How do disconnected macropores in sloping soils facilitate preferential flow?

HYDROLOGICAL PROCESSESHydrol. Process. 24, 1582–1594 (2010)Published online 29 March 2010 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.7633

How do disconnected macropores in sloping soils facilitatepreferential flow?

John L. Nieber1* and Roy C. Sidle2

1 Department of Bioproducts and Biosystems Engineering, University of Minnesota, 1390 Eckles Ave, St Paul, MN 55108, USA2 Department of Geology, Environmental Science Program, Appalachian State University, PO Box 32067, Boone, NC 28608, USA

Abstract:

Runoff from hillslopes is generated by processes such as infiltration-excess and saturation overland flow, subsurface stormflowand subsurface return flow. Preferential flow through macropores can affect any one of these runoff generation processes.Field studies at the Hitachi Ohta Experimental Watershed in Japan have noted that self-organization processes may manifestthe connectivity of such subsurface flow paths, particularly via macropores, from hillslopes to stream channels. It is wellestablished from soil physics principles that the connectivity of macropore networks depends on soil wetness and this has beenshown experimentally at Hitachi Ohta where subsurface flow and streamflow respond to thresholds of wetness. Numericalsolutions to the three-dimensional Richards equation are derived for a sloping soil block containing a population of disconnectedmacropores of various sizes, shapes and orientations. Solutions for the case of steady water flux applied to the surface of thesoil block are evaluated to determine the conditions where the disconnected macropores become active in the flow process.Results show that subsurface flow is directed through the preferential flow network in the saturated portion of the soil butbypasses the macropores in the drier regions. The preferential flow network expands as the degree of saturation increases.The expanding network of active macropores leads to less resistance to overall flow in the domain and access to increasedvolumes of the flow domain. Although the individual macropores are disconnected, it is argued that large localized hydraulicgradients can potentially lead to preferred zones of subsurface erosion. In addition to the importance of these findings relatedto stormflow generation in catchments, they add support to the concept of self-organization of subsurface flow systems insoils. Copyright 2010 John Wiley & Sons, Ltd.

KEY WORDS preferential flow; subsurface flow paths; hillslope hydrology; self-organization; subsurface erosion; subsurfaceflow modelling; Richards equation

Received 1 September 2009; Accepted 26 January 2010

INTRODUCTION

Understanding the complexity of hydrological pathwaysat the hillslope and small catchment scales during peri-ods of water input has remained elusive and controversialin spite of extensive investigation during the past severaldecades. Thorough knowledge of how small-scale hydro-logical processes affect larger scale phenomena, such asflood generation, fate and transport of nutrients and otherchemicals within ecosystems as well as downstream, sus-tained streamflow during periods between precipitationevents, groundwater recharge, transpiration by vegetationand erosion and related sediment transport, is lacking(Sidle, 2006). As such it is important that hydrologistsgain additional insights into the processes of water flowthrough hillslopes of headwater catchments to be able topredict the response to precipitation and snowmelt undera variety of conditions and to better manage the associ-ated landscape.

Numerous studies conducted in headwater catchmentsor individual hillslopes during the past five decades haveadded to our collective knowledge of the hydrologic

* Correspondence to: John L. Nieber, Department of Bioproducts andBiosystems Engineering, University of Minnesota, 1390 Eckles Ave, StPaul, MN 55108, USA. E-mail: [email protected]

behaviour of these landscape components. Many of theearliest studies (Hursh and Brater, 1941; Amerman, 1965;Hewlett and Hibbert, 1967; Ragan, 1967) measured thedischarge from individual hillslopes or small catchmentareas without detailing the prevailing processes withinthe areas monitored; explanations for ongoing internalprocesses were inferred from the measured hydrologicresponses. These studies showed that subsurface flowand saturation overland flow were the dominant con-tributors to runoff generation during rainstorms in well-vegetated landscapes. To advance knowledge about sub-surface stormflow and saturation overland flow, a numberof important studies have collected detailed data aboutthese hydrological processes within hillslopes and catch-ments. Key representative studies include those of Whip-key (1965), Dunne and Black (1970), Weyman (1973),Mosley (1979, 1982), Tanaka et al. (1988), Tsukamotoand Ohta (1988), Terajima and Moroto (1990), Wil-son et al. (1990), McDonnell et al. (1991), Tsuboyamaet al. (1994), Sidle et al. (1995a, 2000), Montgomeryet al. (1997), Tani (1997), Uchida et al. (2005) andTromp-van Meerveld and McDonnell (2006). In addi-tion to monitoring outflow from hillslopes or catchments,these investigations articulated specific flow pathways,residence times and active hydrological processes byuse of detailed and spatially distributed hydrological

Copyright 2010 John Wiley & Sons, Ltd.

DISCONNECTED MACROPORES FACILITATE PREFERENTIAL FLOW 1583

monitoring, tracers, tensiometric and piezometric mea-surements and field observations during storms.

