Water Transport in an Unsaturated Medium to Roots
with Differing Local Geometries
C. L. Petric,* Z. J. Kabala, A. E. Hall, and J. Simunek
ABSTRACTPearl millet \Pennisetum glaucum (L.) R. Br.: syn. P. americanum(L;) Leeke] develops substantially lower predawn leaf water potentials
than cowpea [Vigna unguiculata (L.) Walp.] under conditions of soil
dehydration, but overall root length densities are greater in millet
than in cowpea. The hypothesis was tested that, due to differences inroot geometry and root length density, root systems of millet becomeless efficient in water uptake than those of cowpea as soil dries. Plants
were grown in an artificial medium in pots or 1.15-m tubes in agreenhouse and subjected to soil drying. Root geometries, root lengthdensities, and water potentials of cowpea and millet were measured
as the rooting medium dried. Distribution of roots was quantified by
calculating a clumping ratio (total root length density divided by theroot length density in the region surrounding individual root segmentaxes) at several depths within the profile. Overall root length densities
were higher in millet than in cowpea, and root distributions differed.
Millet roots were clumped locally along the root axis and globally
within the profile, with the highest root length densities at the surface.
Cowpea root lengths were one-half as dense as millet along the rootaxis and distributed fairly uniformly throughout the profile. These
data were used in two-dimensional modeling of horizontal water transport from soil to branched segments of root axes of cowpea and millet
to test the hypothesis. Simulations in which millet water potential
became substantially lower than cowpea, and millet root length was
twice as great as cowpea, predicted little difference in cumulative
water flow to the roots of millet or cowpea during soil dehydration.
This prediction is consistent with observations. Simulated water velocity vector fields indicated that millet's dense and clumped local root
geometry became less efficient in water uptake under soil water deficits than cowpea's less dense and more evenly distributed root geometry because water uptake by millet was mainly at the root tips; incontrast, most of the root length of cowpea contributed to water uptake. Simulations conducted with millet root axes having different root
length densities indicated that the low efficiency of millet in water
uptake was due to the combined effects of local clumping and highroot length density. When input parameters for roots and soil werevaried, conclusions from the simulations remained the same.

U

NDER CONDITIONS of progressive soil dehydration, pearl millet developed lower predawn leaf
water potentials than cowpea, even when the plants
were grown together in the same pot (Petric and Hall,
1992a). Total resistance to water movement became
greater in millet than in cowpea as the soil dried, but
predawn water potential of roots (^R) was approximately the same as the water potential of leaves (^L)
in both cowpea and millet, indicating that the difference between the total resistances was not within theplants (Petric and Hall, 1992c). Application of a mannitol solution eliminated the differential effects of theresistance in that ^L, in both plants, came into an
approximate equilibrium at predawn with the osmotic
potential of the solution (Petrie and Hall, 1992c).
Consequently, the higher resistance to water flow inmillet than cowpea under dry soil conditions may havebeen in the region of soil surrounding the roots.
Lower root length densities result in a longer average path length for water movement to roots andthat longer path length is expected to increase the resistance to water flow in the soil (Newman, 1969).
Overall root length density, however, was at least twice
as great in millet as in cowpea (Petrie and Hall, 1992b).
Predawn ^L was lower for millet than for cowpea
even when millet roots grew deeper than cowpea (Petrie and Hall, 1992b). When plants are grown primarily
C.L. Petric, USDA-ARS Crop Simulation Research Unit, P.O.
Box 5367, Mississippi State, MS 39762-5367; A.E. Hall, Dep.
of Botany and Plant Sciences, and Z.J. Kabala, Dep. of Soil andEnvironmental Sciences, Univ. of California, Riverside, CA 92521;
and J. Simunek. Research Institute for Soil Reclamation and Protection, Prague, Czechoslovakia. Received 16 Mar. 1992. *Corresponding author.
Published in Soil Sci. Soc. Am. J. 56:1686-1694 (1992).

