Water Resources Research

Evolution of unsaturated hydraulic conductivity of aggregated soils due to compressive forces



[1] Prediction of water flow and transport processes in soils susceptible to structural alteration such as compaction of tilled agricultural lands or newly constructed landfills rely on accurate description of changes in soil unsaturated hydraulic conductivity. Recent studies have documented the critical impact of aggregate contact characteristics on water flow rates and pathways in unsaturated aggregated soils. We developed an analytical model for aggregate contact size evolution as a basis for quantifying effects of compression on saturated and unsaturated hydraulic conductivity of aggregated soil. Relating confined one-dimensional sample strain with aggregate deformation facilitates prediction of the increase in interaggregate contact area and concurrent decrease in macropore size with degree of sample compression. The hydrologic component of the model predicts unsaturated hydraulic conductivity of a pack of idealized aggregates (spheres) on the basis of contact size and saturation conditions under prescribed sample deformation. Calculated contact areas and hydraulic conductivity for pairs of aggregates agreed surprisingly well with measured values, determined from compaction experiments employing neutron and X-ray-radiography and image analysis. Model calculations for a unit cell of uniform spherical aggregates in cubic packing were able to mimic some of the differences in saturated and unsaturated hydraulic conductivity observed for aggregates and bulk soil.

1. Introduction

[2] Characterization of fluid fluxes through deformable structured porous media is of interest to agricultural, engineering, civil and environmental engineering, geophysics, and hydrology at multiple scales. Surface soils featuring pronounced secondary structures of relatively dense aggregates or “peds” separated by large interaggregate pores are of particular interest because of their role in agricultural practices. The so-called interaggregate macropores with diameters considerably larger than textural particle sizes (for clays and silts, particle diameter <50 μm) play critical roles in water and air exchange as well as in preferential contaminant transport in soils. The pore size distribution of structured soils containing macropores is often bimodal (interaggregate macropores and intra-aggregate micropores) with two well separated maxima (also referred to as dual-porosity media [Barenblatt et al., 1960]). For nearly saturated conditions which develop during intense water inputs, the large interaggregate pores form the primary pathways for rapid infiltration, causing preferential flow and transport [Jarvis, 2002]. Because interaggregate pores rapidly drain at the end of infiltration, soil water is mostly stored within the aggregates and it redistributes slowly through a network of contacting aggregates.

[3] Because of their high porosity and relatively low stiffness and strength, especially when wet, aggregated soils are prone to compaction and subsequent changes of their hydraulic properties, e.g., decrease in macropore volume and hydraulic conductivity. The effects of compaction on soil productivity through its influence on soil strength and various transport properties has been a subject of research for decades [e.g., Assouline et al., 1997; Baumgartl and Horn, 1991; Bekker et al., 1961; Berli et al., 2004; Braunack and Dexter, 1978; Gräsle et al., 1995; Gysi et al., 1999; Hadas, 1994; Horn, 1988; Horn et al., 1995; Kirby and Blunden, 1991; Peth and Horn, 2006; Richard et al., 2001; Söhne, 1958; Veenhof and McBride, 1996]. Until recently, the complexity of interactions among solid, liquid and gas phases in deforming aggregated soils, and limited observability of pore space and fluid distribution necessitated bulk representation of mechanical and hydraulic properties at characteristic lengths larger than a single pore size. Although sufficient for many practical applications, bulk approaches lack explanatory power that requires understanding of pore-scale processes to explain vast differences in hydromechanical responses of soils with essentially the same macroscopic porosities and other bulk properties. These microscale responses offer a basis for a unified framework useful for several disciplines dealing with similar fluid-filled porous geomaterials.

[4] Advances in imaging techniques such as X-ray and neutron microtomography and increasing computing power enable observation and simulation of mechanical and hydraulic processes at the pore scale. These capabilities offer alternatives to bulk representation in which pore-scale description of solid and fluid mechanics form basic building blocks, that are upscaled to reconstruct (observable) macroscopic bulk deformation and flow properties. A variety of studies already showed the role of individual mineral particles and microaggregates (diameter <50 μm) for soil compaction and hydraulics properties [e.g., Delage, 2005, 2007; Simms and Yanful, 2001, 2005]. Recent studies addressed the deformation of larger soil aggregates (diameter >50 μm) due to capillary and external stresses [Ghezzehei and Or, 2000], deformation of individual macropores within elastoviscoplastic matrix [Berli et al., 2006; Berli and Or, 2006; Ghezzehei and Or, 2003], and the influence of macropore deformation on permeability of water-saturated rock and soil [Eggers et al., 2006; 2007; Sisavath et al., 2000].