Based on field evidence from many of the previouslycited and other studies, it is clear that a significant portionof the subsurface stormflow within hillslopes is transmit-ted via preferential pathways. While the network structureof these preferential flow paths has been investigated(Jones, 1997; Tsuboyama et al., 1994; Noguchi et al.,1999), and analogies can be made to the natural streamchannel hierarchy observed in watersheds (Deurer et al.,2003), a formal concept for how such networks couldoperate dynamically to transport water, solutes and sedi-ment was first presented by Sidle et al. (2000) and Sidleet al. (2001) in which the concept of the self-organizingnature of these networks was introduced. Since the publi-cation of the articles by Sidle et al. (1995a, 2000, 2001),both experimental and modelling studies have been con-ducted to attempt to identify the network structures orself-organized patterns of flow networks. Fundamentalto the concept of self-organization related to preferentialflow networks is that more or less isolated macroporesegments in the soil are able to ‘connect’ via a systemof nodes, such as buried pockets of organic matter, loosesoil, interaction of macropores and mesopores, flow intoand out of bedrock fractures and topographic influences(Sidle et al., 2000, 2001). As the macropore flow andnodes become conditioned by increasing moisture con-tent, either during an extended storm or over a rainyseason, they are able to interconnect via switching func-tions at the node, thus providing a backbone that allowsthe preferential flow system to expand. Earlier work byLuxmoore and Ferrand (1993) proposed the concept ofpercolation theory to simulate a similar ‘switching mech-anism’ for preferential flow at the pore scale.

Later studies, including the development and applica-tion of the so-called fill and spill hypothesis (Spence andWoo, 2003; Tromp-van Meerveld and McDonnell, 2006),virtual hydrology experiments in hillslopes and smallcatchments (Seibert and McDonnell, 2002; Weiler andMcDonnell, 2007), and related discussions of the impor-tance of hydrologic thresholds for triggering rapid subsur-face flow (James and Roulet, 2007; Zehe and Sivapalan,2009) have evoked similar concepts of self-organizationand networking of hydrological flow paths. Nieber et al.(2006) noted that the self-organization concept proposedby Sidle et al. (2000, 2001) may be described by percola-tion theory (Hunt, 2005). Later, others have discussed theapplication of percolation theory to connectivity of sub-surface flow paths (Lehmann et al., 2007; Troch et al.,2009).

Hydrological models have been slower to embrace thisconcept of dynamic, self-organizing flow paths, likelydue to the complexity of the phenomena. For exam-ple, the few larger scale, process-based hydrology modelsthat have included preferential flow simply incorporate itas a modification of the hydraulic conductivity function(Mulungu et al., 2005; Zhang et al., 2006; Tani, 2008). Atsmaller scales, preferential flow has been typically mod-elled as vertical fluxes in very simplified systems (i.e. not

complex networks of macropores as encountered in thefield) (Castiglione et al., 2003; Kohne et al., 2006; Akayet al., 2008). The process-based models that have beenrecently developed at the fine scale to simulate slope-parallel preferential flow (Kosugi et al., 2004; Tsutsumiet al., 2005) have not yet incorporated the distributednature of soil macropores as observed in field sites.Other field and modelling studies that have attemptedto incorporate the spatially dynamic nature of preferen-tial flow systems have articulated such networking atscales of meters, requiring spatial averaging of smallerscale processes (Western et al., 2001; Weiler and Naef,2003; Weiler and Fluhler, 2004; Lin, 2006; Tromp-vanMeerveld and McDonnell, 2006; Weiler and McDonnell,2007; Tromp-van Meerveld and Weiler, 2008; Andersonet al., 2009). In spite of this progress in conceptualiz-ing and modelling preferential flow, little has been doneto examine the details of realistic flow networks at thescale of individual macropores or soil pipes. Hillslopeexcavations at 10-cm intervals conducted by Noguchiet al. (1999) in Japan, as well as more recent unpublisheddata, clearly show the need to incorporate such fine-scaledetail into the process conceptualization of preferentialflow paths.

Although it is imperative that operational models willneed to be based on some level of spatial averaging (Siva-palan, 2003), the upscaling has yet to be accomplishedappropriately, and before that occurs it is necessary tounderstand the processes at the fine scale (Sidle, 2006).This provides motivation for our study, in which we usea physically based model to examine the process of waterflow in a slope segment populated with disconnectedmacropores. The specific objectives of our investigationare to: (i) quantify the role of the macropores in theconveyance of water within the slope segment and (ii)illustrate in some detail the concept of self-organizationof flow pathways within sloping soils.

MODEL FORMULATION

Here, we formulate a model of soil moisture flow inan isolated block of soil extracted from a hillslope.The hillslope is illustrated in Figure 1 with the isolatedblock outlined. Presumably the outlined soil block isrepresentative of other blocks of equal size within thehillslope. In a real field experiment, this block wouldbe isolated by a trench around all sides; the upperand lateral boundaries would be covered with plastic,whereas the lower boundary would be instrumentedto measure subsurface and surface runoff. Within thesloping block, a population of macropores is distributedthroughout the soil profile and along the entire slopelength. None of the macropores are directly connected toone another. The analysis formulation applied is similarto that presented by Nieber et al. (2006) where water flowin a vertically oriented cylinder of soil was simulatedfor the case where disconnected macropores are present.The main difference between the previous study and

Copyright 2010 John Wiley & Sons, Ltd. Hydrol. Process. 24, 1582–1594 (2010)

1584 J. L. NIEBER AND R. C. SIDLE

Soilpipes

Isolated soil blockwith macropores

Figure 1. Illustration of a full hillslope with the isolated soil block designated. The isolated soil block is presumably representative of the entirehillslope except for the large features such as the soil pipes shown

M2

M1

M3 M4 M5

Figure 2. Illustration of the hypothetical soil block containing a population of disconnected macropores. The slope gradient is 15° and slope dimensionsare: 3 m long, 1 m wide and 1 m deep soil. Macropores M1, M2 and M3 originate from decayed root channels or faunal channels. Macropore M4is formed along a plane of separation (discontinuity) between the underlying bedrock and the soil matrix. The ellipsoidal-shaped macropores, like

macropore M5, would originate from animal burrows

the model applied here lies in the function used todescribe the unsaturated hydraulic conductivity, and thefact that in the present case we examine the flow in asloping soil block rather than a vertically oriented soilcolumn.

