def test_biot(self):
self.setup()
g, g_larger = self.g, self.g_larger
specified_data = {"inverter": "python", "biot_alpha": 1}
mechanics_keyword = "mechanics"
flow_keyword = "flow"
data_small = pp.initialize_default_data(
g, {}, mechanics_keyword, specified_parameters=specified_data)
def add_flow_data(g, d):
d[pp.DISCRETIZATION_MATRICES][flow_keyword] = {}
d[pp.PARAMETERS][flow_keyword] = {
"bc": pp.BoundaryCondition(g),
"second_order_tensor":
pp.SecondOrderTensor(np.ones(g.num_cells)),
"bc_values": np.zeros(g.num_faces),
"inverter": "python",
"mass_weight": np.ones(g.num_cells),
"biot_alpha": 1,
}
discr = pp.Biot(mechanics_keyword=mechanics_keyword,
flow_keyword=flow_keyword)
add_flow_data(g, data_small)
discr.discretize(g, data_small)
# Discretization on a small problem
# Perturb one node
g_larger.nodes[0, 2] += 0.2
# Faces that have their geometry changed
update_faces = np.array([2, 21, 22])
# Perturb the permeability in some cells on the larger grid
mu, lmbda = np.ones(g_larger.num_cells), np.ones(g_larger.num_cells)
high_coeff_cells = np.array([7, 12])
stiff_larger = pp.FourthOrderTensor(mu, lmbda)
specified_data_larger = {
"fourth_order_tensor": stiff_larger,
"biot_alpha": 1
}
# Do a full discretization on the larger grid
data_full = pp.initialize_default_data(
g_larger, {},
mechanics_keyword,
specified_parameters=specified_data_larger)
add_flow_data(g_larger, data_full)
discr.discretize(g_larger, data_full)
# Cells that will be marked as updated, either due to changed parameters or
# the newly defined topology
update_cells = np.union1d(self.new_cells, high_coeff_cells)
updates = {
"modified_cells": update_cells,
# "modified_faces": update_faces,
"map_cells": self.cell_map,
"map_faces": self.face_map,
}
# Data dictionary for the two-step discretization
data_partial = pp.initialize_default_data(
g_larger, {},
mechanics_keyword,
specified_parameters=specified_data_larger)
add_flow_data(g_larger, data_partial)
data_partial["update_discretization"] = updates
self._update_and_compare(
data_small,
data_partial,
data_full,
g_larger,
keywords=[flow_keyword, mechanics_keyword],
discr=discr,
)
def test_one_cell_a_time_node_keyword(self):
# Update one and one cell, and verify that the result is the same as
# with a single computation. The test is similar to what will happen
# with a memory-constrained splitting.
g = pp.CartGrid([3, 3])
g.compute_geometry()
# Assign random permeabilities, for good measure
np.random.seed(42)
mu = np.random.random(g.num_cells)
lmbda = np.random.random(g.num_cells)
stiffness = pp.FourthOrderTensor(mu=mu, lmbda=lmbda)
stress = sps.csr_matrix((g.num_faces * g.dim, g.num_cells * g.dim))
bound_stress = sps.csr_matrix(
(g.num_faces * g.dim, g.num_faces * g.dim))
faces_covered = np.zeros(g.num_faces, np.bool)
bnd = pp.BoundaryConditionVectorial(g)
specified_data = {
"fourth_order_tensor": stiffness,
"bc": bnd,
"inverter": "python",
}
keyword = "mechanics"
data = pp.initialize_default_data(g, {},
keyword,
specified_parameters=specified_data)
discr = pp.Mpsa(keyword)
discr.discretize(g, data)
stress_full = data[pp.DISCRETIZATION_MATRICES][keyword][
discr.stress_matrix_key]
bound_stress_full = data[pp.DISCRETIZATION_MATRICES][keyword][
discr.bound_stress_matrix_key]
cn = g.cell_nodes()
for ci in range(g.num_cells):
ind = np.zeros(g.num_cells)
ind[ci] = 1
nodes = np.squeeze(np.where(cn * ind > 0))
data[pp.PARAMETERS][keyword]["specified_nodes"] = nodes
discr.discretize(g, data)
partial_stress = data[pp.DISCRETIZATION_MATRICES][keyword][
discr.stress_matrix_key]
partial_bound = data[pp.DISCRETIZATION_MATRICES][keyword][
discr.bound_stress_matrix_key]
active_faces = data[pp.PARAMETERS][keyword]["active_faces"]
if np.any(faces_covered):
del_faces = self.expand_indices_nd(
np.where(faces_covered)[0], g.dim)
# del_faces is already expanded, set dimension to 1
pp.fvutils.remove_nonlocal_contribution(
del_faces, 1, partial_stress, partial_bound)
faces_covered[active_faces] = True
stress += partial_stress
bound_stress += partial_bound
self.assertTrue((stress_full - stress).max() < 1e-8)
self.assertTrue((stress_full - stress).min() > -1e-8)
self.assertTrue((bound_stress - bound_stress_full).max() < 1e-8)
self.assertTrue((bound_stress - bound_stress_full).min() > -1e-8)
def _test_one_cell_a_time_node_keyword(self):
