example_5_2_1.py

import numpy as np
import scipy.sparse as sps
import porepy as pp

from porepy.numerics.vem import vem_dual, vem_source
from porepy.numerics.fv import mass_matrix, upwind

# ------------------------------------------------------------------------------#


def add_data_darcy(gb, domain, tol):
    gb.add_node_props(["param", "is_tangent"])

    apert = 1e-2

    km = 2.5 * 1e-11
    kf_t = 5 * 1e-6
    kf_n = 1e2 * km

    for g, d in gb:
        param = pp.Parameters(g)

        rock = g.dim == gb.dim_max()
        kxx = km if rock else kf_t
        d["is_tangential"] = True
        perm = pp.SecondOrderTensor(g.dim, kxx * np.ones(g.num_cells))
        param.set_tensor("flow", perm)

        param.set_source("flow", np.zeros(g.num_cells))

        param.set_aperture(np.power(apert, gb.dim_max() - g.dim))

        bound_faces = g.tags["domain_boundary_faces"].nonzero()[0]
        if bound_faces.size != 0:
            bound_face_centers = g.face_centers[:, bound_faces]

            left = bound_face_centers[0, :] < domain["xmin"] + tol
            right = bound_face_centers[0, :] > domain["xmax"] - tol
            boundary = np.logical_or(left, right)

            labels = np.array(["neu"] * bound_faces.size)
            labels[boundary] = ["dir"]

            bc_val = np.zeros(g.num_faces)
            bc_val[bound_faces[left]] = 30 * 1e6

            param.set_bc("flow", pp.BoundaryCondition(g, bound_faces, labels))
            param.set_bc_val("flow", bc_val)
        else:
            param.set_bc("flow", pp.BoundaryCondition(g, np.empty(0), np.empty(0)))

        d["param"] = param

    # Assign coupling permeability
    gb.add_edge_prop("kn")
    for e, d in gb.edges_props():
        g = gb.sorted_nodes_of_edge(e)[0]
        d["kn"] = kf_n / gb.node_prop(g, "param").get_aperture()


# ------------------------------------------------------------------------------#


def add_data_advection(gb, domain, tol):

    # Porosity
    phi_m = 1e-1
    phi_f = 9 * 1e-1

    # Density
    rho_w = 1e3  # kg m^{-3}
    rho_s = 2 * 1e3  # kg m^{-3}

    # heat capacity
    c_w = 4 * 1e3  # J kg^{-1} K^{-1}
    c_s = 8 * 1e2  # J kg^{-1} K^{-1}

    c_m = phi_m * rho_w * c_w + (1 - phi_m) * rho_s * c_s
    c_f = phi_f * rho_w * c_w + (1 - phi_f) * rho_s * c_s

    for g, d in gb:
        param = d["param"]

        rock = g.dim == gb.dim_max()
        source = np.zeros(g.num_cells)
        param.set_source("transport", source)

        param.set_porosity(1)
        param.set_discharge(d["discharge"])

        bound_faces = g.tags["domain_boundary_faces"].nonzero()[0]
        if bound_faces.size != 0:
            bound_face_centers = g.face_centers[:, bound_faces]

            bottom = bound_face_centers[1, :] < domain["ymin"] + tol
            left = bound_face_centers[0, :] < domain["xmin"] + tol
            right = bound_face_centers[0, :] > domain["xmax"] - tol
            boundary = np.logical_or(left, right)
            labels = np.array(["neu"] * bound_faces.size)
            labels[boundary] = ["dir"]

            bc_val = np.zeros(g.num_faces)
            bc_val[bound_faces[left]] = 1

            param.set_bc("transport", pp.BoundaryCondition(g, bound_faces, labels))
            param.set_bc_val("transport", bc_val)
        else:
            param.set_bc("transport", pp.BoundaryCondition(g, np.empty(0), np.empty(0)))
        d["param"] = param

