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Soil Physics

Measuring the Unsaturated Hydraulic Conductivity of Growing Media with a Tension Disc

^a Département des Sols et Génie agroalimentaire, Centre de Recherche en Horticulture, Université Laval, Sainte-Foy, Québec Canada, G1K 7P4
^b Land Resource Science Department, University of Guelph, Guelph, Ontario, Canada N1G 2W1

^* Corresponding author (Jean.caron{at}sga.ulaval.ca)

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 
In greenhouse and nursery production, there is an increasing interest in water conservation and environmental quality. The use of closed and semi-closed subirrigation systems to grow plants potted in organic growing media is an important step in this direction. However, the design of efficient subirrigation systems requires a detailed characterization of the unsaturated hydraulic conductivity of the organic substrates on rewetting. In many cases, organic substrates have extremely high-saturated hydraulic conductivities and also exhibit a dual porosity. Given the lack of a suitable technique for measuring the unsaturated hydraulic conductivity of these substrates near saturation, as well as at saturation in pots (the equivalent of "field conditions" in nursery and greenhouse production), a new in situ procedure has been developed. It is based on an analytical solution to steady state upward flow and assumes a single (SEA) or piecewise exponential (PEA) relationship between the unsaturated hydraulic conductivity and the soil water potential. The proposed method was tested for a sand and an organic growing medium. The results indicate that the unsaturated hydraulic conductivity curve may be obtained from water flux measurements using a specifically designed tension disc placed on top of the substrate, and that the estimates on rewetting are much more accurate, particularly at water contents close to saturation, than those obtained using the instantaneous profile method. Moreover, the proposed procedure is easy to carry out, requires only an inexpensive tension disc and is based on a sound physical representation of the rewetting process. Results indicated that the PEA is appropriate for most substrates and that the SEA can be considered as a special case of the PEA if only one exponential "piece" is required for the entire range (e.g., the sand).

Abbreviations: MWD, mean weight diameter • PEA, piecewise exponential approach • SEA, single exponential approach

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 
THERE IS AN INCREASING interest in closed and semi-closed subirrigation systems for conserving water and environmental quality in greenhouse and nursery production (Lea-Cox and Ross, 2002; Uva et al., 1998). In these systems, there is also a need to examine alternative substrates to recycle industrial and agricultural by-products and to make these production systems more sustainable (Chong, 1999; Wilson et al., 2003). In both cases, the design of efficient subirrigation systems requires an adequate characterization of the unsaturated hydraulic conductivity on rewetting to model capillary rise and water uptake. Caron et al. (2005) derived a capillary rise model to establish guideline values for the proper operation of capillary mats and of sand beds, for specific use in nursery and greenhouse systems. The approach is mainly based on a detailed knowledge of the hydraulic conductivity function on rewetting, to predict the head profile in a pot during evapotranspiration.

To describe such a function, it is mathematically advantageous to choose the exponential model of Gardner (1958) to describe the relationship between the hydraulic conductivity, K, and the soil water pressure head, . The unsaturated hydraulic conductivity, K(), is linked to the water potential, , where has dimensions of pressure head (commonly expressed in units of centimeter water), through the following relationship:

[1]

where K(_w) (L T^–1), is the saturated hydraulic conductivity predicted by Eq. [1] at the water entry value, _w, is a constant called either the parameter or the sorptive number (also the inverse of the characteristic length, L^–1), and is the soil water pressure head (L).

However, the experimental characterization of these substrates is not a simple task. First, growing media used in nursery and greenhouse productions are extremely hysteretic in the relationship between the unsaturated hydraulic conductivity and the water potential (Otten, 1994); that is, the relationship is different on drying from saturation compared with wetting up from a dry condition. Second, the relationship is sometimes piecewise exponential, with increasing extremely rapidly close to saturation due to the presence of macropores (Caron et al., 2005). Third, such substrates have a very sensitive and deformable structure, which is easily affected by time and handling, for example, the sieving, potting, and watering procedures (Heiskanen et al., 1996; Paquet et al., 1993). Fourth, such substrates may possess important hydrophobic properties at low potentials (Michel, 1998).

