epl draft Flow-induced channelization in a porous medium A. Mahadevan 1, A.V. Orpe 2, A. Kudrolli 3, L. Mahadevan 4

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1 epl draft Flow-induced channelization in a porous medium. Mahadevan 1,.V. Orpe,. Kudrolli 3, L. Mahadevan 4 1 Woods Hole Oceanographic Institution, National Chemical Laborator, Pune, 3 epartment of Phsics, Clark Universit, 4 School of Engineering and pplied Sciences, epartment of Phsics, Harvard Universit PCS PCS 81..Rm Porous materials; granular materials 9.4.Gc Erosion and sedimentation; sediment transport bstract Flow through a saturated, granular, porous medium can lead to internal erosion, preferential flow enhancement and the formation of channels within the bulk of the medium. We eamine this phenomenon using a combination of eperimental observations, continuum theor and numerical simulations in a minimal setting. Our eperiments are carried out b forcing water through a Hele-Shaw cell packed with bidisperse grains. When the local fluid flow-induced stress eceeds a critical threshold, the smaller grains are dislodged and transported, thus changing the porosit of the medium and thence the local hdraulic conductivit and the development of erosional channels. The erosion is ultimatel arrested due to the drop in the mean pressure gradient, while most of the flow occurs through the channels. These observations are consistent with a simple theoretical model for channelization in terms of a macroscopic multiphase description of erosion. We model a dnamical porosit field that evolves along with the volume fraction of the mobile and immobile grains in response to fluid flow. Numerical solutions of the resulting initial boundar value problem ield results for the dnamics and morpholog that are in qualitative agreement with our eperiments. In addition to providing a basis for channelization in porous media, our stud highlights how heterogeneit in porous media ma arise from flow as a function of the erosion threshold, and thus potentiall offers the abilit to control channelization. The dnamics of fluid flow through porous continua is relevant over man orders of magnitude in length scale with applications that range from large scale flow through fractured rock in aquifers and oil reservoirs to small scale flows in natural and artificiall engineered sstems [1, ]. In all these cases, flows are characterized b large variations in hdraulic conductivit of the medium. This heterogeneit is usuall ascribed to processes associated with the formation and consolidation of the porous medium via the agglomeration of grains (in geolog) and cells (in biolog). But heterogeneit and channelization ma also arise due to selective erosion of material in non-cohesive porous media. Indeed, flow-induced erosion on the surface of porous substrates has been implicated in the formation of patterns on planetar [3, 4], littoral [] and laborator [6, 9] scales that involve both unconsolidated and consolidated media [7]. issolution and liquefaction can also arise from reactive instabilities as seen in melt migration [11, 1] and cave formation [13]. Here we eplore the purel phsical erosive instabilities occurring in the bulk of fluid saturated materials where the can lead to internal channelization via the dnamic coupling of flow and changes in hdraulic conductivit. To show how this happens, we start b describing some eperiments that demonstrate how erosion and channelization can occur in a saturated porous medium (See Movie1 in SI). The eperiments were carried out in a fluidsaturated porous medium confined to a vertical quasi-two dimensional chamber based on a Hele-Shaw cell filled with bi-disperse miture of glass beads as shown in Fig. 1. The width between the walls of the apparatus is such that gravit is unimportant owing to the formation of arches between walls [14]. s our porous medium, we chose to fill 6% of the initial volume with large beads (diameter d 1 = 4±.1 mm) and 4% small beads (d =.7±.1 mm). The width of the chamber, chosen to be 1. d 1, enables visualization of the porosit pattern, and allows the larger beads to be rearranged locall without being transported b the flow, while the smaller beads can be dislodged and transported through the matri of large beads. Since onl p-1