# EK: this test makes less sense after the notion of partial updates were
# changed (git commit 41f754940). Decactive the test for now, pending an update
# of the notion of discretization updates for FV methods.
# Update one and one cell, and verify that the result is the same as
# with a single computation. The test is similar to what will happen
# with a memory-constrained splitting.
g = pp.CartGrid([3, 3])
g.compute_geometry()
# Assign random permeabilities, for good measure
np.random.seed(42)
mu = np.random.random(g.num_cells)
lmbda = np.random.random(g.num_cells)
stiffness = pp.FourthOrderTensor(mu=mu, lmbda=lmbda)
nd = g.dim
nf = g.num_faces
nc = g.num_cells
grad_p = sps.csr_matrix((nf * nd, nc))
div_u = sps.csr_matrix((nc, nc * nd))
bound_div_u = sps.csr_matrix((nc, nf * nd))
stab = sps.csr_matrix((nc, nc))
bound_displacement_pressure = sps.csr_matrix((nf * nd, nc))
faces_covered = np.zeros(g.num_faces, np.bool)
cells_covered = np.zeros(g.num_cells, np.bool)
bnd = pp.BoundaryConditionVectorial(g)
specified_data = {
"fourth_order_tensor": stiffness,
"bc": bnd,
# "inverter": "python",
"biot_alpha": 1,
}
keyword_mech = "mechanics"
keyword_flow = "flow"
data = pp.initialize_default_data(g, {},
keyword_mech,
specified_parameters=specified_data)
data = pp.initialize_default_data(g, data, keyword_flow)
discr = pp.Biot()
discr.discretize(g, data)
div_u_full = data[pp.DISCRETIZATION_MATRICES][keyword_flow][
discr.div_u_matrix_key]
bound_div_u_full = data[pp.DISCRETIZATION_MATRICES][keyword_flow][
discr.bound_div_u_matrix_key]
stab_full = data[pp.DISCRETIZATION_MATRICES][keyword_flow][
discr.stabilization_matrix_key]
grad_p_full = data[pp.DISCRETIZATION_MATRICES][keyword_mech][
discr.grad_p_matrix_key]
bound_pressure_full = data[pp.DISCRETIZATION_MATRICES][keyword_mech][
discr.bound_pressure_matrix_key]
cn = g.cell_nodes()
for ci in range(g.num_cells):
ind = np.zeros(g.num_cells)
ind[ci] = 1
nodes = np.squeeze(np.where(cn * ind > 0))
data[pp.PARAMETERS][keyword_mech]["specified_nodes"] = nodes
discr.discretize(g, data)
partial_div_u = data[pp.DISCRETIZATION_MATRICES][keyword_flow][
discr.div_u_matrix_key]
partial_bound_div_u = data[pp.DISCRETIZATION_MATRICES][
keyword_flow][discr.bound_div_u_matrix_key]
partial_grad_p = data[pp.DISCRETIZATION_MATRICES][keyword_mech][
discr.grad_p_matrix_key]
partial_stab = data[pp.DISCRETIZATION_MATRICES][keyword_flow][
discr.stabilization_matrix_key]
partial_bound_pressure = data[pp.DISCRETIZATION_MATRICES][
keyword_mech][discr.bound_pressure_matrix_key]
active_faces = data[pp.PARAMETERS][keyword_mech]["active_faces"]
if np.any(faces_covered):
del_faces = self.expand_indices_nd(
np.where(faces_covered)[0], g.dim)
del_cells = np.where(cells_covered)[0]
pp.fvutils.remove_nonlocal_contribution(
del_cells, 1, partial_div_u, partial_bound_div_u,
partial_stab)
# del_faces is already expanded, set dimension to 1
pp.fvutils.remove_nonlocal_contribution(
del_faces, 1, partial_grad_p, partial_bound_pressure)
faces_covered[active_faces] = True
cells_covered[ci] = True
div_u += partial_div_u
bound_div_u += partial_bound_div_u
grad_p += partial_grad_p
stab += partial_stab
bound_displacement_pressure += partial_bound_pressure
self.assertTrue((div_u_full - div_u).max() < 1e-8)
self.assertTrue((bound_div_u_full - bound_div_u).min() > -1e-8)
self.assertTrue((grad_p_full - grad_p).max() < 1e-8)
self.assertTrue((stab_full - stab).min() > -1e-8)
self.assertTrue(
(bound_displacement_pressure - bound_pressure_full).min() > -1e-8)
def conv_test(n):
# Analytical solution
def mandel_solution(g, Nx, Ny, times, F, B, nu_u, nu, c_f, mu_s):
# Some needed parameters
a = np.max(g.face_centers[0]) # a = Lx
x_cntr = g.cell_centers[0][:Nx] # [m] vector of x-centers
y_cntr = g.cell_centers[1][::Nx] # [m] vector of y-centers
# Solutions to tan(x) - ((1-nu)/(nu_u-nu)) x = 0
"""
This is somehow tricky, we have to solve the equation numerically in order to
find all the positive solutions to the equation. Later we will use them to
compute the infinite sums. Experience has shown that 200 roots are more than enough to
achieve accurate results. Note that we find the roots using the bisection method.
"""
f = lambda x: np.tan(x) - (
(1 - nu) /
(nu_u - nu)) * x # define the algebraic eq. as a lambda function
n_series = 200 # number of roots
a_n = np.zeros(n_series) # initializing roots array
x0 = 0 # initial point
for i in range(0, len(a_n)):
a_n[i] = opt.bisect(
f, # function
x0 + np.pi / 4, # left point
x0 + np.pi / 2 -
10000 * 2.2204e-16, # right point (a tiny bit less than pi/2)
xtol=1e-30, # absolute tolerance
rtol=1e-15 # relative tolerance
)
x0 += np.pi # apply a phase change of pi to get the next root
# Creating dictionary to store analytical solutions
mandel_sol = dict()
mandel_sol['p'] = np.zeros((len(times), len(x_cntr)))
mandel_sol['u_x'] = np.zeros((len(times), len(x_cntr)))
mandel_sol['u_y'] = np.zeros((len(times), len(y_cntr)))
mandel_sol['sigma_yy'] = np.zeros((len(times), len(x_cntr)))
# Terms needed to compute the solutions (these are constants)
p0 = (2 * F * B * (1 + nu_u)) / (3 * a)
ux0_1 = ((F * nu) / (2 * mu_s * a))
ux0_2 = -((F * nu_u) / (mu_s * a))
ux0_3 = F / mu_s
uy0_1 = (-F * (1 - nu)) / (2 * mu_s * a)
uy0_2 = (F * (1 - nu_u) / (mu_s * a))
sigma0_1 = -F / a
sigma0_2 = (-2 * F * B * (nu_u - nu)) / (a * (1 - nu))
sigma0_3 = (2 * F) / a
# Saving solutions for the initial conditions
mandel_sol['p'][0] = ((F * B * (1 + nu_u)) / (3 * a)) * np.ones(Nx)
mandel_sol['u_x'][0] = (F * nu_u * x_cntr) / (2 * mu_s * a)
mandel_sol['u_y'][0] = ((-F * (1 - nu_u)) / (2 * mu_s * a)) * y_cntr
# Storing solutions for the subsequent time steps
for ii in range(1, len(times)):
# Analytical Pressures
p_sum = 0
for n in range(len(a_n)):
p_sum += (((np.sin(a_n[n])) /
(a_n[n] - (np.sin(a_n[n]) * np.cos(a_n[n])))) *
(np.cos((a_n[n] * x_cntr) / a) - np.cos(a_n[n])) *