Because of this sensitive structure, growing media should be characterized in situ (Caron and Rivière, 2003), in conditions close to nursery or greenhouse use. Transient state methods, such as the one step and multistep outflow procedures, the evaporative method (Brandyk et al., 2003), and the internal drainage method of Hillel et al. (1972) have been used successfully in many situations. However, such methods face limitations when close to saturation, as our experience indicates, these methods give inaccurate results for the hydraulic properties on rewetting. Moreover, on rewetting from underneath, K() estimates at potentials above container capacity (i.e., the water content at quasi-equilibrium after saturation and drainage of the pot, where the average potential expressed in pressure head terms roughly corresponds to half the container height) are impossible to obtain with the above methods, as the potentials within the pot reach hydrostatic equilibrium when approaching container capacity. Alternatively, one might think that steady-state methods using horizontal infiltration could perhaps be used. However, the sensitivity of growing media to handling and the need to characterize the substrate at the pot scale severely limits the use of these steady-state methods, particularly with the pot set horizontally for unsaturated hydraulic conductivity measurements. Indeed, because of particular properties of these organic substrates, large potential differences can exist between the top and the bottom surfaces of 5-, 9-, and 13-L pots. This can result in calculating an average unsaturated hydraulic conductivity based on a linearized Darcy's law over a too large a range of pressure heads. The same limitation also applies to vertically set pots where an average hydraulic conductivity is not accurate enough. This is inadequate for the precise modeling of capillary rise and water uptake where conditions close to saturation exist (Caron et al., 2005).

A new steady-state flow method is therefore proposed for a potted medium in a vertical pot, subjected to rewetting from underneath, and in which small pressure head intervals close to saturation can be established using a tension disc. The objective of this paper is to present this new procedure for measuring the hydraulic conductivity in potted substrates and to compare the results obtained using this procedure with results obtained using a conventional technique; namely, a transient flow measurement procedure.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 
Preparation of Substrates
Two different substrates were prepared. The first consisted of silica sand (sand), the other was made with a brown sphagnum peat of H4-H5 of degree of decomposition on the Von Post scale (Von Post and Granlund, 1926) and composted fir bark and sand, mixed in a 3:6:1 volume ratio (Sp30B60). Particle-size distribution was measured on three replicates of a 500-mL sample of air-dried material (which were first sieved for 1 min though a 40-mm sieve), sieved again manually for 1 min on a nest of sieves (0.25, 1, 2, 4, 8, 16, and 25 mm and a pan). The average particle size of the different materials was expressed using the mean weight diameter (MWD) calculated from (Kemper and Rosenau, 1986):

[2]

where x_i is the weight retained on the sieve divided by the total medium mass, f_i is the average particle size, and n is the number of size classes, here equal to 8. Substrates were potted into 4-L pots (15 cm in diameter, 18.5 cm high), wetted from underneath, drained and resaturated and drained two more times to achieve a more stable volume and conform more closely to the assumption of rigidity. Three replicates were used for the sand and the Sp30B60 treatment but one set of data for the sand was dropped, because of a malfunctioning of the data acquisition system.