2 . Mahadevan1,.V. Orpe,. Kudrolli3, L. Mahadevan4 the smaller beads can be eroded, we scale the porosit b the maimum attainable absolute porosit,.4. Then, the scaled porosit (or liquid volume fraction), is denoted b φl [, 1]. To observe the evolution of the porosit induced b fluid flow, we use a back-lighting technique and an intensit-densit calibration that allows us to obtain the volume fraction of liquid (void fraction) or porosit as a function of space and time at a grid resolution of d1 d1 (4 4 mm ) in -dimensions (Fig. 1C). We control the inlet flow rate Q, which is applied uniforml to the lower boundar of the eperimental domain to provide a uniform specific discharge q = Q/ over the cross-sectional area. Since the sudden imposition of a large flow rate (and pressure gradient) can cause the entire medium to fluidize, we graduall ramp up the flow rate at the inlet from to cm3 /min over 3 hours so that q increases from.6 and 3.7 cm/s. The increase in flow rate is carried out smoothl, as well as in steps of.13 cm/s over time intervals of 6 s to allow equilibration intermittentl, but the behavior is qualitativel the same in both cases (Fig. 1). The outlet at the upper boundar is maintained at atmospheric pressure. B measuring the pressure at the inlet, we can determine the average pressure gradient. It increases linearl with the flow rate for a constant porosit material. This confirms that the flow within the porous medium obes arc s law even after channels are formed. With each increment in the flow rate, erosion occurs for a few minutes before stopping. The porous medium which is initiall fairl uniform in porosit, becomes increasingl heterogeneous as the flow rate is increased, as erosion generates regions of enhanced porosit that connect to form channels with high conductivit (Fig. 1B,C). Eventuall, the erosion patterns coarsen with time and just a few channels preferentiall conduct most of the flow. s the flow rate is graduall increased, the flow-induced stress, i.e. the magnitude of the local pressure gradient, can eceed the local threshold required to dislodge and mobilize particles. The microscopic phsics of failure is complicated in general owing to the formation of fragments of varing sizes and shapes. However, for our simple bidisperse medium, the process is somewhat simpler since the dominant erosive process involves the dislodgment and movement of the smaller grains from the interstices between the larger grains. This leads to a local increase in the hdraulic conductivit and a readjustment of the flow. s the material erodes and porosit increases, the pressure gradient drops below the critical erosive stress threshold and erosion stops. For ever increment in the flow rate, the pressure gradient is raised above the threshold and leads to further erosion; this increases the porosit and thus reduces the pressure gradient, which falls below the threshold so that erosion stops. Thus, for a given flow rate, the hdraulic conductivit evolves in space and time until it reaches a stead state that is a function the initial heterogeneit in the porosit of the medium. lthough deposition can act to mitigate the erosion patterns b decreasing B Water Out Water Out Outlet 1. d1 7 d1 Miture of beads of dia d1, d 7 d1 d1 /d = 6 z Water In Water In Water Inlet C (cm) φl (cm) 1 1 (cm) E q=.6 cm/s q (cm/s) 3 F 1 1 p(φ)l q= 3.7 cm/s.. φl φl.7 1. Fig. 1: Eperiments : Schematic of the eperimental set up showing face-on and sectional views. B: n instantaneous snap shot showing the spatiall varing porosit generated b the flow. Light color indicates high porosit where the small beads have been removed b erosion. Small beads transported out of the porous medium are seen to pile up at the interface between the porous bed and clear water. C: Maps of the scaled porosit φl (absolute porosit normalized b the maimum attainable absolute porosit) for specific discharge rates q=.6, 1.3, 1.96 and.6 cm/s after the sstem has reached a stead state. : omain-averaged φl is plotted as a function of specific discharge q as it is ramped up over 3 hours (red and green curves) or stepped incrementall ever 1 min intervals (black points) so as to achieve a near stead stead for each flow rate. Histogram or probabilit distribution of relative porosit p(φl ) shown after stead state is achieved for a specific discharge rate E: q =.6cm/s and F:q = 3.7cm/s. For low discharge rates (q =.6cm/s in E), the peak in p(φl ) is slightl broadened with respect to the initial condition but remains at the initialized value of φl =.4. t higher discharge rates (q = 3.7cm/s in F) the distribution broadens further and a second peak that represents completel eroded channels is developed at φl = 1. the porosit and the hdraulic conductivit downstream, this effect is strongl dependent on whether the outflow region serves to sieve the mobile particles or trap them. The spatial pattern of porosit can var significantl from one eperiment to another due to the slight inhomogeneit in the initial conditions of the medium. However, in all cases, the basic phenomena is robust; preferential erosion, cou- p-

