Relaxation of a fluid-filled blister on a porous substrate

The relaxation dynamics of a fluid-filled blister between an elastic sheet and a porous substrate are controlled by the deformation of the elastic sheet, the viscous stresses in the pores, and the capillary pressure at the liquid-air interface due to imbibition. We develop a mathematical model to study the effects of varying the permeability of the porous substrate, the bending stiffness of the elastic sheet, and the blister size on the relaxation dynamics. Experiments are conducted by injecting a finite volume of viscous fluid between a porous substrate and an elastic sheet, where fluid first invades the pores, and subsequently, as the pressure in the fluid increases, the elastic sheet is peeled and uplifted from the substrate to form a fluid-filled blister. After injection is stopped, the fracture front is static, and the elastic stresses in the overlying sheet and the capillary pressure at the liquid-air interface drive the drainage of the blister into the pores. We identify two regimes of drainage. For thick sheets and more permeable substrates, drainage is primarily due to the stresses in the deformed elastic sheet. For thin sheets and less permeable substrates, drainage is driven by the imbibition of the liquid into the pore space. Our model and experiments are relevant to the drainage of fluid-driven fractures in porous media.

Figure 1

(a) Schematic of experimental setup and theoretical model. (b) Time sequence images of the experimental relaxation dynamics where fluid from the blister (dark blue) drains into the porous substrate (light blue). The blister and fluid front in the pores are outlined.

Figure 2

(a) Pillar array geometry for three substrates. (b) Out-of-plane curvature, ${\kappa }_1$, between the bottom of the porous substrate and the elastic top sheet. (c) In-plane curvature, ${\kappa }_2$, between two adjacent pillars. (d) Solutions of Eq. (4) for varying ${\tilde{p}}_c$ (solid lines) to determine best fit of ${\tilde{p}}_c$ for an experiment (black dots with representative error bars). (e) Best fit of ${\tilde{p}}_c$ for experiments of varying characteristic elastic pressure, $P_e$, and permeability, $k$.

Figure 3

(a) Image processing to determine the regions of the blister and of the fluid in the pores. (b) Normalized image intensity, $I/I_0$, profiles through the centroid of the blister are shown for 90 s intervals. Noise is due to the spatial structure of the pillars. Note that at the center, the signal is blocked due to the injection hole. (c) Normalized intensities from (b) are filtered using a Gaussian filter, $G(I/I_0)$, to remove noise. The profiles through the centroid are plotted for 90 s intervals.

Figure 4

(a) Time sequence of images showing blister height measured using the dye intensity calibration method. The centroid of the blister is shown by the white dot near the injection hole. The subsequent plots are profiles of the horizontal line through the centroid. (b) Blister shape as a function of time. Theoretical blister profiles [black curves, Eqs. (17) and (18)] are plotted using the volume of fluid in the blister calculated from taking the sum of the height of each pixel within the blister at each time step compared to experimental data (colored symbols) (c) Dimensionless blister shape where $h$ is rescaled by the maximum theoretical height, $h_{\max }$, and $r$ is rescaled by $R$. $R=11\text{mm}$, $D=5.7\times {10}^{-4}\mathrm{Pa}{\mathrm{m}}^3$.

Figure 5

(a) Blister volume $V$ as a function of time $t$ for varying permeability of the porous substrate, $k$, bending modulus of the elastic sheet, $D$, blister radius, $R$, initial blister volume, $V_0$, and total volume, $V_{\mathrm{tot}}$. Symbols show experimental results with representative error bars. Solid lines show the solution to the thick elastic sheet model, Eq. (19). (b) Rescaled blister volume $\mathcal{V}$ as a function of rescaled time $\tau$. Solid lines show the solution to the dimensionless thick elastic sheet model, Eq. (21).

Figure 6

(a) Blister volume $V$ as a function of time $t$ for varying permeability of the porous substrate, $k$, bending modulus of the elastic sheet, $D$, blister radius, $R$, initial blister volume, $V_0$, and total volume, $V_{\mathrm{tot}}$. Dashed lines show the solution to the model [Eqs. (5) and (6)] neglecting tension in the sheet. The solid lines show the solution to the model including contributions from both bending and stretching. (b) Rescaled blister volume $\mathcal{V}$ as a function of rescaled time $\tau$.