PETRIE ET AL.: WATER TRANSPORT IN AN UNSATURATED MEDIUM

on stored soil water, greater rooting depth and root
length density can be detrimental if they cause plants
to use water too quickly (Ludlow and Muchow, 1988),
but millet did not use more water than cowpea by thetime when differences in predawn ^L developed (Petrie and Hall, 1992b). Apparently there is no obvious
explanation for the higher resistance to water movement in the soil surrounding the roots of millet compared with cowpea.
Computer simulation models have been used to study
water uptake by roots and to evaluate the magnitude
of resistances to water movement in the soil and inroots (Molz, 1981b). Root models are typically microscopic or macroscopic (Molz, 198la). The microscopic model generally evaluates radial water transport
to a single root cylinder of infinite length, while themacroscopic model examines the root system as a wholewithout considering effects of individual roots (Hillel
et al., 1975, 1976). In macroscopic models, roots are
assumed to be distributed uniformly over the entire
soil profile, or root length densities that differ between
layers are assumed to be uniformly distributed withineach soil layer (Taylor and Klepper, 1978). Models
can be static or can include root growth (Hillel andTalpaz, 1976). Steady state and transient flow havebeen modeled using analytic and iterative numericalmethods, and numbers of dimensions modeled andinput parameters have varied (Philip, 1966; Molz,
1981a,b). In macroscopic models, root water uptake
is simulated with a root extraction function, and theforms chosen for this function vary. Molz (1981b) has
noted that some of _the controversy concerning themagnitudes of root arid soil resistances may have resulted from different assumptions used in defining theextraction function. Application of the extraction
function in a model results in spatial averaging, which
ignores effects of water depletion around individualroots, but Hillel et al. (1975) found that a much lower
plant water potential is required to maintain water flux
into a plant when the model includes water uptake byindividual rbots. They stated that the neglect of flow
patterns around individual roots is a major flaw inmacroscopic models. A computerized tomography scanof the single root of a seedling showed experimentallythat a water content gradient formed around the root
as transpiration proceeded, and soil resistance in the
sand-clay soil mix used in that study became important even when bulk soil water potential was high(Hainsworth and Aylmore, 1986, 1989).
Microscopic computer simulation models have beenused to evaluate water transport to single, unbranchedroot segments, but most root axes have branches. Differences in root branching patterns can produce different geometries, which may affect root function bychanging the shape of the root-soil interface and thespace-filling properties of roots. Root geometry has
been described and related to root growth and function
with an approach from topology (Fitter, 1985, 1986).
The two contrasting root geometries considered in thisapproach were the herringbone system, which has short,closely spaced primary laterals emanating from theaxis, and the dichotomous system, which has secondary branches spaced at greater intervals (Fitter, 1986).
Grass root systems were found to be more herring-1687

bone-like than legume root systems (Fitter, 1987). The
herringbone system was hypothesized to be the mostexpensive in terms of root mass but more effective insoil exploration than the dichotomous system (Fitter,1987). The effects of prolonged drought were not tested
in experiments or simulations, however (Fitter, 1986,
1987). We are not aware of any computer simulationsof water uptake by root systems with contrasting geometries under conditions of progressive soil dehydration.
The root geometries of cowpea (a legume) and millet (a grass) appear to differ. Millet has primary root
branches that are generally short and densely distributed along the axis, similar to the herringbone system.
Cowpea root branches are longer and less dense along
the axis, similar to the dichotomous system. Preliminary studies indicated that millet root branches may
be clumped along the root axis, while cowpea roots
are spaced more widely and seem to be more uniformly distributed along the axis. A clumped distribution of branches along a root axis could cause
localized drying of small regions of soil, encasing asubstantial portion of the root surface in dry soil andincreasing the apparent soil resistance to water flow
to the axis. Such a root could be very efficient if water
is applied frequently but becomes progressively lesseffective in water uptake when plants are subjected toprolonged soil dehydration.
The observed differences in cowpea and millet rootaxis geometry and root length density in the region
surrounding the axis are on a local scale, a scale often
considered unimportant because of its variability. Some
aspects of root axis geometry, however, may be undergenetic control and largely fixed for each species. If
each segment of root axis is a module that is similar
to other root segments, water transport to each local
module may represent a pattern that is repeated at
different locations in the root system. A more complete understanding of the local pattern could provide
an improved understanding of water transport to theentire root system.
Water movement to branched root segments is threedimensional. Even in a two-dimensional model, however, convergence of the numerical solution to a watertransport problem requires a fine mesh and small time
steps, particularly when steep water potential gradients develop. Many hours of computer time are required to solve the equations in each iteration of eachtime step. The number of equations solved in a threedimensional model is orders of magnitude greater.
Models require careful interpretation, and adjustments, such as time scaling, may be necessary in twodimensional representations of three-dimensional
problems. An advantage in two-dimensional modeling, however, is that a greater number of models canbe studied economically, so that parameters of the
problem can be varied individually and in combination, and their effects can be studied more closely.
The primary objectives of this study were to: (i)quantify differences in geometry and root length density of millet and cowpea roots, and (ii) model twodimensional horizontal water flow to roots under drying
conditions to determine whether the differences in root
geometry and root length density could be responsible