np.exp((-(a_n[n]**2) * c_f * times[ii]) / (a**2)))
mandel_sol['p'][ii] = p0 * p_sum
# Analytical horizontal displacements
ux_sum1 = 0
ux_sum2 = 0
for n in range(len(a_n)):
ux_sum1 += ((np.sin(a_n[n]) * np.cos(a_n[n])) /
(a_n[n] - np.sin(a_n[n]) * np.cos(a_n[n])) *
np.exp((-(a_n[n]**2) * c_f * times[ii]) / (a**2)))
ux_sum2 += ((np.cos(a_n[n]) /
(a_n[n] - (np.sin(a_n[n]) * np.cos(a_n[n])))) *
np.sin(a_n[n] * (x_cntr / a)) * np.exp(
(-(a_n[n]**2) * c_f * times[ii]) / (a**2)))
mandel_sol['u_x'][ii] = (
ux0_1 + ux0_2 * ux_sum1) * x_cntr + ux0_3 * ux_sum2
# Analytical vertical displacements
uy_sum = 0
for n in range(len(a_n)):
uy_sum += (((np.sin(a_n[n]) * np.cos(a_n[n])) /
(a_n[n] - np.sin(a_n[n]) * np.cos(a_n[n]))) *
np.exp((-(a_n[n]**2) * c_f * times[ii]) / (a**2)))
mandel_sol['u_y'][ii] = (uy0_1 + uy0_2 * uy_sum) * y_cntr
# Analitical vertical stress
sigma_sum1 = 0
sigma_sum2 = 0
for n in range(len(a_n)):
sigma_sum1 += (((np.sin(a_n[n])) /
(a_n[n] - (np.sin(a_n[n]) * np.cos(a_n[n])))) *
np.cos(a_n[n] * (x_cntr / a)) * np.exp(
(-(a_n[n]**2) * c_f * times[ii]) / (a**2)))
sigma_sum2 += (((np.sin(a_n[n]) * np.cos(a_n[n])) /
(a_n[n] - np.sin(a_n[n]) * np.cos(a_n[n]))) *
np.exp(
(-(a_n[n]**2) * c_f * times[ii]) / (a**2)))
mandel_sol['sigma_yy'][ii] = (sigma0_1 + sigma0_2 * sigma_sum1) + (
sigma0_3 * sigma_sum2)
return mandel_sol
# Computing the initial condition
def get_mandel_init_cond(g, F, B, nu_u, mu_s):
# Initialing pressure and displacement arrays
p0 = np.zeros(g.num_cells)
u0 = np.zeros(g.num_cells * 2)
# Some needed parameters
a = np.max(g.face_centers[0]) # a = Lx
p0 = ((F * B * (1 + nu_u)) / (3 * a)) * np.ones(g.num_cells)
u0[::2] = (F * nu_u * g.cell_centers[0]) / (2 * mu_s * a)
u0[1::2] = ((-F * (1 - nu_u)) / (2 * mu_s * a)) * g.cell_centers[1]
return p0, u0
# Getting the time-dependent boundary condition
def get_mandel_bc(g, y_max, times, F, B, nu_u, nu, c_f, mu_s):
# Initializing top boundary array
u_top = np.zeros((len(times), len(y_max)))
# Some needed parameters
a = np.max(g.face_centers[0]) # a = Lx
b = np.max(g.face_centers[1]) # b = Ly
y_top = g.face_centers[1][
y_max] # [m] y-coordinates at the top boundary
# Solutions to tan(x) - ((1-nu)/(nu_u-nu)) x = 0
"""
This is somehow tricky, we have to solve the equation numerically in order to
find all the positive solutions to the equation. Later we will use them to
compute the infinite sums. Experience has shown that 200 roots are more than enough to
achieve accurate results. Note that we find the roots using the bisection method.
"""
f = lambda x: np.tan(x) - (
(1 - nu) /
(nu_u - nu)) * x # define the algebraic eq. as a lambda function
n_series = 200 # number of roots
a_n = np.zeros(n_series) # initializing roots array
x0 = 0 # initial point
for i in range(0, len(a_n)):
a_n[i] = opt.bisect(
f, # function
x0 + np.pi / 4, # left point
x0 + np.pi / 2 -
10000 * 2.2204e-16, # right point (a tiny bit less than pi/2)
xtol=1e-30, # absolute tolerance
rtol=1e-15 # relative tolerance
)
x0 += np.pi # apply a phase change of pi to get the next root
# Terms needed to compute the solutions (these are constants)
uy0_1 = (-F * (1 - nu)) / (2 * mu_s * a)
uy0_2 = (F * (1 - nu_u) / (mu_s * a))
# For the initial condition:
u_top[0] = ((-F * (1 - nu_u)) / (2 * mu_s * a)) * b
for i in range(1, len(times)):
# Analytical vertical displacements at the top boundary
uy_sum = 0
for n in range(len(a_n)):
uy_sum += (((np.sin(a_n[n]) * np.cos(a_n[n])) /
(a_n[n] - np.sin(a_n[n]) * np.cos(a_n[n]))) *
np.exp((-(a_n[n]**2) * c_f * times[i]) / (a**2)))
u_top[i] = (uy0_1 + uy0_2 * uy_sum) * y_top
# Returning array of u_y at the top boundary
return u_top
# Getting mechanics boundary conditions
def get_bc_mechanics(g, u_top, times, b_faces, x_min, x_max, west, east,
y_min, y_max, south, north):
# Setting the tags at each boundary side for the mechanics problem
labels_mech = np.array([None] * b_faces.size)
labels_mech[west] = 'dir_x' # roller
labels_mech[east] = 'neu' # traction free
labels_mech[south] = 'dir_y' # roller
labels_mech[
north] = 'dir_y' # roller (with non-zero displacement in the vertical direction)
# Constructing the bc object for the mechanics problem
bc_mech = pp.BoundaryConditionVectorial(g, b_faces, labels_mech)
# Constructing the boundary values array for the mechanics problem
bc_val_mech = np.zeros((
len(times),
g.num_faces * g.dim,
))
for i in range(len(times)):
# West side boundary conditions (mech)
bc_val_mech[i][2 * x_min] = 0 # [m]
bc_val_mech[i][2 * x_min + 1] = 0 # [Pa]
# East side boundary conditions (mech)
bc_val_mech[i][2 * x_max] = 0 # [Pa]
bc_val_mech[i][2 * x_max + 1] = 0 # [Pa]
# South Side boundary conditions (mech)
bc_val_mech[i][2 * y_min] = 0 # [Pa]
bc_val_mech[i][2 * y_min + 1] = 0 # [m]
# North Side boundary conditions (mech)
bc_val_mech[i][2 * y_max] = 0 # [Pa]
bc_val_mech[i][2 * y_max + 1] = u_top[i] # [m]
return bc_mech, bc_val_mech
# Getting flow boundary conditions
def get_bc_flow(g, b_faces, x_min, x_max, west, east, y_min, y_max, south,
north):
# Setting the tags at each boundary side for the mechanics problem
labels_flow = np.array([None] * b_faces.size)
labels_flow[west] = 'neu' # no flow
labels_flow[east] = 'dir' # constant pressure
labels_flow[south] = 'neu' # no flow
labels_flow[north] = 'neu' # no flow
# Constructing the bc object for the flow problem
bc_flow = pp.BoundaryCondition(g, b_faces, labels_flow)
# Constructing the boundary values array for the flow problem
bc_val_flow = np.zeros(g.num_faces)
# West side boundary condition (flow)
bc_val_flow[x_min] = 0 # [Pa]
# East side boundary condition (flow)
bc_val_flow[x_max] = 0 # [m^3/s]
# South side boundary condition (flow)
bc_val_flow[y_min] = 0 # [m^3/s]
# North side boundary condition (flow)
bc_val_flow[y_max] = 0 # [m^3/s]
return bc_flow, bc_val_flow
# ## Setting up the grid
# In[7]:
Nx = 40
Ny = 40
Lx = 100
Ly = 10