Transient Flow Measurement
This method is a variation of the well-known instantaneous profile method (Hillel et al., 1972; Klute and Dirksen, 1986). Tensiometers, made of a 38 mm long by 8 mm OD porous ceramic cups glued to a PVC tubing (40 mm long by 8 mm OD) instrumented with differential pressure transducers (PX-26, Omega Stamford, CT) were installed horizontally at two depths (2.5 and 13 cm from the bottom of the pot) within the substrates (Fig. 1). Next, 12-cm long CS-615 probes (Campbell Scientific, Logan, UT) were inserted vertically and softly pushed into the substrate to approximately fit between the tensiometers (within l cm accuracy). They were used to obtain oscillation period measurements, later converted to volumetric water content (), using a calibration curve derived independently for each substrate following the procedure outlined by Caron et al. (2002). Once instrumented, the pots were placed onto a tension table (Topp and Zebchuk, 1979) at zero water potential and allowed to drain to a water potential of –60 cm (Fig. 1) and covered at the top with a polyethylene sheet to prevent surface evaporation. To run these measurements, the water potential at the bottom of the pot (the top of the tension table) was gradually decreased from 0 to –60 cm at a rate of 15 cm d^–1. Rewetting was then performed by increasing the water potential at the same rate, to bring the substrate back to container capacity. During the draining and wetting cycles, measurements were taken every 15 min using a CR-10 datalogger (Campbell Scientific, Logan, UT). At the conclusion of this experiment the CS-615 probes and tensiometers were removed and the holes left were filled with the same substrate. The sides of the containers were then sealed with tape. The flux density [J_w(0, t_avg)] was calculated from

[3]

where z is the vertical coordinate, t = t₂ – t₁ is the time interval where t₁ is Time 1 and t₂ is Time 2, and L_s represents the position of the top of the substrate. Also, = _t1 – _t2, represents the change in water content, where is the average water content over the length of the CS615 probe, measured at t₁ and t₂. Changes in water content were assumed to extend throughout the whole substrate height (15 cm). The flux density applies for t_avg = . Then, K() was estimated from

[4]

where H is the average hydraulic head, obtained from two consecutive measurements in time of matric potential at the two heights, that is H = and L_t represents the distance between the center of the porous cups of the two tensiometers. The average matric potential (_avg) is the average of the measured water potential at the two heights, that is _avg = where H1 = 1 + z1 is the hydraulic head at Tensiometer 1, 1 is the pressure head at Tensiometer 1, H2 = 2 + z2 is the hydraulic head at Tensiometer 2, 2 is the pressure head at Tensiometer 2, and t₁ and t₂ as defined above.

View larger version (17K): [in this window] [in a new window]   Fig. 1. Schematic view of the experimental set up used to measure the unsaturated hydraulic conductivity under transient flow with a CS-615 and tensiometers.

1. The substrate in the pot is assumed to be of a known height, L_s = L₁ + L₂ (L), with the origin set at the bottom of the substrate and with z (L), the vertical space coordinate, defined as positive upward.
   
2. The substrate material is assumed to be rigid, homogeneous and isotropic. This assumption may be questionable given the sensitivity of these substrates, but is necessary to solve the differential equation and is reasonable for short-time measurement intervals within a limited range of matric heads.
   
3. The total volume flux density of water, J_w (L T^–1), is constant with time; that is, steady-state flow conditions apply.
   
4. The water pressure head, , in the substrate is zero at a position determined by both the outside water level and the Darcy drop in potential due to the upward flow of water in the saturated substrate.
   
5. The well-known approach of Gardner (1958) was used to obtain the steady-state relationship between soil water pressure head, (negative in unsaturated soil), and height z.
   

View larger version (36K): [in this window] [in a new window]   Fig. 2. Schematic view of the experimental apparatus used to measure the unsaturated hydraulic conductivity under steady-state flow with a tension disc. The glass bead height is not to scale, and is shown thicker for representation purposes only. A Mariotte bottle (not shown) is used to maintain a constant water level within the box.

[5]

Gardner (1958) showed that the Darcy steady-state flow equation (J_w = constant) can be linearized by use of Eq. [1] and the Kirchoff transform :

[6]

giving

[7]

The following boundary conditions (see Fig. 2) are applicable to 1-D steady-state flow in the unsaturated zone only of the pot:

[8]

[9]

where A is the difference in height between the substrate level and that of the outflow of the tension disc device. It corresponds approximately to the value of the suction applied at the surface by the tension disc, when neglecting membranes resistances. We also make the assumption that the thickness of the tension disc system (disc plus glass beads) is negligible. The value of L corresponds to

[10]

It is a correction factor and L is the increase in depth of the unsaturated zone as a result of the drop in head during upward water flow from the bottom of the pot. The variable K_ss is the saturated hydraulic conductivity of the substrate and L₁ is the distance measured from the bottom of the substrate in the pot to the water level outside the pot in the acrylic box.