3 Flow-induced channelization in a porous medium pled to conductivit and flow through positive feedback leads to the formation of channels of highl variable conductivit from a relativel uniform porous medium. The average porosit over the region is observed to increase linearl with flow rate for scaled porosit in the range.4.6 as shown in Fig. 1C,. n eample of the distribution (pdf) in porosit values over the region is shown in Figs. 1E,F for small and large flow rates, respectivel. Initiall, the distribution is broadl peaked at φ l.4 corresponding to 6% large and 4% small beads. s the flow increases, a second peak develops at φ l 1 corresponding to regions containing onl large beads where complete erosion of the finer beads has occurred, a process that is concomitant with channelization. Motivated b these qualitative observations, we now turn to a continuum theor for the dnamical evolution of channels via flow-induced erosion within a saturated porous medium. Our theor for the active co-evolution of the phases in a porous medium is qualitativel different from the single phase diffusive models for the evolution of free surfaces b aggregation and erosion [1] or multiphase bulk theories for multiple reactive fluids interacting with each other [11, 13] but et combines features of both in considering the fluid-induced erosion and deposition processes acting in the bulk of a solid skeleton. To correctl.3 account for the onset of erosion as well as its evolution as vφ observed in the quasi-two dimensional bidisperse granular medium, we need to consider a multiphase theor that. involves fluid, granular and immobile phases interacting.1 with each other. To enable a continuum field description of the process, we consider a representative volume much larger than the grain/pore size with φ s (,, z, t) the volume fraction of the immobile solid phase, φ g (,, z, t), the volume fraction of the granular mobile phase, and φ l (,, z, t), the liquid volume fraction in the medium so that φ s + φ g + φ l = 1. We describe the time-evolution of these phases b accounting for the transport of the liquid and granular phases b velocities u l and u g, respectivel, and a transformation between phases b erosion and deposition. The erosion rate, e, is the rate of transformation of the immobile phase to mobile phase, d, is the rate of deposition or conversion of mobile phase to immobile phase. Volume conservation for the individual phases (each of which are assumed to be incompressible) implies that t φ s = e + d (1) t φ g = +e d (φ g u g ) () t φ l = t (φ s + φ g ) = (φ l u l ) (3) We note that adding equations (1)-(3) ields the global continuit equation (φ g u g + φ l u l ) =. (4) For simplicit, we will assume that u g = u l = u, i.e. the granular and liquid phases have the same velocit and the φ φ at t=1 C (γ 1 P) t=1 σ (γ 1 P) t=6 σ E.3.3. (γ 1 P) 3 1 B φ at t=6 a a a a t=6 t=1. t= t Fig. : Numerical solution to equations (1) (7) with a stead specific discharge q = q prescribed at the lower boundar =, and constant pressure at the upper boundar =.3. () Spatial distribution of porosit φ at t = 1 and (B) at t=6. (C) (γ 1 P ) (red) and σ (black) are plotted along a a (at =.16) at t = 1 and () at t = 6 corresponding to the panels above. Erosion occurs where (γ 1 P ) > σ. t earl times, this occurs at several locations. s erosion progresses, the pressure gradient drops, heterogeneit in σ increases, and erosion is limited onl to the channels. (E) The flu in the - direction, v(φ g + φ l ) plotted at section a a at t =. (black), t = 1. (red), and at t = 6 (blue). (F) s the flu increases in eroded regions, it decreases in non-channelized regions. effects of inertia and bod forces associated with sedimentation are negligible, a reasonable approimation for slow flows of nearl jammed grains. Then the continuit equation reduces to φu =, where φ φ g +φ l. To describe the rates of phase transformation e, from solid to mobile phase, and d, from mobile to solid phase. In a saturated porous medium, we postulate that erosion of φ s can occur onl when the fluid-induced stress eceeds a critical threshold σ. Then we ma write the local convservation F σ p-3

