# Annex 5. Capillary rise and data set for soil hydraulic functions

Capillary rise

To calculate the capillary recharge, it is possible to apply Darcy's Law, which can be written for the unsaturated zone as:

(1)

where:

q = soil water flux (positive upward) (cm/d);
K(q) = unsaturated hydraulic conductivity (cm/d);
h = soil pressure head (cm); and
z = vertical coordinate (positive upward) (cm).

Water balance considerations of an infinitely small soil volume result in the continuity equation for soil water:

(2)

Combination of Equations 1 and 2 results in the general equation for flow through the unsaturated soil:

(3)

To calculate the capillary rise, the unsaturated hydraulic conductivity, K(q), needs to be known. The hydraulic conductivity is a function of the moisture content and the moisture content is a function of the pressure head. The soil-water retention curve is the graph representing the relationship between pressure head and water content. Each soil has a unique soil-water retention curve. Wösten et al. (2001) published a series of physical soil characteristics for functional soil physical horizons (Tables A5.1, A5.2 and A5.3). These were based on measured values of soil-water retention and hydraulic conductivity. This information can be extrapolated with caution to other soil layers that have identical soil textures. In a similar manner, Mualem (1976) developed a catalogue of hydraulic properties of soils.

Table A5.1. Dutch nomenclature known as Staring Series based on texture, organic matter content, and sand fraction - topsoils

 Dutch nomenclature1 Clay-silt(< 50 mm)(%) Clay(< 2 mm)(%) Organic matter(%) M50 (mm) Number curves(-) Topsoils Sand B1 Fine to moderately fine sand 0-10 0-15 105-210 32 B2 Loamy sand 10-18 0-15 105-210 27 B3 Sandy loam 18-33 0-15 105-210 14 B4 Sandy clay loam 35-50 0-15 105-210 9 B5 Coarse sand 0-15 210-2 000 26 Silt B7 Silt 8-12 0-15 6 B8 Light silt loam 12-18 0-15 43 B9 Heavy silt loam 18-25 0-15 29 Clay B10 Silty clay loam 25-35 0-15 12 B11 Silty clay 35-50 0-15 13 B12 Clay 50-100 0-15 9 Loam B13 Loam 50-85 0-15 10 B14 Silt loam 85-100 0-15 67

1 Translation of the Dutch nomenclature by means of FAO textural classes based on the clay, silt and sand fractions (FAO, 1990a).

Source: Wösten et al., 2001.

Table A5.2. Dutch nomenclature known as Staring Series based on texture, organic matter content, and sand fraction - subsoils

 Dutch nomenclature1 Clay-Silt(< 50 mm)(%) Clay(< 2 mm)(%) Organic matter(%) M50(mm) Number curves(-) Subsoils Sand O1 Fine to moderately fine sand 0-10 0-3 105-210 109 O2 Loamy sand 10-18 0-3 105-210 14 O3 Sandy loam 18-33 0-3 105-210 23 O4 Sandy clay loam 33-50 0-3 105-210 9 O5 Coarse sand 0-3 210-2 000 17 Silt O8 Silt 8-12 0-3 14 O9 Light silt loam 12-18 0-3 30 O10 Heavy silt loam 18-25 0-3 25 Clay O11 Silty clay loam 25-35 0-3 11 O12 Silty clay 35-50 0-3 25 O13 Clay 50-100 0-3 19 Loam O14 Loam 50-85 0-3 9 O15 Silt loam 85-100 0-3 53

1 Translation of the Dutch nomenclature by means of FAO textural classes based on the clay, silt and sand fractions (FAO, 1990a).

Source: Wösten et al., 2001.

The unsaturated hydraulic conductivity can also be calculated based on a model developed by Van Genuchten (1980) in which Mualem's model is combined with an empirical S-shaped curve for the soil-water retention function to predict the unsaturated hydraulic conductivity curve. The Van Genuchten model contains six unknowns and can be written as follows:

(4)

where:

Ks = saturated hydraulic conductivity (cm/d);
h = pressure head (cm);
l = dimensionless shape parameter depending on dK/dh (-);
a = shape parameter (cm-1);
n = dimensionless shape parameter (-); and
m = 1-1/n (-).