A close-up view of the model of the sloping soil blockis illustrated in Figure 2. The block is 3 m long, 1 mwide and 1 m deep, lies on a 15° slope, and is under-lain by an impervious layer. A number of macroporefeatures appear in the soil profile. Most of the macrop-ore features are cylindrical in cross-section, but there area few ellipsoid-shaped macropores that might representabandoned animal burrows. All the macropores dead-endwithin the soil block. Even macropores M1 and M2 whichappear to connect to the downslope face actually do notoutlet at that face. The rectangular ‘macropore’ (M4) atthe base of the soil profile represents a gap or disconti-nuity between the soil and underlying impervious layer,a common feature reported in detailed headwater catch-ment studies which facilitates preferential flow and evensupports subsurface erosion (Montgomery et al., 1997;Noguchi et al., 1999; Buttle and MacDonald, 2002). Thelocation, orientation and size of the macropores in the

soil profile were placed deterministically with no consid-eration for macropore morphology. An improvement forplacing macropores in the soil would be to use a macro-pore assignment procedure that is based on a stochasticprocess such as that used by Sidle et al. (2001) in devel-oping the macropore population shown in Figure 10 ofthat article.

The population of macropores used in our example(total porosity of the flow domain 0Ð37%) is rather sparsecompared with macropore densities reported in manyother field studies, particularly forest soils. Other detailedinvestigations of macropore density in both columnsand hillslopes indicate that our assumed value may beat least an order of magnitude lower than many sites(Kitahara, 1989; Mori et al., 1999; Perret et al., 1999;Botschek et al., 2002; Mooney, 2002; Shi et al., 2007).Nevertheless, the example we use falls within the lowerend of the density range of visible macropores (½2 mmdiameter) found in soil pits within the Hitachi Ohtacatchment, Japan (0Ð31–0Ð51%; Noguchi et al., 1997),which contributed significantly to subsurface stormflowduring wet antecedent conditions (Sidle et al., 1995a,2000).



The subsurface flow processes are assumed to be gov-erned by the steady-state form of the Richards equation,which can be expressed as:

∂

∂x

(K

∂h

∂x

)C ∂

∂y

(K

∂h

∂y

)C ∂

∂z

(K

∂h

∂z

)C ∂K

∂zD 0

�1�where h is the water pressure head (m), K is theunsaturated hydraulic conductivity (m/s) and x, y and zare Cartesian coordinates (m). Equation (1) is subject toboundary conditions of specified pressure head, h D h,and/or specified flux, �K ∂h

∂n � K ∂z∂n D q, where h is

the pressure head specified on the boundary, q is the fluxnormal to the boundary and n is the unit vector normalto the boundary.

The solution to Equation (1) is implemented throughthe Earth Science module of the commercial softwareCOMSOL MP3Ð5a (COMSOL, 2009). This numerical solu-tion is based on a Galerkin finite element procedure.

The boundary conditions for the flow domain shownin Figure 2 are the following:

ž vertical lateral sides, bottom, and upslope boundaries;impermeable boundary, q D 0;

ž top surface; rainfall infiltration, q D rain (where rainis the intensity of rainfall at the soil surface), or forsurface saturation, h D 0;

ž downslope boundary; seepage face, h D 0 for q < 0,or impermeable surface, �K ∂h

∂n � K ∂z∂n D q D 0 for

h < 0.

An explanation is in order to demonstrate how COM-SOL handles the seepage face and saturated soil surfaceconditions. As the position of the seepage face and thesaturated soil surface conditions are not known a priori, itis necessary to determine them by iteration. The appro-priate a priori condition is achieved within COMSOLby representing the boundary condition with the mixedcondition,

�K∂h

∂n� K

∂z

∂nD N0 C Rb[�h � hR� C �z � zR�] �2�

where N0 is the inward flux on the boundary, Rb is theconductance between the boundary and some referencepoint R, hR is the reference pressure head, zR is theelevation of the reference point, h is the pressure head atthe point on the seepage surface and z is the elevation ofthe point on the surface. This type of boundary condition(mixed condition) is generally used for simulating theflow between a remote water source and local flowdomain without having to actually discretize the spacebetween the remote source and the local flow domain.

In the present application, Equation (2) is used tosimulate the switching between specified flux (Neumann)and specified pressure head, hR, conditions. For this, thefollowing conditions are set:

ž zR D z

ž hR D 0

ž N0 D rain, Rb D 0, h < 0; N0 D 0, Rb D 1 ð 106,h ½ 0 for the soil surface

ž N0 D 0, Rb D 0, h < 0; N0 D 0, Rb D 1 ð 106, h ½0 for the downslope seepage face.

The value of hR is set to zero on these boundariesbecause then the solution will give h

¾D 0 along thesurface of soil saturation or along the seepage face. Onecan think of any grid point on the actual physical surfaceof the boundary as having an imaginary reference pointthat exits on an imaginary plane parallel to the actualphysical surface, and at an infinitesimal distance fromthe physical surface. Because the conductance term isso large, the pressure on the physical surface will beinfinitesimally close to the pressure on the referencesurface when the boundary pressure is greater than orequal to zero.

Rather than having the macropores as hollow vol-umes inside the matrix, we represented them as beingcomposed of very coarse porous media, such that theywould essentially behave like hollow volumes. Treatingthe macropores as being filled with porous media makesthe numerical solution somewhat easier to set up andachieve because we do not then need to deal with compli-cated internal seepage faces along the boundaries betweenmacropores and matrix. A justification for this treatmentof the macropores is given in the discussion.