1688

SOIL SCI. SOC. AM. J., VOL. 56, NOVEMBER-DECEMBER 1992

for the lower predawn water potentials observed inmillet than in cowpea.
MATERIALS AND METHODS
Experiments
Cowpea and millet were grown in a greenhouse in tubes
1.15 m long and 0.2 m in diam. filled with a 50:50 (v/v)
mixture of Turface Plus (AIMCOR, Applied Industrial Materials Corp., Deerfield, IL) and a finesilica sand (maximum particle diameter of 2.5 x 104 or 60 mesh). Plants
were subjected to progressive dehydration of the rooting
medium (dry treatment) as described by Petrie and Hall
(1992b); duration of the dry treatment was between 12 and35 d. Samples of rooting medium were collected at 10
different depths from five tubes each of cowpea and millet
at three different times after initiation of the dry treatment
by inserting a coring tool horizontally through holes predrilled in the tubes (Petrie and Hall, 1992b). Roots wereseparated from the medium with a sieve and stored for latermeasurement (Petrie and Hall, 1992b).
Five root axes were selected at random from each sample
of roots, placed on a glass plate, and photocopied; 1-cm
axis segments were marked and axis root length densities
within 0.5 cm on each side of the axis were measured(Petrie, 1992). In addition, total root length density wascalculated for each depth at which roots were detected. The
entire root sample was used to determine total root length
by a modification of the Newman line intersect method(Tennant, 1975).
Although predawn ¥L was lower in millet than in cowpea
as soil dried, values of predawn ¥L and WR were approximately the same within individual plants (Petrie and Hall,
1992c). Therefore, values of ^R used in the simulations
were assumed to be equivalent to values of predawn ^Lmeasured on cowpea and millet. Five replicate pots consisting of one cowpea and one millet plant growing together
in a 25-L pot of Turface Plus-sand mix in a greenhousewere subjected to a dry treatment, and predawn measurements of ^PL were made with psychrometers on four different days after initiation of the dry treatment (Petrie andHall, 1992a). A second-degree polynomial was fitted by
the least squares method to the predawn ^PL values for cowpea and for millet, curve-fitting equations were derived,
and a smooth curve of water potential over time was developed for each species.
Computer Simulation Model
Assuming that water flow in the vapor phase is insignificant, transient water flow in an unsaturated porous medium
is described by the Richards equation (Richards, 1931),
which is ordinarily solved by numerical means. The computer program used to simulate water transport in thesemodels was the SWMII-ST program, an earlier version of
SWMS-2D developed by Simunek et al. (1992). The program combined a Galerkin-type finite element approach for
spatial discretization with a finite difference method for
temporal discretization to solve a modified form of the
Richards equation,

dt

- S,

where C is the soil water capacity function (m-1), h is1 soilpressure head (m), K is hydraulic conductivity(m s- ), S
is a sink term for uptake by roots(s-1), Jt, are spatial coA
ordinates (m), t is time (s), and K is a dimensionless tensor
that describes conductivity anisotropy (Simunek et al., 1992).
Since horizontal water transport was simulated in thesestudies, the vertical component of conductivity anisotropy,

K\, was set to zero. The sink term, S, represents either auniform or a nonuniform spatial distribution of the water
uptake rate across a root zone of arbitrary shape. Since inour studies we wanted to evaluate water transport to individual roots with nonuniform distributions, S was set tozero and a different method (described below) was devised
to simulate nonuniform uptake.
The expressions of van Genuchten (1980) were used todescribe the unsaturated soil hydraulic properties:

h <0
h >0

(0. -

e(h) =
where

=
Kr =

h <0
h>0
- (I - Se1/m)m]2

and= (0- 0r)/(B. - 0,),where 6 is volumetric water content, 6, is residual moisture
content, 0S is saturated watercontent, Ks is saturated hydraulic conductivity (m s-1), a and n are parameters of theretention curve, and 5e is the degree of saturation. For thesesimulations, the soil hydraulic parameters for the A horizonof a grass field described in Simunek et al. (1992) wereassumed representative of a soil that effectively supported
plant growth. Values of the soil parameters
were: 0, =0.001, 0S =6 0.399,
a = 1.74 m- 1 , n = 1.3757, Ks =1
3.33 x 10- m s- , and m = 1 - (1/n), n > 1. Two setsof simulations were conducted to test the effect of differences in hydraulic conductivity on water transport to theroots. For the
first 1set of simulations, Ks was reduced to
1.67 x 10-6 m s, and for another set of simulations Ks
was 3.33 x 1Q-6 m s-1.
Representative, symmetrical, branched root segment axes
were described for cowpea and millet using observations ofroots and experimental measurements. Each root segment
was compressed along the axis into a two-dimensional model.
Each root axis became a point, and horizontal water flow
to the axis point and root branches was simulated. Experimental measurements (presented below) indicated that millet overall root length density was twice as great as cowpea,
and millet lateral roots were observed to be generally shorterand more densely distributed along the root axes than cowpea lateral roots. The root geometry used in the millet model,
therefore, was locally clumped, with 16 short branches,
each 0.25 cm long, radiating from the point representing
the root axis. The root geometry used for the cowpea model
had four longer branches, each 0.5 cm in length. Total root
length densities in the two-dimensional models were 4 cm
cm- 2 for millet and 2 cm cm~ 2 for cowpea (Fig. la andIb). Each simplified, branched root segment axis was assumed to be contained within a 1-cm2 pot with a no-flow
boundary (no water flowing in or out of the region except
out of roots), no macropores, and no thermal gradients.
Values of ^R assumed for cowpea and millet were obtained by fitting a second-degree polynomial to ^PL values
measured experimentally. These ^R values were input into
the model at prescribed times, and differences in watercontent distribution and water velocity that resulted fromthe differences in root geometry were calculated and plotted. The ^PR values for cowpea and millet were based on
predawn ¥L measurements taken during a 12-d period. In
the root models, root length densities measured within a 1e