4 . Mahadevan1,.V. Orpe,. Kudrolli3, L. Mahadevan4 rate from solid to granular phases as e = k e φ s ( (γ 1 p) σ ), ().3 q=.3 B q=1 C q=3.3.3 where k e is a characteristic rate and γ = q /, the ratio of the characteristic specific discharge (q = 1 m/s) to characteristic hdraulic conductivit ( = 1 6 m 3 s kg 1 ), is used to normalize the pressure gradient. The form of e follows from considerations of smmetr: a hdrostatic pressure p cannot lead to erosion, but a gradient can. However, the sign of the gradient is not important, so that we have chosen the simplest analtic dependence consistent with this smmetr (using the asmptoticall correct but non-analtic form p instead of ( p) ields qualitativel similar results - see Figure S1 in SI). The rate k e is taken equal to the characteristic specific discharge q divided b a characteristic length scale L; i.e. k e = q /L (where q =.1m/s and L = 1m as later described). The critical stress for erosion σ should be a function of the solid fraction φ s. However, the non-local nature of elastic stress distribution in the porous medium warrants σ be more appropriatel cast in terms of φ s = V 1 V φ sdv, a regional average of φ s over a region V associated with a few grain volumes. This is because forces applied at a point deca over a length scale larger than the individual grain size. lthough the form of σ is difficult to gauge from our eperiments, we use the function σ = (tanh(π(φ s.6))+1), where φ s 1 which mimics the sharp dependence of the failure stress σ on the volume fraction simple model for d, the rate at which the mobile granular material is converted back to the immobile solid phase, is given b d = k d (φ s φ s) φ g. (6) The form of the deposition, with rate k d, is based on a binar collision picture mobile grains will come to rest onl if the interact with immobile grains; hence the dependence on φ s with a threshold φ s. The rate k d is taken as the characteristic specific discharge divided b a characteristic length scale; k d = q /L. In the porous medium, we assume that the volumetric flow rate per unit cross-sectional area, i.e. the specific discharge q u(φ l + φ g ), is given b arc s law 1 q uφ = (φ) p, where = φ 3 l p µ (1 φ ). (7) Here, (φ), the hdraulic conductivit, is assumed to follow the Carman-Kozen relation [] and is in general a nonlinear function of the local fluid (pore) volume fraction φ = φ l + φ g. Here l p is the nominal pore size (which scales with the grain diameter), µ is the dnamic viscosit of the interstitial fluid, and the dimensionless constant = 18 for spherical grains []. For l p = 1mm, µ = 1 3 kg m 1 s 1, ranges from m 3 s kg 1 for φ between This can be generalized to a Brinkman-like equation if necessar in regions of high porosit. <φ> q q.3q. 1 t.3 <φ> E. 1 3 q.3..1 p(φ) F.3q q 3q Fig. 3: The final distribution of porosit φ at t = 6 depending on the specific discharge q (scaled b q ) specified at = ; () q=.3, (B) q=1, (C) q=3. () For each q, the average porosit increases with time, until as stead state is achieved. (E) The final porosit (at t = 6) is a function of q at the inlet. (F) Histograms of the porosit distribution for q =.3, 1 and 3 shows bimodalit and a peak at φ = 1 due to channelization. We use the domain size, L = 1m, specific discharge q = 1cm/s, and time scale T = L/q to make the problem dimensionless, so that the dimensionless parameters in the problem include the thresholds for erosion and deposition σ and φ s as well as the ratio of deposition to erosion rates Π 1 = k d k e, and the ratio of advection to erosion rate Π = q k. Our choice of k el e and k d implies that Π 1 = Π = 1. We solve equations (1 7) numericall in -dimensions (, ) using a finite volume method. In the direction, we use periodic boundar conditions at = and = L, with a square domain of dimension L =.3, L =.3 and a uniform grid resolution =.. Prescribing the inlet pressure instead of the inlet discharge leads to either no erosion (if (γ 1 p) < σ ) or catastrophic erosion if the pressure gradients are larger than the threshold. Thus, we prescribe a constant scaled specific discharge q = 1 at the inflow boundar =, while at the outflow boundar = L we set pressure p = (atmospheric pressure). The pressure is determined b iterativel solving the Poisson equation ((φ) p) = obtained b substituting (7) into (4), then calculating the erosion rate e and the deposition rate d, and finall evolving equations (1) (3) to update the volume fraction of the phases φ s, φ g, φ l from one time step to the net, with time step t = 1 4. Starting with an initial volume fraction of mobile grains φ g = throughout the domain, and a mean liquid volume fraction φ l =. with an additive white Gaussian noise (standard deviation sd = φ l φ l 1/ =.1) that leads to weak heterogeneit in σ, we allow the sstem to evolve until it reaches a quasi-stead state. The non-local form of the erosion threshold σ is based on φ s, the weighted spatial average of φ s, which is calculated φ.3 p-4