g = pp.CartGrid([Nx, Ny], [Lx, Ly])
#g.nodes = g.nodes + 1E-7*np.random.randn(3, g.num_nodes)
g.compute_geometry()
V = g.cell_volumes
# Physical parameters
# Skeleton parameters
mu_s = 2.475E+09 # [Pa] Shear modulus
lambda_s = 1.65E+09 # [Pa] Lame parameter
K_s = (2 / 3) * mu_s + lambda_s # [Pa] Bulk modulus
E_s = mu_s * ((9 * K_s) / (3 * K_s + mu_s)) # [Pa] Young's modulus
nu_s = (3 * K_s - 2 * mu_s) / (2 * (3 * K_s + mu_s)
) # [-] Poisson's coefficient
k_s = 100 * 9.869233E-13 # [m^2] Permeabiliy
# Fluid parameters
mu_f = 10.0E-3 # [Pa s] Dynamic viscosity
# Porous medium parameters
alpha_biot = 1. # [m^2] Intrinsic permeability
S_m = 6.0606E-11 # [1/Pa] Specific Storage
K_u = K_s + (alpha_biot**2) / S_m # [Pa] Undrained bulk modulus
B = alpha_biot / (S_m * K_u) # [-] Skempton's coefficient
nu_u = (3 * nu_s + B *
(1 - 2 * nu_s)) / (3 - B *
(1 - 2 * nu_s)) # [-] Undrained Poisson's ratio
c_f = (2 * k_s * (B**2) * mu_s * (1 - nu_s) *
(1 + nu_u)**2) / (9 * mu_f * (1 - nu_u) *
(nu_u - nu_s)) # [m^2/s] Fluid diffusivity
# Creating second and fourth order tensors
# Permeability tensor
perm = pp.SecondOrderTensor(g.dim, k_s * np.ones(g.num_cells))
# Stiffness matrix
constit = pp.FourthOrderTensor(g.dim, mu_s * np.ones(g.num_cells),
lambda_s * np.ones(g.num_cells))
# Time parameters
t0 = 0 # [s] Initial time
tf = 100 # [s] Final simulation time
tLevels = 100 # [-] Time levels
times = np.linspace(t0, tf, tLevels + 1) # [s] Vector of time evaluations
dt = np.diff(times) # [s] Vector of time steps
# Boundary conditions pre-processing
b_faces = g.tags['domain_boundary_faces'].nonzero()[0]
# Extracting indices of boundary faces w.r.t g
x_min = b_faces[g.face_centers[0, b_faces] < 0.0001]
x_max = b_faces[g.face_centers[0, b_faces] > 0.9999 * Lx]
y_min = b_faces[g.face_centers[1, b_faces] < 0.0001]
y_max = b_faces[g.face_centers[1, b_faces] > 0.9999 * Ly]
# Extracting indices of boundary faces w.r.t b_faces
west = np.in1d(b_faces, x_min).nonzero()
east = np.in1d(b_faces, x_max).nonzero()
south = np.in1d(b_faces, y_min).nonzero()
north = np.in1d(b_faces, y_max).nonzero()
# Applied load and top boundary condition
F_load = 6.8E+8 # [N/m] Applied load
u_top = get_mandel_bc(g, y_max, times, F_load, B, nu_u, nu_s, c_f,
mu_s) # [m] Vector of imposed vertical displacements
# MECHANICS BOUNDARY CONDITIONS
bc_mech, bc_val_mech = get_bc_mechanics(g, u_top, times, b_faces, x_min,
x_max, west, east, y_min, y_max,
south, north)
# FLOW BOUNDARY CONDITIONS
bc_flow, bc_val_flow = get_bc_flow(g, b_faces, x_min, x_max, west, east,
y_min, y_max, south, north)
# Initialiazing solution and solver dicitionaries
# Solution dictionary
sol = dict()
sol['time'] = np.zeros(tLevels + 1, dtype=float)
sol['displacement'] = np.zeros((tLevels + 1, g.num_cells * g.dim),
dtype=float)
sol['displacement_faces'] = np.zeros(
(tLevels + 1, g.num_faces * g.dim * 2), dtype=float)
sol['pressure'] = np.zeros((tLevels + 1, g.num_cells), dtype=float)
sol['traction'] = np.zeros((tLevels + 1, g.num_faces * g.dim), dtype=float)
sol['flux'] = np.zeros((tLevels + 1, g.num_faces), dtype=float)
sol['iter'] = np.array([], dtype=int)
sol['time_step'] = np.array([], dtype=float)
sol['residual'] = np.array([], dtype=float)
# Solver dictionary
newton_param = dict()
newton_param['tol'] = 1E-6 # maximum tolerance
newton_param['max_iter'] = 20 # maximum number of iterations
newton_param['res_norm'] = 1000 # initializing residual
newton_param['iter'] = 1 # iteration
# Discrete operators and discrete equations
# Flow operators
F = lambda x: biot_F * x # Flux operator
boundF = lambda x: biot_boundF * x # Bound Flux operator
compat = lambda x: biot_compat * x # Compatibility operator (Stabilization term)
divF = lambda x: biot_divF * x # Scalar divergence operator
# Mechanics operators
S = lambda x: biot_S * x # Stress operator
boundS = lambda x: biot_boundS * x # Bound Stress operator
divU = lambda x: biot_divU * x # Divergence of displacement field
divS = lambda x: biot_divS * x # Vector divergence operator
gradP = lambda x: biot_divS * biot_gradP * x # Pressure gradient operator
boundDivU = lambda x: biot_boundDivU * x # Bound Divergence of displacement operator
boundUCell = lambda x: biot_boundUCell * x # Contribution of displacement at cells -> Face displacement
boundUFace = lambda x: biot_boundUFace * x # Contribution of bc_mech at the boundaries -> Face displacement
boundUPressure = lambda x: biot_boundUPressure * x # Contribution of pressure at cells -> Face displacement
# Discrete equations
# Generalized Hooke's law
T = lambda u, bc_val_mech: S(u) + boundS(bc_val_mech)
# Momentum conservation equation (I)
u_eq1 = lambda u, bc_val_mech: divS(T(u, bc_val_mech))
# Momentum conservation equation (II)
u_eq2 = lambda p: -gradP(p)
# Darcy's law
Q = lambda p: (1. / mu_f) * (F(p) + boundF(bc_val_flow))
# Mass conservation equation (I)
p_eq1 = lambda u, u_n, bc_val_mech, bc_val_mech_n: alpha_biot * (divU(
u - u_n) + boundDivU(bc_val_mech - bc_val_mech_n))
# Mass conservation equation (II)
p_eq2 = lambda p, p_n, dt: (p - p_n) * S_m * V + divF(Q(
p)) * dt + alpha_biot * compat(p - p_n)
# Creating AD variables
# Retrieve initial conditions
p_init, u_init = get_mandel_init_cond(g, F_load, B, nu_u, mu_s)
# Create displacement AD-variable
u_ad = Ad_array(u_init.copy(), sps.diags(np.ones(g.num_cells * g.dim)))
# Create pressure AD-variable
p_ad = Ad_array(p_init.copy(), sps.diags(np.ones(g.num_cells)))
# The time loop
tt = 0 # time counter
while times[tt] < times[-1]:
################################
# Initializing data dictionary #
################################
d = dict() # initialize dictionary to store data
################################
# Creating the data objects #
################################
# Mechanics data object
specified_parameters_mech = {
"fourth_order_tensor": constit,
"bc": bc_mech,
"biot_alpha": 1.,
"bc_values": bc_val_mech[tt],
