In this tutorial, we show how Reaktoro can be used for sequential reactive transport calculations of the injected H2S-rich brine into the siderite bearing reservoir with subsequent pyrrhotite, or otherwise called mackinawite, (FeS) precipitation.
First, we need to import a few Python packages to enable the numerical calculations and plotting.
from reaktoro import *
import numpy as np
from natsort import natsorted
from tqdm.notebook import tqdm
import os
# Import components of bokeh library
from bokeh.io import show, output_notebook
from bokeh.layouts import column
from bokeh.plotting import figure
from bokeh.models import Range1d, ColumnDataSource
from bokeh.layouts import gridplot
We import the reaktoro Python package so that we can use its classes and methods for performing chemical reaction calculations, numpy for working with arrays, tqdm for the progress bar functionality and os, to provide a portable way of using operating system dependent functionality. For plotting capabilities of obtained results, we use bokeh library.
In this step, we initialize auxiliary time-related constants from seconds up to days used in the rest of the code.
second = 1
minute = 60
hour = 60 * minute
day = 24 * hour
Next, we define reactive transport and numerical discretization parameters. In particular, we specify the considered rock domain by setting coordinates of its left and right boundaries to 0.0 m and 100.0 m, respectively. The discretization parameters, i.e., the number of cells and steps in time, are set to 100 and 1000, respectively. The reactive transport modeling procedure assumes a constant fluid velocity of 1.05 · 10-5 m/s and the zero diffusion coefficient for all fluid species. The size of the time-step is set to 0.05 days (1.2 hours). Temperature and pressure are set to 25 °C and 1 atm (1.01325 bar), respectively, throughout the tutorial. The porosity fo the rock is set to 10%.
# Discretization parameters
xl = 0.0 # x-coordinate of the left boundary
xr = 100.0 # x-coordinate of the right boundary
ncells = 100 # number of cells in the discretization
nsteps = 1000 # number of steps in the reactive transport simulation
dx = (xr - xl) / ncells # length of the mesh cells (in units of m)
dt = 0.05*day # time step
# Physical parameters
D = 0 # diffusion coefficient (in units of m2/s)
v = 1.05e-5 # fluid pore velocity (in units of m/s)
T = 25.0 + 273.15 # temperature (in units of K)
P = 1.01325 * 1e5 # pressure (in units of Pa)
phi = 0.1 # the porosity
Next, we generate the coordinates of the mesh nodes (array xcells
) by equally dividing the interval [xr, xl] with
the number of cells ncells
. The length between each consecutive mesh node is computed and stored in dx
(the
length of the mesh cells).
xcells = np.linspace(xl, xr, ncells + 1) # interval [xl, xr] split into ncells
The boolean variable dirichlet
is set to True
or False
depending on which boundary condition is considered in
the numerical calculation. False
corresponds to imposing the flux of the injected fluid; otherwise, True
means
imposing the fluid composition on the left boundary.
dirichlet = False # parameter that determines whether Dirichlet BC must be used
To make sure that the applied finite-volume scheme is stable, we need to keep track of Courant–Friedrichs–Lewy (CFL) number, which should be less than 1.0.
CFL = v * dt / dx
assert CFL <= 1.0, f"Make sure that CFL = {CFL} is less that 1.0"
Before running the reactive transport simulations, we create the list of parameters we are interested in outputting.
In this case, it is pH
, molality of H+
, HS-
, S2--
, CO3--
, HSO4-
, H2S(aq)
, Fe++
, as well as a phase
amount/volume of minerals pyrrhotite and siderite.
output_quantities = """
pH
speciesMolality(H+)
speciesMolality(HS-)
speciesMolality(S2--)
speciesMolality(CO3--)
speciesMolality(HSO4-)
speciesMolality(H2S(aq))
speciesMolality(Fe++)
phaseAmount(Pyrrhotite)
phaseAmount(Siderite)
phaseVolume(Pyrrhotite)
phaseVolume(Siderite)
""".split()
Then, we define the list of names for the DataFrame columns. Note, that they must correspond
to the order of the properties defined in the output_quantities
list:
column_quantities = """
pH
Hcation
HSanion
S2anion
CO3anion
HSO4anion
H2Saq
Fe2cation
pyrrhotite_phase_amount
siderite_phase_amount
pyrrhotite_phase_volume
siderite_phase_volume
""".split()
Create the list of columns and initialized with it the instance of dataframe:
columns = ['step', 'x'] + column_quantities
import pandas as pd
df = pd.DataFrame(columns=columns)
The main part of the program (at the bottom of this tutorial) consists of three parts, each represented by a Python function and documented in the following sections:
make_results_folders()
),simulate()
), andUsing os package, we create required folders for outputting the obtained results and for the plot and video files later.