The water retention properties and hydraulic conduc-tivity of the matrix and macropore material were quanti-fied with the van Genucthen (1980) relations given by:

Se D � � �r

�s � �rD

{1[

1 C �˛h�n]}m

�3a�

K�h� D KsS2e

[1 � (

1 � S1/me

)m]2

�3b�

where � is the volumetric water content, �s is the volumet-ric saturated water content, �r is the volumetric residualwater content and ˛, n and m are empirical parameters.Equation (3b) is obtained from a more general unsatu-rated hydraulic conductivity function for the case wherem D 1 � 1/n. For the simulations conducted here, thevalues of the parameters for the matrix and the macrop-ores are:

Matrix : �s D 0Ð35, �r D 0Ð05,

Ks D 0Ð0862 m/day, ˛ D 0Ð5 m�1, n D 2Ð5Macropores : �s D 0Ð98, �r D 0Ð0,

Ks D 86 200 m/day, ˛ D 200 m�1, n D 10.

The value of saturated hydraulic conductivity for themacropore may seem to be inordinately large; however,by applying the Poiseuille flow principle to a 0Ð005-mradius macropore the saturated hydraulic conductivity isabout 2Ð6 ð 106 m/day, giving a ratio of macropore tomatrix saturated hydraulic conductivity of about 30 ð106. From our numerical solutions, we found that resultsdid not change for ratios greater than 1 ð 103.



Figure 3. Illustration of the water table elevation (shaded surface) and streamline distribution for the soil block assuming that the water retentionand hydraulic properties of the macropores and the matrix are identical (i.e. no macropore influence). This case is for the flux (rainfall) rate equal to

0Ð025 m/day

For the matrix and macropore parameters used, waterwill not enter the macropores from the matrix if thewater pressure in the matrix is less than �9 mm. Itshould be noted however that Oldenburg and Pruess(1993) demonstrated that some numerical leakage acrossboundaries does occur and can be reduced only byrefining the local grid at the interface between the fineand coarse textured materials.

Finite element grids that were generated used lineartetrahedral elements. For most of the cases considered,the element grids had 419 642 elements and 74 070 gridpoints. The finite element solution has a number ofequation solver options. For the simulations reportedhere, we selected damped Newton iteration for the non-linear steps of the solution, and Generalized MinimalResidual (GMRES) iteration (Saad, 1996) for the linearmatrix solver with incomplete LU decomposition for thematrix preconditioner.

RESULTS

Here, we examine the effect of a population of dis-connected macropores on the flow of water within thesoil block illustrated in Figure 2. Simulations are con-ducted by comparing the sloping soil block with andwithout macropores, and comparing the flow patternswhen macropores are present but where the magnitudeof the flow or degree of wetness of the soil block is con-trolled. We also discuss the potential effect of progressivepipe erosive formation on the contribution of preferentialflow to discharge from the block.

The first case is the situation where macropores arenot present within the soil block. The applied flux (rain-fall) on the top surface of the block is 0Ð025 m/day. Thesteady-state flow regime for this model is illustrated inFigure 3. The images of the macropores are superim-posed in this illustration only for the comparison withthe cases that follow, i.e. the superimposed macroporescontain matrix material identical to the surrounding soil.For this case, the water table reaches the soil surface

within the middle part of the soil block, thereby reduc-ing the actual infiltration flux into the block while at thesame time generating surface runoff on the saturated area.The average infiltration into the block is 0Ð0217 m/day,whereas the mean surface runoff is 0Ð0033 m/day. Theflow within the soil block is observed to follow theexpected pattern for throughflow for a sloping homo-geneous soil profile. The maximum hydraulic gradientnormal to the seepage face for this case is 3Ð05 (Table I).

By introducing a population of disconnected macrop-ores into the flow domain, the flow pattern is significantlyaltered (Figure 4) leading to an increase in the effectivehydraulic conductivity of the soil block. This increasedeffective hydraulic conductivity is manifested in a largedrop in the water table within the soil profile. The pres-ence of the macropores increases the effective hydraulicconductivity of the soil by approximately 40%. This esti-mate for the increase in effective hydraulic conductivitywas determined by simulating the case with no macrop-ores using several trial runs with various increased val-ues of the matrix saturated hydraulic conductivity. Thisincrease in effective hydraulic conductivity is very sim-ilar to the result found by Nieber et al. (2006) for flowin a saturated cylinder of soil containing a population ofdisconnected macropores. The hydraulic gradient alongthe seepage face in line with macropores M1 and M2 is31Ð2 and 10Ð6, respectively (Table I).

Before showing additional simulations with the macro-pore arrangement shown in Figure 2, we momentarilyconsider a case with all macropores oriented perpendicu-lar to the bottom of the block. The purpose for consider-ing this case is to emphasize the importance of macroporeorientation on the flow pattern and the effective resistanceto flow. One can readily see that the resulting water table(Figure 5) is very similar to the case with no macropores(Figure 3). The vertical macropore orientation does affectthe water flow in the vertical direction as indicated by thevertical ‘steps’ in the flow pathlines. On the contrary, thecase with arbitrary orientation of macropores (Figure 2)



Table I. Mean and variance of Darcy flux and seepage surface hydraulic gradient as a function of surface flux (rainfall rate) andwater content of the soil

Surface flux(m/day)

Mean water content (m3/m3), meanwater pressure head (m)

Mean Darcy flux(10�3 m/day)

Variance in Darcyflux (10�3 m2/ day2)