"mass_weight": S_m
}
pp.initialize_default_data(g, d, "mechanics",
specified_parameters_mech)
# Flow data object
specified_parameters_flow = {
"second_order_tensor": perm,
"bc": bc_flow,
"biot_alpha": 1.,
"bc_values": bc_val_flow,
"mass_weight": S_m,
"time_step": dt[tt - 1]
}
pp.initialize_default_data(g, d, "flow", specified_parameters_flow)
################################
# CALLING MPFA/MPSA ROUTINES #
################################
# Biot discretization
solver_biot = pp.Biot("mechanics", "flow")
solver_biot.discretize(g, d)
# Mechanics discretization matrices
biot_S = d['discretization_matrices']['mechanics']['stress']
biot_boundS = d['discretization_matrices']['mechanics']['bound_stress']
biot_divU = d['discretization_matrices']['mechanics']['div_d']
biot_gradP = d['discretization_matrices']['mechanics']['grad_p']
biot_boundDivU = d['discretization_matrices']['mechanics'][
'bound_div_d']
biot_boundUCell = d['discretization_matrices']['mechanics'][
'bound_displacement_cell']
biot_boundUFace = d['discretization_matrices']['mechanics'][
'bound_displacement_face']
biot_boundUPressure = d['discretization_matrices']['mechanics'][
'bound_displacement_pressure']
biot_divS = pp.fvutils.vector_divergence(g)
# Flow discretization matrices
biot_F = d['discretization_matrices']['flow']['flux']
biot_boundF = d['discretization_matrices']['flow']['bound_flux']
biot_compat = d['discretization_matrices']['flow'][
'biot_stabilization']
biot_divF = pp.fvutils.scalar_divergence(g)
################################
# Saving Initial Condition #
################################
if times[tt] == 0:
sol['pressure'][tt] = p_ad.val
sol['displacement'][tt] = u_ad.val
sol['displacement_faces'][tt] = (
boundUCell(sol['displacement'][tt]) +
boundUFace(bc_val_mech[tt]) +
boundUPressure(sol['pressure'][tt]))
sol['time'][tt] = times[tt]
sol['traction'][tt] = T(u_ad.val, bc_val_mech[tt])
sol['flux'][tt] = Q(p_ad.val)
tt += 1 # increasing time counter
################################
# Solving the set of PDE's #
################################
# Displacement and pressure at the previous time step
u_n = u_ad.val.copy()
p_n = p_ad.val.copy()
# Updating residual and iteration at each time step
newton_param.update({'res_norm': 1000, 'iter': 1})
# Newton loop
while newton_param['res_norm'] > newton_param['tol'] and newton_param[
'iter'] <= newton_param['max_iter']:
# Calling equations
eq1 = u_eq1(u_ad, bc_val_mech[tt])
eq2 = u_eq2(p_ad)
eq3 = p_eq1(u_ad, u_n, bc_val_mech[tt], bc_val_mech[tt - 1])
eq4 = p_eq2(p_ad, p_n, dt[tt - 1])
# Assembling Jacobian of the coupled system
J_mech = np.hstack(
(eq1.jac, eq2.jac)) # Jacobian blocks (mechanics)
J_flow = np.hstack((eq3.jac, eq4.jac)) # Jacobian blocks (flow)
J = sps.bmat(np.vstack((J_mech, J_flow)),
format='csc') # Jacobian (coupled)
# Determining residual of the coupled system
R_mech = eq1.val + eq2.val # Residual (mechanics)
R_flow = eq3.val + eq4.val # Residual (flow)
R = np.hstack((R_mech, R_flow)) # Residual (coupled)
y = sps.linalg.spsolve(J, -R) #
u_ad.val = u_ad.val + y[:g.dim * g.num_cells] # Newton update
p_ad.val = p_ad.val + y[g.dim * g.num_cells:] #
newton_param['res_norm'] = np.linalg.norm(R) # Updating residual
if newton_param['res_norm'] <= newton_param[
'tol'] and newton_param['iter'] <= newton_param['max_iter']:
print('Iter: {} \t Error: {:.8f} [m]'.format(
newton_param['iter'], newton_param['res_norm']))
elif newton_param['iter'] > newton_param['max_iter']:
print('Error: Newton method did not converge!')
else:
newton_param['iter'] += 1
################################
# Saving the variables #
################################
sol['iter'] = np.concatenate(
(sol['iter'], np.array([newton_param['iter']])))
sol['residual'] = np.concatenate(
(sol['residual'], np.array([newton_param['res_norm']])))
sol['time_step'] = np.concatenate((sol['time_step'], dt))
sol['pressure'][tt] = p_ad.val
sol['displacement'][tt] = u_ad.val
sol['displacement_faces'][tt] = (boundUCell(sol['displacement'][tt]) +
boundUFace(bc_val_mech[tt]) +
boundUPressure(sol['pressure'][tt]))
sol['time'][tt] = times[tt]
sol['traction'][tt] = T(u_ad.val, bc_val_mech[tt])
sol['flux'][tt] = Q(p_ad.val)
# Calling analytical solution
sol_mandel = mandel_solution(g, Nx, Ny, times, F_load, B, nu_u, nu_s, c_f,
mu_s)
# Creating analytical and numerical results arrays
p_num = (Lx * sol['pressure'][-1][:Nx]) / F
p_ana = (Lx * sol_mandel['p'][-1]) / F
# Returning values
return p_num, p_ana
def set_mechanics_parameters(self):
""" Mechanical parameters.
Note that we divide the momentum balance equation by self.scalar_scale.
A homogeneous initial temperature is assumed.
"""
gb = self.gb
for g, d in gb:
if g.dim == self.Nd:
# Rock parameters
rock = self.rock
lam = rock.LAMBDA * np.ones(g.num_cells) / self.scalar_scale
mu = rock.MU * np.ones(g.num_cells) / self.scalar_scale
C = pp.FourthOrderTensor(mu, lam)
bc = self.bc_type_mechanics(g)
bc_values = self.bc_values_mechanics(g)
sources = self.source_mechanics(g)
# In the momentum balance, the coefficient hits the scalar, and should
# not be scaled. Same goes for the energy balance, where we divide all
# terms by T_0, hence the term originally beta K T d(div u) / dt becomes
# beta K d(div u) / dt = coupling_coefficient d(div u) / dt.
coupling_coefficient = self.biot_alpha(g)
pp.initialize_data(
g,
d,
self.mechanics_parameter_key,
{
"bc": bc,
"bc_values": bc_values,
"source": sources,
"fourth_order_tensor": C,
"biot_alpha": coupling_coefficient,
"time_step": self.time_step,
},
)
pp.initialize_data(
g,
d,
self.mechanics_temperature_parameter_key,
{
"biot_alpha": self.biot_beta(g),
"bc_values": bc_values
},
)
elif g.dim == self.Nd - 1:
pp.initialize_data(
g,
d,
self.mechanics_parameter_key,
{
"friction_coefficient": self._friction_coefficient(g),
"contact_mechanics_numerical_parameter": 10000,
},
)
for e, d in gb.edges():
mg = d["mortar_grid"]