PETRIE ET AL.: WATER TRANSPORT IN AN UNSATURATED MEDIUM
(a)

1 cm

(b)

1 cm

Millet full grid

Cowpea full grid

(c)

1/2

Cowpea quarter grid

(d)

1/2

Millet quarter gridFig. 1. Models for local water transport to a root axis with ageometry with a smaller root length density (2 cm cm"2)that is not clumped (cowpea) and to a root axis with ageometry with a greater root length density (4 cm cm~ 2 )that is clumped (millet). Grids used in computer simulation:(a) cowpea full grid (1 cm2), (b) millet full grid, (c) cowpea

quarter grid (1/4 cm2) and (d) millet quarter grid.

1689

To test the convergence of the numerical model, the program was run for coarse meshes of 441 evenly spaced nodes(21 by 21 grid) and for fine meshes of 1681 evenly spaced
nodes (41 by 41 grid) for the cowpea and millet root geometries. Initial time steps of 0.6 s were used, with a dynamic
adjustment via a multiplier of 1.3 when the number of iterations required for convergence was less than or equal to three.
Water content, cumulative flow, and instantaneous flow were
determined for cowpea and millet. Results were similar forcoarse and fine meshes, implying convergence. Approximately 3 h of computer time on a SUNSPARK workstation(Sun Microsystems, Mountain View, CA) were required to
run the computer program on each of the fine meshes; < 1 h
was required to run the program on the coarse mesh.
The basic millet model was varied by removing two root
branches from the quarter grid (equivalent to eight branches
removed from the full millet grid) to separate effects of root
density from root geometry. The same number of root nodes
as cowpea (21) was therefore arranged in the clumped,
short-branched geometry of millet to establish a low-density
millet model. The program was run on the grid for lowdensity millet using the ^R values for both millet and cowpea so that the effects of differences in root geometry androot length density could be determined alone and in combination with differences in ^PR.
Since root length densities at the soil surface are generally at least four times as great in millet as in cowpea (Petrieand Hall, 1992b), a high-density millet root model containing 82 root nodes (double-axis millet) was created byadding a second one-quarter root axis to the lower rightcorner of the one-quarter millet grid shown in Fig. Id. The
program was run on the grid for double-axis millet withboth the millet and cowpea ^R values.
Computer simulations of water transport to each root geometry and root length density were run; values for total
amount of water within the profile, outflow, mean head,

cm3 soil volume were compressed into a 1-cm2 area, so thatmuch less water was available to roots in the 1-cm2 3simulations than would have been available within a 1-cm rooting volume. Time was scaled, therefore, so that 1 d of realtime was equivalent to 1 h of computer simulated time.
Symmetry was assumed in the model, and water movement
to only one quadrant of the 1-cm2 area containing the cowpea and millet root axes was simulated (see Fig. Ic and d).
The sink term in the Richards equation solved by the
SWMII-ST program assumes that the potential transpiration
demand by plants is distributed across a two-dimensionalroot zone. Since the purpose of these simulations was toevaluate the effect of nonuniform local root distribution,
the sink term was set to zero, boundary nodes corresponding to the two-dimensional geometric pattern of root branches
within the mesh were designated as root nodes, and theprogram was modified to read changes in ^ at the root
nodes at the appropriate times (Dirichlet boundary conditions). One corner node represented the root axis, 10 root
nodes were allocated to each 0.25 cm of root branch, androot nodes were spaced as evenly as possible along the line
on a the grid that represented the root branch. By the property of symmetry, the root lengths on the perimeter of themesh take up equal amounts of water from each side of thequadrant perimeter, and only five root nodes for every 0.25
cm of root length were spaced along the perimeter. The
total number of root nodes, including the root axis, was 21
for cowpea and 41 for millet. All root nodes within the gridwere at the same ¥R at any given time. The SWMII-ST
program was modified to calculate cumulative and instantaneous flow to all root boundary nodes in the interior andon the perimeter. Output files were created for plotting
water content distribution and water velocity vector fields
using NCAR Graphics programs (University Corporation
for Atmospheric Research, 1989).