5 Flow-induced channelization in a porous medium.3 B C Fig. : The evolution of the porosit is sensitive to the functional form of the erosion threshold σ. Here the final distribution of porosit φ is shown for other choices of σ. () σ = φs, (B) σ = φs. These results can be compared with Fig. 1, where σ =.(tanh(π(φs.6)) + 1) (subtracting.6 instead of. provides a slight asmmetr to the tanh profile.) B Fig. 4: The final distribution of porosit (at t = 6) is shown to be sensitive to the initial heterogeneit in σ arising from φs. The upper row shows the porosit distribution resulting when the initial distribution of φs is set to.8 everwhere, ecept in specific places where it is decreased b 1% from the uniform background value () along two lines, and (B) along 1 lines, each being one grid cell wide. In the lower row, the standard deviation (sd) in the initial perturbation to φs is varied from its previous value of sd =.1. (C) sd =.1. () sd =.3. numericall as φs (i, j) =.φs (i, j) +.1φs (i ± 1, j ± 1). veraging over the 8 surrounding neighbors of a grid point amounts to a radius of influence of the stress that etends a few grain diameters, in contrast to using a local value of φs that would lead to runawa erosion at individual grid cells. In Fig.,B we show two snapshots of the initiall homogeneous porous medium as it erodes and channelizes (see Movie in SI). This process involves positive feedback: flow is enhanced through the regions of low solid fraction (high hdraulic conductivit), while regions with a higher solid fraction and lower hdraulic conductivit are circumvented b the flow. Indeed, in Fig. C, we see the interpla between the heterogeneit in the erosion threshold σ and the squared pressure gradient (γ 1 p). The material is transformed from solid to mobile phase where (γ 1 p) > σ and transported b the flow, reducing φs, lowering σ, and augmenting erosion. The channelization leads to enhanced flow through regions of high hdraulic conductivit at the epense of low reduced flow through other regions (Fig. E) even as the total flow rate remains the same. In Fig. F, we provide a global view of the process. s erosion progresses, the average porosit and hdraulic conductivit over the entire domain increases. The pressure gradient required to sustain the same flow rate drops until it falls below the threshold for erosion almost everwhere and the sstem asmptoticall approaches a stead state. To understand how this stead state depends on the dimensionless parameters, we first var the scaled specific discharge at the inlet q/q. When q < qc, a critical scaled flow rate that depends on the initial porosit distribution in the medium, no erosion or channelization occurs, because the pressure gradient is everwhere smaller than the erosion threshold. For q = qc, a single narrow channel and secondar partial channel are formed as shown in Fig. 3. s q is increased further, the number of channels as well as the width of channels increases (Fig. 3B); for even higher flow rates, the entire medium begins to erode awa as shown in Fig. 3C. In all cases, the porosit increases with time after initiation of the flow (Fig. 3), linearl at first, and then slowl approaching a stead state value that depends on the inlet specific discharge. Much like the eperiments, the average porosit of the medium is a function of the inflow q (Fig. 3E). Histograms of the porosit for varing inflow rates are shown in Fig. 3F; the case q = 1, produces the most bimodal porosit due to channelization. Varing the erosion and deposition rates also leads to variations in the erosion patterns. In Fig. 4 we show the effect of a 1-fold increase in erosion rate ke (Π1 =.1), which leads to a faster evolution of channels. Increasing kd 1-fold so that Π1 = 1 increases the deposition rate and causes blockages that leads to termination and reinitiation of channels (Fig. 4B). In this model, deposition remains small even when kd = ke, because φg φl and a significant (1-fold) increase in kd is needed before we begin to see the effects of deposition. The average number or size of channels does not change in either case relative to when Π1 = 1 (corresponding to Fig. B). question that naturall arises is the mechanism for the selection of channel spacing and width when fluid flows p-