[5] Notwithstanding progress in microscale hydromechanical modeling, the extension of these concepts to unsaturated conditions and directly linking effects of aggregate deformation on unsaturated hydraulic conductivity remain a challenge. Ghezzehei and Or [2000] introduced a model for aggregate coalescence due to negative pore water pressure in unsaturated soil. Or et al. [2006] proposed a simple analytical model for a soil unit cell that links deformation with hydraulic conductivity of unsaturated soil on the basis of the findings by Eggers et al. [2006, 2007]. Recent studies by Carminati et al. [2007, 2008] demonstrated that contacts between aggregates control unsaturated water flow through serially arranged stack of aggregates.

[6] The primary objective of this paper was to develop a hydromechanical model for the evolution of aggregate contacts and their impact on interaggregate as well as intra-aggregate fluid flow. We sought to link the model with recent results from microscale aggregate hydraulics presented by Carminati [2007] to quantify effects of soil compaction on soil unsaturated hydraulic conductivity. The developed micromechanical model was expected to (1) describe the growth of the interaggregate contact between a pair of aggregates due to an axial load and the influence of interaggregated contact area on the unsaturated hydraulic conductivity of the pair of aggregates, (2) expand the results to a soil unit cell consisting of uniform spherical aggregates in cubic packing as introduced by Or et al. [2006], and, finally, (3) compare hydraulic conductivity simulations with measurements from compression tests on pairs of aggregates on which water flow has been monitored using neutron and X-ray tomography.

2. Theoretical Considerations

[7] Loose aggregated soils are formed during tillage to create desirable conditions for plant growth, or during other mechanical disturbances or packing of excavated earth materials. These loose soils are susceptible to compaction by internal or external stresses because of their high porosity and relatively low stiffness and strength, particularly at high water content. Compression and associated volume change decreases the total amount of pore space and affects soil hydraulic properties. Large interaggregate pores and aggregate micropores are not deformed uniformly and the large pores are considered to be more readily compacted than fine pores [see, e.g., Assouline et al., 1997; Bruand and Cousin, 1995; Koliji et al., 2006; Richard et al., 2001]. The same applies to the hydraulic properties of the two pore domains. We expect a decrease in the saturated hydraulic conductivity of the interaggregate pores, while the water flux through the micropore region may increase because of enhanced aggregate contacts that expand at the expense of larger voids of the intra-aggregate pore space.

2.1. Contact Mechanics of a Pair of Soil Aggregates: An Analytical Model

[8] In this study, we proposed a simple analytical model to describe deformation of a pair of uniform spherical aggregates with initial radius, r0, stacked on top each other and undergoing uniaxial load within a rigid cylinder (with inner diameter equal to initial aggregate radius as seen in Figure 1). Applying confined one-dimensional compression in z direction results in an axial sample strain, ɛz = Δz/r0 and increases the contact areas between the aggregates, characterized by the radius of the interaggregate contact, rcz (Figure 1). For small strains, we assumed that the general shape of the aggregates remains spherical, except for interaggregate and aggregate-cylinder wall contacts (Figure 1b). The volume of a deformed sphere, V, is then given as

equation image

with r0 the initial sphere radius, rcz the radius of the interaggregate contact area, l half the height of the contact between aggregate and cylinder wall and

equation image

the height of the spherical segment defined by r0 at the bottom and rcz on the top (Figure 1b). Assuming the aggregate material is incompressible, the aggregate volume remains constant for all sample strains, such that the volume of the deformed aggregate, V, is equal to its initial volume, V0, which is given by

equation image

Combining equations (1)(3), the radius of interaggregate contact area, rcz, is given implicitly as

equation image
equation image

with l derived from geometrical considerations (Figure 1). It was easier to solve equations (4) and (5) first for l which yielded, after some algebraic manipulations,

equation image

The solution for l substituted into equation (4) and the latter solved for rcz provides

equation image

From equation (7) the area of interaggregate contact, Acz, was calculated immediately as Acz = πrcz2.

Figure 1.

(a) A pair of uniform spherical soil aggregates before and after undergoing confined, one-dimensional strain of ɛz = 10% and (b) definition of geometric variables. The sample height decreases while the interaggregate contact area increases with increasing strain.

