Spontaneous imbibition in parallel layers of packed beads

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1 Eur. Phys. J. E (2017) 40: 39 DOI /epje/i Regular Article THE EUROPEAN PHYSICAL JOURNAL E Spontaneous imbibition in parallel layers of packe beas S. Ashraf 1, G. Visavale 1, S.S. Bahga 2, an J. Phirani 1,a 1 Department of Chemical Engineering, Inian Institute of Technology Delhi, Hauz Khas, New Delhi, , Inia 2 Department of Mechanical Engineering, Inian Institute of Technology Delhi, Hauz Khas, New Delhi, , Inia Receive 16 December 2016 an Receive in final form 15 February 2017 Publishe online: 5 April 2017 c EDP Sciences / Società Italiana i Fisica / Springer-Verlag 2017 Abstract. The imbibition of a wetting flui in a homogeneous porous meium follows the iffusion-like behavior escribe by Washburn. The impregnation of a two-layere porous meium by a wetting flui ue to capillary action has been previously escribe to have two fronts, one saturating the meium an the other, leaing front, which propagates in finer pores. Here, we report that the leaing front is governe by the porous structure an is not always in the finer pores. Base on the experiments in a layere porous meium of permeability varying perpenicular to the irection of flow, we show that the permeability of the ajacent layers plays a significant role in etermining the leaing front amongst the layers. We have also evelope an analytical moel which escribes the flow ynamics in the layere porous meium. The moel preicts the conition for which the leaing front in the larger pores is followe by the front in the finer pores. This conition is also verifie experimentally. 1 Introuction The flow of viscous fluis in porous meia ue to capillary forces is of great importance in the fiels of enhance oil recovery [1,2], geological CO 2 sequestration [3,4], fabric esign [5], chemical reactors [6] an meical iagnostics [7]. Lucas [8] an Washburn [9] gave a lubrication approximation moel, showing the iffusive ynamics of imbibe length l of a perfectly wetting flui of viscosity )t, in a capillary of iameter. It is important to note that flui traverses more istance in a larger iameter capillary, in the same time interval t because the permeability is proportional to 2 an the capillary force is proportional to μ, surface tension γ with time t as l 2 =( γ 4μ 1. Several moifications to Washburn s law for iffusive ynamics have been reporte for ifferent cross-section shapes of capillary [10 12], axially varying geometries [13, 14] an in the presence of a viscous non-wetting flui in the capillary [15 18] showing that the meniscus isplacement eviates from Washburn s law (l 2 t). The spontaneous imbibition ue to capillary forces in a homogenous porous meium has also been shown to follow the iffusion-like response escribe by Washburn. This is because the permeability of a porous meium is irectly proportional to the square of the particle iameter accoring to the classical Kozeny-Carman moel [19], whereas the capillary pressure is inversely proportional to the particle iameter [6, 20]. This is analogous to the relationship between the permeability an Young-Laplace pressure in a jphirani@chemical.iit.ac.in a capillary an explains the observe Washburn-like behavior seen in the porous meia. The front isplacement square is also proportional to the particle iameter of a homogeneous porous meium. Reyssat et al. [20] showe that the iffusive ynamics vary in a porous meium with permeability varying in the irection of flow. Bico an Quéré [21] an Dong et al. [22] showe that in a bi-isperse meium, flui rise is more in the layer of narrow channels an this is attribute to the flui transfer from large channels to narrow channels. Bico an Quéré [21] also showe that the front measure by the weight imbibe an the visible front in the finer pores, both follow the iffusive law. In this work we explore spontaneous imbibition an its ynamics in layere porous meia using experiments an a one-imensional lubrication approximation moel. Our experiments show that ifferent imbibition patterns emerge in a layere porous meium for a capillary-riven flow. The leaing front of the imbibing flui is not always in the narrow channels an is epenent on the relative position an permeability of the layers. Our oneimensional moel is able to explain the conitions for which the leaing front is not in the narrowest channel. At very early time scales, the inertial effects are also important [23 25]. However, the inertial time scales in the present work are of the orer of 1 milisecon an can be neglecte for the time scales use in the moel an in the experiments. We also show that the sum of squares of the iniviual lengths imbibe in the layers is proportional to time, rather than the square of the total volume imbibe being proportional to time.