folder_results = 'results-rt-scavenging'
def make_results_folders():
os.system('mkdir -p ' + folder_results)
The reactive transport simulation is performed in the function simulate
, which consists of several building blocks
(functions):
The preparatory initialization step consists of the following sub-steps:
define_chemical_system()
,define_initial_condition()
,define_initial_condition()
,partition_indices()
and elements' partitioningcorresponding to fluid and solid species with function partition_elements_in_mesh_cell()
, and finally
The simulation of the reactive transport problem is represented by the loop over discretized time interval until the final time is reached. On each step of this loop, the following functionality of performed:
transport()
,reactive_chemistry()
function, andoutputstate()
.Performing the transport and reactive chemistry sequentially is possible due to the operator splitting
procedure, in which we first update the amounts of elements b
. These updated amounts of
elements in the cell are used to evaluate its new chemical equilibrium state, thus producing new amounts of the
species in both the fluid and solid phases (available in the list states
of
ChemicalState objects). This chemical reaction
equilibrium calculation step, at each mesh cell, permits, for example, aqueous species and minerals to react,
and thus causes mineral dissolution or precipitation, depending on how much the amount of mineral species changes.
This can then be used, for example, to compute new porosity value for the cell.
def simulate():
# Construct the chemical system with its phases and species
system = define_chemical_system()
# Define the initial condition of the reactive transport modeling problem
state_ic = define_initial_condition(system)
# Define the boundary condition of the reactive transport modeling problem
state_bc = define_boundary_condition(system)
# Generate indices of partitioning fluid and solid species
nelems, ifluid_species, isolid_species = partition_indices(system)
# Partitioning fluid and solid species
b, bfluid, bsolid, b_bc = partition_elements_in_mesh_cell(ncells, nelems, state_ic, state_bc)
# Create a list of chemical states for the mesh cells (one for each cell, initialized to state_ic)
states = [state_ic.clone() for _ in range(ncells + 1)]
# Create the equilibrium solver object for the repeated equilibrium calculation
solver = EquilibriumSolver(system)
# Running the reactive transport simulation loop
step = 0 # the current step number
t = 0.0 # the current time (in seconds)
# Output the initial state of the reactive transport calculation
outputstate_df(step, system, states)
with tqdm(total=nsteps, desc="Reactive transport simulations") as pbar:
while step < nsteps:
# Perform transport calculations
bfluid, bsolid, b = transport(states, bfluid, bsolid, b, b_bc, nelems, ifluid_species, isolid_species)
# Perform reactive chemical calculations
states = reactive_chemistry(solver, states, b)
# Increment time step and number of time steps
t += dt
step += 1
# Output the current state of the reactive transport calculation
outputstate_df(step, system, states)
# Update a progress bar
pbar.update(1)
Subsections below correspond to the methods responsible for each of the functional parts of simulate()
method.
To define the chemical system, we need to initialize the class Database that provides operations to retrieve physical and thermodynamic data of chemical species. To achieve that, we use supcrt07.xml database file.
Note: If filename does not point to a valid database file or the database file is not found, then a default built-in database with the same name will be tried. If no default built-in database exists with a given name, an exception will be thrown.
In addition to the database, we also need to initialize parameters in the Debye-Huckel activity model used for aqueous
mixtures. Method setPHREEQC
allows setting parameters å and b of the ionic species according to those used
in PHREEQC v3.
Reaktoro is a general-purpose chemical solver that avoids as much as possible presuming specific assumptions about
considered problems. Thus, one needs to specify how the chemical system of interest encompasses all phases
as well as the chemical species in each phase. Using the
ChemicalEditor class, one can conveniently achieve
this, as shown below in method define_chemical_system()
.
In this step, we create an object of class ChemicalEditor and specify three phases, an aqueous and a
mineral that should be considered in the chemical system. The aqueous phase is defined by specifying the list of
chemical species. Function setChemicalModelDebyeHuckel()
helps to set the chemical model of the phase with
the Debye-Huckel equation of state, providing specific parameters dhModel
defined earlier. The mineral phases
are defined as two mineral species: pyrrhotite (FeS) and siderite (FeCO3).
Finally, we create an object of class ChemicalSystem
using the chemical system definition details stored in the object editor
.
