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Wall Effect Mitigation Techniques for Experiments with Planar Walls

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Abstract

Experiments in porous media suffer from preferential flow along apparatus walls—called the wall effect—which results from higher porosity and therefore higher permeability near the walls. Through a theoretical analysis of porosity in a three-dimensional rectangular apparatus containing spherical beads with hexagonal close packing, this study shows that porosity, and therefore wall effects, exhibits different behavior along each of the orthogonal walls. This study also experimentally evaluates two techniques for mitigating wall effects in experiments of solute transport in porous media using monodisperse, spherical beads in hexagonal close packing as the bulk porous medium. The first mitigation technique adds a sublayer of smaller beads of one third of the diameter of the primary beads between the wall and the bulk media. The second mitigation technique applies a half-bead-diameter-thick layer of silicone to the wall and embeds one layer of beads into the silicone, creating a wall of hemispheres. Both techniques seek to impose more uniform porosity up to the wall. Velocity profiles indicate that both techniques eliminate preferential flow along the wall and therefore are effective at mitigating the wall effect.

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The authors gratefully acknowledge the two anonymous reviewers whose comments improved the paper. Funding was provided by the National Science Foundation under Grants EAR-1417005 and EAR-1417107.

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Correspondence to John P. Crimaldi.

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Appendix: Areal Porosity Near Walls

Appendix: Areal Porosity Near Walls

Figure 12 shows a horizontal projection of two layers of spheres packed with hexagonal close packing in a three-dimensional, rectangular prismatic box, with the light-colored spheres (gray in online version) on the bottom (Layer 1) and the dark-colored spheres (tan in online version) in the layer above (Layer 2). The white triangles in Fig. 12 are used in Fig. 13 to determine the offsets between spheres in the same layer and between two spheres in adjacent layers, which are used below to calculate areal porosity. Let the coordinates of the center of Spheres G (in Layer 1) and T (in Layer 2) be (xG, yG, zG) and (xT, yT, zT), respectively. By inspection, xT − xG = d/2. From Fig. 13, yT − yG = \(d/\left( {2\sqrt 3 } \right)\). Setting the distance between the centers of Spheres G and T to d leads to zT − zG = \({2d/{\sqrt 6}}\). Although not shown in the figure, the alternating of gray and tan layers continues in the z direction (as shown in Fig. 1b), with the bottom elevation of Layer j equal to \(z ={2\left( {j - 1} \right)d}/ {\sqrt 6 }\).

Fig. 12
figure 12

Horizontal projection of two layers of spheres with HCP packing in a rectangular box. Light circles (gray in online version) represent odd-numbered layers of spheres, numbered sequentially from the bottom. Dark circles (tan in online version) represent the even-numbered layers of spheres. The angles and side lengths of triangles ABC and ACD are presented in Fig. 13. Line JK is a projection of a control surface in the yz plane; Line LM is a projection of a control surface in the xz plane; and Trapezoid NOPQ is a control surface in the xy plane

Fig. 13
figure 13

Trigonometry of triangles ABC and ACD in Fig. 12. Vertices A, C, and D represent centers of spheres in the odd-numbered layers (light circles in Fig. 12), and Vertex B represents a point of contact between two such spheres. Point E is equidistant from vertices A, C, and D, and is situated directly below the center of a sphere in the even-numbered layers (dark circles in Fig. 12). The lengths of AE, CE, and DE are found from the law of sines as \(\overline{\text{AE}} = \sin 30^{^\circ } d/\sin 120^{^\circ } = d/\sqrt 3\)

The areal porosity, n(s), in a plane parallel to the s-normal wall is given by

$$n\left( s \right) = \frac{{A_{\text{CS}} - A_{\text{S}} \left( s \right)}}{{A_{\text{CS}} }}$$
(3)

where s is the distance from the wall, ACS is the area of a control surface parallel to the wall, and AS(s) is the total area of intersection of the spheres and the control surface at a distance s from the wall. The shape and size of the control surface are chosen such that a tessellation of the shape covers the plane parallel to the wall, and each tile of the tessellation contains the same sphere geometry. The area of intersection of a single sphere with the control surface is a circle of radius, ρ, given by

$$\rho = \sqrt {\frac{{d^{2} }}{4} - \eta^{2} }$$
(4)

where η is the distance in the s direction between the sphere center and the control surface.

For the porosity in the x direction relative to the left wall in Fig. 12, let the control surface be a rectangle represented by line JK in Fig. 12, which has length \(\ell = d\sqrt 3\) in the y direction and height \(h = 4d/ {\sqrt 6 }\) in the z direction, from the center of one layers of light-colored spheres to the center of the next layer of light-colored spheres. The area of this control surface in the yz plane is \(A_{\text{CS}} = \ell h = {{4d^{2} }/{\sqrt 2 }}\).