With the above boundary conditions, the solution of Eq. [7] follows a similar pattern to the solution of Elrick et al. (1994) but differs in that z is now defined as positive upward and that the boundary conditions are different.

Separating the variables in Eq. [7], and applying the conditions established by Eq. [8] and [9] gives:

[11]

Carrying out the integration and rearranging gives:

[12]

where J_w on the right-hand side of Eq. [12] is a part of the correction factor given by Eq. [10]. The contribution of the correction factor is generally small, but not negligible. Note that if we set K(_w) = K_ss (and K_ss is known), then Eq. [12] contains only the one unknown parameter, , since J_w, L₁, L₂, and A are measured. Solving Eq. [12] for and knowing K_ss would then allow K() to be calculated from Eq. [1]. We define this procedure as the single exponential approach (SEA).

A second and more general approach assumes that Eq. [12] contains the two unknowns, and K(_w) as defined in Eq. [1]. If, however, the value of a particular substrate is not constant over the range from = –A to = _w, there may be some concern about the use of Eq. [12]. This important issue is discussed in the Appendix below where a theoretical analysis is performed to show that the following procedure is correct for analyzing the experimental data. It extends considerably the applicability of the approach as it can be used with soils and substrates having variable values over a wide range. Hence, the SEA is a special case of this second approach.

Similar to the approach of Reynolds and Elrick (1991) for analyzing data from the tension disc infiltrometer, we assume that K() is piecewise exponential between adjacent measurements i and j realized at two subsequent pressure heads; that is, _i = –A_i and _j = –A_j. The measurement of J_wi can be obtained using _i = –A_i and J_wj can be obtained using _j = –A_j. However, taking the ln of Eq. [12], ln (J_w), and removing J_w on the right-hand side of Eq. [12] by setting the correction term L (given by Eq. [10]) equal to zero does not give a linear plot with = –A, whereas the expression used by Reynolds and Elrick (1991) does lead to a linear relationship between ln(flux) and . Therefore, the proposed approach had to be modified from Reynolds and Elrick (1991) in that the numerical FIND procedure in Mathcad (Mathsoft Inc., Cambridge, MA) was used (other numerical packages can also be used) to solve the following two equations in the two unknowns, and K(_w), where K(_w) is the piecewise intercept at = _w:

[13]

[14]

Note that dividing Eq. [13] by [14] eliminates K(_w) giving the following equation in the single unknown :

[15]

The parameter is a function only of the ratio J_wi/J_wj (see Appendix below) and it can be determined using the numerical root function in Mathcad. Once is known, K(_w) can then be calculated by substituting the known value of in either Eq. [13] or [14] and solving for K(_w). This procedure provides an alternative numerical method for calculating and K(_w) with this second approach.

The second approach, using Eq. [13] and [14], or Eq. [15] is called the piecewise exponential approach (PEA).

The saturated hydraulic conductivity of the apparatus (tension disc and substrate) is then obtained by applying Darcy's law for saturated flow to the last measurement where the water level is established above the surface of the substrate; that is, L₁ > L_s.

Because of the very high hydraulic conductivity of the substrates, the saturated hydraulic conductivity of the disc system (disc including membrane plus glass beads) may have to be taken into account. The disc system plus substrate can be considered as a layered system where the saturated hydraulic conductivity of the substrate, K_ss, is given by:

[16]

where L_s is the thickness of the substrate, L_R is the thickness of the disc system (the thickness of the glass beads plus membrane is assumed to be negligible), K_T is the saturated hydraulic conductivity of the combined system, K_sR is the saturated hydraulic conductivity of the disc system, (H)_T is the measured head drop across the substrate plus the disc system (the combined system), J_T is the measured flux through the combined system and R_R is the independently measured resistance to flow through the disc system only.

R_R can be obtained from flow experiments conducted only on the disc system:

[17]

where (H)_R is the measured head drop and J_R is the measured flux, both through the disc system only (disc including membrane plus glass beads).