2.2. Unsaturated Hydraulic Conductivity of a Pair of Deforming Aggregates

[9] Carminati et al. [2008] showed that unsaturated hydraulic conductivity of a pair of aggregate, Kpair(h), with h being the matric head of the aggregate matrix, highly depends on the contacts between the aggregates. They found that for matric heads h <−1 m, Kpair(h) decreases proportionally to the water-filled interaggregate contact area. The reason is that as long as the contacts are narrow and partially drained (h < −1 m), the contacts act as bottle necks for the flow, and therefore determine the conductivity of the entire pack of aggregates, similar to the case of a series of resistors with low resistance (aggregates) separated by a resistor of high resistance (contacts). The key idea of this study was to combine the results by Carminati et al. [2008] on interaggregate hydraulic properties with above described contact mechanics to predict hydraulic conductivity of a deforming pack of aggregates.

[10] For the model, we assumed that changes in hydraulic conductivity of the soil aggregate matrix, Km (h), by the applied load has little affect on the hydraulic conductivity of a pair of aggregates since the latter is mainly controlled by the hydraulic properties of the contacts as shown by Carminati et al. [2008]. Assuming a linear relation between conductivity of aggregate pairs and contact area, we defined Kpair(h, ɛz) as

equation image

with Acz the interaggregated contact area and A0 the cross-section area of the confining cylinder. Since for a pair of aggregates, stacked on each other as in Figure 1, Acz = rcz2π and A0 = r02π, Kpair (h, ɛz) can be written as

equation image

with rcz according to equation (7). Note that Kpair(h, ɛz) includes the effect of both, aggregate geometry and hydraulic properties of the aggregate matrix. Dividing Acz by the total cross section of the cylinder A0, we obtained the macroscopic hydraulic conductivity [m s−1] of the entire pair of aggregates. Kpair (h, ɛz) is therefore a macroscopic property of the material, which is determined by interaggregate and intra-aggregate pores.

[11] The proposed model reaches its limits if (1) the pair of aggregates is wet (>−1 m matric head) and underwent just little deformation (rcz ≈ 0) or (2) the pair of aggregates is strongly deformed; that is, the radius of the interaggregate contact is close to the aggregate radius, rczr0. For unsaturated conditions (matric heads <−1 m) and moderately deformed aggregates 0 < rcz < r0, the capillary bridges between facing aggregates are small and can be considered as part of the interaggregated contact, as shown by Carminati et al. [2008]. Closer to saturation, however, capillary bridges become increasingly important for water flow between the aggregates and, particularly for the case of small “mechanical” contact areas, control unsaturated hydraulic conductivity for a pair of aggregates. For the case of a strongly deformed pair of aggregates (rczr0), hydraulic conductivity of the aggregate matrix material becomes increasingly important compared to the interaggregate contact area and therefore, changes in hydraulic conductivity of the aggregate matrix cannot be neglected.

2.3. Hydraulic Conductivity of an Axially Compressed Aggregate Unit Cell

[12] On the basis of insights regarding pore deformation and fluid flow behavior learned from numerical analyses of 2-D compressed sample cross section [Eggers et al., 2006, 2007], we suggested a simple analytical model to describe the influence of one-dimensional compression on unsaturated hydraulic conductivity of a three-dimensional unit cell consisting of uniform spherical aggregates with initial aggregate radius, r0, and unit cell side length, 2r0 (Figure 2). Applying one-dimensional compression in the z direction while confining the unit cell in x and y direction results in an axial sample strain, ɛz, which reduces the size of interaggregate pore (expressed in terms of interaggregated pore radius, rpz) while increasing the contact areas between the aggregates, characterized by the radius of the aggregate contacts, rcz (Figure 2).

Figure 2.

Half of a unit cell of (a) nondeformed and (b) deformed uniform spherical soil aggregates undergoing a one-dimensional strain of ɛz = 0%. The interaggregate pore space decreases, while interaggregate contact area increases with increasing strain.