Figure 14a shows the control surface and the areas of intersection of spheres for various values of x. For x < d/2, the control surface intersects four light-colored spheres whose intersection with the control surface sum to form a complete circle and one dark-colored sphere entirely intersecting the control surface. The radius of the circles is found using (4) with η = d/2 − x. For \(x \ge {d \mathord{\left/ {\vphantom {d 2}} \right. \kern-0pt} 2}\), the control surface intersects nine spheres—six light-colored spheres and three dark-colored spheres as shown in Fig. 14a (for some x, the circles have a radius of 0 based on the value of η for that x). From trigonometry and Eqs. (3) and (4), and summation of the areas of each circle and partial circle in Fig. 14a, the areal porosity relative to the left wall is given by

$$n\left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{2d^{2} - \pi \sqrt 2 \left( {xd - x^{2} } \right)}}{{2d^{2} }}} \hfill & {x < {d \mathord{\left/ {\vphantom {d 2}} \right. \kern-0pt} 2}} \hfill \\ {\frac{{8d^{2} - \pi \sqrt 2 \left( {d^{2} + 4d{\kern 1pt} \left| {x - kd} \right| - 8\left( {x - kd} \right)^{2} } \right)}}{{8d^{2} }}} \hfill & {\left( {k - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)d \le x < \left( {k + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)d} \hfill \\ \end{array} } \right.$$
(5)

for k = 1, 2, … This porosity distribution is plotted in Fig. 3. Beyond x = d/2, the pattern repeats at intervals of d.

Fig. 14
figure 14

Intersection of spheres with the control surfaces in a the yz plane for various values of x; b the xz plane for various values of y; and c the xy plane for various values of z. In subplot a, the shapes with dashed outlines represent spheres centered at x = kd, and the shapes with solid outlines represent spheres centered at x = (2 k − 1) d/2, where k = 1, 2, 3, … In subplot b, the light (gray) and dark (tan) shapes with solid outlines represent spheres centered at y = 0.5d and \(y = 0.5d + d / ({2\sqrt 3 })\), respectively. The light (gray) and dark (tan) shapes with dashed outlines represent spheres centered at \(y = 0.5d\left( {1 + \sqrt 3 } \right)\) and \(y = ( 3 + 4\sqrt 3)d/ 6\), respectively. In subplot c, the spheres are centered at \(z = \left( {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2} + 2j\sqrt 6 } \right)d\), where j is the layer number

For the porosity in the y direction relative to the front wall in Fig. 12, let the control surface be a rectangle represented by line LM in Fig. 12, which has length d in the x direction and again extends in the z direction from the center of one layers of light-colored spheres to the center of the next layer of light-colored spheres. The area of this control surface in the xz plane is \(A_{\text{CS}} = 4d^{2}/{\sqrt 6 } .\) Figure 14b shows the control surface and the areas of intersection of the spheres for various values of y. For \(y < d/\left( {2\sqrt 3 } \right)\) (y < 0.289d), the control surface partially intersects two light-colored spheres, whose areas sum to form a complete circle. For \(d/\left( {2\sqrt 3 } \right) \le y < d{\sqrt 3 }/2 \left( {0.289d \le y < 0.866d} \right),\) the control surface partially intersects two light-colored spheres and partially intersects two dark-colored spheres, whose areas sum to form one complete light-colored circle and one complete dark-colored circle, each with a different radius. For \(y \ge {d\sqrt 3 } / 2 \left( {y \ge 0.866d} \right),\) the control surface may partially intersect two to six light-colored spheres and may wholly or partially intersect one to three dark-colored spheres, depending on the value of y.

From trigonometry and Eqs. (3) and (4), and summation of the areas of each circle and partial circle in Fig. 14b, the areal porosity relative to the front wall is given by

$$n\left( y \right) = 1 - A_{ylw}^{*} \left( y \right) - A_{ydp}^{*} \left( y \right) - A_{ylp}^{*} \left( y \right) - A_{ydw}^{*} \left( y \right)$$
(6)

where \(A_{yK}^{*} \left( y \right)\) is the normalized area (normalized by the control section area) for the different groups of spheres in Fig. 14b, where K = lw for the whole light-colored spheres, K = dp for the partial dark-colored spheres, K = lp for the partial light-colored spheres, and K = dw for the whole dark-colored spheres. The normalized areas are

