Obtaining Relative Diffusivity Curves#

This is an example of calculating relative diffusivity in OpenPNM. The term relative indicates the existence of another phase in the medium. The concept of relative property (permeability) in multiphase transport was applied in relative permeability example for a flow problem. In this example the transport mechanism is defined as diffusion.

import numpy as np
import openpnm as op
import matplotlib.pyplot as plt
%matplotlib inline

Create network and phases#

First, we create a network and assign phases and properties in a similar way that we used to do for the other examples. Note that op.models.collections.phase is added to assign fluid properties such as diffusivity, surface tension, etc, and op.models.collections.physics is added to assign pore-scale models such as entry pressure, conduit conductance. Surface tension info will be used in Invasion percolation in the next step. Here, we assumed a user-defined value for air surface tension and contact angle, as they are not included in collections.phase.air. Similarly, a user-defined value for water diffusivity was assumed, as it is not included in collections.phase.water.

pn = op.network.Cubic(shape=[15, 15, 15], spacing=1e-6)
air = op.phase.Air(network=pn,name='air')
air['pore.surface_tension'] = 0.072
air['pore.contact_angle'] = 180.0
water = op.phase.Water(network=pn,name='water')
water['pore.diffusivity'] = 1e-9

Apply ordinary percolation#

To simulate a partially saturated material we first invade some non-wetting phase. This will be accomplished using the InvasonPercolation or OrdinaryPercolation algorithm. In both algorithms the invading phase invades the network based on the capillary pressure of the throats in the network. These two algorithms have difference in their percolation algorithm and results. As Invasion percolation was implemented in the relative permeability example, here we choose ordinary percolation to demonstrate how we can use its results. It is assumed that the invading phase is water:

drn = op.algorithms.Drainage(network=pn, phase=water)

In ordinary percolation, occupancies and saturation can be extracted from the results for a capillary pressure point. For example:

water['pore.occupancy'] = drn['pore.invasion_pressure'] < 10000
water['throat.occupancy'] = drn['throat.invasion_pressure'] < 10000
air['pore.occupancy'] = 1 - water['pore.occupancy']
air['throat.occupancy'] = 1 - water['throat.occupancy']

Saturation and rate functions#

To update the saturation based on occupancy from each pressure point, we created a custom function:

def sat_update(network, nwp, wp):
        Calculates the saturation of each phase using occupancy information from ordinary percolation.
        network: network
        nwp : phase
            non-wetting phase
        wp : phase
            wetting phase
    pore_mask = nwp["pore.occupancy"] == 1
    throat_mask = nwp["throat.occupancy"] == 1
    sat_p = np.sum(network['pore.volume'][pore_mask])
    sat_t = np.sum(network['throat.volume'][throat_mask])
    sat1 = sat_p + sat_t
    bulk = network['pore.volume'].sum() + network['throat.volume'].sum()
    sat = sat1/bulk
    return sat

Similar to the relative permeability example, the geometrical parameters in relative diffusivity can cancel out. Therefore, the remaining fraction includes diffusion rates as follows:

\[ D_{r}=\frac {D_\text{eff-multiphase}}{D_\text{eff-single phase}}=\frac {Rate_\text{multiphase}}{Rate_\text{single phase}} \]

Where \(Rate_\text{multiphase}\) is the inlet diffusion rate of the phase (water/air) for a Fickian diffusion algorithm with multiphase conduit_conductance assigned as conduit diffusive conductance model to account for the existence of the other phase. Whereas \(Rate_\text{single phase}\) is the inlet diffusion rate of the phase (water/air) for a Fickian diffusion algorithm with generic_diffusive_conductance assigned as diffusive conductance model for a single phase flow.

Note that \(D_\text{eff-single phase}\) is a property of the porous material, not the fluid. However, here to cancel out diffusivity in the \(D_{r}\) fraction, the same phase must be used to calculate \(Rate_\text{single phase}\). Therefore, we only need to find the rate values for calculating the relative permeability. We define a function Rate_calc, which simulates a Fickian diffusion given the diffusive conductance keyword and returns the rate diffusion at inlet pores.

def Rate_calc(network, phase, inlet, outlet, conductance):
        Fd = op.algorithms.FickianDiffusion(network=network, phase=phase)
        Fd.settings._update({'conductance' : conductance})
        Fd.set_value_BC(pores=inlet, values=1)
        Fd.set_value_BC(pores=outlet, values=0)
        val = np.abs(Fd.rate(pores=inlet, mode='group'))
        return val

In this example we are focusing on finding the relative diffusivities in x direction. A similar procedure can be applied on y and z directions. Let’s define inlet and outlet pores:

diff_in = pn.pores('left')
diff_out = pn.pores('right')

Define multiphase conductance#

Similar to the relative permeability example, multiphase conductance models are assigned to account for the presence of the other phase:

model_mp_cond = op.models.physics.multiphase.conduit_conductance
air.add_model(model=model_mp_cond, propname='throat.conduit_diffusive_conductance',
              throat_conductance='throat.diffusive_conductance', mode='medium', regen_mode='deferred')
water.add_model(model=model_mp_cond, propname='throat.conduit_diffusive_conductance',
              throat_conductance='throat.diffusive_conductance', mode='medium', regen_mode='deferred')

Calculate relative diffusivities#

Now that we have defined the required models and functions, we can find the relative diffusivity of phases over a range of saturation. The saturation points are found at a range of invasion pressure:

Snwparr = []
reldiff_nwp = []
reldiff_wp = []
for Pc in np.unique(drn['pore.invasion_pressure']):
    water['pore.occupancy'] = drn['pore.invasion_pressure'] < Pc
    water['throat.occupancy'] = drn['throat.invasion_pressure'] < Pc
    air['pore.occupancy'] = 1 - water['pore.occupancy']
    air['throat.occupancy'] = 1 - water['throat.occupancy']
    sat_point = sat_update(pn, water, air)
    Rate_single_nwp = Rate_calc(pn, air, diff_in, diff_out, conductance = 'throat.diffusive_conductance')
    Rate_single_wp = Rate_calc(pn, water, diff_in, diff_out, conductance = 'throat.diffusive_conductance')
    Rate_mult_nwp = Rate_calc(pn, air, diff_in, diff_out, conductance = 'throat.conduit_diffusive_conductance')
    Rate_mult_wp = Rate_calc(pn, water, diff_in, diff_out, conductance = 'throat.conduit_diffusive_conductance')

Plotting the curves:

plt.plot(Snwparr, reldiff_nwp, '*-', label='Dr_nwp')
plt.plot(Snwparr, reldiff_wp, 'o-', label='Dr_wp')
plt.title('Relative Diffusivity in x direction')
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