The activity coefficients of the aqueous species in this tutorial are calculated using the Debye-Huckel model for solvent water and ionic species. The standard chemical potentials of the species are calculated using the equations of state of Helgeson and Kirkham (1974), Helgeson et al. (1974), Tanger and Helgeson (1988), Shock and Helgeson (1988), and Shock et al. ( 1992). The database file slop07.dat from the software SUPCRT07 is used to obtain the parameters for the equations of state. The equation of state of Wagner and Pruss (2002) is used to calculate the density of water and its temperature and pressure derivatives. Kinetics of dissolution and precipitation of both pyrrhotite and siderite is neglected, i.e., the local equilibrium assumption is employed.
def define_chemical_system():
# Construct the chemical system with its phases and species
db = Database('supcrt07.xml')
dhModel = DebyeHuckelParams()
dhModel.setPHREEQC()
editor = ChemicalEditor(db)
editor.addAqueousPhase(['Ca(HCO3)+', 'CO3--', 'CaCO3(aq)', 'Ca++', 'CaSO4(aq)', 'CaOH+', 'Cl-',
'FeCl++', 'FeCl2(aq)', 'FeCl+', 'Fe++', 'FeOH+', 'FeOH++', 'Fe+++',
'H2S(aq)', 'H2(aq)', 'HS-', 'H2O(l)', 'H+', 'OH-', 'HCO3-', 'HSO4-',
'KSO4-', 'K+',
'Mg++', 'Mg(HCO3)+', 'MgCO3(aq)', 'MgSO4(aq)', 'MgOH+',
'Na+', 'NaSO4-',
'O2(aq)',
'S5--', 'S4--', 'S3--', 'S2--', 'SO4--']).\
setChemicalModelDebyeHuckel(dhModel)
editor.addMineralPhase('Pyrrhotite')
editor.addMineralPhase('Siderite')
system = ChemicalSystem(editor)
return system
We have defined and constructed the chemical system of interest, enabling us to move on to the next step in Reaktoro's modeling workflow: defining our chemical reaction problems. Below, we define its initial condition with already prescribed equilibrium conditions for temperature, pressure, and amounts of elements consistent with modeling reactive transport of injected hydrogen sulfide brine into the rock-fluid composition of siderite at 25 °C and 1.01325 bar. The resident fluid in the rock is obtained by the mixture of the aqueous species summarized in the following table:
Aqueous species | Amount (kg) |
---|---|
H2 | 58.0 |
Cl- | 1122.3 · 10-3 |
Na+ | 624.08 · 10-3 |
SO42- | 157.18 · 10-3 |
Mg2+ | 74.820 · 10-3 |
Ca2+ | 23.838 · 10-3 |
K+ | 23.142 · 10-3 |
HCO3- | 8.236 · 10-3 |
O2(aq) | 58 · 10-12 |
For that purpose, the class EquilibriumInverseProblem is used, where specific fixed pH and pE are be prescribed to 8.951 and 8.676, respectively.
def define_initial_condition(system):
problem_ic = EquilibriumInverseProblem(system)
problem_ic.setTemperature(T)
problem_ic.setPressure(P)
problem_ic.add("H2O", 58.0, "kg")
problem_ic.add("Cl-", 1122.3e-3, "kg")
problem_ic.add("Na+", 624.08e-3, "kg")
problem_ic.add("SO4--", 157.18e-3, "kg")
problem_ic.add("Mg++", 74.820e-3, "kg")
problem_ic.add("Ca++", 23.838e-3, "kg")
problem_ic.add("K+", 23.142e-3, "kg")
problem_ic.add("HCO3-", 8.236e-3, "kg")
problem_ic.add("O2(aq)", 58e-12, "kg")
problem_ic.add("Pyrrhotite", 0.0, "mol")
problem_ic.add("Siderite", 0.5, "mol")
problem_ic.pH(8.951)
problem_ic.pE(8.676)
# Calculate the equilibrium states for the initial conditions
state_ic = equilibrate(problem_ic)
# Scale the volumes of the phases in the initial condition
state_ic.scalePhaseVolume('Aqueous', 0.1, 'm3') # 10% of porosity
state_ic.scaleVolume(1.0, 'm3')
return state_ic
Note: After providing the amounts of substances H2O, aqueous species, pyrrhotite, and siderite in the above code, Reaktoro parses these chemical formulas (using the thermodynamic database) and determines the elements and their coefficients. Once this is done, the amount of each element stored inside the object
problem_ic
is incremented according to the given amount of substance and its coefficient in the formula. The provided element amounts are then used as constraints for the Gibbs energy minimization calculation when computing the state of chemical equilibrium (i.e., when we try to find the amounts of all species in the system that corresponds to a state of minimum Gibbs energy while satisfying the element amounts constraints).