[13] Invoking a similar assumption as for a pair of aggregates, water flow through the unit cell in z direction is controlled (1) by interaggregate pore throat while it remains water filled (matric potential higher than air entry value), and (2) by interaggregate contact areas based on Buckingham's [1907] approach. The unsaturated hydraulic conductivity Km (h) of the soil matrix is unaffected by aggregate contact deformation and remains just a function of the hydraulic head h. Water flow in the unit cell under unsaturated conditions is thus controlled by the radius of the interaggregate pore, rpz, and the radius of the contact area between aggregates, rcz. Both radii can be readily determined assuming that (1) the soil matrix material is incompressible and (2) individual aggregates remain spherical, except at the interaggregate contacts, with

equation image
equation image

where r is the current aggregate radius for a given strain ɛz

equation image


equation image

During one-dimensional compaction, the shape of the interaggregate pore throat can be approximated as a square (see Figure 2) with side length 2rpz [see also Eggers et al., 2006]. The hydraulic conductivity Kp of a cylindrical duct of length L and cross-sectional area of 4rpz2 is given by [Berker, 1963]

equation image

with f = 2.253 the shape factor for rectangular cross sections and η the dynamic viscosity of water. The hydraulic conductivity of the intra-aggregate flow domain is controlled by the contacts between aggregates and is calculated according to equation (8). Hydraulic conductivity of the entire deforming unit cell, Kcell (h, ɛz), is the sum of the conductivity of the interaggregate and intra-aggregate domains and is calculated as

equation image

with Ap the cross section are of the interaggregate pore throat, Acz the interaggregated contact area and Auc the cross section area of the entire unit cell. Since for the unit cell Ap = 4rpz2, Acz = rcz2π and Auc = 4r02, Kcell(h, ɛz) can be written as

equation image

with rpz, rcz according to equations (10) and (11). Equation (16) holds for matric heads larger than the head at air entry value, hair, given as

equation image

(σ the tension at the air-water interface) when all pores are water filled which, together with equations (10)(13), basically describes the change in saturated hydraulic conductivity of the unit cell due to one-dimensional strain, ɛz. Since for saturated conditions, Ksat,mrcz2rpz4/η, equation (16) can be simplified to

equation image

with rpz according to (10). For h < hair the interaggregate pore drains and water flow through the unit cell is therefore confined to the matrix and controlled by the size of the interaggregated contact area Acz which simplifies equation (16) to

equation image

3. Material and Methods

[14] Direct observations of interaggregate contact growth and subsequent change in sample unsaturated hydraulic conductivity with increasing axial sample strain were obtained from laboratory experiments using pairs of aggregates undergoing confined one-dimensional compaction. Aggregates were collected from a loam soil [Soil Science Society of America, 2007] at “Abist,” a forest area northeast of Zurich, Switzerland [Richard and Lüscher, 1983]. Samples were take from the [E]B horizon at 20–30 cm depth (texture: 21% clay, 30% silt, 49% sand; bulk soil total porosity ϕb = 0.46; as determined by Richard and Lüscher [1983]), air dried and sieved to obtain aggregates with diameters smaller than 5 mm.

[15] The water retention curve of the aggregates, θm(h), was determined by stepwise draining a layer of aggregates on a porous plate in a pressure apparatus according to Richards and Fireman [1943]. The water retention curve was fitted using the van Genuchten approach [van Genuchten, 1980]:

equation image

The fitted parameters for the aggregates are θsat = 0.39, α = 0.44, n = 1.57. Residual water content was set to θres = 0.

[16] The unsaturated hydraulic conductivity of the aggregate matrix material, Km(h), and the interaggregate contact area, Acz, were determined by simulating observed water redistribution within and water exchange between aggregates, equilibrating at different matric heads as described by Carminati et al. [2007]. The water content distribution was monitored by means of neutron radiography at high spatial and temporal resolution [Pleinert and Lehmann, 1997], [Hassanein et al., 2006]. A total number of 15 replicates, each consisting of two aggregates were stacked in a cylindrical Plexiglas container (15 mm long, 5 mm inner diameter) so that the aggregates were in contact (similar to Figure 3a, where three aggregates are present). A volume of water of 0.01 to 0.02 mL was injected onto the top of the air-dry aggregate with a syringe. The cylinder was then closed with a piston, gently pressed onto the top aggregate in order to avoid evaporation (Figure 3a). Then water redistribution within and between the aggregates was monitored by means of neutron radiography. At equilibrium, the pair of aggregates was tomographed and the interaggregate contact area Acz was determined by image analysis. To calculate the interaggregate contact area, the facing aggregates were first segmented using a double threshold method [Soille, 2003] and then separated using the watershed algorithm [Vincent and Soille, 1991]. After distinguishing the facing aggregates, contact voxels were identified from which the contact area was calculated. For more details on the image processing for interaggregated contact analysis, we refer to Carminati et al. [2007].

Figure 3.