    # Assign coupling discharge
    gb.add_edge_prop("param")
    for e, d in gb.edges_props():
        g = gb.sorted_nodes_of_edge(e)[1]
        discharge = gb.node_prop(g, "param").get_discharge()
        d["param"] = pp.Parameters(g)
        d["param"].set_discharge(discharge)


# ------------------------------------------------------------------------------#
# ------------------------------------------------------------------------------#


tol = 1e-4
export_folder = "example_5_2_1"

T = 40 * np.pi * 1e7
Nt = 20  # 10 20 40 80 160 320 640 1280 2560 5120 - 100000
deltaT = T / Nt
export_every = 1
if_coarse = True

mesh_kwargs = {"mesh_size_frac": 500, "mesh_size_min": 20, "tol": tol}
domain = {"xmin": 0, "xmax": 700, "ymin": 0, "ymax": 600}
gb = pp.importer.from_csv("network.csv", mesh_kwargs, domain)
gb.compute_geometry()
if if_coarse:
    pp.coarsening.coarsen(gb, "by_volume")
gb.assign_node_ordering()


# Assign parameters
add_data_darcy(gb, domain, tol)

# Choose and define the solvers and coupler
solver_flow = vem_dual.DualVEMMixDim("flow")
A_flow, b_flow = solver_flow.matrix_rhs(gb)

solver_source = vem_source.DualSourceMixDim("flow")
A_source, b_source = solver_source.matrix_rhs(gb)

up = sps.linalg.spsolve(A_flow + A_source, b_flow + b_source)
solver_flow.split(gb, "up", up)

gb.add_node_props(["p", "P0u", "discharge"])
solver_flow.extract_u(gb, "up", "discharge")
solver_flow.extract_p(gb, "up", "p")
solver_flow.project_u(gb, "discharge", "P0u")

# compute the flow rate
total_flow_rate = 0
for g, d in gb:
    bound_faces = g.tags["domain_boundary_faces"].nonzero()[0]
    if bound_faces.size != 0:
        bound_face_centers = g.face_centers[:, bound_faces]
        left = bound_face_centers[0, :] < domain["xmin"] + tol
        flow_rate = d["discharge"][bound_faces[left]]
        total_flow_rate += np.sum(flow_rate)

save = pp.Exporter(gb, "darcy", export_folder, binary=False)
save.write_vtk(["p", "P0u"])

#################################################################

physics = "transport"
advection = upwind.UpwindMixedDim(physics)
mass = mass_matrix.MassMatrixMixedDim(physics)
invMass = mass_matrix.InvMassMatrixMixDim(physics)

# Assign parameters
add_data_advection(gb, domain, tol)

gb.add_node_prop("deltaT", prop=deltaT)

U, rhs_u = advection.matrix_rhs(gb)
M, _ = mass.matrix_rhs(gb)
OF = advection.outflow(gb)
M_U = M + U

rhs = rhs_u

# Perform an LU factorization to speedup the solver
IE_solver = sps.linalg.factorized((M_U).tocsc())

theta = np.zeros(rhs.shape[0])

# Loop over the time
time = np.empty(Nt)
file_name = "theta"
i_export = 0
step_to_export = np.empty(0)

production = np.zeros(Nt)
save.change_name("theta")

for i in np.arange(Nt):
    print("Time step", i, " of ", Nt, " time ", i * deltaT, " deltaT ", deltaT)
    # Update the solution
    production[i] = np.sum(OF.dot(theta)) / total_flow_rate
    theta = IE_solver(M.dot(theta) + rhs)

    if i % export_every == 0:
        print("Export solution at", i)
        advection.split(gb, "theta", theta)
        save.write_vtk(["theta"], i_export)
        step_to_export = np.r_[step_to_export, i]
        i_export += 1

save.write_pvd(step_to_export * deltaT)

times = deltaT * np.arange(Nt)
np.savetxt(
    export_folder + "/production.txt", (times, np.abs(production)), delimiter=","
)