The disc resistance will also influence the calculation of the unsaturated hydraulic conductivity, particularly at values close to saturation where A is small in value and the resistance of the disc system and that of the substrate become more equal. To take the disc resistance into account in these calculations, the boundary conditions at z = L_s (Eq. [8]) need to be adjusted for the pressure drop across the disc system, J_TR_R. Equation [8] then needs to be replaced by:

[18]

Where the corrected pressure head (–A_c) is given by:

[19]

Equation [13] and [14] are subsequently replaced by:

[20]

[21]

Equation [15] can then be replaced by:

[22]

The other correction factor, J_TR_R, was found to be important only at pressure heads close to saturation. The K()– relationship can be derived by determining the average matric head potential _avg and the corresponding K(_avg), as defined in the appendix, where this overall procedure is validated. The analysis in the appendix shows that using the PEA procedure, the calculated and K(_w) are not correct individually. This occurs because the calculated value for the two-line exponential model (see Appendix) where ₁ > ₂, reflects the increased flow (the calculated is larger than the input in Table A1 and the calculated [extrapolated] intercept, K(0), is smaller than the K_ss in Table A1). However, when the calculated and K(0) values are inserted into Eq. [1], the calculated K(_avg) returns the input value within the approximation error.

View this table: [in this window] [in a new window]   Table A1. Water flux values (Jw) generated theoretically. The substrate properties are given by Kss = 0.146 cm s–1, 1 = 0.85 cm–1, 2 = 0.23 cm–1, and b = –7.5 cm. The substrate in the pot was assumed to have a depth of Ls = 16 cm.

View larger version (17K): [in this window] [in a new window]   Fig. 3. The simple tension disc. The bottom surface is covered with nylon cloth (15-µm pore size), which is glued along the disc periphery. The bottom surface is made of acrylic through which large holes are made.

Suction was applied to the water-filled tension disc by lowering the bottom end of the plastic tubing shown in Fig. 3 to a predetermined distance A below the level of the membrane (Fig. 2). An approximately 2 cm difference in height was then applied between the water exit level from the plastic tubing and the water level in the box (A – L₂ = 2 cm), ensuring that the unsaturated hydraulic conductivity was measured using a small hydraulic gradient during upward flow through the substrate. Steady-state flow was established very quickly in these substrates, and steady-state flow was assumed when three measurements gave approximately the same flow rate. The water level in the box, L₁ was then raised by approximately 2 cm, the level of A similarly raised so that A – L₂ = 2 cm and the flow measurements repeated. This procedure was repeated until L₂ was equal to –2 cm (2 cm above the substrate [or membrane] surface), the container side still preventing water from flowing directly into the pot, to measure K_ss. The series of measurements using the tension disc can easily be performed within a 2- to 4-h period if the substrate has already been equilibrated at container capacity after rewetting from underneath. If not, the substrate should be rewetted for a longer period from underneath to obtain hydrostatic equilibrium at container capacity.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 
Particle-Size Distribution
Dry sieving indicated that particles sizes differed considerably between the two substrates, as substantial differences were found in many classes between substrates (Table 1). The MWD also differed significantly. The sand particle size was very uniform and dominated by particles ranging between 0.25 and 1 mm, as the MWD of particles was around 0.71 mm on average for the three replicates. Meanwhile, the Sp30b60 had a higher proportion of coarser material (larger than 4 mm), resulting in a MWD of 5.13 mm.

View larger version (26K): [in this window] [in a new window]   Fig. 4. Unsaturated hydraulic conductivity curves for the two substrates. The curve parameters are in Table 2.

View this table: [in this window] [in a new window]   Table 2. Hydraulic conductivity data measured on the substrate Sp30b60 and the sand using tension disc measurements and the piecewise exponential approach. The water entry value was set equal to zero (w = 0). RR = 20.59 s for Sp30b60 and RR = 122 s for the sand.