Note: Please note that we are not condemning the input form shown above in terms of element amounts, but only telling you to be attentive with the values you input. If you are using Reaktoro as a chemical reaction solver in a reactive transport simulator, for example, you will most likely need to work directly with given amounts of elements, which shows that this input form is required in certain cases. For such time-dependent modeling problems, you often only need to ensure that the initial conditions for elements amounts result in feasible initial species amounts.
To calculate the system's chemical equilibrium state with the given initial conditions, we use the method
equilibrate, the numerical
solution of which is written in the objects state_ic
. It is an instance of the class
ChemicalState that stores the temperature,
pressure, and the amounts of every species in the system.
For this calculation, Reaktoro uses an efficient Gibbs energy minimization computation to determine the species
amounts that correspond to a state of minimum Gibbs energy in the system, while satisfying the prescribed amount
conditions for temperature, pressure, and element amounts. In an inverse equilibrium problem, however, not all
elements have known molar amounts. Their amount's constraints are replaced by other equilibrium constraints such as
fixed pH and pE.
See tutorials [EquilibriumInverseProblem](eq.inverse-equilibrium.ipynb) for more detailed explanation of capabilities of this class.
The function ends with scaling the volume to 1 m3. Moreover, we specify the 10% porosity of the rock
by calling state_ic.scalePhaseVolume('Aqueous', 0.1, 'm3')
.
Next, we define the boundary condition of the constructed chemical system with its temperature, pressure, and amounts of elements. We prescribe the amount of injected hydrogen sulfide brine, in particular, 0.0196504 mol of hydrosulfide ion (HS-) and 0.167794 mol of aqueous hydrogen sulfide (H2S(aq)). Here, the ph is lowered in comparison to the initial state to 5.726.
After equilibration, the obtained chemical state representing the boundary condition for the injected fluid composition, we scale its volume to 1 m3. This is done so that the amounts of the species in the fluid are consistent with a mol/m3 scale.
def define_boundary_condition(system):
# Define the boundary condition of the reactive transport modeling problem
problem_bc = EquilibriumInverseProblem(system)
problem_bc.setTemperature(T)
problem_bc.setPressure(P)
problem_bc.add("H2O", 58.0, "kg")
problem_bc.add("Cl-", 1122.3e-3, "kg")
problem_bc.add("Na+", 624.08e-3, "kg")
problem_bc.add("SO4--", 157.18e-3, "kg")
problem_bc.add("Mg++", 74.820e-3, "kg")
problem_bc.add("Ca++", 23.838e-3, "kg")
problem_bc.add("K+", 23.142e-3, "kg")
problem_bc.add("HCO3-", 8.236e-3, "kg")
problem_bc.add("O2(aq)", 58e-12, "kg")
problem_bc.add("Pyrrhotite", 0.0, "mol")
problem_bc.add("Siderite", 0.0, "mol")
problem_bc.add("HS-", 0.0196504, "mol")
problem_bc.add("H2S(aq)", 0.167794, "mol")
problem_bc.pH(5.726)
problem_bc.pE(8.220)
# Calculate the equilibrium states for the boundary conditions
state_bc = equilibrate(problem_bc)
# Scale the boundary condition state to 1 m3
state_bc.scaleVolume(1.0, 'm3')
return state_bc
Only species in fluid phases are mobile and transported by advection and diffusion mechanisms. The solid phases are
immobile. The code below identifies the indices of the fluid and solid species. We use methods
indicesFluidSpecies
and
indicesSolidSpecies
of class ChemicalSystem to get the indices of the
fluid and solid species stored in the lists ifluid_species
and isolid_species
, respectively.
def partition_indices(system):
nelems = system.numElements()
ifluid_species = system.indicesFluidSpecies()
isolid_species = system.indicesSolidSpecies()
return nelems, ifluid_species, isolid_species
In this function, we create arrays to track the amounts of elements in the fluid and solid partition
(i.e., the amounts of elements among all fluid phases, here only an aqueous phase, and the amounts of elements among
all solid phases, here the mineral phases). The arrays b
, bfluid
, and bsolid
will store, respectively,
the concentrations (mol/m3) of each element in the system, in the fluid
partition, and in the solid partition at every time step.