water content at each node, and velocity vectors in the x
and y directions were printed for the first 0, 600, and 900s, and every 1 h thereafter until 12 h of simulated time werecompleted. Mass balances were calculated and found toaccount for at least 92% of the water in simulations using
the coarse meshes and at least 96% for the fine meshes,
indicating that computational and rounding errors were small.
Changes with time in head, average water content, andcumulative flow to root nodes were plotted for cowpea root
geometry run with the cowpea ^R values, and for millet
root geometry run with the millet ^R values.
Water content output data generated at each node was used
to plot water content distribution around cowpea, millet, lowdensity millet, and double-axis millet root geometries.
Contour intervals of 0.005 (corresponding to 0.005 m2 m-2 watercontent) were established for all contour plots. Water velocity
vector fields were generated from velocity vector data at eachnode, and a lower limit of velocity was set at 8.33 x 10~10
m s~' for easier comparison of root models.
RESULTS AND DISCUSSION
Experiments
Root length densities were measured to quantify
root clumping. Axis root length density within a 0.5cm radius immediately surrounding root segment axes
was twice as great in millet as in cowpea (3.5 x 104
m m~ 3 for millet and 1.7 x 104 m m~ 3 for cowpea
in Fig. 2). In cowpea, total root length densities wereconsistently lower than densities measured in the region surrounding individual root axes (Fig. 2). In con-trast, millet total root length densities at the surface
were significantly higher than the root length densities
along the axes, but no significant differences weredetected at lower depths. A dimensionless ratio of

1690

SOIL SCI. SOC. AM. J., VOL. 56, NOVEMBER-DECEMBER 1992

total root length density to axis root length density(clumping ratio) was calculated to quantify global
clumping of roots within the profile (Table 1). A
clumping ratio of 1.0 indicates that approximately
oneroot axis is contained within each 1 cm3 of soil, a ratio<1.0 means that each root axis has access to a soilvolume greater than 1 cm3, and a ratio > 1.0 indicatesthat morethan one root axis is contained within each
1-cm3 volume.
Clumping ratios in cowpea were <1.0 throughout
the profile and were similar at all levels by Day 35 of
the dry treatment (Table 1), indicating uniform root
distribution on a global scale. In millet, the average
clumping ratio at 0.1-m depth was 2.07 (indicatingthat approximately two millet axes were contained
within each 1-cm3 volume at the surface), and millet
clumping ratios decreased with depth (Table 1). Root
distribution in millet, therefore, was clumped globally, from the top to the bottom of the rooting profile,
as well as locally along the root axis. In a homogeneous rooting medium, a greater soil volume and alarger quantity of stored soil water are potentiallyavailable to each root when clumping ratios are lower.
Computer Simulation Models
Root Models
The root clumping values derived from experiments
were used to develop representative root axis models
for use in water transport simulations. For the millettype2 geometry, three models were evaluated: (i) a 1cm model containing one millet axis and branches(41 root nodes) representing root distributions at lower
depths; (ii) a 1-cm2 model containing two millet axes
and branches (82 root nodes) representing the greater
degree of root clumping at the surface; and (iii) a lowdensity millet geometry (21 root nodes) that compared

the effects on water transport of the differences in root
geometry between cowpea and millet at the same root
length density. For cowpea, a single model consistingof 21 root nodes was developed, since root lengthdensity in the region surrounding the root axis did notvary in cowpea and clumping ratios were relatively
constant throughout the profile, indicating a relatively
constant number of root axes per soil volume.
For these models, it was assumed that roots do not
grow in each 1-cm2 pot as soil dries. Root growth canincrease the efficiency of water uptake, but root segments that have developed branches may not continue
to grow, and overall rooting densities can decrease as
soil dries (Klepper et al., 1973). In millet and othercereals, root growth may stop at anthesis (Gregory
and Squire, 1979). When root growth ceases, water
uptake depends on soil conductance of water to root
surfaces (Klute and Peters, 1969), and the water potential gradient between the root and soil that drives
water movement can be largely dissipated within 5 or
6 mm from the root surface (Cowan, 1965; Hainsworth and Aylmore, 1986). In a drying soil, a highroot length density around the root axis can increase
the rate of local soil dehydration and slow the rate ofwater uptake. Local root length density, and the spatial arrangement of that density, will determine thesize and shape of the root-soil interface and affectwater transport to the root.