Tension Disc Measurements
Examples of the data sets obtained (using corrections for resistance where necessary) of the two media using the Laval tension disc approach are given in Table 2 for one single replicate of each medium. The water entry value was assumed to be zero for both the organic substrate and the sand. The resulting K() data for all replications obtained from the tension disc procedure are reported in Fig. 4 to show the repeatability of the method, relative to the transient state measurements.

The K() values increased monotonically from low to high potentials, a trend theoretically expected but not always observed with transient data. Also, the K() estimates obtained with the tension disc were consistent with those obtained from the transient state method (Fig. 4). Note that the K() data for the Sp30b60 substrate obtained under wetting conditions using the tension disc lined up well with the K() data obtained under wetting conditions using the transient analysis. The K() data for the sand obtained under wetting conditions using the tension disc also lined up well with the K() data using the transient analysis for draining conditions.

Data for both methods were compared using t tests in the potential range where a full set or data was available for all replicates and methods. This restricted the range to –17 to –7 cm for the Sp30b60 treatment and to –14 to –10 cm for the sand. For the sand, data at potentials higher than –10 cm were not used as they were all linked to one single rep and hence, would have biased the estimation. The analysis indicated no significant differences between methods for both the Sp30b60 (p = 0.97) and the sand (p = 0.26).

The analysis also indicated significantly higher K() values with the tension disc relative to the transient flow method on rewetting (p 0.0001) for the Sp30b60 treatment. This was somewhat expected as the rewetting curve for the transient measurements was obtained from rewetting from –60 cm, whereas estimates on rewetting from the tension disc started from –15 cm, after a period of equilibrium. The time of equilibrium therefore differs between both methods and a less uniform rewetting may have been reached with the transient method relative to the steady state of the tension disc method. Moreover, some rewetting occurred from above when the tension disc was put in contact with the top surface of the substrate. This may favor the reactivation of some pores, emptied during drainage, and difficult to rewet from underneath, because of bottleneck effects (Hillel, 1980).

Measurements of K() close to saturation have always been difficult to obtain. As mentioned previously, data from the transient flow procedure in a substrate 15 cm high are valid only for –7.5 cm (approximately). The tension disc procedure can, however, provide data in the region closer to saturation ( –2 cm [approximate]), but the error in the estimation of K near saturation becomes larger because of errors in measurement of the smaller numbers applicable to A and L₂, and because of problems in the estimation of L and R_R. Despite this, the tension disc procedure produced smooth estimates of K() and a general observation of the tension disc data in Fig. 4 indicates much less variability than with transient state estimates. Replicated experiments on both media also showed good agreement.

The two substrates, one primarily organic and the other a coarse sand, exhibited extremely different characteristics. The organic substrate Sp30b60 exhibited, as expected, a very sharp rise in hydraulic conductivity when approaching saturation, consistent with others (Wallach et al., 1992). The sand medium, in contrast, revealed a nearly single exponential increase when approaching saturation. Estimates for the organic substrate Sp30b60 were also consistent with others on drainage (Brandyk et al., 2003; Caron et al., 1998; Otten, 1994). An extremely sharp drop in hydraulic conductivity from saturation is the result of the very large initial value of 3.56 cm^–1. This reflects the large macropore effect on this organic medium.

Practical Implications
The results obtained have important implications for the design and characterization of growing media used with subirrigation devices (ebb and flow, capillary mats). First of all, the suggested analysis and methodology is fast, relative to other approaches. It requires about 1 d for the whole series of measurements (2–4 h of intensive measurements and usually an overnight sample preparation and equilibrium), relative to the 1 or 2 wk necessary with the transient state approach. Second, it provides in situ estimates. This is critical for growing media as their structure is highly sensitive to any disturbance. Pots from the field, with the plants rooted in them, could be taken to the lab and once the top part of the plant is cut and removed, measurements can be performed directly on the potted substrates. Third, they provide consistent estimates at potentials above container capacity on rewetting, estimates otherwise impossible with previous techniques. As these estimates are critical for subirrigation models (Caron et al., 2005), this methodology is of great importance for the design of subirrigation systems. A fourth important impact of these results is that only simple equipment is needed. A basic tension disc and an acrylic box are all that is necessary, while a transient state measurement requires at least two tensiometers and pressure transducer devices, and possibly CS-615 or time domain reflectometry probes.