Experimental setup used to measure hydraulic conductivity and interaggregate contact area during compression tests. (a) Sample used to study water infiltration through series of aggregates. A piston is used to apply load. Water redistribution was imaged with neutron radiography, and the contact area was obtained by neutron tomography and image processing. (b) Cylinder containing aggregates with a porous plate at the bottom for controlling the water potential. The aggregates were equilibrated at h = −1.2 m and were compacted with increasing loads of 10, 20, 50, and 100 g. In this case, the region around the interaggregate was scanned with X-ray radiography at each load step.

[17] The unsaturated hydraulic conductivity of the aggregates, Km(h), was calculated from the water content distribution within the aggregates using the model by Mualem [1976]:

equation image

where h is the matric head, Θ = (θ − θres)/(θsat − θres) is the degree of water saturation, Ksat the saturated hydraulic conductivity of the aggregates, and τ the tortuosity of the pores. Tortuosity τ was kept constant at τ = 0.5 on the basis of values published in literature, Saturated hydraulic conductivity, Ksat, was used as the fitting parameter to match measured values with three-dimensional simulations of water distribution within and between the aggregates. The best fit was found for Ksat = 10−7 [m s−1].

[18] The unsaturated hydraulic conductivity of the aggregate pairs Kpair (h) was calculated by applying the Darcy-Buckingham law to the observed water flow rate Q [m3 s−1]:

equation image

where Δz is the distance between the centers of the facing aggregates, Δh is the difference of matric head between the aggregates and Acc is the cross-sectional area of the cylinder. The difference Δh was estimated from the water content θ within the aggregates, as it was determined by high spatial resolution neutron radiography. The radiograms showed that during water redistribution, the water content within each aggregate remained quite uniform, except of a steep gradient in water content at the interaggregate contacts. Therefore it was possible to assign a water content value to each aggregate at any time. Inverting the water retention curve of the aggregates θm(h), we calculated the matric head within the aggregates. The conductivity of the pair of aggregates, Kpair, was directly related to the matric head, h, in the wet-top aggregate, where water flows under unit gradient conditions, i.e., −h(z) = constant. This approach is analogous to the Green and Ampt model for infiltration [Green and Ampt, 1911]. In our case, h in the wet region decreases with time. As water infiltrates into the dry-bottom aggregate, the matric head into this aggregate as well as the position of the wetting front are ill defined. To overcome this difficulty, we evaluate equation (23) only at time t = 0. Note that with this approach we do not loose relevant information, because the trend of the water exchange during time remained rather constant [Carminati et al., 2007]. Rearranging equation (22) we calculated Kpair for each sample:

equation image

where hi is the matric head in the top aggregate immediately after applying the drops of water, i.e., t = 0.

[19] Following the infiltration tests for the noncompressed samples, the same 15 samples were gently compacted by manually pressing a piston with outer diameter slightly smaller than the inner diameter of the cylinder onto the top aggregate (similar to Figure 3a). The load has been applied for 30 min. The piston was then removed and the water in the aggregates was allowed to evaporate. When the aggregates were dry, a few drops of water were applied onto the upper aggregate. After 3 hours the load was applied again, followed by the removal of the piston to permit evaporation. This procedure was repeated three times. At the end, the total sample height had decreased by approximately 0.5 mm, corresponding to an average axial sample strain of ɛz = 5%. After this treatment, a few drops of water were again applied onto the top aggregate and the water redistribution was monitored by means of neutron radiography. The interaggregate contact area and the unsaturated conductivity of the compacted samples were obtained by the same procedure as described for the nondeformed sample. Eventually, we obtain Acz and Kpair(h) for 15 replicates, each before and after compaction at ɛz = 5%.

[20] In a second experimental setup, we used X-ray tomography to image the deformation of the aggregate contact region at high spatial resolution. Aim of this experiment was to verify the assumptions that (1) the contacts undergo larger deformation than the aggregate interior, and (2) capillary bridges between aggregates are small for matric heads h < −1.2 m. Similar to the previously described conductivity tests, two aggregates of “Abist” soil were placed in a cylindrical Plexiglas container (25 mm long, 5 mm inner diameter,) and a porous plate at the bottom of the cylinder connected to a water reservoir was used to control matric head h of the sample (Figure 3b). Before the first compression step, the sample was equilibrated at h = −1.2 m and the second contact from the top was scanned by means of X-ray radiography. X-ray tomography was carried out at the TOMCAT facility of the Swiss Light Source (SLS) at the Paul Scherrer Institute, Villigen [Lehmann et al., 2005], providing a field of view of 2 × 5 mm2 and spatial resolution of 5.92 μm pixel size. The sample was subsequently stepwise compacted by applying pistons of 10, 20, 50 and 100 g weight. For each load step, the sample was let to equilibrate for 30 min before the interaggregate contact was scanned. During the compaction process, the matric head was kept constant at h = −1.2 m.