Coefficient of variationfor Darcy flux

Hydraulicgradient

0Ð00125 0Ð333, �0Ð713 2Ð4 9Ð8 ð 10�3 1Ð28 1Ð3b

0Ð00313 0Ð34, �0Ð546 6Ð0 0Ð04 1Ð08 1Ð8b

0Ð0050 0Ð343, �0Ð434 9Ð5 0Ð23 1Ð57 (2Ð1, 0)c

0Ð00625 0Ð345, �0Ð375 12Ð2 0Ð71 2Ð18 (6Ð0, 0)c

0Ð0125 0Ð348, �0Ð144 23Ð9 4Ð2 2Ð72 (19Ð1, 2Ð5)c

0Ð020 0Ð350, C0Ð074 35Ð6 10Ð0 2Ð81 (26Ð8, 8Ð3)c

0Ð025 0Ð350, C0Ð205 42Ð2 16Ð1 2Ð98 (31Ð2, 10Ð6)c

0Ð025a 0Ð350, C0Ð319 37Ð0 0Ð59 0Ð65 3Ð05b

0Ð025e 0Ð350, C0Ð182 42Ð5 15Ð2 2Ð90 (30Ð3, 7Ð3)d

0Ð025f 0Ð350, C0Ð148 42Ð1 14Ð4 2Ð84 (32Ð7, 8Ð6)d

0Ð025g 0Ð350, C0Ð108 41Ð6 13Ð9 2Ð83 (33Ð9, 8Ð1)d

a No macropores present.b Maximum hydraulic gradient on seepage face.c Mean hydraulic gradient over a 0Ð005-m radius area on the seepage face in line with macropores M1 and M2, respectively.d Mean hydraulic gradient along the upslope end of the soil pipe and over a 0Ð005-m radius area on the seepage face in line with macropore M2,respectively.e case of soil pipe with length 0.51 m.f case of soil pipe with length 0.65 m.g case of soil pipe with length 0.78 m.

Figure 4. Illustration of the water table elevation (shaded surface) and streamline distribution for the soil block with disconnected macropores. Thiscase is for the flux (rainfall) rate equal to 0Ð025 m/day

shows a preferential flow that is enhanced in the slopedirection (Figure 4).

Returning now to the case of macropores shown inFigure 2, it is clear that the resulting flow patternsfor the case of the highest applied steady-state rainfallflux (0Ð025 m/day) manifest a wide variability in flowdirection and flow magnitude (Figure 4). The manifestcomplexity of flow is directly induced by the presence ofthe macropores (i.e. compare flow paths in Figure 3 withthose in Figure 4). In this case, most of the macropores,that is, those macropores that are below the water tablesurface, contribute to the flow. A few of the macroporeslie either completely or partially above the water table.Those macropores are excluded from the flow that occurswithin the unsaturated soil above the water table. Formacropores that are in the saturated portion of the soilprofile, the water flow is preferentially directed towardstheir location to find the path of least resistance within the

soil profile. These deflected, least resistive flow pathwaysare observed throughout the saturated portion of the soilprofile. The flow pattern in Figure 4 evolves as the degreeof saturation of the soil mass increases, and is reminiscentof the self-organization of flow pathways in forestedhillslopes described in the conceptual model of Sidleet al. (2000) and Sidle et al. (2001). To demonstrate theprocess of evolution of this flow pattern, the effects ofan increasing series of applied fluxes (rainfall inputs) onthe flow domain are modelled and graphically illustratedin Figure 6.

The lowest flow considered in our analysis is shownin Figure 6a. For this condition, the applied flux at thesoil surface is 0Ð00125 m/day. Overall, the flow patternis very regular, except that upon close examination offlow pathways it can be seen that flow avoids enteringmacropores within the unsaturated zone. This diversionaround macropores in the unsaturated zone is like the



Figure 5. Illustration of the water table elevation (shaded surface) and streamline distribution for the soil block with disconnected macropores orientedvertically. This case is for the flux (rainfall) rate equal to 0Ð025 m/day

(a) (b)

(c) (d)

(e) (f)

Figure 6. Illustration of the water table elevation (shaded surface) and streamline distribution for the soil block with disconnected macropores with vari-able applied surface flux, R. (a) R D 0Ð00125 m/day; (b) R D 0Ð00313 m/day; (c) R D 0Ð005 m/day; (d) R D 0Ð00625 m/day; (e) R D 0Ð0125 m/day;and (f) R D 0Ð020 m/day. Hydraulic gradient values on the seepage face are summarized in Table I. The rise in pore water pressure at the blocked

end of macropores M1 and M2 are illustrated in Figure 7a and b for some of the cases

result for unsaturated flow around embedded texturallycontrasting material shown by Bakker and Nieber (2005).For this case, the saturated zone occupies only a smallvolume near the toe of the slope, and all the macroporeslie within the overlying unsaturated zone (Figure 6a). Themaximum hydraulic gradient along the seepage face forthis case is 1Ð3 (Table I).

An increase of the applied flux to 0Ð00313 m/dayyields a flow pattern (Figure 6b) similar to that shown

in Figure 6a. The saturated zone has expanded whencompared with the case with the lower applied flux(i.e. in Figure 6a), but all macropores still reside withinunsaturated soil. The maximum hydraulic gradient alongthe seepage face for this case is 1Ð8 (Table I).

Increasing the applied flux to 0Ð005 m/day causesthe water table to intersect ‘macropore’ M4 as well asportions of macropores M1 and M3 (Figure 6c). Theflow pattern now shows that the macropores, or portions



(a)

(b)

Figure 7. Distribution of water pressure head along the centerline ofmacropores M1 and M2 for several cases of surface applied flux: (a) M1and (b) M2. For the case of no macropores, the pressure head is plotted

along the same transect as the corresponding macropore

thereof, located within the saturated zone become activeconduits for flow, whereas previously the flow divertedaround these features. The water table slightly risesabove macropore M1 resulting in some flow into thismacropore and a resultant buildup of pressure at theblocked downslope end of the macropore. This spikein pore pressure near the downslope end of macroporeM1 is clearly seen in a plot of the pressure along thecentral axis of this macropore (Figure 7a). The hydraulicgradients along the seepage face in line with macroporesM1 and M2 are 2Ð1 and 0, respectively (Table I).