$$\begin{aligned} A_{ylw}^{*} \left( y \right) & = \frac{\pi \sqrt 6 }{4}\left[ {\frac{1}{4} - \left( {\frac{y}{d} - \frac{1}{2} - \left( {m_{1} - 1} \right)\sqrt 3 } \right)^{2} } \right]\quad {\text{for}}\;\left( {m_{1} - 1} \right)d\sqrt 3 \le y < d + \left( {m_{1} - 1} \right)d\sqrt 3 \\ A_{ydp}^{*} \left( y \right) & = \frac{\pi \sqrt 6 }{4}\left[ {\frac{1}{4} - \left( {\frac{y}{d} - \frac{1}{2} - \left( {m_{2} - \frac{5}{6}} \right)\sqrt 3 } \right)^{2} } \right]\quad {\text{for}}\;\left( {m_{2} - \frac{5}{6}} \right)d\sqrt 3 \le y < d + \left( {m_{2} - \frac{5}{6}} \right)d\sqrt 3 \\ A_{ylp}^{*} \left( y \right) & = \frac{\pi \sqrt 6 }{4}\left[ {\frac{1}{4} - \left( {\frac{y}{d} - \frac{1}{2} - \left( {m_{3} - \frac{1}{2}} \right)\sqrt 3 } \right)^{2} } \right]\quad {\text{for}}\;\left( {m_{3} - \frac{1}{2}} \right)d\sqrt 3 \le y < d + \left( {m_{3} - \frac{1}{2}} \right)d\sqrt 3 \\ A_{ydw}^{*} \left( y \right) & = \frac{\pi \sqrt 6 }{4}\left[ {\frac{1}{4} - \left( {\frac{y}{d} - \frac{1}{2} - \left( {m_{4} - \frac{1}{3}} \right)\sqrt 3 } \right)^{2} } \right]\quad {\text{for}}\;\left( {m_{4} - \frac{1}{3}} \right)d\sqrt 3 \le y < d + \left( {m_{4} - \frac{1}{3}} \right)d\sqrt 3 \\ \end{aligned}$$
(7)

where mi = 1, 2, 3, … This porosity distribution is plotted in Fig. 3. For \(y \ge (\sqrt 3 - 1)d /{\sqrt 3 }\), the pattern repeats at intervals of \({{d\sqrt 3 }/ 2}\).

For the porosity in the z direction relative to the bottom wall, let the control surface be a trapezoid NOPQ in Fig. 12, of length d on each side. The area of \(A_{\text{P}} = {d^{2} \sqrt 3 } / 2\) in the xy plane. Figure 14c shows the control surface and the areas of intersection of the spheres for various values of z. For all z, the control surface partially intersects four light-colored spheres whose areas sum to form a complete circle; however, depending on η, the circle may have a radius of ρ = 0. For \(z \ge {{2d} / {\sqrt 6 }} \left( {z \ge 0.816d} \right),\) the control surface also intersects one or three dark-colored spheres whose areas sum to form a complete circle (depending on η, the circle may have a radius of ρ = 0). All spheres are centered at \(z = \left( {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2} + 2j\sqrt 6 } \right)d\), where j is the layer number. From trigonometry and Eqs. (3) and (4), and summation of the areas of each circle and partial circle in Fig. 14c, the areal porosity relative to the bottom is given by

$$n\left( z \right) = 1 - \sum\limits_{j} {A_{zj}^{*} } \left( z \right)$$
(8)

where \(A_{zj}^{*} \left( z \right)\) is the normalized area (normalized by the control section area) of the spheres in Layer j as a function of z, given by

$$A_{zj}^{*} \left( z \right) = \frac{\pi }{2\sqrt 3 }\left[ {1 - \left( {\frac{2z}{d} - 1 - \frac{{4\left( {j - 1} \right)}}{\sqrt 6 }} \right)^{2} } \right]\quad {\text{for}}\quad \frac{{2\left( {j - 1} \right)}}{\sqrt 6 }d \le z < d + \frac{{2\left( {j - 1} \right)}}{\sqrt 6 }d$$
(9)

This porosity distribution is plotted in Fig. 3. For \(z \ge {{\left( {3 - \sqrt 6 } \right)d} / 3}\), the pattern repeats at intervals of \({{2d} /{\sqrt 6 }}\).

As drawn in Fig. 12, the box dimensions are \(N_{x} d\, \times \;d + \left( {N_{y} - 1} \right)d{{\sqrt 3 } /2}\) in the xy plane, where Nx and Ny are the number of spheres in the x- and y-directions, respectively, in an odd-numbered layer. With these box dimensions, the odd-numbered layers are perfectly contained within the box. Nevertheless, the porosity is higher near the back wall than near the front wall, because the distance between the back wall and the last row of dark-colored spheres is greater than the distance between the front wall and the first row of dark-colored spheres. Thus, even for a perfectly dimensioned container, different wall effects would be observed along the two different walls in the yz plane.

Furthermore, any rectangular prismatic box is likely not to be perfectly dimensioned, further increasing the wall effects. For example, if the box in Fig. 12 is slightly larger in the x direction, the spheres would be more loosely packed at the right wall, assuming that during packing the spheres are placed tightly along the left wall. Thus, n(x) near the right wall would be higher than n(x) near the left wall, leading to higher preferential flows and magnified wall effects along the right wall.

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Roth, E.J., Neupauer, R.M., Mays, D.C. et al. Wall Effect Mitigation Techniques for Experiments with Planar Walls. Transp Porous Med 132, 423–441 (2020). https://doi.org/10.1007/s11242-020-01399-9

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