4 Results and Discussion

4.1. Interaggregate Contact Between a Pair of Deforming Aggregates

[21] In a first step, we observed the evolution of the contact area between a pair of aggregates undergoing uniaxial compression by X-ray, microtomography. Figure 4 shows a vertical cross section through the center of the contact region unloaded (top) and at after application of a load of 100 g (bottom). The latter corresponds to a total sample strain of approximately 5%. The unloaded interaggregate contacts were rather narrow and the contact region contained large air-filled voids. Under unsaturated conditions, the unloaded contacts were expected to limit water flow across the aggregates and reduce the unsaturated conductivity of the sample. With increasing load, the contacts flattened and the contact area increased. Figure 4 depicts considerable growth of the diameter of the interaggregate contact due to a relatively small increase in sample axial strain of ɛz = 5%. Note the initially small mechanical contacts between the aggregates for the nonloaded stage and formation of small water bridges (light gray areas) which connect the water phase between the two aggregates. Figure 4 supports our assumptions that for h < −1.2 m capillary bridges are small and that the contacts undergo a higher deformation than the core of the aggregates.

Figure 4.

Vertical cross section of a real interaggregate contact as imaged with X-ray tomography. (top) Contact of the nonloaded sample. (bottom) Contact at a sample strain ɛz = 5%. The sample was kept at a matric head h = −1.2 m. The view shows a 3 × 1 mm2 cut of the interaggregated contact at a spatial resolution of 5.92 μm.

[22] Simulated growth of interaggregate contact using equation (7) showed a strong increase in interaggregate contact area at low-sample axial strains (Figure 5a), similar to contact growth observed by microtomography (Figure 4). It is interesting to note that while the interaggregate contact radius evolved nonlinearly, the contact area increased almost linearly with increasing sample axial strain, at least up ɛz = 0.2 (Figure 5b). The increase in contact area, however, became also nonlinear close to maximum axial strain of ɛz = 1/3.

Figure 5.

(a) Calculated radius and (b) area of the interaggregate contact as a function of the sample axial strain, ɛz.

[23] The calculated interaggregate contact areas were compared with contact areas measured by means of neutron radiography and image analysis. We observed that calculated values provide a lower limit for measured values, (Figure 6), at least for axial strains up to ɛz = 0.1 for which data from compression experiments were available. This underestimation of measured values by the model calculations could be explained by the infinitely small initial contact area assumed in the model whereas in reality there is a finite (even small, see Figure 6) initial contact area between the aggregates. As subsequent analyses of neutron radiography images showed, the initial interaggregate contact area may reach up to 10% of aggregate cross-section area [Carminati, 2007, p. 98]. Model calculations assuming an initial relative contact area Acz/A0 = 0.1 provide a reasonable upper limit for the interaggregate contact size within the range the observed sample deformation.

Figure 6.

Measured and calculated interaggregate contact area as a function of sample axial strain ɛz.

4.2 Interaggregate Pore and Contact Deformation of the Unit Cell

[24] Unit cell deformation calculations showed that with increasing strain, ɛz, the size of the macropores between the aggregates (i.e., the interaggregate pore throat, rp/r0) decreased (Figure 7a), while the radius of the interaggregate contact area, rcz/r0 increased (Figure 7a). Similar to calculations for a pair of aggregates (Figure 7b), the interaggregate contact area increased linearly with increasing unit cell strain (Figure 7b). For the unit cell, the range of linear contact area growth expanded almost up to 30% strain. The cross-sectional area of the interaggregate pore throat, however, decreased exponentially with increasing cell strain. Figure 7 also shows that when the radius of the interaggregate contact, rcz, reached the value of the initial aggregate radius, r0, the radius of the pore throat, rpz was reduced by 40% and its cross-sectional area, Apz, by roughly 60%. Hence, under saturated conditions, a considerable amount of fluid may still flow through the throat, even if interaggregate contacts are fully developed. The limit of rcz reaching r0 is also the reason why unit cell axial strains larger than ɛz = 0.41 were not considered. For rcz > r0 the interaggregate contact cannot be modeled as circular anymore and a more complicated contact mechanics applies. A similar problem for the transition from a pack of aggregates to the case of isolated pores within a “continuous” solid matrix was also identified by Ghezzehei and Or [2003].

