Spontaneous Imbibition and Drainage of Water in a Thin Porous Layer: Experiments and Modeling

Abstract

The typical characteristic of a thin porous layer is that its thickness is much smaller than its in-plane dimensions. This often leads to physical behaviors that are different from three-dimensional porous media. The classical Richards equation is insufficient to simulate many flow conditions in thin porous media. Here, we have provided an alternative approach by accounting for the dynamic capillarity effect. In this study, we have presented a set of one-dimensional in-plane imbibition and subsequent drainage experiments in a thin fibrous layer. The X-ray transmission method was used to measure saturation distributions along the fibrous sample. We simulated the experimental results using Richards equation either with classical capillary equation or with a so-called dynamic capillarity term. We have found that the standard Richards equation was not able to simulate the experimental results, and the dynamic capillarity effect should be taken into account in order to model the spontaneous imbibition. The experimental data presented here may also be used by other researchers to validate their models.

Article Highlights

• We have presented a set of one-dimensional in-plane imbibition and subsequent drainage experiments in a thin fibrous layer.

• The experimental data were simulated using the Richards equation either with classical capillary equation or with a so-call dynamic capillarity term.

• The dynamic capillarity effect should be taken into account in order to model the spontaneous imbibition.

Appendices

Appendix 1: Including Evaporation Effect During Imbibition

The evaporation rate r was measured experimentally. In the experiment, a fully saturated fibrous layer was placed on a stainless-steel porous plate. The temporal changes of weight were recorded using a three-digit precision balance (Kern & Sohn GmbH, Germany). Figure 

Fig. 7

Saturation changes in the fibrous layer as a function of time during evaporation

7 shows the saturation changes in the fibrous layer as a function of time. An evaporation rate of r = 0.165 h⁻¹ was obtained by fitting the data linearly. We performed a set of simulations including evaporation during spontaneous imbibition process. The governing equations were modified by adding the evaporation term, − φr, to the right-hand side of Eq. (4). They are almost identical to the ones shown in Fig. 3a; this indicates that the influence of evaporation is negligible during spontaneous imbibition process here. This is to be expected as the time scale of evaporation was much larger than the one for our experiments (Fig. 

Fig. 8

Observed saturation profiles (open circles) and simulated curves (solid lines) obtained using the classical Richards equation including evaporation effect

8).

Appendix 2: Simulations Obtained by Adjusting Intrinsic and/or Relative Permeabilities

Figure 

Fig. 9

Observed saturation profiles (open circles) and simulated curves (solid lines) obtained using the classical Richards equation with adjusting intrinsic and/or relative permeabilities

9a shows the simulation results obtained using the classical Richards equation with the intrinsic permeability value reduced by a factor of 9.2/4.2 = 2.19 compared to the value used in Simulations I–III. In an additional set of simulations, the relative permeability was also reduced by increasing the value of parameter l in Eq. (3) from 0.5 to 1.0. Results are shown in Fig. 9a, b, respectively. It is evident that reducing the intrinsic permeability was capable of decelerating the imbibition process, similarly to decreasing the relative permeability. However, the shape of saturation profiles and saturation value at x = 1 cm were unchanged. Clearly, the classical Richards equation was not capable of modeling the spontaneous imbibition process despite adjusting values of intrinsic and/or relative permeabilities.

Appendix 3: Simulations Obtained with Different τ–S Functions

Based on the studies of dynamic capillarity in soils (Zhuang et al. 2019), we have chosen five typical functional forms for the dynamic capillarity coefficient τ. The functions are shown in Table

Table 3 Different expressions for the dynamic capillarity coefficient τ.

3. The constant τ₀ in each function was optimized and may vary in different functional forms. The simulation results using the quadratic power function τ₄ are shown in Fig. 3. The selected simulation results obtained using the remaining four τ–S functions are shown in Fig. 

Fig. 10

Observed saturation profiles (open circles) and simulated curves (solid lines) obtained using the extended Richards equation with different τ–S functional forms

10. Obviously, regardless of τ–S functional forms, including the dynamic capillarity term always improved the agreement with the wetted length of the layer at different times. Specifically, as shown in Fig. 10a, b, for τ₁ and τ₂, the saturation profiles were relatively steep with unchanged saturation at x = 1 cm. These simulations curves are not anywhere close to the measured values. The simulations using τ₃ (linear) and τ₅ (cubic power) gave a better agreement with the observations (see Fig. 10c, d). However, these results still showed either faster or slower temporal saturation changes at x = 0. The function τ₄ (quadratic power function), therefore, was chosen as the best form for modeling dynamic capillarity in this work.