The array b
is initialized with the concentrations of the elements at the initial chemical state, state_ic
,
using method
elementAmounts
of class ChemicalState. The array b_bc
stores
the concentrations of each element on the boundary in mol/m3fluid.
def partition_elements_in_mesh_cell(ncells, nelems, state_ic, state_bc):
# The concentrations of each element in each mesh cell (in the current time step)
b = np.zeros((ncells, nelems))
# Initialize the concentrations (mol/m3) of the elements in each mesh cell
b[:] = state_ic.elementAmounts()
# The concentrations (mol/m3) of each element in the fluid partition, in each mesh cell
bfluid = np.zeros((ncells, nelems))
# The concentrations (mol/m3) of each element in the solid partition, in each mesh cell
bsolid = np.zeros((ncells, nelems))
# Initialize the concentrations (mol/m3) of each element on the boundary
b_bc = state_bc.elementAmounts()
return b, bfluid, bsolid, b_bc
This step updates in the fluid partition bfluid
using the transport equations (without reactions).
The transport_full_implicit()
function below is responsible for solving an advection-diffusion equation, that is
later applied to transport the concentrations (mol/m3) of elements in the fluid partition (a
simplification that is possible because of common diffusion coefficients and velocities of the fluid species,
otherwise the transport of individual fluid species would be needed).
To match the units of concentrations of the elements in the fluid measure in mol/m3bulk and the
imposed concentration b_bc[j]
mol/m3fluid, e need to multiply it by the porosity phi_bc
on the boundary cell m3fluid/m3bulk. We use function
properties
of the class ChemicalState to retrieve fluid volume
m3fluid and total volume m3bulk in the inflow boundary cell.
The updated amounts of elements in the fluid partition are then summed with the amounts of elements in the solid
partition bsolid
, which remained constant during the transport step), and thus updating the amounts of elements
in the chemical system b
. Reactive transport calculations involve the solution of a system of
advection-diffusion-reaction equations.
def transport(states, bfluid, bsolid, b, b_bc, nelems, ifluid_species, isolid_species):
# Collect the amounts of elements from fluid and solid partitions
for icell in range(ncells):
bfluid[icell] = states[icell].elementAmountsInSpecies(ifluid_species)
bsolid[icell] = states[icell].elementAmountsInSpecies(isolid_species)
# Get the porosity of the boundary cell
bc_cell = 0
phi_bc = states[bc_cell].properties().fluidVolume().val / states[bc_cell].properties().volume().val
# Transport each element in the fluid phase
for j in range(nelems):
transport_full_implicit(bfluid[:, j], dt, dx, v, D, phi_bc * b_bc[j])
# Update the amounts of elements in both fluid and solid partitions
b[:] = bsolid + bfluid
return bfluid, bsolid, b
The function transport()
expects a conservative property (argument u
) (e.g., the concentration mol/m3
of jth element in the fluid given by bfluid[j]
), the time step (dt
), the mesh cell length (dx
),
the fluid velocity (v
), the diffusion coefficient (D
), and the boundary condition of the conservative property
(g
) (e.g., the concentration of the jth element in the fluid on the left boundary).
The transport equations are solved with a finite volume method, where diffusion and convection are treated implicitly.
Its discretization in space and time (implicit) results in the constants alpha
and beta
. These correspond to
the diffusion and advection terms in the equation: D*dt/dx**2
and v*dt/dx
, respectively.
Arrays a
, b
, c
are the diagonals in the tridiagonal matrix that results by writing all discretized equations
in a matrix equation. This system of linear equations is solved by the tridiagonal matrix algorithm, also known
as the Thomas algorithm.
def transport_full_implicit(u, dt, dx, v, D, ul):
# Number of DOFs
n = len(u)
alpha = D * dt / dx ** 2
beta = v * dt / dx
# Upwind finite volume scheme
a = np.full(n, -beta - alpha)
b = np.full(n, 1 + beta + 2 * alpha)
c = np.full(n, -alpha)
# Set the boundary condition on the left cell
if dirichlet:
# Use Dirichlet BC boundary conditions
b[0] = 1.0
c[0] = 0.0
u[0] = ul
else:
# Flux boundary conditions (implicit scheme for the advection)
# Left boundary
b[0] = 1 + alpha + beta
c[0] = -alpha # stays the same as it is defined -alpha
u[0] += beta * ul # = dt/dx * v * g, flux that we prescribe is equal v * ul
# Right boundary is free
a[-1] = - beta
b[-1] = 1 + beta
# Solve a tridiagonal matrix equation
thomas(a, b, c, u)
The tridiagonal matrix equation is solved using the Thomas algorithm (or the TriDiagonal Matrix Algorithm (TDMA)). It is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations.