Figure 7.

Calculated relative radii of (a) interaggregate pore throat and contact areas and (b) the area of the interaggregate pore throat and interaggregate contacts as a function of the unit cell axial strain, ɛz.

4.3. Unsaturated Hydraulic Conductivity of a Pair of Aggregates

[25] In this section, hydraulic conductivity of a pair of aggregates, Kpair (h, ɛz), as predicted by equations (8) and (9), were compared to measured values for nondeformed and deformed (axial strain ɛz = 0.05) pairs of aggregates. To calculate conductivity of the nondeformed stack of aggregates, ɛz = 0.001 was chosen. Measurements and simulations showed the expected increase in hydraulic conductivity due to aggregate pair deformation, which agreed well for hydraulic heads h < −1 m (Figure 8). Closer to saturation (h > −1 m), model calculations underestimated the measured conductivity values. The reason is that capillary bridges between aggregates contribute considerably to the flow across the interaggregate contact close to saturation [see, e.g., Carminati et al., 2008] and becomes dominant, particularly if the mechanical contacts between aggregates are small. The role of the capillary bridges becomes less significant as the samples are dryer and more compacted, as shown in Figure 4, where at h = −1.2 m no large capillary bridges between aggregates are visible. Therefore, the model presented in this study applies mainly to unsaturated conditions with matric heads h < −1 m. Considering the simplicity of the model derived from first principles and that there is no parameter fitting involved, however, measured and calculated unsaturated hydraulic conductivity values for a deforming pair of aggregates agree surprisingly well.

Figure 8.

Measured (symbols) and calculated (lines) unsaturated hydraulic conductivity of a sample consisting of a pair of “Abist” aggregates as a function of sample axial strain, ɛz. For nondeformed conditions, ɛz = 0.001 was used as an approximation for the conductivity calculations. For ɛz = 1/3 the conductivity of a pair of aggregates becomes the hydraulic conductivity measured for the aggregate matrix. Hydraulic conductivity of the aggregates was determined according to Carminati et al. [2007].

4.4. Unsaturated Hydraulic Conductivity of a Unit Cell

[26] The final step in this study focused on the effect of one-dimensional deformation of a soil unit cell on its saturated and unsaturated hydraulic conductivity. Model calculations predicted a decrease in saturated hydraulic conductivity of up to 3 orders of magnitude and an increase in unsaturated hydraulic conductivity of 1 order of magnitude with increasing unit cell strain ɛz up to 45% (Figure 9). It is interesting to note that under saturated conditions, even at ɛz = 0.4, hydraulic conductivity of the interaggregate pore throats remained higher than intra-aggregate hydraulic conductivity, controlled by the interaggregate contacts. The increase in hydraulic conductivity due to interaggregate contacts is larger for lower ɛz whereas the decrease in conductivity due to pore throat closure accelerates with increasing ɛz. The air entry value of the unit cell, controlled by the diameter of the largest interaggregate pore throat and expressed as absolute matric head, increased because of the decreasing radii of the interaggregate pore throats with increasing unit cell strain, ɛz.

Figure 9.

Hydraulic conductivity of a soil unit cell as a function of matric potential for different unit cell strain ɛz of 10, 40, and 45%. Hydraulic conductivity of the “Abist” aggregates was determined according to Carminati et al. [2007]. Values for the bulk soil were taken from Richard and Lüscher [1983].

[27] To evaluate the simulated hydraulic conductivities for a unit cell, Kcell(h, ɛz), measured hydraulic conductivity values of aggregate material and bulk soil from “Abist” were compared with the outcome of model calculations employing equations (18) and (19) (Figure 9). Values for aggregate hydraulic conductivity were determined by neutron tomography as described previously whereas bulk soil data taken from Richard and Lüscher [1983] determined on undisturbed core samples (1000 cm3 volume) from the same horizon. For the comparison, the bulk soil was considered a pack of aggregate equivalent to a moderately compacted unit cell whereas a hypothetical densest packing of aggregates (just aggregated material without interaggregate pore space) was considered equivalent to a unit cell at maximum compaction.