CowpeaDay 7

Day 21

Day 35

Total /
-0.2

Total

Total

E" -0.4

Axis<-S£ -0.6a.

v
Q -0.8
-1.0

0

1

2

0

1

2

0

Root length density (m.m

1

2

x 10

)

MilletDay 7-0.1

Day 35

Day 21

Axis
Axis

-0.2

Total

Axis
Total
Total

S" -0.3

a

-0.4

5

10

5

10

Root length density (m.m

_~

5 1 0
_

x 10

)

Fig. 2. Root length density in the total soil volume (total) andwithin a 0.5-cm radius around the root axis (axis) in cowpea
and millet grown under a drying treatment in separate
polyvinyl chloride tubes containing a Turface Plus-sandmixture (experiment planted 25 Aug. 1990). Five replicates
destructively sampled on Days 7, 21, and 35 after initiationof dry treatment. Error bars indicate least significant
differences at the 0.05 level [LSD(O.OS)].

Initial Simulations
In the first set of computer simulations, water transport to root axes with cowpea root geometry (21 root
Table 1. Clumping ratios (dimensionless ratios of total root
length density to root length density within a 0.5-cm radiusaround root axes) in tube depths containing roots. Drytreatment cowpea and millet were planted 25 Aug. 1990 inseparate tubes containing a Turface Plus-sand mixture.
Measurements were based on five replicates destructively
sampled 7,21, and 35 d after initiation of the dry treatment.
Clumping ratio
Day 21

Day 7

Depth

m
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Mean
SD

CowpeaMillet

0.79
0.71

2.21
1.20
0.90

0.55
0.45
0.37

0.57
±0.18

1.44
±0.69

Day 35

CowpeaMilletCowpeaMillet

0.79

2.11
1.27
1.12
0.83

0.63
0.60
0.67
0.58

1.37
1.29
0.71

0.55
0.66
0.63
0.36
0.46
0.35
0.68
0.65
0.07
0.52
±0.21

1.33
±0.55

0.43
0.43
0.56
0.68
0.62
0.61
0.55
±0.11

1.89

1.32
±0.48

1691

PETRIE ET AL.: WATER TRANSPORT IN AN UNSATURATED MEDIUM

nodes) and millet root geometry (41 root nodes) was
evaluated. The water potential values obtained experimentally declined from 0 to -0.8 MPa in cowpea andfrom 0 to -1.9 MPa in millet. Mean cumulative flow
to root nodes was predicted to be similar for cowpea
and millet ( — 1% higher for millet) in the simulations,comparable to results obtained in greenhouse experiments with cowpea and millet grown in pots (Petrieand Hall, 1992b).
Throughout the simulation for cowpea, water content was lower in the region around the roots than in
the regions furthest from the roots, and overall watercontent in the profile progressively decreased (Petrie,1992). The change in water content contours between
4 and 12 h of simulated water transport indicated thata steeper water content gradient had developed between the region of soil around the root and the outer
grid edges (Petrie, 1992; Fig. 3). Early in the simulation, water movement to the entire cowpea root length
was >8.33 x 10"10 m s-1, but water movement was
slower in the region furthest from the root and in theregion where the cowpea root branches joined the axis
(Petrie, 1992). As the simulation continued, the rateof water movement
to most of the root length was >8.33 x 10-10 m s-1, indicating that most of the root
was involved in water uptake, but the area of slower
water movement in the region farthest from the root
became larger. By the 12th hour of simulation, thelength of root involved in water uptake (Fig. 4) decreased slightly compared with the earlier simulations(Petrie, 1992).
For millet, water content contours by the 12th hourof simulation indicated that a larger zone of dry soil
surrounded the root surface than for cowpea (Fig. 5).
As the simulation progressed, the water content gradient from the soil region around the roots to the outeredges of the grid became steeper and water movement
to refill the root zone became slower (Petrie, 1992).

The rate of water movement was greatest at the roottips during the entire simulation (Petrie, 1992) and,by the 12th hour of simulated water transport, only
the outer portion of the root zone, representing theroot tips, was exhibiting substantial water uptake (Fig.
6).
Effects of Varying Water Potential
and Root Length Density
The effect on a water transport of the plant ^ differences that developed between cowpea and millet
was tested with simulations using values of ^R for
both cowpea and millet. Patterns of water content distribution and water velocity vectors were the same,but the gradient of water content from the region surrounding the root to the outer edge of the profile wassteeper, and the water content in the region immediately surrounding the root was lower when the lower^R values based on measurements of millet were usedwith either cowpea or millet root models (Petrie, 1992).
Water uptake in millet, however, still occurred primarily at the root tips (Petrie, 1992).
When water transport was simulated to a root axis
with a root length density equal to that of cowpea butwith a millet-type geometry (low-density millet), watercontent contours indicated that a smaller area of very
dry soil surrounded the root branches (Fig. 7), andthat water movement was faster in a larger area of theroot zone (Fig. 8), than in millet with a higher root
length density (Fig. 5 and 6). Patterns of results anddifferences between cowpea and millet were similar
when the program was run on the same models using
the cowpea ^R values (Petrie, 1992). With both root
length densities, the millet root geometry was lessefficient than the cowpea root geometry, in that more
Distance from root axis node (cm)0.0Distance from root axis node (cm)
0.0