def thomas(a, b, c, d):
n = len(d)
c[0] /= b[0]
for i in range(1, n - 1):
c[i] /= b[i] - a[i] * c[i - 1]
d[0] /= b[0]
for i in range(1, n):
d[i] = (d[i] - a[i] * d[i - 1]) / (b[i] - a[i] * c[i - 1])
x = d
for i in reversed(range(0, n - 1)):
x[i] -= c[i] * x[i + 1]
return x
The chemical equilibrium calculations performed in each mesh cell, using Gibbs energy minimization algorithm ( provided by the class EquilibriumSolver).
def reactive_chemistry(solver, states, b):
# Equilibrating all cells with the updated element amounts
for icell in range(ncells):
solver.solve(states[icell], T, P, b[icell])
return states
Function outputstate_df
is the auxiliary function to add data to the DataFrame at each time step.
def outputstate_df(step, system, states):
# Define the instance of ChemicalQuantity class
quantity = ChemicalQuantity(system)
# Create the list with empty values to populate with chemical properties
values = [None] * len(columns)
for state, x in zip(states, xcells):
# Populate values with number of reactive transport step and spacial coordinates
values[0] = step
values[1] = x
# Update the
quantity.update(state)
for quantity_name, i in zip(output_quantities, range(2, len(states))):
values[i] = quantity.value(quantity_name) * (100 / (1 - phi) if "phaseVolume" in quantity_name else 1)
df.loc[len(df)] = values
The last block of the main routine is dedicated to the plotting of the results in a Jupyter app generated by the library bokeh. It is an interactive visualization library that provides elegant, concise construction of versatile graphics, and affords high-performance interactivity over large or streaming datasets.
Below, we list auxiliary functions that we use in plotting. Function titlestr
returns a string for the title
of a figure in the format Time: #h##m
def titlestr(t):
t = t / minute # Convert from seconds to minutes
h = int(t) / 60 # The number of hours
m = int(t) % 60 # The number of remaining minutes
return 'Time: %2dh %2dm' % (h, m)
Routines plot_figures_ph()
, plot_figures_pyrrhotite_siderite_volume()
,
plot_figures_pyrrhotite_siderite_amount()
, and 'plot_figures_aqueous_species()' are dedicated to drawing the
plots with chemical properties on the selected steps that are specified by the user below.
def plot_figures_ph(steps):
# Plot ph on the selected steps
plots = []
for i in steps:
print("On pH figure at time step: {}".format(i))
t = i * dt
source = ColumnDataSource(df[df['step'] == i])
p = figure(plot_width=600, plot_height=250)
p.line(x='x', y='pH', color='teal', line_width=2, legend_label='pH', source=source)
p.x_range = Range1d(-1, 101)
p.y_range = Range1d(4, 9.0)
p.xaxis.axis_label = 'Distance [m]'
p.yaxis.axis_label = 'pH'
p.legend.location = 'bottom_right'
p.title.text = titlestr(t)
plots.append([p])
grid = gridplot(plots)
show(grid)
def plot_figures_pyrrhotite_siderite_volume(steps):
plots = []
for i in steps:
print("On pyrrhotite-siderite figure at time step: {}".format(i))
t = i * dt
source = ColumnDataSource(df[df['step'] == i])
p = figure(plot_width=600, plot_height=250)
p.line(x='x', y='pyrrhotite_phase_volume', color='orange', line_width=2, legend_label='Pyrrhotite',
muted_color='orange', muted_alpha=0.2, source=source)
p.line(x='x', y='siderite_phase_volume', color='steelblue', line_width=2, legend_label='Siderite',
muted_color='steelblue', muted_alpha=0.2, source=source)
p.x_range = Range1d(-1, 101)
p.y_range = Range1d(-0.001, 0.018)
p.xaxis.axis_label = 'Distance [m]'
p.yaxis.axis_label = 'Phase Volume [m3]'
p.legend.location = 'center_right'
p.title.text = titlestr(t)
p.legend.click_policy = 'mute'
plots.append([p])
grid = gridplot(plots)
show(grid)
def plot_figures_pyrrhotite_siderite_amount(steps):
plots = []
for i in steps:
print("On pyrrhotite-siderite figure at time step: {}".format(i))
t = i * dt
source = ColumnDataSource(df[df['step'] == i])
p = figure(plot_width=600, plot_height=250)
p.line(x='x', y='pyrrhotite_phase_amount', color='orange', line_width=2, legend_label='Pyrrhotite',
muted_color='orange', muted_alpha=0.2, source=source)
p.line(x='x', y='siderite_phase_amount', color='steelblue', line_width=2, legend_label='Siderite',
muted_color='steelblue', muted_alpha=0.2, source=source)