2 Page 2 of 6 Eur. Phys. J. E (2017) 40: 39 2 Experiments an analytical moel 2.1 Two-layere porous meium A porous meium is constructe using a pair of microscopic glass slies of length 75 mm, place 1 mm apart. Three glass bea sizes of iameter 0.1 mm, 0.5 mm an 1 mm are use to construct the porous meia. The glass beas are washe with acetone an rinse several times to remove the impurities ahearing onto the beas. Three separate porous meia having two layers are constructe using i) 0.1 mm an 0.5 mm, ii) 0.5 mm an 1 mm an iii) 0.1 mm an 1 mm iameter beas as shown in fig. 1. The with of the slies use in these cases is 16 mm (8 mm for each layer). The layers are continuous an can hyroynamically communicate with each other through the pore spaces. A reservoir is constructe on one en of the moel an paraffin oil is fille in the reservoir, which is the wetting phase an imbibes into the porous meium. The viscosity of the paraffin oil is Pa s, the surface tension is N m 1, an the ensity is 875 kg m 3.The other en of the porous meium is left open for outflow. A high-resolution camera (Nikon D 750) is use for recoring the imbibition phenomena an a customize image analysis software in MATLAB is use for image analysis. The front isplacement snapshots an length vs. time graphs of the three cases are shown in fig. 1(a), (b), (c). For the two-layere meia, the flui front isplacement is always more in the smaller permeability layer of the two layers, for all the three arrangements. This is in agreement with the results by Bico an Quéré [21]. We now use a one-imensional lubrication approximation moel to explain the pattern of the observe front isplacement. We assume that for a layere porous meium forme by beas of iameters an D ( <D), the layers have permeability k = Λ 2 an K = ΛD 2. This scaling for permeability hols for small beas, where Λ can be estimate by the Kozeny-Carman equation [19]. On the other han, for large beas which are of the size of gap between the top an the bottom walls, the same scaling is vali. This is because for large beas, even the gap between the top an the bottom walls scales with the iameter of the beas. However, we note that Λ in such a case cannot be estimate by the Kozeny-Carman equation. The orer of magnitue of Λ is assume to be same for the single layer of beas, as the resistance offere by the beas is similar. Therefore, same Λ is use in eveloping the oneimensional moel. The front isplacements of a completely wetting flui of viscosity μ an surface tension γ are l an L, respectively, at time t (l >L), as shown in fig. 2(a). The viscous loss in the large permeability layer is given by Darcy s equation as P (L, t) P 0 = μul(t) K, (1) where U is the velocity of the flui in the large permeability layer (K), P 0 is the atmospheric pressure, an u is the velocity of the flui in the small permeability layer Fig. 1. Interacting two-layere porous meia. The images of porous meia show the imbibition at ifferent times. The flui exchange between the layers is represente by arrows in the snapshots. (a), (b) an (c) show the ifferent cases for a twolayere porous meia mae of 0.1 mm, 0.5 mm, an 1 mm glass beas. The l i (mm) vs. t (s) graphs show the experimental ata for each of the arrangements. () shows the sum of squares of imbibe lengths ( P l 2 i = L 2 + l 2 )inmm 2 vs. timeinsfor the cases (a), (b) an (c). The slopes for the lines shown in () for the cases (a), (b) an (c) are 2.8, 7.2 an 6.8 mm 2 s 1, respectively. (k) till L. We now hypothesize that for length L, the pressure graient in both the layers is same which implies U K = u k, an flui transfer happens between the ajoining layers only at the front assuming lubrication approximation. The flui from the large permeability layer goes to the small permeability layer at the front [21], increasing the flow rate in the latter after the front at L. The viscous loss in the small permeability layer after the transfer q

