Microscopic Validation of a Pore Network Model for Hygric Properties of Porous Materials

Abstract

In a pore network model, microscopic moisture storage and transport processes are modeled at pore level after which this information is extrapolated to obtain macroscopic properties describing the material’s moisture behavior. Such a model is typically validated by comparing measured and simulated macroscopic properties. However, due to the uncertainty associated with the experimental property determination, a possibly incorrect description of the model’s microscopic processes can be overlooked. Assessing the model’s ability to correctly simulate the moisture behavior at pore level is therefore required for its complete validation. To this aim, the moisture stored in the individual pores of unsaturated materials is imaged with the help of X-ray computed tomography images and compared to the moisture distribution simulated with a pore network model. The acquired X-ray computed tomography images clearly show the evolution of the drying process, wherein emptied pores retain water in their irregularly shaped corners. While some corners do not store any moisture, others allow a maximum of 10 % of the pore’s volume to be filled with corner islands. By comparing these images with the pore network model, however, it becomes clear that the amount of water, trapped in these pore corners is heavily overestimated in the model’s current implementation. Since this implementation is commonly used in existing pore network models, this paper proves the need of a detailed investigation of the corner islands in real porous media to formulate a different way of calculating moisture storage in pore corners.

Article Highlights

• Visualising the distribution of moisture in unsaturated material using X-ray computed tomography

• Microscopic validation of a pore network model by comparing the simulated moisture storage in the individual pore bodies and throats with X-ray computed tomography images of unsaturated material.

• Proof of the limitations of the pore network model with regards to moisture storage in pore corners and recommendations for possible model improvement.

Appendices

Appendix A Invasion Percolation Algorithm

In Islahuddin and Janssen (2019), 4 different percolation algorithms are defined which all describe a different method of moisture invasion or withdrawal: adsorption, desorption, imbibition and drainage. The percolation algorithm used in this paper closely resembles the drainage algorithm, but is slightly adapted for preconditioning in the centrifuge. Assume the pore network in Fig. 16. From a previous study (Deckers and Janssen 2023), it is known that moisture is forced outward in the direction of the centrifugal force (F_centr) and that it expels from the sample at the right surface, indicated by the blue line. The other surfaces are covered with vapor tight tape to eliminate moisture evaporation. Due to imperfect contact between the tape and the sample, some air can still penetrate the left surface, indicated by the dotted line.

An invasion percolation algorithm attempts to replicate the centrifuge drainage process by stepwise emptying the different elements in the pore network. In every step, only the elements connected via a continuous airway to the air boundary (dotted line) are candidates to be emptied next. In other words, pores 1 and 2 in Fig. 16 are candidates to be emptied first. Whenever there are multiple candidates, the element with the highest threshold capillary pressure, as calculated by Eq. 1 (i.e. the biggest element), empties first. This results in pore 1 in Fig. 16 emptying in the first step of the algorithm. This first step is subsequently characterised by pore 1’s threshold capillary pressure (p_1,thres). In other words, the algorithm remembers which pores are filled at this capillary pressure for further calculations. Since element 1 empties, element 5 becomes viable to be emptied next. If its threshold capillary pressure is even bigger than p_1,thres, it is also emptied in step 1 and element 11 becomes viable to be emptied next. If the threshold capillary pressure of element 5 is lower than p_1,thres, it is not emptied in step 1, but simply added to the list of candidates to be emptied in step 2. This drainage algorithm continues until full desaturation.

This discussion describes the drainage algorithm as developed in Islahuddin and Janssen (2019). Since drainage here is governed by the centrifugal force, the algorithm is slightly changed to only allow elements to desaturate in the direction of the centrifugal force. This prohibits water movement against the direction of the driving force. Element 9, which has no connecting throats in the direction of this centrifugal force therefore does not empty.

Fig. 16

Illustration of the percolation algorithm

Appendix B Calculation of the Liquid Permeability in the PNM

This appendix handles the permeability calculation of the PNM. A more detailed explanation and several examples are given in Islahuddin and Janssen (2019). After the percolation algorithm, the distribution of moisture throughout the pore network is known at different capillary pressures. The algorithm knows which pores are capillary filled (and hence only transport moisture through liquid flow) and which pores are partially empty with liquid corner islands and liquid films (and hence transport moisture both in vapor and liquid form). The next step is to calculate the moisture conductance of each element and to do so, liquid and vapor flow are assumed to be two parallel processes. This means that in empty elements with liquid flow in corner islands and in films adhered to the surfaces, the vapor (g_v) and liquid (g_l) conductances are calculated separately and summed up to form the total conductance (g_tot). In reality, these transport processes will interact (Zhao et al. 2020b), however, simulations with much higher complexity and increased computation time are required to simulate this. The vapor (g_v) and liquid (g_l) conductances are calculated based on Fickian diffusion processes and Hagen-Poiseuille flow, respectively, using following Equations:

$$\begin{aligned}{} & {} g_l = \frac{c_G \rho _l G A^2}{\mu L} \end{aligned}$$

(3)

$$\begin{aligned}{} & {} g_v = \frac{1}{1+K_N} \frac{\delta _vp_v}{\rho _lR_vT}\frac{A^*}{L} \end{aligned}$$

(4)

$$\begin{aligned}{} & {} K_N = \frac{\lambda }{2r} \end{aligned}$$

(5)

$$\begin{aligned}{} & {} \lambda = \frac{RT}{\sqrt{1+(M_v/M_a)\, \pi N_a ((d_v + d_a)/2)^2P}} \end{aligned}$$

(6)

with c_G [m²] a constant dependent on the shape factor (Patzek and Kristensen 2001), $\mathrm {\rho _l}$ [kg/m$^{3}$] the density of water, G [-] the shape factor, $\mathrm {\mu }$ [Pa s] the dynamic water viscosity, L [m] the length of the element, $\mathrm {\delta _v}$ [kg/msPa] is the air vapor permeability, p_v [Pa] the vapor permeability calculated using Kelvin’s law, A* the element’s cross section which is reduced to account for the adhered water films and possible corner islands, R_v [J/kgK] the vapor constant, T [K] the temperature, $\mathrm {\lambda }$ [m] the mean-free-path length, M_v and M_a [g/mol] the molar masses of water vapor and air, N_a [1/mol] the Avogadro constant and d_v [m] and d_a [m] the collision diameters of vapor and air. For more information about these parameters, the reader is referred to Islahuddin and Janssen (2019).

Once the conductances of all element are known, the pore body (i) to pore body (j) conductances (g_ij) are calculated by assuming that the conductances of half of pore body i, the entire throat between i and j and half of pore body j are placed in series. The total pore body to pore body conductance is hence calculated by reciprocal addition of these three conductances. A detailed example is given in Islahuddin and Janssen (2019).

Finally, a set of mass balance equations is composed in all pores i by assuming mass conservation in each pore body, as shown in Eq. 7. This allows calculating the capillary pressure distribution across the elements of the pore network.

$$\begin{aligned} \sum g_{ij}(p_{c,j}-p_{c,i}) = 0 \end{aligned}$$

(7)

The network permeability k [kg/msPa] is then calculated from Darcy’s law in Eq. 8 where Q is the total moisture flow into or out the pore network, which can be derived from the calculated conductances and capillary pressures. A_n and L_n are the cross-section and length of the pore network.

$$\begin{aligned} Q = k \Delta p_c \frac{A_n}{L_n} \end{aligned}$$

(8)