6 . Mahadevan1,.V. Orpe,. Kudrolli3, L. Mahadevan4 through a nominall homogeneous porous medium. The spacing and width of channels is insensitive to the domain size. oubling the model domain size does not change the picture. The natural length scales in the problem are the sstem size L the nominal pore size l p which evolves with time, but remains a microscopic length, and the length scales q /k e, q /k d ; the latter control the dnamical evolution of the channels but not their final state. What remains is the threshold for erosion σ; since the onset of channelization is strongl influenced b fluctuations in the porosit (and thus the fragilit) of the medium, we epect that linear stabilit analsis of the base state should predict that channels form at locations where σ is smallest initiall. Thus for the same inlet specific discharge, the size and number of channels is a function of σ(,, ). In Fig. 4 we show that for a given inlet specific discharge, if σ(,, ) f() has a single minimum, a single channel forms and grows until the pressure gradient falls below the erosion threshold, while in Fig. 4B, we see that if σ(,, ) has multiple minima, multiple channels form. Of course, it is not sufficient to consider the mean value of the threshold; instead one must account for the complete probabilit distribution of the erosion threshold. For our simple Gaussian model of disorder, if the variance in the threshold for erosion (or equivalentl the porosit fluctuations) is also changed, this leads to variations in the patterns as well. In Fig. 4C,, we show that an increase in the standard deviation of the initial white noise leads to greater heterogeneit in the channel number and spacing. Finall, we consider the functional form of the erosion threshold σ(φ s ). In Fig. we see that for σ = φ s the final morpholog of the erosion patterns is more uniform compared to that shown in Fig. B for σ = φ s, which is itself less variable than for the case when σ follows a tanh profile (Fig. 1-4). We thus see that the form of the erosion threshold function, and its initial, possibl heterogeneous spatial distribution are crucial in determining the growth and form of the channels, which a simple linear analsis alone cannot capture. Our model qualitativel captures the basic features of our phsical eperiments: below a critical flow rate, little or no erosion occurs. bove this threshold, the porous medium starts to erode heterogeneousl at locations where the critical threshold is lowest; positive feedback then enhances erosion locall in other regions leading to oriented regions of higher porosit that are the hallmarks of channels. The modeled channels are not onl able to branch, but also end and initiate, leaving dislocations in the medium. For a given flow rate, as channels form, the pressure gradient across the domain decas with time and the rate of erosion follows a similar trend in our numerical simulations. Consistent with this, we find an increase in the densit of eroded channels and average porosit of the region for higher flow rates. In the eperiments, the rate of erosion ceases abruptl when the pressure gradient falls below a threshold and small perturbations in the flow rate do not lead to further erosion. This is likel due to the effects of a nonlocal jamming process that does not have a counterpart in our theoretical description, where the erosion decas asmptoticall for a constant flow rate. With increasing flow rates, our eperiments show a linear increase in the average final porosit (Fig. 1), a result that is mirrored in the numerical simulations (Fig. 3) as well. The strong dependence of the channelization patterns on the heterogeneit of the initial porosit is, in addition, a feature of both our eperiments and our numerical simulations. Indeed this is wh we cannot predict the wavelength of the channelization patterns, which are instead determined b a combination of the prescribed flow rate and the initial distribution of variations in porosit. Taken together, our qualitative eperiments and continuum models allow us to move beond the description of flow through porous media as just a passive response to its a priori prescribed hdraulic properties, and account for processes such as erosion, deposition, and dnamic permeabilit changes, to not onl understand natural phenomena such as channelization, but also start to understand how one might control it in particular materials processing application. We thank OE NICCR (M, K), Harvard-NSF MR- SEC (LM) and the Macrthur Foundation for partial support, and ndrew Fowler for comments on an earlier draft. 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