Recommendations

1. Substrates should be brought from a dry state to container capacity by wetting them from underneath. Also, because drying of the very top surface may have occurred, a mist rewetting of the top surface is recommended. Indeed, organic growing media may sometimes be hydrophobic below –1000 cm of pressure head (Michel, 1998). If hydrophobic at their surface, the tension disc may not rewet the top surface. Therefore, a mist rewetting should be performed to change the water content of the surface if dry and equilibration to hydrostatic conditions performed by covering the pot surface with a plastic film a couple of hours or more if necessary. This step will necessarily delay the procedure but remains compulsory to get valid measurements and respect the basic assumptions needed for adequate K() calculations.
   
2. Glass beads, just fine enough to maintain the maximum pressure head imposed at the top surface should be applied in a layer just thick enough to allow good contact between the tension disc and the substrate. This contact material is necessary as fines in coarse media sometimes migrate from the top surface to below 1 cm, resulting in a shallow layer of very coarse material at the top of the substrate. If this occurs, the very coarse material should be removed and then the glass beads applied in a shallow layer (5 mm at most). Then, the tension disc should be placed on the surface of the substrate and the substrate left to equilibrate overnight.
   
3. Measurements should be run in at least five steps (water level at the bottom of the pot, ¹/₄, ¹/₂, ³/₄, and full height) minimum to identify the breakpoint pressure head _b (see Appendix), which in these substrates is generally above container capacity, as this point is critical in subirrigation models (Caron et al., 2005).
   
4. When carrying out the flow measurements, we suggest using hydraulic head differences ([A – L₂] in Fig. 2) of about 2 cm as the flow is extremely rapid in these growing media, particularly near saturation where L₂ is small. For good accuracy, use of the outflow device as illustrated in Fig. 2 (Reynolds, 1993) will give a better estimation of (A – L₂).
   
5. Results indicated that the PEA is appropriate for most substrates and that the SEA can be considered as a special case of the PEA if only one exponential "piece" is required for the entire range (e.g., the sand). Therefore, because of its more general application, the PEA approach should be preferred to the SEA, as long as the true shape of the K()– relationship remains unknown.
   
6. Finally, when the tension disc is used to determine K_ss, corrections for resistances induced by the tension disc device, the membrane and the glass beads (the tension disc system) should be performed.
   

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 
A fast (about 1 d), inexpensive and simple method is described to characterize the K() curve in potted substrates, which is the equivalent of "field conditions" in nursery and greenhouse production. It is based on a PEA for the K()– relationship, a model that may be particularly appropriate for organic substrates. It has provided estimates, which are consistent with transient state measurements, and, most importantly, has given hydraulic conductivity data in the –7.5- to 0-cm pressure head range on rewetting. The data also indicated that for measurements of K_ss and K() very close to saturation, it is important to take into account the resistance of the disc system.

	APPENDIX

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 
Solving the Flow Equation for the Two-line Exponential Hydraulic Function
An assumption inherent in the integration of the left-hand side of Eq. [11] is that Eq. [1], where is assumed to be constant, is applicable over the entire range from = –A to saturation at = _w. But because of the fibrous structure of the organic substrates and the regular inclusion of coarse fragments (bark, perlite, peat fragments, etc.), they possess a large number of macropores, which drain rapidly very close to saturation. This results in a rapid increase in the value of close to saturation, given that the exponential form of K() given by Eq. [1] is applicable. To determine the correct procedure to determine K() from experimental data, K() was assumed to have the following form:

[A1]

[A2]

where K_b is the hydraulic conductivity at the break-point pressure head, _b and _i is the initial pressure head in the substrate:

[A3]

This particular form of K() (the two-line exponential model [Jarvis and Messing, 1995]) was chosen because it has been shown to be appropriate for porous media having significant macropore structure. The advantage of using Eq. [A1] and [A2] is that it is possible to obtain an analytical solution of Eq. [5] (modified for the two-line exponential model) with the appropriate boundary conditions. A validation process can therefore be established to determine the correct procedure for obtaining K() from experimental data.