p.x_range = Range1d(-1, 101)
p.y_range = Range1d(-0.5, 5.5)
p.xaxis.axis_label = 'Distance [m]'
p.yaxis.axis_label = 'Phase Amount [mol]'
p.legend.location = 'center_right'
p.title.text = titlestr(t)
p.legend.click_policy = 'mute'
plots.append([p])
grid = gridplot(plots)
show(grid)
def plot_figures_aqueous_species(steps):
plots = []
for i in steps:
print("On aqueous-species figure at time step: {}".format(i))
source = ColumnDataSource(df[df['step'] == i])
t = dt * i
p = figure(plot_width=600, plot_height=300, y_axis_type = 'log',)
p.line(x='x', y='HSanion', color='deepskyblue', line_width=2, legend_label='HS-', source=source)
p.line(x='x', y='S2anion', color='darkorange', line_width=2, legend_label='S2--', source=source)
p.line(x='x', y='CO3anion', color='seagreen', line_width=2, legend_label='CO3--', source=source)
p.line(x='x', y='HSO4anion', color='indianred', line_width=2, legend_label='HSO4-', source=source)
p.line(x='x', y='H2Saq', color='gray', line_width=2, legend_label='H2S(aq)', source=source)
p.line(x='x', y='Hcation', color='darkviolet', line_width=2, legend_label='H+', source=source)
p.line(x='x', y='Fe2cation', color='darkblue', line_width=2, legend_label='Fe++', source=source)
p.x_range = Range1d(-1, 101)
p.y_range = Range1d(1e-12, 1e-1)
p.xaxis.axis_label = 'Distance [m]'
p.yaxis.axis_label = 'Concentration [molal]'
p.legend.location = 'top_right'
p.title.text = titlestr(t)
p.legend.click_policy = 'mute'
plots.append([p])
grid = gridplot(plots)
show(grid)
First, we create folders for the results:
make_results_folders()
Run the reactive transport simulations:
simulate()
To inspect the collected data, one can run:
df
To save the results in csv-format, please execute:
df.to_csv(folder_results + '/rt.scavenging.csv', index=False)
Select the steps, on which results must plotted:
selected_steps_to_plot = [60, 120, 960]
assert all(step <= nsteps for step in selected_steps_to_plot), f"Make sure that selected steps are less than " \
f"total amount of steps {nsteps}"
Outputting the plots to the notebook requires the call of output_notebook()
that specifies outputting the plot
inline in the Jupyter notebook:
output_notebook()
Plot ph on the selected steps:
plot_figures_ph(selected_steps_to_plot)
Plot pyrrhotite and siderite phase amounts on the selected steps:
plot_figures_pyrrhotite_siderite_amount(selected_steps_to_plot)
One can also call plot_figures_pyrrhotite_siderite_volume(selected_steps_to_plot)
instead of the function
plot_figures_pyrrhotite_siderite_amount()
.
Plot aqueous species on the selected steps:
plot_figures_aqueous_species(selected_steps_to_plot)
We see on the plots above that the main chemical reactions can be divided into two parts. On the first step, the iron ions Fe2+ are being released by the siderite (FeCO3) reacting with the H2S-brine. On the second one, free Fe2+ ions are reacting with hydrogen sulfide precipitating iron sulfide (pyrrhotite): Fe2+ + H2S → FeS2.
The minerals' dissolution and precipitation are accompanied by the formation and dilution of aqueous species. For instance, we see the sharp increase of Fe2+ at the point of the phase transformation from the siderite phase to pyrrhotite and followed by the gradual decrease as it gets used during the formation of FeS2. Both curves representing HS- and H2S(aq) have two points of the sharp decrease. The first one is where both species are involved in the dissolution of FeCO3, and the second one is where they are being exhausted by the reaction with iron ions to form iron sulfide. The CO32- anion line has an interesting shape: it first locally drops but then increases due to continuous dissolution of siderite.
To study the time-dependent behavior of the chemical properties, we create a Bokeh application using the function
modify_doc(doc)
. It creates Bokeh content and adds it to the app. The speed of streaming of reactive transport
data can be controlled by the parameter step
defined below (bigger the step, faster we will run through available
data set):
step = 10
The data streaming is looped, i.e., we will return to the initial time step when reaching the end of the reactive transport simulations.