0.1

0.2

0.3

0.4

0.5
i i i,i\ 1,1,1 fi\i

0.5

i \ 1\ \ \ 1ififtwftiViftwfc
ii n t ( I t ILo0.1

-7.0'"

c

.—
'x

D

X

o

0.2

O

o

•*-*

o

xz-:

E
E

0.3

o

0.4

o

.4-1

xir:
*?«

»<Er:

tn

5

0.5.
0.5

Fig. 3. Water content distribution (x 102) after 12 h ofcomputer-simulated water transport to a root axis with
cowpea geometry and cowpea water potential values. Boldlines indicate roots.

Fig. 4. Water velocity vectors a8.33 x 10-'° m s-1 after 12h of computer-simulated water transport to a root axis with
cowpea geometry and cowpea water potential values.
Maximum and minimum velocities were 3.50 x 10~9 and3.00 x 10-11 m s-1.

1692

SOIL SCI. SOC. AM. J., VOL. 56, NOVEMBER-DECEMBER 1992
Distance from root axis node (cm)Distance from root axis node (cm)
0.0

0.1

0.2

0.3

0.4

0.0

0.5

0.1

0.2

0.3

0.4

0.5

E

u

^_x

o

a>
-a
o
c

TJ
O

c

E
o
V

u
c
a

-M

CO

u

0.4

c

-

D
-*-*

n
5

b
0.5

Fig. 5. Water content distribution (x 102) after 12 h ofcomputer-simulated water transport to a root axis with millet
geometry and millet water potential values. Bold lines indicateroots.

Distance from root axis node (cm)Fig. 7. Water content distribution (x 102) after 12 h ofcomputer-simulated water transport to a root axis with low
root length density millet geometry and millet water potential
values. Bold lines indicate roots. Overlapping circles represent
on artifact of the method of representing root branches byroot nodes in a simulation with sharp changes in water
potential.

0.5

0.0

Distance from root axis node (cm)
E

0.0

u

0
.. . . K__
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 J .1 1 1 1 1 F

1 1 1 1 1 1 1 1 1 1 1 1 1 1, » \Nv-s•a, \ \ N V «•

o

c.

_

_
_
_
-

o

_
_

-

^

f \ \ \ V

o
E
o

:

x
|_\

Q>
U

N \

%

»

^ ,,*

v

v

x

v

^ x

,, , . v. . , . . . .

-

E

C

\

o

O
-t-«
CO

_

b

_
_

u

0.5

' ' I ' ' ' ' 'iiiiiiiiiiiii

Fig. 6. Water velocity vectors >8.33 x 10-'° m s-' after 12h of computer-simulated water transport to a root axis withmillet geometry and millet water potential values. Maximum
and minimum velocities were 7.17 x 10-'and 5.00 x 10-"m s~'.

of the root length of millet than of cowpea was surrounded by a region of soil in which water movement
was very slow. Higher root length densities caused
millet roots to be even less efficient.
Clumping ratios for millet (Table 1) indicated thatat least
two millet root axes were present in each 1cm3 soil volume at the surface. When water transport
was simulated using the double-axis millet model, water

_
_
_

-

c

—

-

-

in
b

;

n ^

Fig. 8. Water velocity vectors >8.33 x lO"10 m s~' after 12h of computer-simulated water transport to a root axis witha low root length density millet geometry and millet water
potential values. Maximum and minimum velocities were1.10 x 10-8 and 6.33 x 10-12 m s-1.

content contours indicated that drier soil surrounded
the roots than for simulations run with the single-axis
millet model (Petrie, 1992). Velocity vector fields indicated that regions of slow water movement had developed between the root axes and between the root