The appropriate analytical solution is developed for substrates having a two-line (two values) exponential model of K() given by Eq. [A1], [A2], and [A3]. The boundary conditions given by Eq. [8] and [9] are modified to take into account the two-line model. Note that ₁ (close to saturation) is chosen (see Table A1) to have a considerably larger value than ₂ to account for the presence of macropores.

For the two-line exponential model (Eq. [A1], [A2], and [A3]), Eq. [7] is replaced by:

[A4]

[A5]

where C₃ is a constant of integration.

The boundary conditions of Eq. [8] and [9] are subsequently replaced by:

[A6]

[A7a]

[A7b]

Following the pattern established earlier (see Eq. [12]), the solution is given by:

[A8]

[A9]

For steady-state flow:

[A10]

and equating Eq. [A8] and [A9] allows us to solve numerically for z_b, when the other parameters are known.

J_w1 and J_w2 numbers (J_w1 = J_w2) were generated for values of the hydraulic properties (₁, ₂, _b, and K_ss) and the pot parameters (A, L₁, and L₂) that previous experiments deemed appropriate (Table A1). Because the parameter J_w1 is present in the correction factor (Eq. [10]) on the right-hand side of Eq. [A8], we first solve Eq. [A8] and [A9] for J_w1 = J_w2 with the correction factor L = 0. We then substitute the calculated value of J_w1 = J_w2 obtained with L = 0 into Eq. [A8], solve for a new value of J_w1 = J_w2 and the iteration process is continued until J_w1 = J_w2 is stable.

Two different procedures for processing the data were developed (_w = 0 in both cases). The water entry value, _w, was included in the analysis to be comprehensive, but set at 0 because Caron et al.'s (2002) results suggest this value to be between –1 and 0 cm for a wide range of coarse media. In fine media where _w is not negligible, this parameter would have to be measured independently (Fallow and Elrick, 1996).

Single Exponential Approach. Set K(_w) = K(0) = K_ss in Eq. [20] so that is the only unknown. Then solve Eq. [20] for using a single value of J_w. Calculate K(–A) using Eq. [1]. This solution is appropriate for a linear ln K()– relationship.

Piecewise Exponential Approach. Solve Eq. [20] and [21] simultaneously (numerically) for both and K(_w) = K(0). Then determine K(_avg), where

[A11]

and

[A12]

This solution is appropriate for a piecewise linear relationship between lnK() and . Using the J_w numbers generated theoretically, the SEA and PEA procedures were applied and the K(–A) values calculated.

The results are presented in Table A1 and Fig. A1. The results show (Fig. A1) that PEA returns an excellent approximation of the input value of K(–A), for < _b. The SEA procedure, which is based on a single value, returns K(–A) values that are too low because the calculated values are too high. The PEA does not return the correct value of either or K(0), but their insertion into Eq. [1] does give values of K(–A) very close to the true or input values, which is the result that we seek. Both the SEA and PEA procedures return the true or input value of the K(–A) for > _b because only ₁ (constant ) is applicable in this range.

View larger version (17K): [in this window] [in a new window]   Fig. A1. Hydraulic conductivity values calculated with the single exponential approach and the piecewise exponential approach (PEA) procedures from the theoretically generated Jw values, themselves using the theoretical K()– values.

Checking the assumption of being dependent only on the flux ratio with the PEA

View this table: [in this window] [in a new window]   Table A2. Alpha () values calculated using the piecewise exponential approach (Eq. [20] and [21]). In this example, Ai = 13 cm, Aj = 11 cm, Kss = 0.241, KT = 0.089 and w = 0.

	ACKNOWLEDGMENTS

Received for publication March 16, 2004.

	REFERENCES

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX REFERENCES

 

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