As the applied flux (rainfall rate) is increased incre-mentally from 0Ð005 m/day to the final value of 0Ð025 m/

day, there is a corresponding increase in the numberof macropores that contribute to preferential subsurfaceflow. These results are illustrated for fluxes of 0Ð00625,0Ð0125 and 0Ð020 m/day in Figures 6d–f, respectively.The result for the highest flux (0Ð025 m/day) is shownin Figure 4. For these cases, the excess pressure thatbuilds up in macropore M1, and eventually in macroporeM2 for input fluxes ½0Ð0125 m/day, also increases withincreasing applied flux (Figure 7a and b). The hydraulicgradients along the seepage face in line with macroporesM1 and M2 are summarized in Table I for these cases. Asthe flux increases, the corresponding hydraulic gradientsincrease as well.

Now, we consider the case where macropore M1 isprogressively eroded out by a sapping process, leadingto the development of a soil pipe. Our analysis for ero-sion does not model the dynamics of the erosion processbut assumes progressive erosional advance with end-stateconditions determined from some quantification of poten-tial seepage erosion. In the analysis, we assume that thesapping process will be promoted by particle detachmentat locations having sufficiently elevated hydraulic gradi-ents. Howard and McLane (1988) presented an expres-sion for the seepage force acting on individual soil parti-cles along seepage surfaces. When this force exceeds theinter-particle forces tending to hold particles in place, theparticles will detach, thus producing seepage erosion orthe sapping process. In the expression for the seepageforce the local hydraulic gradient is a key variable. Inour analysis we determined the location where the localhydraulic gradient is maximum, and then we assumedthat particle detachment by seepage forces will occur atthat location thus leading to soil pipe growth.

The end-state condition is the point where the soil justdownslope of macropore M1 is sapped away so that themacropore now opens at the seepage face. From Table I,we see that when M1 dead-ends the hydraulic gradienton the seepage face just downslope of the macropore is31Ð2. Without doing a balance of the seepage and inter-particle forces, we assume that this gradient is sufficientto cause the soil along the seepage face to be detached.

For this new end-state condition, we now refer to thismacropore as being a soil pipe. The entire boundary ofthis soil pipe is treated as a surface of seepage rather thansimulating it as a porous medium. To treat this boundarycondition, we use the condition given by Equation (2) andthe constraints associated with that boundary condition.We do not model the hydraulics of the flow of waterwithin the soil pipe, but assume that the pipe is flowingpartially full because the flow capacity of a pipe with thegiven slope and cross-section for laminar flow conditionswould be 55 m3/ day. The finite element grid for this casecontains 425 246 linear tetrahedral elements and 75 415node points.

The result for the first end-state condition of subsurfacepipe erosion is illustrated in Figure 8a. The flow patternappears to be only slightly modified from the original casewith the dead-ended macropore M1 present (Figure 4),but the water table has dropped several centimetres on



(a)

(c)

(b)

Figure 8. Illustration of the water table elevation (shaded surface) and streamline distribution for the soil block with applied flux (rainfall) of0Ð025 m/day for three cases where macropore M1 has been eroded to become a soil pipe open to the seepage face. For all cases the pipes have adiameter of 0Ð01 m. (a) Soil pipe with length of 0Ð51 m; (b) Soil pipe eroded to a length of 0Ð65 m; (c) Soil pipe eroded to a length of 0Ð78 m. Thehydraulic gradient along the upper entry point of the soil pipe and along the seepage face in line with macropore M2 is summarized in Table I for

each case

average over the domain. The outflow from the soilmatrix at the seepage face is 0Ð059 m3/ day, whereasthe outflow from the pipe is 0Ð016 m3/ day, indicatingthat about 20% of the hillslope domain discharge iscontributed by the soil pipe. The hydraulic gradientsalong the seepage face of soil pipe and along thedownslope seepage face in line with macropore M2 aregiven in Table I. The hydraulic gradient at the upslopeend of the soil pipe is 30Ð3, which we assume is stillsufficient to cause particle detachment by seepage forces.

Further erosional development of the soil pipe ismanifested here as the lengthening of the pipe (Figure 8b)to 0Ð65 m. The finite element grid for this case had428 088 linear tetrahedral elements, and 75 946 nodepoints. From the solution, the water table has droppedseveral centimetres compared with that of the shortestpipe length (Figure 8a). Outflows from the soil matrixand pipe are 0Ð055 and 0Ð021 m3/ day, respectively,indicating that about 28% of the flow is contributed bythe soil pipe. The hydraulic gradients along the upperend of the soil pipe and along the downslope seepageface in line with macropore M2 are given in Table I.The hydraulic gradient at the upper end of the pipeis 32Ð7, which we assume will support continued pipegrowth.

In the final case, the soil pipe has grown further ups-lope to reach a length equal to 0Ð78 m. The finite ele-ment grid for this case had 429 669 linear tetrahedralelements and 76 204 node points. The simulated flowdomain shows that the water table has reduced in eleva-tion even further (Figure 8c). The subsurface fluxes con-tributed 0Ð049 m3/ day for the matrix and 0Ð026 m3/ dayfor the soil pipe, or about 35% of the total hillslope dis-charge for the soil pipe. The hydraulic gradients at theupslope end of the soil pipe and along the seepage face

in line with macropore M2 are given in Table I. Thehydraulic gradient at the upper end of the soil pipe isstill large and presumably the soil pipe would continueto migrate upslope.