Wösten et al. (2001) have also published parameters for the different soil types (Table A5.4).

Although water movement in the unsaturated zone is in reality unsteady, calculations can be simplified by assuming steady-state flow during a certain period of time. The steady-state flow equation can be written as:

(5)

Equation 5 can be transformed to:

(6)

Where is the hydraulic conductivity for which is (h1 + h2)/2. With this equation, the soil pressure head profiles for stationary capillary rise fluxes can be calculated. From these pressure head profiles, the contribution of the capillary rise under shallow groundwater table management can be estimated. The following section provides an example to show the calculation procedures to derive at the soil pressure head profiles for a given stationary capillary flux.

Example to calculate the pressure head profiles for a silty soil for stationary capillary rise fluxes

Given is a uniform silty soil with a clay percentage of 10 percent and a low organic matter content (< 3 percent). As in semi-arid climates, the organic matter content is in general low Table A5.2 for subsoils will be used to establish the nomenclature. Table A5.2 shows that this soil can be classified as O8. Six capillary fluxes will be calculated: q = 10 mm/d; q = 7.5 mm/d; q = 5 mm/d; q = 2.5 mm/d; q = 1 mm/d; and q = 0 mm/d.

Table A5.3. Unsaturated hydraulic conductivity K(q) (cm/d) and q (cm3/cm3) in relation to the soil pressure head úhú (cm) and pF for topsoils and subsoils