[28] Whereas bulk soil and aggregate material have considerably different saturated hydraulic conductivity, the difference in unsaturated conductivity decreases with increasing absolute head (Figure 9). The model could capture this difference, particularly the decrease in saturated conductivity with increasing compaction, whereas the data did not reflect the simulated increase in unsaturated conductivity with increasing compaction. This is due to the fact that the presented values are for a hypothetical aggregate packing and not from an actual compaction experiment on a pack of aggregates, which should be addressed in a future study. Considering the many simplifying assumptions regarding aggregate size and arrangement the model given by equation (18) predicts the measured saturated hydraulic conductivity of the bulk soil fairly well. This result also confirms the findings by Eggers et al. [2006, 2007] that details of pore cross section geometry play a small role in estimation of saturated hydraulic conductivity as long as the geometry of the key flow path can be reasonably well modeled with idealized shape such as an ellipse, triangle or a rectangle.

[29] The rather poor performance of the model for unsaturated hydraulic conductivity close to the air entry value is mainly due to the fact that (1) the macropore throats have just one single diameter and (2) the effect of capillary bridges is neglected in the current model. Both simplifications cause the very drastic transition from saturated to unsaturated conductivity, which is much smoother in reality as the measured values in Figure 9 for bulk soil showed. Therefore improved simulations for unsaturated hydraulic conductivity of deforming soil close to the air entry value could be achieved by considering flow through capillary bridges, which form around the interaggregate contacts at lower absolute hydraulic head ∣h∣ [see, e.g., Ghezzehei and Or, 2000], and support flow at the interaggregate contacts, particularly if the mechanical contacts are still small [Carminati et al., 2008].

5. Summary and Conclusions

[30] Optimization of tillage operations, erosion control, prevention and remediation of soil compaction necessitate improved understanding of relationships between soil mechanical status and hydraulic transport properties at various scales. In this study, we developed a conceptual model for predicting the evolution of interaggregate contacts and macropore size to quantify effects of compression on saturated and unsaturated hydraulic conductivity of aggregated soils. For saturated conditions, the assumption was that hydraulic conductivity is determined by the radius of the interaggregate pore throat, as proposed by Berker [1963]. As interaggregate pores become drained because of decreasing hydraulic head, water flow becomes restricted to the network of aggregates and hydraulic conductivity of a pack of aggregates becomes controlled by interaggregate contacts as shown by Carminati et al. [2008]. Contact areas between aggregates were calculated analytically assuming that aggregate volume remains constant during compression and porosity changes in a pack of aggregates are due to a decrease in interaggregate macropore volume. Hydraulic conductivity was simulated for a pair of aggregates and a unit cell consisting of eight uniform spherical aggregates in cubic packing deforming under a one-dimensional load. Model calculations were compared with measured values for pairs aggregates, determined from compaction experiments employing neutron radiography and image analysis, as well as bulk soil values from literature.

[31] Simulations showed a linear increase in interaggregate contact area over for applied strains of up to 10% for the pair of aggregates, and up to 20% for the unit cell. The size of the interaggregate pore throat, however, decreased exponentially with increasing sample strain. For aggregate pairs, model calculations of interaggregate contact areas form a lower limit relative to measured values for applied strains of up to 5%. Measured and modeled unsaturated hydraulic conductivity of aggregate pairs agreed well for hydraulic heads h < −1 m. For heads from 0 to −1 m, model calculations underestimated measured values, which could be explained by the neglect of capillary bridges between aggregates. Simulated hydraulic conductivity values for a deforming unit cell were between measured values for bulk soil and aggregates in densest packing for a sandy loam under forest in northeastern Switzerland. Model calculations could mimic the difference in saturated hydraulic conductivity due to increasing compaction. Similar to the case of the aggregate pair, also the model for the unit cell underestimated unsaturated hydraulic conductivity. Besides the lack of capillary bridges, the unit cell also assumes that macropores between the aggregates are of just one single size, which leads to a very sharp transition from saturated to unsaturated hydraulic conductivity.

[32] The study demonstrates the potential of pore-scale modeling for an improved understanding of coupled mechanics and hydraulics processes in soils with pronounced macropore structure (pore diameter > 50 μm). In a next step, incorporating capillary bridges into the presented hydromechanical model will be necessary to simulate deformation and hydraulic conductivity response of a real pack of soil aggregates. Considering the simplicity of the presented models (aggregate arrangement, pore geometries and matrix material properties) derived from first principles, the simulations are already in good qualitative agreement with measurements from real soil material.


[33] The research underlying this paper has been sponsored by the U.S. Department of Agriculture (USDA-NRI, grant 2003-35107-13598). We are particularly grateful to R. Hassanein, P. Vontobel, and E. Lehmann of NEUTRA-PSI group for their advice and support in neutron radiography and to M. Stampanoni and A. Groso from SLS-PSI for the technical support during X-ray tomography.