# Parameters for the surface diffusion. Not used as of now.
pp.initialize_data(
mg,
d,
self.mechanics_parameter_key,
{
"mu": self.rock.MU,
"lambda": self.rock.LAMBDA
},
)
def biot_convergence_in_space(N):
# coding: utf-8
# ### Source terms and analytical solutions
# In[330]:
def source_flow(g, tau):
x1 = g.cell_centers[0]
x2 = g.cell_centers[1]
f_flow = tau*(2*np.sin(2*np.pi*x2) - \
4*x1*np.pi**2*np.sin(2*np.pi*x2)*(x1 - 1)) - \
x1*np.sin(2*np.pi*x2) - \
np.sin(2*np.pi*x2)*(x1 - 1) + \
2*np.pi*np.cos(2*np.pi*x2)*np.sin(2*np.pi*x1)
return f_flow
def source_mechanics(g):
x1 = g.cell_centers[0]
x2 = g.cell_centers[1]
f_mech = np.zeros(g.num_cells * g.dim)
f_mech[::2] = 6*np.sin(2*np.pi*x2) - \
x1*np.sin(2*np.pi*x2) - \
np.sin(2*np.pi*x2)*(x1 - 1) - \
8*np.pi**2*np.cos(2*np.pi*x1)*np.cos(2*np.pi*x2) - \
4*x1*np.pi**2*np.sin(2*np.pi*x2)*(x1 - 1)
f_mech[1::2] = 4*np.pi*np.cos(2*np.pi*x2)*(x1 - 1) + \
16*np.pi**2*np.sin(2*np.pi*x1)*np.sin(2*np.pi*x2) + \
4*x1*np.pi*np.cos(2*np.pi*x2) - \
2*x1*np.pi*np.cos(2*np.pi*x2)*(x1 - 1)
return f_mech
def analytical(g):
sol = dict()
x1 = g.cell_centers[0]
x2 = g.cell_centers[1]
sol['u'] = np.zeros(g.num_cells * g.dim)
sol['u'][::2] = x1 * (1 - x1) * np.sin(2 * np.pi * x2)
sol['u'][1::2] = np.sin(2 * np.pi * x1) * np.sin(2 * np.pi * x2)
sol['p'] = sol['u'][::2]
return sol
# ### Getting mechanics boundary conditions
# In[331]:
def get_bc_mechanics(g, b_faces, x_min, x_max, west, east, y_min, y_max,
south, north):
# Setting the tags at each boundary side for the mechanics problem
labels_mech = np.array([None] * b_faces.size)
labels_mech[west] = 'dir'
labels_mech[east] = 'dir'
labels_mech[south] = 'dir'
labels_mech[north] = 'dir'
# Constructing the bc object for the mechanics problem
bc_mech = pp.BoundaryConditionVectorial(g, b_faces, labels_mech)
# Constructing the boundary values array for the mechanics problem
bc_val_mech = np.zeros(g.num_faces * g.dim)
return bc_mech, bc_val_mech
# ### Getting flow boundary conditions
# In[332]:
def get_bc_flow(g, b_faces, x_min, x_max, west, east, y_min, y_max, south,
north):
# Setting the tags at each boundary side for the mechanics problem
labels_flow = np.array([None] * b_faces.size)
labels_flow[west] = 'dir'
labels_flow[east] = 'dir'
labels_flow[south] = 'dir'
labels_flow[north] = 'dir'
# Constructing the bc object for the flow problem
bc_flow = pp.BoundaryCondition(g, b_faces, labels_flow)
# Constructing the boundary values array for the flow problem
bc_val_flow = np.zeros(g.num_faces)
return bc_flow, bc_val_flow
# ### Setting up the grid
# In[333]:
Nx = Ny = N
Lx = 1
Ly = 1
g = pp.CartGrid([Nx, Ny], [Lx, Ly])
g.compute_geometry()
V = g.cell_volumes
# ### Physical parameters
# In[334]:
# Skeleton parameters
mu_s = 1 # [Pa] Shear modulus
lambda_s = 1 # [Pa] Lame parameter
K_s = (2 / 3) * mu_s + lambda_s # [Pa] Bulk modulus
E_s = mu_s * ((9 * K_s) / (3 * K_s + mu_s)) # [Pa] Young's modulus
nu_s = (3 * K_s - 2 * mu_s) / (2 * (3 * K_s + mu_s)
) # [-] Poisson's coefficient
k_s = 1 # [m^2] Permeabiliy
# Fluid parameters
mu_f = 1 # [Pa s] Dynamic viscosity
# Porous medium parameters
alpha_biot = 1. # [m^2] Intrinsic permeability
S_m = 0 # [1/Pa] Specific Storage
# ### Creating second and fourth order tensors
# In[335]:
# Permeability tensor
perm = pp.SecondOrderTensor(g.dim, k_s * np.ones(g.num_cells))
# Stiffness matrix
constit = pp.FourthOrderTensor(g.dim, mu_s * np.ones(g.num_cells),
lambda_s * np.ones(g.num_cells))
# ### Time parameters
# In[336]:
t0 = 0 # [s] Initial time
tf = 1 # [s] Final simulation time
tLevels = 1 # [-] Time levels
times = np.linspace(t0, tf, tLevels + 1) # [s] Vector of time evaluations
dt = np.diff(times) # [s] Vector of time steps
# ### Boundary conditions pre-processing
# In[337]:
b_faces = g.tags['domain_boundary_faces'].nonzero()[0]
# Extracting indices of boundary faces w.r.t g
x_min = b_faces[g.face_centers[0, b_faces] < 0.0001]
x_max = b_faces[g.face_centers[0, b_faces] > 0.9999 * Lx]
y_min = b_faces[g.face_centers[1, b_faces] < 0.0001]
y_max = b_faces[g.face_centers[1, b_faces] > 0.9999 * Ly]
# Extracting indices of boundary faces w.r.t b_faces
west = np.in1d(b_faces, x_min).nonzero()
east = np.in1d(b_faces, x_max).nonzero()
south = np.in1d(b_faces, y_min).nonzero()
north = np.in1d(b_faces, y_max).nonzero()
# Mechanics boundary conditions
bc_mech, bc_val_mech = get_bc_mechanics(g, b_faces, x_min, x_max, west,
east, y_min, y_max, south, north)
# FLOW BOUNDARY CONDITIONS
bc_flow, bc_val_flow = get_bc_flow(g, b_faces, x_min, x_max, west, east,
y_min, y_max, south, north)
# ### Initialiazing solution and solver dicitionaries
# In[338]:
# Solution dictionary
sol = dict()
sol['time'] = np.zeros(tLevels + 1, dtype=float)
sol['displacement'] = np.zeros((tLevels + 1, g.num_cells * g.dim),
dtype=float)
sol['displacement_faces'] = np.zeros(
(tLevels + 1, g.num_faces * g.dim * 2), dtype=float)
sol['pressure'] = np.zeros((tLevels + 1, g.num_cells), dtype=float)
sol['traction'] = np.zeros((tLevels + 1, g.num_faces * g.dim), dtype=float)
sol['flux'] = np.zeros((tLevels + 1, g.num_faces), dtype=float)
sol['iter'] = np.array([], dtype=int)
sol['time_step'] = np.array([], dtype=float)
sol['residual'] = np.array([], dtype=float)
# Solver dictionary
newton_param = dict()
newton_param['tol'] = 1E-8 # maximum tolerance
newton_param['max_iter'] = 20 # maximum number of iterations
newton_param['res_norm'] = 1000 # initializing residual
