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# Snapshots from network simulations with Bond numbers comparable to those set by the inclinations in the experiments. Invading air is in white; dense wetting defending fluid is in black. The first two columns show the displacement structure after, respectively, 8% and 16% of air has saturated the medium; the two last ones show the breakthrough and final configurations.

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We experimentally and numerically study the influence of gravity and finite-size effects
on the pressure-saturation relationship in a given porous medium during slow drainage.
The effect of gravity is systematically varied by tilting the system relative to the horizontal configuration. The use of a quasi two-dimensional porous media allows for dire...

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We present a theoretical and experimental investigation of drainage in porous media. The study is limited to stabilized fluid fronts at moderate injection rates, but it takes into account capillary, viscous, and gravitational forces. In the theoretical framework presented, the work applied on the system, the energy dissipation, the final saturation...

## Citations

... 2 of 16 notable role (Ayaz et al., 2020;Kuang et al., 2011;Luo et al., 2020). Moreover, there are now published datasets consisting of stacks of many tomograms taken at different elevations of the sample (Jackson, 2019) resulting in very tall images. ...

... As long as the vertical extent H of the domain of immiscible displacement satisfies , fluid invasion patterns are not affected by gravitational forces, with the result that capillary pressure and saturation distribution are spatially uniform. In practice this condition is often not met, such that fluid saturation is not uniform but varies with vertical position in a way determined by the equilibrium between capillary and gravity forces (Ayaz et al., 2020;Clement et al., 1987;Ioannidis et al., 1996;Shahidzadeh-Bonn et al., 2004;Zhao et al., 2013). Until now there did not exist a means of including gravity effects in porous media drainage simulations via an image-based modeling technique. ...

... The Python package, PoreSpy (J. Gostick et al., 2019) was used to generate 2D and 3D porous material images with specified properties by stochastically generating overlapping disks (2D) or spheres (3D), as well as non-overlapping spheres to compare to the micromodel experiments of Ayaz et al. (2020). For 2D, a 3 cm wide by 8 cm tall (1,500 px by 4,000 px) domain was generated with disks of diameter 1 mm to create a porous medium with a target porosity of 65% and a resolution of = 0.02 (i.e., one pixel = 0.02 mm). ...

Simulating drainage in volumetric images of porous materials is a key technique for studying multiphase flow and transport. Image‐based techniques based on sphere insertion are popular due to their computational efficiency and reasonable predictions, though they lack physical rigor. Since most tomograms are small, the impact of gravity on the fluid distributions has not been previously considered. With the advent of stochastically generated images of arbitrary size, and ever larger field‐of‐view images, the validity of neglecting gravity is becoming questionable. In this work, an image‐based technique that includes the effect of gravity during gravity stabilized displacements was developed and validated. Results compared favorably with analytical solutions of capillary rise in tubes, and to micromodel experiments in terms of the pseudo‐capillary pressure curves. The compactness of the invasion front was also shown to vary linearly with the inverse Bond number. Finally, a contour map of expected error as a function of image size and Bond number was generated to help identify when gravitational effects cannot be ignored. The presented algorithm utilizes only basic image processing tools and offers the same computational advantage as other image‐based sphere insertion methods.

... Let L be the length of the porous model and assume L > w. The final saturation behind the front of the invading fluid and its dependence on the pressure across the model has been studied in Ref. [81]. In this study, we considered the volume of the invaded fluid in boxes with size corresponding to the width of the invading front. ...

... On length scales below this size, the structure of the invading fluid is fractal, while on length scales larger than this size, the structure is homogeneous. For sufficiently large F, when η is the characteristic length scale of the front, the saturation S F nw of the nonwetting fluid behind the front becomes [81]. ...

... (50) Figure 11 shows a two-dimensional invasion percolation simulation with a gravitational field together with drainage experiments performed by Ayaz et al. [81]. The red dash-dotted line confirms the predictions of the theoretical scaling in Eq. 49. ...

We present a theoretical and experimental investigation of drainage in porous media. The study is limited to stabilized fluid fronts at moderate injection rates, but it takes into account capillary, viscous, and gravitational forces. In the theoretical framework presented, the work applied on the system, the energy dissipation, the final saturation and the width of the stabilized fluid front can all be calculated if we know the dimensionless fluctuation number, the wetting properties, the surface tension between the fluids, the fractal dimensions of the invading structure and its boundary, and the exponent describing the divergence of the correlation length in percolation. Furthermore, our theoretical description explains how the Haines jumps’ local activity and dissipation relate to dissipation on larger scales.

... Let L be the length of the porous model and assume L > w. The final saturation behind the front of the invading fluid and its dependence on the pressure across the model has been studied in reference [62]. In those studies we considered the volume of the invaded fluid in boxes corresponding to the width of the invading front. ...

... On length scales below this length scale, the structure of the invading fluid will be fractal, while on length scales larger than this length scale the structure will be homogeneous. For sufficient large F , when η is the characteristic length scale of the front, the saturation S F nw of the non-wetting fluid behind the front will be [62] S F nw ∝ where d is the spatial dimension (2 or 3). Hence using Eq. ...

... However, when the fluctuation number F is sufficient small, η > w, and the width of the model w will be the characteristic length scale in the problem. Then Fig. 11 shows two dimensional invasion percolation simulation with a gravitational field and experiments, by Ayaz et al. [62]. In the equations above we have neglected the boundary effects at the inlet and outlet since L/w 1. ...

We present a theoretical and experimental investigation of drainage in porous media. The study is limited to stabilized fluid fronts at moderate injection rates, but it takes into account capillary, viscous, and gravitational forces. In this theory the work applied on the system, the energy dissipation, the final saturation and the width of the stabilized fluid front can all be calculated if we know the dimensionless fluctuation number, the wetting properties, the surface tension between the fluids, the fractal dimensions of the invasion front and the invading structure, and the exponent describing the divergence of the correlation length in percolation. This theoretical description explains how the Haines jumps' local activity and dissipation relate to dissipation on larger scales.

... Let L be the length of the porous model and assume L > w. The final saturation behind the front of the invading fluid and its dependence on the pressure across the model has been studied in Ref. [82]. In this study, we considered the volume of the invaded fluid in boxes with size corresponding to the width of the invading front. ...

... On length scales below this size, the structure of the invading fluid is fractal, while on length scales larger than this size, the structure is homogeneous. For sufficiently large F , when η is the characteristic length scale of the front, the saturation S F nw of the nonwetting fluid behind the front becomes [82] S F nw ∝ ...

... However, when the fluctuation number F is sufficient small, η > w, and the width of the model w will be the characteristic length scale in the problem. Then Fig. 11 shows a two-dimensional invasion percolation simulation with a gravitational field together with drainage experiments performed by Ayaz et al. [82]. The red dash-dotted line confirms the predictions of the theoretical scaling in Eq. (49). ...