DISCUSSION

Our treatment of the macropores as being voids filled withcoarse textured porous media is not new. For example,previous treatments include Kosugi et al. (2004), Podgor-ney and Fairley (2008) and Nieber et al. (2006). A moreexact formulation would treat the macropores as voidsand would simulate water flow in the macropores usingthe Navier–Stokes equations along with handling multi-ple seepage faces. Although this more exact formulationwould be tractable if there were only one macropore,dealing with many macropores simultaneously is over-whelming. Fortunately, the types of flows expected toexist in the macropores should be approximated well byDarcy’s law because those flows will be of low Reynoldsnumber and inertia should be very small. With these con-ditions, the main difference between the Navier–Stokesequations treatment and the Darcy’s law treatment willbe that boundary layers within the macropores will not betreated explicitly with Darcy’s law. However, this effectwill be small considering the expected low Reynoldsnumber flows.

Although the macropore population used in this analy-sis represented only about 0Ð37% of the total soil volume,it exerted a disproportional influence on subsurface waterflux, particularly during higher rates of simulated rainfall.As noted, this macropore density is considerably lowerthan many reported studies, thus the effect of these pref-erential flow paths on the overall flow within the soil



block, their effect on the equivalent hydraulic conductiv-ity and their influence on flow patterns, and thereby theefficiency to move subsurface water downslope, wouldbe expected to be much greater in most field applicationsthan what has been demonstrated in our simulations.

In addition to the fact that the macroporosity consid-ered in our analysis is smaller than in typical undisturbedsoils, we did not account for the possibility that dis-connected macropores might be separated by loose soilor pockets of decayed organic materials (i.e. not densematrix material). Such conditions were observed anddescribed by Noguchi et al. (1999) in their field obser-vations and later incorporated into a three-dimensionalmodel of macropore distribution (Sidle et al., 2001). Suchloose material would certainly provide much less resis-tance to flow between macropores than currently repre-sented in our model, and this of course would increase theefficiency of ‘connections’ between the discrete macrop-ore segments and thereby dramatically increase the effec-tive hydraulic conductivity of the soil block.

The buildup of water pressures in the blocked macrop-ores M1 and M2 becomes significant as the applied fluxincreases. This accretion occurs once the macropore isinundated by the expanding saturated zone allowing waterto readily enter the macropore; however, the downslopemovement of water is interrupted at the blocked endof the macropore. This blocking mechanism, and theassociated pressure buildup, was first reported by Pier-son (1983), who indicated that this pore water pressureincrease could be the cause for initiation of mass failureson hillslopes. Such a pressure buildup has been analysedexperimentally (Sidle et al., 1995b) and via numericalsimulation (Tsutsumi et al., 2004, 2005). Although thepressure buildup at the blocked ends of macropores M1and M2 as shown Figure 6a and b would not likely initi-ate a landslide, it might be sufficient to promote seepageerosion at the downslope seepage face.

The evolution of the simulated hillslope flow patternsshows an increase in connectivity of the macropore path-ways as the mean flux and the degree of saturationincreases. This phenomenon was first described by Sidleet al. (2000) (also Sidle et al., 2001) as a process ofself-organizing of preferential flow pathways. Based onour simulated flow patterns, as the degree of saturationincreases along with the applied flux, macropores in satu-rated regions become active, increasing flow in the slop-ing soil block; this activation then affects the pathwaysconnecting the matrix volume with the macropores andthe discharge point on the downslope end of the block.These results are qualitative; we illustrate using path-line plots that as the degree of saturation increases, flowspassing through the soil matrix converge onto the macro-pores, thereby connecting together individual macroporefeatures, even though those features are not physicallyconnected.

One way to indirectly quantify the effect of macroporeactivity on the flow might be to calculate the variancein the Darcy flux field. Observing the flow fields shownin Figures 4 and 6 indicates that as macropores become

more active the flow field becomes more heterogeneous,that is, there is more convergence and divergence of flowoccurring with increasing subsurface fluxes. We expectthat as the flow field becomes more heterogeneous dueto macropore activity, the variance of the flow shouldincrease. The mean, variance and coefficient of variation(COV) for the Darcy fluxes were calculated for each ofthe cases considered (Table I), including those shown inFigures 3, 4, 6 and 8.

As the degree of saturation increases, both the meanflux and flux variances also increase indicating anincreased participation of macropores in the flow regime(Table I). Flow in the domain without macropores is veryregular according to the velocity COV, whereas the flowin the domain with macropores is very irregular. Thevisual images shown in Figures 3, 4 and 6 indicate thatunder moist conditions the macropores affect flow pat-terns, and the measure of flow regularity given in Table Iquantifies the influence of macropores on the flow struc-ture. The images shown for the cases with the soil pipe,Figure 8, also indicate the irregular flow produced by thepresence of the macropores and the soil pipe. It is interest-ing that the presence of the soil pipe seems to decrease therelative variance of the flows compared with the highestflow case shown in Figure 4.

Our findings related to subsurface erosion processesand associated soil piping are relevant because they relateto the evolution of preferential subsurface pathways, andpossibly the development of streambank instability andsurface erosion features such as gullies (Jones, 1987).Mechanisms of subsurface erosion have been describedby Dunne (1990), and detailed field investigations ofthese erosion and sediment transport processes have beenconducted in various settings (Hagerty, 1991b; Terajimaet al., 1997; Sayer et al., 2006).