newton_param['iter'] = 1 # iteration
# ### Discrete operators and discrete equations
# ### Flow operators
# In[339]:
F = lambda x: biot_F * x # Flux operator
boundF = lambda x: biot_boundF * x # Bound Flux operator
compat = lambda x: biot_compat * x # Compatibility operator (Stabilization term)
divF = lambda x: biot_divF * x # Scalar divergence operator
# ### Mechanics operators
# In[340]:
S = lambda x: biot_S * x # Stress operator
boundS = lambda x: biot_boundS * x # Bound Stress operator
divU = lambda x: biot_divU * x # Divergence of displacement field
divS = lambda x: biot_divS * x # Vector divergence operator
gradP = lambda x: biot_divS * biot_gradP * x # Pressure gradient operator
boundDivU = lambda x: biot_boundDivU * x # Bound Divergence of displacement operator
boundUCell = lambda x: biot_boundUCell * x # Contribution of displacement at cells -> Face displacement
boundUFace = lambda x: biot_boundUFace * x # Contribution of bc_mech at the boundaries -> Face displacement
boundUPressure = lambda x: biot_boundUPressure * x # Contribution of pressure at cells -> Face displacement
# ### Discrete equations
# In[341]:
# Source terms
f_mech = source_mechanics(g)
f_flow = source_flow(g, dt[0])
# Generalized Hooke's law
T = lambda u: S(u) + boundS(bc_val_mech)
# Momentum conservation equation (I)
u_eq1 = lambda u: divS(T(u))
# Momentum conservation equation (II)
u_eq2 = lambda p: -gradP(p) + f_mech * V[0]
# Darcy's law
Q = lambda p: (1. / mu_f) * (F(p) + boundF(bc_val_flow))
# Mass conservation equation (I)
p_eq1 = lambda u, u_n: alpha_biot * divU(u - u_n)
# Mass conservation equation (II)
p_eq2 = lambda p, p_n, dt: (p - p_n) * S_m * V + divF(Q(
p)) * dt + alpha_biot * compat(p - p_n) * V[0] - (f_flow / dt) * V[0]
# ## Creating AD variables
# In[343]:
# Create displacement AD-variable
u_ad = Ad_array(np.zeros(g.num_cells * 2),
sps.diags(np.ones(g.num_cells * g.dim)))
# Create pressure AD-variable
p_ad = Ad_array(np.zeros(g.num_cells), sps.diags(np.ones(g.num_cells)))
# ## Performing discretization
# In[344]:
d = dict() # initialize dictionary to store data
# Mechanics data object
specified_parameters_mech = {
"fourth_order_tensor": constit,
"bc": bc_mech,
"biot_alpha": 1.,
"bc_values": bc_val_mech
}
pp.initialize_default_data(g, d, "mechanics", specified_parameters_mech)
# Flow data object
specified_parameters_flow = {
"second_order_tensor": perm,
"bc": bc_flow,
"biot_alpha": 1.,
"bc_values": bc_val_flow
}
pp.initialize_default_data(g, d, "flow", specified_parameters_flow)
# Biot discretization
solver_biot = pp.Biot("mechanics", "flow")
solver_biot.discretize(g, d)
# Mechanics discretization matrices
biot_S = d['discretization_matrices']['mechanics']['stress']
biot_boundS = d['discretization_matrices']['mechanics']['bound_stress']
biot_divU = d['discretization_matrices']['mechanics']['div_d']
biot_gradP = d['discretization_matrices']['mechanics']['grad_p']
biot_boundDivU = d['discretization_matrices']['mechanics']['bound_div_d']
biot_boundUCell = d['discretization_matrices']['mechanics'][
'bound_displacement_cell']
biot_boundUFace = d['discretization_matrices']['mechanics'][
'bound_displacement_face']
biot_boundUPressure = d['discretization_matrices']['mechanics'][
'bound_displacement_pressure']
biot_divS = pp.fvutils.vector_divergence(g)
# Flow discretization matrices
biot_F = d['discretization_matrices']['flow']['flux']
biot_boundF = d['discretization_matrices']['flow']['bound_flux']
biot_compat = d['discretization_matrices']['flow']['biot_stabilization']
biot_divF = pp.fvutils.scalar_divergence(g)
# Saving initial condition
sol['pressure'][0] = p_ad.val
sol['displacement'][0] = u_ad.val
sol['displacement_faces'][0] = (boundUCell(sol['displacement'][0]) +
boundUFace(bc_val_mech) +
boundUPressure(sol['pressure'][0]))
sol['time'][0] = times[0]
sol['traction'][0] = T(u_ad.val)
sol['flux'][0] = Q(p_ad.val)
# ## The time loop
# In[345]:
tt = 0 # time counter
while times[tt] < times[-1]:
tt += 1 # increasing time counter
# Displacement and pressure at the previous time step
u_n = u_ad.val.copy()
p_n = p_ad.val.copy()
# Updating residual and iteration at each time step
newton_param.update({'res_norm': 1000, 'iter': 1})
# Newton loop
while newton_param['res_norm'] > newton_param['tol'] and newton_param[
'iter'] <= newton_param['max_iter']:
# Calling equations
eq1 = u_eq1(u_ad)
eq2 = u_eq2(p_ad)
eq3 = p_eq1(u_ad, u_n)
eq4 = p_eq2(p_ad, p_n, dt[tt - 1])
# Assembling Jacobian of the coupled system
J_mech = np.hstack(
(eq1.jac, eq2.jac)) # Jacobian blocks (mechanics)
J_flow = np.hstack((eq3.jac, eq4.jac)) # Jacobian blocks (flow)
J = sps.bmat(np.vstack((J_mech, J_flow)),
format='csc') # Jacobian (coupled)
# Determining residual of the coupled system
R_mech = eq1.val + eq2.val # Residual (mechanics)
R_flow = eq3.val + eq4.val # Residual (flow)
R = np.hstack((R_mech, R_flow)) # Residual (coupled)
y = sps.linalg.spsolve(J, -R) #
u_ad.val = u_ad.val + y[:g.dim * g.num_cells] # Newton update
p_ad.val = p_ad.val + y[g.dim * g.num_cells:] #
newton_param['res_norm'] = np.linalg.norm(R) # Updating residual
if newton_param['res_norm'] <= newton_param[
'tol'] and newton_param['iter'] <= newton_param['max_iter']:
print('Iter: {} \t Error: {:.8f} [m]'.format(
newton_param['iter'], newton_param['res_norm']))
elif newton_param['iter'] > newton_param['max_iter']:
print('Error: Newton method did not converge!')
