp https://link.springer.com/book/10.1007/978-3-642-27177-9

Lewis RW, Schrefler BA (1998): The finite element method in the static and dynamic deformation and consolidation of porous media. Wiley, New York https://www.wiley.com/en-us/The+Finite+Element+Method+in+the+Static+and+Dynamic+Deformation+and+Consolidation+of+Porous+Media%2C+2nd+Edition-p-9780471928096

Liakopoulos AC (1964): Transient flow through unsaturated porous media. PhD thesis. University of California, Berkeley, USA. sources: OGS BMB1 (sec. 13.2

This article was written by Norbert Grunwald, Olaf Kolditz. If you are missing something or you find an error please let us know.
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