3 Eur. Phys. J. E (2017) 40: 39 Page 3 of 6 shoul be atmospheric for which the larger permeability layer must have a higher capillary pressure than the small permeability layer. This is a contraiction an proves that the front will always be ahea in the small permeability layer in the two-layere system as shown in fig. 2(a). Aing eqs. (7) an (8), we get Fig. 2. (a) A schematic of the flui transfer from the large permeability layer to the small permeability layer. (b) A schematic of the flui transfer from the small permeability layer to a large permeability layer which is a contraiction. (c) Pressure vs. istance graph. The soli an the ashe lines represent the large an the small permeability layers shown in the case (a); the circles represent the large permeability layer shown in case (b). can be written as μ(l L) P (l, t) P (L, t) = k ( u + q ), (2) A where A is the area of the cross-section. The velocities of the flui fronts after the flui transfer are φ L t = U q A, (3) φ l t = u + q A, (4) where φ is the porosity of the layers. The Young-Laplace equation for pressure rop in the two layers is P (L) P 0 = cγ D, (5) P (l) P 0 = cγ, (6) where c is a constant. These equations for l an L with time reuce to ( ) cγλ D ( L = L D t + l ), (7) t ( cγλ ) l (D ) =(l L) D t. (8) Equation (8) explains that the leaing front will always be in the small permeability layer. Because ( cγλ(d ) D )is always positive an ( l t ) is also positive uring the imbibition process, so l>l. This can also be explaine by consiering the pressure rop in the layers as shown in fig. 2. Till L, the pressure an the pressure graient in both the layers is same. We now suppose that the flui front has move ahea in the large permeability layer as shown in fig. 2(b). In the small permeability layer, the pressure after the front at L will be atmospheric pressure as shown in fig. 2(c). The large permeability layer has the flui flowing from L to l also an the pressure rops further till l as shown by circles in the graph of fig. 2(c). After the front at l in the large permeability layer, the pressure cγλ ( + D) =1 2 t (L2 + l 2 ). (9) Equation (9) shows that the sum of squares of the lengths imbibe is proportional to the time of imbibition as shown in fig. 1(). This shows that in a layere porous meium (l 2 + L 2 ) (Volume) 2 t, whereas in a homogeneous porous meium the square of the total volume imbibe is proportional to time. For each of the cases (a), (b), (c) shown in fig. 1, the value of the constant ( 2cγΛ ) is 4.6, 4.8, 6.1 mm s 1 respectively. Equation (9) is applicable when the constant Λ use in calculating the permeability is same for all the layers irrespective of the bea iameters. Although the iameter of the large (1 mm) bea packing scales with the wall spacing, the Λ for the large beas is slightly ifferent from that of layers of the small (0.1 mm) an the meium (0.5 mm) beas. We now evaluate the orer of magnitue of the constant ( 2cγΛ ). In all the layers, c has an orer of magnitue ( )1 [20], an ( γ μ ) 103 mm s 1. Λ for all the layers is ifferent because the Kozeny-Carman equation cannot be use when the bea iameter is the same as the spacing between the glass plates. But the orer of magnitue of Λ can be etermine by the Kozeny-Carman equation which is 10 4, for a porosity of 0.3, measure for all the beas. This implies that the orer of magnitue of ( 2cγΛ )is1mms 1 as evaluate for the three cases in fig Three-layere porous meium To make a three-layere porous meium, the beas are fille layer-by-layer for a with of 8 mm each. All the three possible combinations of the interacting layers are consiere for a three-layere system as shown in fig. 3. Spontaneous imbibition experiments are carrie out an the isplacement of the flui fronts with time is shown in fig. 3. In the arrangements shown in fig. 3(a), (b), we observe that the flui front is ahea in the small permeability layer, which