else:
newton_param['iter'] += 1
# Saving variables
sol['iter'] = np.concatenate(
(sol['iter'], np.array([newton_param['iter']])))
sol['residual'] = np.concatenate(
(sol['residual'], np.array([newton_param['res_norm']])))
sol['time_step'] = np.concatenate((sol['time_step'], dt))
sol['pressure'][tt] = p_ad.val
sol['displacement'][tt] = u_ad.val
sol['displacement_faces'][tt] = (boundUCell(sol['displacement'][tt]) +
boundUFace(bc_val_mech) +
boundUPressure(sol['pressure'][tt]))
sol['time'][tt] = times[tt]
sol['traction'][tt] = T(u_ad.val)
sol['flux'][tt] = Q(p_ad.val)
# Determining analytical solution
sol_anal = analytical(g)
# Determining norms
p_norm = np.linalg.norm(sol_anal['p'] - sol['pressure'][-1]) / (
np.linalg.norm(sol['pressure'][-1]))
u_mag_num = np.sqrt(sol['displacement'][-1][::2]**2 +
sol['displacement'][-1][1::2]**2)
u_mag_ana = np.sqrt(sol_anal['u'][::2]**2 + sol_anal['u'][1::2]**2)
u_norm = np.linalg.norm(u_mag_ana -
u_mag_num) / np.linalg.norm(u_mag_num)
return p_norm, u_norm
def assign_parameters(
grid_object,
data_dictionary,
parameter_keyword_flow,
parameter_keyword_mechanics,
boundary_conditions_dictionary,
):
"""
Assign data to the model, which will later be used to discretize
the coupled problem.
Parameters:
grid_object (PorePy object): PorePy grid object
data_dictionary (Dict): Model's data dictionary
parameter_keyword_flow (String): Keyword for the flow parameter
parameter_keyword_mechanics (String): Keyword for the mechanics parameter
boundary_conditions_dictionary (Dict): Dictionary containing boundary conditions
"""
# Renaming input data
g = grid_object
d = data_dictionary
kw_f = parameter_keyword_flow
kw_m = parameter_keyword_mechanics
bc_dict = boundary_conditions_dictionary
# Assing flow data
# Retrieve data for the flow problem
k = d[pp.PARAMETERS][kw_f]["permeability"]
alpha_biot = d[pp.PARAMETERS][kw_f]["alpha_biot"]
S_m = d[pp.PARAMETERS][kw_f]["specific_storage"]
dt = d[pp.PARAMETERS][kw_f]["time_step"]
bc_flow = bc_dict[kw_f]["bc"]
bc_flow_values = bc_dict[kw_f]["bc_values"]
# Create second order tensor object
kxx = k * np.ones(g.num_cells)
perm = pp.SecondOrderTensor(kxx)
# Create specified parameters dicitionary
specified_parameters_flow = {
"second_order_tensor": perm,
"biot_alpha": alpha_biot,
"bc": bc_flow,
"bc_values": bc_flow_values,
"time_step": dt,
"mass_weight": S_m * np.ones(g.num_cells),
}
# Initialize the flow data
d = pp.initialize_default_data(g, d, kw_f, specified_parameters_flow)
# Assign mechanics data
# Retrieve data for the mechanics problem
mu = d[pp.PARAMETERS][kw_m]["lame_mu"]
lmbda = d[pp.PARAMETERS][kw_m]["lame_lambda"]
alpha_biot = d[pp.PARAMETERS][kw_m]["alpha_biot"]
bc_mech = bc_dict[kw_m]["bc"]
bc_mech_values = bc_dict[kw_m]["bc_values"][1]
# Create fourth order tensor
mu_lame = mu * np.ones(g.num_cells)
lambda_lame = lmbda * np.ones(g.num_cells)
constit = pp.FourthOrderTensor(mu_lame, lambda_lame)
# Create specified parameters dicitionary
specified_parameters_mechanics = {
"fourth_order_tensor": constit,
"biot_alpha": alpha_biot,
"bc": bc_mech,
"bc_values": bc_mech_values,
}
# Initialize the mechanics
d = pp.initialize_default_data(g, d, kw_m, specified_parameters_mechanics)
# Save boundary conditions in d[pp.STATE]
pp.set_state(d, {kw_m: {"bc_values": bc_dict[kw_m]["bc_values"][0]}})
def set_parameters(self, g, data_node, mg, data_edge):
"""
Set the parameters for the simulation. The stress is given in GPa.
"""
# Define the finite volume sub grid
s_t = pp.fvutils.SubcellTopology(g)
# Rock parameters
rock = pp.Granite()
lam = rock.LAMBDA * np.ones(g.num_cells) / self.pressure_scale
mu = rock.MU * np.ones(g.num_cells) / self.pressure_scale
F = self._friction_coefficient(g, mg, data_edge, s_t)
k = pp.FourthOrderTensor(g.dim, mu, lam)
# Define boundary regions
top = g.face_centers[g.dim - 1] > np.max(g.nodes[1]) - 1e-9
bot = g.face_centers[g.dim - 1] < np.min(g.nodes[1]) + 1e-9
top_hf = top[s_t.fno_unique]
bot_hf = bot[s_t.fno_unique]
# Define boundary condition on sub_faces
bc = pp.BoundaryConditionVectorial(g, top + bot, 'dir')
bc = pp.fvutils.boundary_to_sub_boundary(bc, s_t)
# Set the boundary values
u_bc = np.zeros((g.dim, s_t.num_subfno_unique))
u_bc[1, top_hf] = -0.002
u_bc[0, top_hf] = 0.005
# Find the continuity points
eta = 1 / 3
eta_vec = eta * np.ones(s_t.num_subfno_unique)
cont_pnt = g.face_centers[:g.dim, s_t.fno_unique] + eta_vec * (
g.nodes[:g.dim, s_t.nno_unique] -
g.face_centers[:g.dim, s_t.fno_unique])
# collect parameters in dictionary
key = 'mech'
key_m = 'mech'
data_node = pp.initialize_data(
g, data_node, key, {
'bc': bc,
'bc_values': u_bc.ravel('F'),
'source': 0,
'fourth_order_tensor': k,
'mpsa_eta': eta_vec,
'cont_pnt': cont_pnt,
'rock': rock,
})
pp.initialize_data(mg, data_edge, key_m, {
'friction_coeff': F,
'c': 100
})
# Define discretization
# For the 2D domain we solve linear elasticity with mpsa.
mpsa = pp.Mpsa(key)
data_node[pp.PRIMARY_VARIABLES] = {'u': {'cells': g.dim}}
data_node[pp.DISCRETIZATION] = {'u': {'mpsa': mpsa}}
# And define a Robin condition on the mortar grid
contact = pp.numerics.interface_laws.elliptic_interface_laws.RobinContact(
key_m, mpsa)
data_edge[pp.PRIMARY_VARIABLES] = {'lam': {'cells': g.dim}}
data_edge[pp.COUPLING_DISCRETIZATION] = {
'robin_disc': {
g: ('u', 'mpsa'),
g: ('u', 'mpsa'),
(g, g): ('lam', contact),
}
}
return key, key_m