def modify_doc(doc):
# Initialize the data by the initial chemical state
source = ColumnDataSource(df[df['step'] == 0])
# Auxiliary function that returns a string for the title of a figure in the format Time: #h##m
def titlestr(t):
t = t / minute # Convert from seconds to minutes
h = int(t) / 60 # The number of hours
m = int(t) % 60 # The number of remaining minutes
return 'Time: %2dh %2dm' % (h, m)
# Plot for ph
p1 = figure(plot_width=500, plot_height=250)
p1.line(x='x', y='pH', color='teal', line_width=2, legend_label='pH', source=source)
p1.x_range = Range1d(-1, 101)
p1.y_range = Range1d(4.0, 9.0)
p1.xaxis.axis_label = 'Distance [m]'
p1.yaxis.axis_label = 'pH'
p1.legend.location = 'bottom_right'
p1.title.text = titlestr(0 * dt)
# Plot for calcite and dolomite
p2 = figure(plot_width=500, plot_height=250)
p2.line(x='x', y='pyrrhotite_phase_volume', color='orange', line_width=2,
legend_label='Pyrrhotite', muted_color='orange', muted_alpha=0.2,
source=source)
p2.line(x='x', y='siderite_phase_volume', color='steelblue', line_width=2,
legend_label='Siderite', muted_color='steelblue', muted_alpha=0.2,
source=source)
p2.x_range = Range1d(-1, 101)
p2.y_range = Range1d(-0.001, 0.018)
p2.xaxis.axis_label = 'Distance [m]'
p2.yaxis.axis_label = 'Phase Volume [%vol]'
p2.legend.location = 'center_right'
p2.title.text = titlestr(0 * dt)
p2.legend.click_policy = 'mute'
p3 = figure(plot_width=500, plot_height=250, y_axis_type='log')
p3.line(x='x', y='HSanion', color='deepskyblue', line_width=2, legend_label='HS-', source=source)
p3.line(x='x', y='S2anion', color='darkorange', line_width=2, legend_label='S2--', source=source)
p3.line(x='x', y='CO3anion', color='seagreen', line_width=2, legend_label='CO3--', source=source)
p3.line(x='x', y='HSO4anion', color='indianred', line_width=2, legend_label='HSO4-', source=source)
p3.line(x='x', y='H2Saq', color='gray', line_width=2, legend_label='H2S(aq)', source=source)
p3.line(x='x', y='Hcation', color='darkviolet', line_width=2, legend_label='H+', source=source)
p3.line(x='x', y='Fe2cation', color='darkblue', line_width=2, legend_label='Fe++', source=source)
p3.x_range = Range1d(-1, 101)
p3.y_range = Range1d(1e-12, 1e-1)
p3.xaxis.axis_label = 'Distance [m]'
p3.yaxis.axis_label = 'Concentration [molal]'
p3.legend.location = 'top_right'
p3.title.text = titlestr(0 * dt)
p3.legend.click_policy = 'mute'
layout = column(p1, p2, p3)
# Function that return the data dictionary with provided index of the file
def update():
if source.data['step'][0] + 1 <= nsteps:
step_number = source.data['step'][0] + step
else:
step_number = 0
new_source = ColumnDataSource(df[df['step'] == step_number])
new_data = dict(index=np.linspace(0, ncells, ncells + 1, dtype=int),
step=new_source.data['step'],
x=new_source.data['x'],
pH=new_source.data['pH'],
pyrrhotite_phase_volume=new_source.data['pyrrhotite_phase_volume'],
siderite_phase_volume=new_source.data['siderite_phase_volume'],
pyrrhotite_phase_amount=new_source.data['pyrrhotite_phase_amount'],
siderite_phase_amount=new_source.data['siderite_phase_amount'],
HSanion=new_source.data['HSanion'],
S2anion=new_source.data['S2anion'],
CO3anion=new_source.data['CO3anion'],
HSO4anion=new_source.data['HSO4anion'],
H2Saq=new_source.data['H2Saq'],
Hcation=new_source.data['Hcation'],
Fe2cation=new_source.data['Fe2cation'])
p1.title.text = titlestr(step_number * dt)
p2.title.text = titlestr(step_number * dt)
p3.title.text = titlestr(step_number * dt)
source.stream(new_data, rollover=ncells+1)
doc.add_periodic_callback(update, 500)
doc.add_root(layout)
Outputting the plots to the notebook requires the call of output_notebook()
that specifies outputting the plot
inline in the Jupyter notebook. Finally, the function modify_doc()
must be passed to show
, so that the app defined
by it is displayed inline.
Important: If you run this tutorial in the localhost, make sure that number provided to the variable
notebook_url
below coincides with the number of the localhost you have in your browser.
In the app below, we refresh the reactive time step in a loop, which automatically updates the data source for the plots for ph, volume phases of calcite and dolomite, and mollalities of aqueous species (in logarithmic scale).
output_notebook()
show(modify_doc, notebook_url="http://localhost:8888")