The onset of erosion at seepage faces is related to thehydraulic gradient (Howard and McLane, 1988; Hagerty,1991a; Fox et al., 2006). In our study, the location ofthe maximum hydraulic gradient on seepage boundarieswas determined, and this indicates the potential locationsof seepage erosion. The hydraulic gradient informationis summarized in Table I. In contrast to the simulationwithout preferential flow (Figure 3), when macroporeswere included the flow became focused at locales alongthe seepage face and these represent the regions ofamplified hydraulic gradient (Figures 4, 6c–f and 8a–c).It is at these locations where we would expect seepageerosion to initiate. Once started, the process produces afeedback in which each step of erosion leads to moreconcentration of flow, and more concentration of seepageforces. To simulate these steps of soil evolution willrequire incorporation of the force balance for particlesexisting on the seepage face into a model similar to that ofHoward and McLane (1988). This should be the subjectof future research efforts.

Perhaps, the self-organizing of flow pathways des-cribed herein can be viewed from the standpoint ofconstructal theory described by Bejan (2000) and laterapplied to a network of macropores (Lorente and Bejan,



2006). According to this theory, a flow domain suchas a sloping soil will adjust to maximize the access ofthe flow to all parts of the system. Such a system willthereby require the minimum energy to pass a given flowrate through the global system. As such, it is expectedthat the erosion of the system will occur in such a wayas to form macropores/soil pipes that will provide thismaximum flow access. This derived system will not havea rectangular or regular structure, rather it will morelikely be dendritic. Perhaps at some future time it willbe possible to mimic through simulation the subsurfaceerosion that forms such networked patterns.

SUMMARY AND CONCLUSIONS

Finite element solutions to the steady-state form of thethree-dimensional Richards equation were derived toevaluate the effect of a population of disconnected macro-pores on the flow within a sloping block of soil underlainby an impervious boundary. The macropores representfeatures originating from decayed root channels, aban-doned animal burrows and breaks or discontinuities atthe bedrock–soil interface. In the numerical solution, themacropores were treated as voids filled with very coarseporous media having a saturated hydraulic conductivity1 ð 106 times that of the surrounding soil matrix. Thenumerical solutions were derived for cases where a steadyflux was applied to the block surface, leading to subsur-face flow that exited from the downslope boundary of theblock. Applied fluxes ranged from a very low level thatproduced a very small perched saturated zone, to higherflux that led to a nearly fully saturated domain.

The simulation results show that at the lowest appliedflux, none of the macropores participate in transportingwater through the soil block, and actually water com-pletely bypasses the macropores. However, as the degreeof wetness of the soil block progressively increases con-currently with increasing applied flux, the number ofactive macropores increases. As more macropores con-tribute to the flow, the effective hydraulic conductivityof the soil block increases significantly.

In conclusion, this study shows that although macro-pores may not be directly connected to a water sourceor not connected to one another, they can still substan-tially decrease the resistance to flow within the soil pro-file. As the degree of wetness in a soil block increases,potential preferential flow pathways, such as macropores,become active and these disconnected macropores thenself-organize the flow into an efficient flow network. Thisfinding supports the concepts of threshold behaviour andself-organization suggested from field staining, tracer andhydrometric investigations in Japan (Sidle et al., 1995a,2000, 2001), as well as findings from later studies thathave implied such types of connectivity at larger scales(Western et al., 2001; James and Roulet, 2007; Tromp-van Meerveld and Weiler, 2008; Anderson et al., 2009).Additionally, active preferential flow pathways in a slop-ing soil tend to focus the flow into localized areas along

seepage faces, and these then become the locations ofamplified hydraulic gradients, and the locations for poten-tial seepage erosion.

The findings herein have strong implications for catch-ment management, particularly with respect to articu-lating pathways of water and solutes through hillslopesoils. Improved knowledge of such pathways is neededfor scheduling management activities in headwater catch-ments, such as timber harvesting, application of agro-chemicals, grazing locations, construction and mainte-nance of roads and cultivation. Furthermore, our resultsindicate the important contribution of macropores to thedevelopment of excess pore water pressure and seepageerosion. This process strongly affects soil piping, sedi-ment transport, gully formation and landslide initiation(Jones, 1987; Terajima et al., 1997; Valentin et al., 2005;Sidle and Ochiai, 2006). The basic simulation method-ology presented herein articulates important small-scaleprocesses, heretofore not described in models of macro-pore flow, and is potentially scalable up to hillslopes andeventually to catchments.

Our results should be viewed as only an initial exami-nation into the mechanisms of preferential flow networkformation within hillslopes. We imagine a number ofinvestigations that should follow, including the evalua-tion of the development of preferential flow networksduring transient response to unsteady rainfall, simula-tions with more realistic macropore distributions withinsloping soils, simulating the interaction of macroporesand surrounding mesopores, evaluation of the transportof solutes through sloping soils containing discontinuousmacropores, modelling the processes of seepage erosionand soil pipe formation, and modelling pore pressureaccretion within dead-end macropores and soil pipes andpotentially ensuing hillslope instability. In addition toexamination of these processes at appropriate scales, itis important that the results of future studies are properlyscaled to provide useful tools for prediction of runoff gen-eration, solute transport, subsurface erosion and potentiallandslide initiation in hillslopes and catchments.

ACKNOWLEDGEMENTS

The authors wish to thank the three anonymous reviewersfor their constructive criticisms that have helped improvethis manuscript. Furthermore, insights into preferentialflow processes based on the second author’s researchwith Japanese colleagues, especially at Hitachi Ohta, aregratefully acknowledged. The first author acknowledgesthe support of the Minnesota Agricultural ExperimentStation Project MIN-12-046, ‘Characterizing mass andenergy transport at different scales’.

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How do disconnected macropores in sloping soils facilitate preferential flow?

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