is similar to the results of the two-layere system. However, in fig. 3(c), the flui front is ahea in the meium permeability layer. To moel this anomalous behavior, we assume that the flui transfer happens from the large permeability layer to the ajacent small permeability layer(s) at the front only, as shown in fig. 3 by arrows in the snapshots. We use Darcy s law in each section between the fronts to arrive at the pressure rop. The total suction pressure in each layer is given by the Young-Laplace equation. Then proceeing as in the two-layere case, for arrangements in fig. 3(a), (b), the governing equations for the rate of front

4 Page 4 of 6 Eur. Phys. J. E (2017) 40: 39 where = 0.1mm, m = 0.5mm an D = 1 mm an l, l m an L are the flui front positions in the small (k = Λ 2 ), the meium (k m = Λ 2 m) an the large (K = ΛD 2 ) permeability layers, respectively, at time t. Aing eqs. (10), (11), an (12) for the arrangements shown in fig. 3(a), (b) we get (D + m + ) cγλ = 1 ( L 2 + lm 2 + l 2). (13) 2 t Figure 3() shows that eq. (13) is satisfie by the experiments, i.e. (l 2 + lm 2 + L 2 ) (Volume) 2 t with a slope of (D + m + ) 2cΛγ =4.4mm2 s 1. As the velocities in all the layers are positive uring the imbibition process an the left-han terms of the equations are also positive, it can be inferre from eq. (11) that l>l m an l m >Lfrom eq. (12). Thus, the flui front positions are arrange as L<l m <lat all times, which is in the orer of ecreasing permeability or ecreasing particle iameter of the layer. For the arrangement shown in fig. 3(c), proceeing as in the earlier case, the governing equations for the rate of front isplacement with time are D m + 2 ( cγλ L D = L t + l m t + l ), t (14) (D m ) m cγλ D =(l m L) l m t, (15) l (D )cγλ =(l L) D t. (16) Fig. 3. Interacting three-layere porous meia. The images of the porous meia show the imbibition at ifferent times. The flui exchange among the layers is represente by arrows in the snapshots. (a), (b) an (c) show the ifferent arrangement of layers in three-layere porous meia mae of 0.1 mm, 0.5 mm, an 1 mm glass beas. The l i (mm) vs. t (s) graphs show the experimental ata for each of the arrangements. () The slope of the sum of squares of imbibe lengths ( P l 2 i = L 2 + l 2 m + l 2 ) in mm 2 vs. time in s for the arrangements in (a), (b) an (c) is 4.4 mm 2 s 1. isplacement in ifferent layers are given by D m + 2 cγλ D = L ( 1 ( 2 m+ 2 ) 1 m D ) cγλ =(l m L) ( L t + l m t + l t ( lm t + l t ), (10) ), (11) m ( m ) cγλ =(l l m) l t, (12) Aing eqs. (14), (15), an (16) for the arrangement shown in fig. 3(c), we get eq. (13). Figure 3() suggests that the moel is able to preict the capillary-riven flow behavior in arbitrarily arrange parallel layers of packe ) for three-layere porous meia is 2.75 mm s 1 (which is again of the orer 1). Now, we try to preict the flui front positions for the three-layere case. From eqs. (15), an (16), L<l m an L<l; but the relative positioning of l an l m is still unclear. Using the hypothesis evelope in the two-layere system, it is clear that the flui transfer occurs from the large permeability layer (K) to the smaller permeability layers (k m an k). Since the small permeability layer an the meium permeability layer are not in contact with each other, their pressure graients can be ifferent after L, unlike in the previous cases. We now use the pressure rop in the layers to arrive at the possible solutions of the front positions. Figure 4(a) shows a conition where beas. The value of ( 2cγΛ ( ) ( ) P P < for z>l. (17) z z m Now, the front in the small permeability layer shown by the otte line in fig. 4(a) has to travel a larger istance (l m l) sothat(p c ) m < (P c ) as shown in fig. 4(a), where P c is the capillary suction pressure. From Darcy s