PETRIE ET AL.: WATER TRANSPORT IN AN UNSATURATED MEDIUM

tips on each axis, most of the root length was encasedin a region in which water movement was slow, andwater uptake occurred mainly at the outer edge of theroot region representing the root tips (Fig. 9). Results
were similar when the model was run using the cowpea *R values (Petrie, 1992).
By the eighth hour of computer simulated watertransport, millet f was ~0.3 MPa lower than cowpea
in the initial simulations. Mean cumulative water uptake was about the same, however, for the cowpea
root geometry (21 root nodes), millet root geometry(41 root nodes), and low-density millet root geometry(21 root nodes), and water uptake was only «1% greaterfor the double-axis millet root geometry (82 root nodes).
When the same ^R values based on measurements ofmillet were used for the cowpea and millet geometries, water uptake throughout the simulation was =1.5
to 2% greater for cowpea than for all of the millet
geometries, even though cowpea had one-half the root
length density of millet and one-quarter that of the
double-axis millet. A substantial increase in root lengthdensity, therefore, does not necessarily increase water
uptake under conditions of soil dehydration.
Effect of Varying Hydraulic Conductivity
An additional set of simulations was conducted inwhich the initial hydraulic conductivity of the rooting
medium was increased from Ks = 1.67 x 10 ~ 6 m
s-1 to Ks = 3.33 x 10~6 m s-1. These simulations
were conducted for cowpea, millet, cowpea root geometry with millet ^R values, and millet root geometry with cowpea ^R values. Results were similar tosimulations using the lower hydraulic conductivity,
but overall water depletion was greater when hydraulic conductivity was higher (Petrie, 1992).
Discussion
According to the two-dimensional simulations, themillet root system was much less efficient than the
cowpea root system due to the development of dry
zones surrounding the roots that permitted little water
uptake except at the root tips of millet. It is necessary
to consider, however, whether the same phenomenon
would occur in the real world, which is three-dimensional. There are technical constraints to conducting
simulations in three dimensions. The computer timerequired to run a three-dimensional model would besubstantially greater than for a two-dimensional model.
The increase in simulated time from 12 h to 12 d,
together with the increased number of equations solved
at each iteration (>265 000 equations in the threedimensional model compared with 1681 in the twodimensional quarter grid) would require a supercomputer. Consequently we can only speculate about theeffects of an additional dimension. In the two-dimensional simulations for cowpea, the 90° angle formedby the joining of branch roots to the axis created asmall region of slower water movement, and water
uptake at the root surfaces near that angle became lessefficient as soil dried. Many such angles would bepresent in a three-dimensional model of a millet rootaxis and would cause many zones of less efficientwater uptake. In contrast, a three-dimensional cowpea

1693

Distance from first root axis node (cm)

p
b

D
3
O
CD

O

3

3
O
CL

^

p
b

——————————^,

pbi

Distance from second root axis node (cm)Fig. 9. Water velocity vectors >8.33 x 10-10 m s~' after 12h of computer-simulated water transport in a profile with
two root axes with millet geometry and millet water potential
values. Maximum and minimum velocities were 2.67 x 10"'and 5.00 x 10-15 m s-1.

model would have fewer branch angles than millet,
and a greater volume of stored soil moisture would beavailable to the root surfaces near those angles.
The root geometry and the high root length densities
of millet are not unique. Hamblin and Tennant (1987)
and Richards (1991) have also reported higher root
length densities in cereals than in legumes grown under the same conditions. The reasons for the high root
length densities and clumping in cereal roots are unclear. They may be evolutionary inefficiencies thathave not been bred out of modern crops (Richards,
1991), ensurance against damage to part of the rootsystem (Hamblin and Tennant, 1987), or a means for
exploiting light rainfalls before water is lost from thesoil surface through evaporation (Taylor, 1983). Highrooting densities at the surface may also allow plants
to exploit immobile nutrients such as P, or mobile
nutrients such as N before they are lost through leaching.
The modeling approach used in these studies was
modular; each root axis and its local branching patternrepresented a pattern similar to that in other portionsof the root system. The models predicted that the root
geometry of millet became progressively less efficientthan cowpea as soil dried, even when input parameters
used in simulations were altered. In simulations thatwere conducted using the lower values of ^R derived
from experimental data on millet, the amount of watertaken up by both cowpea and millet was only slightly
greater than in simulations conducted using WR values
for cowpea. For the cowpea root geometry, water
moved with similar velocities across virtually all of
the root surface. For the millet root geometry, water
movement became very slow in most of the root region except for the root tips. At higher local root length1694

SOIL SCI. SOC. AM. J., VOL. 56, NOVEMBER-DECEMBER 1992

densities, an even greater proportion of the millet root
surface was encased in dry soil in which water movement was slow. Water uptake was not necessarily
greater with higher root length densities, even when
the investment in root structure was two to four times
as great.
ACKNOWLEDGMENTS
This research was partially supported by the Bean/Cowpea Collaborative Research Support Program, U.S. Agency
for International Development (USAID) Grant no. DAN1310-G-SS-6008-00. The opinions and recommendations
are those of the authors and not necessarily those of USAID.
Additional support was provided by the University of California, Riverside, through fellowships and research assistantships.