5 Eur. Phys. J. E (2017) 40: 39 Page 5 of 6 Fig. 4. Possible conitions for pressure vs. istance graph of an interacting three-layere porous meium. equation, the relation between pressure graients can be expresse as μφ( l t ) lm μφ( Λ 2 < t ) Λ 2. (18) m ( 1 This conition hols true if ( l t ) ( lm t )as( 1 ) > 2 ). This implies that l<l 2 m. But (P c ) m < (P c ), im- m plies l m <l, which is a contraiction. This preclues the case where ( P z ) < ( P z ) m.now,weknowthat ) ) ( P z i.e., from Darcy s law, μφ(l m /t) Λ 2 m m < ( P z, (19) < μφ(l/t) Λ 2. (20) This conition is shown in fig. 4(b), (c) for which we explore the possible front positions. Because ( 1 ) < ( 1 2 m ), 2 ( l lm t ) can be greater or less than ( t ) base on the bea iameter ratio. If m + = D, the left-han sie of eqs. (15) an (16) becomes ( mcγλ D ) implying l m = l.if+ m <D, then these equations lea to l<l m as seen in the experiment shown in fig. 3(c) for =0.1mm, m =0.5mm an D = 1 mm. But if + m >D, then eqs. (14), (15) an (16) lea to l>l m. Now, to prove this, we choose bea iameters of =0.5mm, m =0.8mm an D = 1 mm an perform the spontaneous imbibition experiment where beas of iameter D are place between the layers of bea iameters m an as shown in fig. 5(a). The slies are kept 2 mm apart to accommoate more than 1 layer of 0.8 mm beas. In this case, the front in the small permeability layer is ahea of the front in the meium permeability layer unlike for the arrangement in fig. 3(c). The front in the large permeability layer trails the two fronts. Figure 5(b) shows that (l 2 +lm 2 +L 2 ) Volume 2 t. The value of ( 2cγΛ ) for this case is 1.87 mm s 1. As explaine in sect. 2.1, the orer of magnitue of the constant ( 2cγΛ ) is 1 for all the cases escribe in the three-layere porous meia. The conition of + m > D gives a very narrow range of iameters unlike the heterogeneity encountere in the real situations. For the conition + m <D,the front isplacement in the arrangement of fig. 3(c) shows an anomalous behavior. So, for the three-layere system we prove that the leaing front is not always present in the small permeability layer. Fig. 5. The porous meium is mae of bea iameters 0.5 mm, 0.8 mm, an 1 mm. The positioning of layers is the same as in fig. 3(c). (a) The l i (mm) vs. t (s) graphs show the experimental ata an the graph in (b) shows the sum of length squares ( P l 2 i =(l 2 + l 2 m + L 2 )) in mm 2 vs. time in s. 3 Conclusions In summary, we have analyze the imbibition ynamics in layere systems. In a two-layere porous meium, the flui front in the smaller permeability layer always leas, which is in concurrence with the results of Bico an Quéré [21]. But for a layere system, the sum of squares of volume imbibe in iniviual layers is proportional to time, unlike escribe by Bico an Quéré [21] that the (Total Volume) 2 t. In a three-layere porous meium, the positioning of the flui front is highly epenent on the arrangement of the layers an their permeability with respect to each other. An anomalous behavior is seen: the leaing front is in the meium permeability layer (k m ), if the meium (k m ) an the small (k) permeability layers are separate by the large (K) permeability layer an + m <D. But, when + m >D, the flui front in the small (k) permeability layer leas. In all the cases, (Volume) 2 t. This can be use in esigning new porous meia for iagnostics, to stuy imbibition of fluis in naturally occurring porous meia such as woo which helps in choosing a suitable coat to prevent ecay an to analyze the recovery from fracture oil reservoirs where spontaneous imbibition is important. We acknowlege the financial support grante by the Inian Institute of Technology Delhi (Project No. MI01054). Author contribution statement JP an SSB esigne the research. SA performe the experiments. SA an JP evelope the moel. GV assiste in ata analysis. SA prepare the figures. SA, JP, an SSB rafte the manuscript. All authors reviewe the manuscript an gave final approval for publication.

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