 |h|(cm) 1 10 20 31 50 100 250 500 1000 2500 5000 10000 16000 pF 0.0 1.0 1.3 1.5 1.7 2.0 2.4 2.7 3.0 3.4 3.7 4.0 4.2 Topsoils B1 K(q) 23.41 11.38 6.04 3.13 1.14 1.6E-1 7.5E-3 6.5E-4 5.4E-4 2.0E-6 1.6E-7 1.4E-8 2.6E-9 q 0.430 0.417 0.391 0.356 0.302 0.210 0.118 0.077 0.053 0.036 0.029 0.025 0.024 B2 K(q) 12.52 3.18 1.57 0.85 0.38 9.2E-2 1.1E-2 2.1E-3 3.8E-4 4.0E-5 7.3E-6 1.3E-6 4.2E-7 q 0.420 0.402 0.377 0.350 0.311 0.248 0.172 0.130 0.098 0.070 0.056 0.045 0.040 B3 K(q) 15.42 6.56 4.05 2.58 1.33 3.5E-1 3.6E-2 5.2E-3 6.9E-4 4.7E-5 6.1E-6 7.9E-7 2.0E-7 q 0.460 0.452 0.439 0.423 0.393 0.329 0.232 0.171 0.125 0.085 0.065 0.051 0.044 B4 K(q) 29.22 8.49 4.86 2.97 1.48 4.0E-1 4.4E-2 7.0E-3 1.0E-3 8.1E-5 1.2E-5 1.7E-6 4.4E-7 q 0.460 0.451 0.438 0.423 0.397 0.345 0.263 0.208 0.163 0.119 0.095 0.077 0.067 B5 K(q) 52.91 17.54 5.97 2.14 0.54 5.6E-2 2.3E-3 2.0E-4 1.8E-5 6.9E-7 6.0E-8 5.2E-9 1.0E-9 q 0.360 0.329 0.272 0.219 0.159 0.094 0.046 0.029 0.020 0.014 0.012 0.011 0.011 B7 K(q) 14.07 1.78 0.93 0.55 0.28 8.3E-2 1.3E-2 2.9E-3 6.0E-4 7.4E-5 1.5E-5 3.1E-6 1.0E-6 q 0.400 0.390 0.379 0.367 0.350 0.315 0.263 0.224 0.190 0.151 0.127 0.107 0.095 B8 K(q) 2.36 0.59 0.38 0.27 0.17 7.0E-2 1.7E-2 4.9E-3 1.4E-3 2.4E-4 6.3E-5 1.7E-5 6.7E-6 q 0.430 0.425 0.419 0.412 0.399 0.370 0.314 0.268 0.225 0.176 0.146 0.122 0.108 B9 K(q) 1.54 0.55 0.39 0.29 0.20 9.5E-2 2.6E-2 8.1E-3 2.3E-3 4.0E-4 1.0E-4 2.7E-5 1.1E-5 q 0.430 0.427 0.423 0.418 0.409 0.385 0.331 0.280 0.229 0.173 0.138 0.111 0.095 B10 K(q) 0.70 0.14 0.10 0.07 0.05 2.4E-2 7.8E-3 2.9E-3 1.0E-3 2.4E-4 8.1E-5 2.7E-5 1.3E-5 q 0.430 0.427 0.424 0.420 0.414 0.398 0.362 0.327 0.289 0.243 0.212 0.185 0.169 B11 K(q) 4.53 0.15 0.08 0.05 0.03 1.2E-2 3.3E-3 1.2E-3 4.0E-4 9.5E-5 3.2E-5 1.1E-5 5.2E-6 q 0.590 0.581 0.573 0.565 0.553 0.529 0.490 0.459 0.428 0.389 0.362 0.336 0.320 B12 K(q) 5.37 0.12 0.06 0.04 0.02 7.7E-3 1.9E-3 6.3E-4 2.0E-4 4.5E-5 1.4E-5 4.6E-6 2.1E-6 q 0.540 0.531 0.523 0.516 0.505 0.485 0.453 0.427 0.402 0.370 0.348 0.327 0.313 B13 K(q) 12.98 5.86 4.13 3.05 1.98 8.4E-1 1.8E-1 4.5E-2 1.0E-2 1.4E-3 3.0E-4 6.4E-5 2.3E-5 q 0.420 0.417 0.411 0.403 0.390 0.354 0.280 0.220 0.168 0.117 0.089 0.068 0.057 B14 K(q) 0.80 0.29 0.21 0.16 0.11 5.0E-2 1.2E-2 2.7E-3 5.4E-4 5.4E-5 9.2E-6 1.5E-6 4.5E-7 q 0.420 0.418 0.415 0.412 0.405 0.388 0.345 0.300 0.253 0.197 0.162 0.133 0.117 Subsoils O1 K(q) 15.22 11.17 6.88 3.64 1.15 9.6E-2 1.8E-3 7.6E-5 3.2E-6 5.0E-8 2.2E-9 1.2E-9 1.0E-9 q 0.360 0.354 0.332 0.296 0.229 0.124 0.048 0.026 0.016 0.012 0.011 0.010 0.009 O2 K(q) 12.68 7.60 4.38 2.35 0.84 1.0E-1 3.2E-3 2.0E-4 1.2E-5 2.9E-7 1.7E-8 1.1E-9 1.0E-9 q 0.380 0.372 0.351 0.321 0.269 0.179 0.092 0.058 0.040 0.028 0.024 0.022 0.021 O3 K(q) 10.87 5.71 3.48 2.10 0.96 1.9E-1 1.2E-2 1.2E-3 1.1E-4 4.8E-6 4.5E-7 4.2E-8 8.3E-9 q 0.340 0.334 0.321 0.303 0.271 0.206 0.123 0.80 0.053 0.032 0.024 0.018 0.016 O4 K(q) 9.86 3.93 2.34 1.44 0.71 1.7E-1 1.6E-2 2.1E-3 2.7E-4 1.7E-5 2.0E-6 2.4E-7 5.8E-8 q 0.350 0.343 0.332 0.318 0.295 0.244 0.170 0.124 0.090 0.060 0.045 0.034 0.029 O5 K(q) 25.00 10.08 2.43 0.55 0.08 3.2E-3 4.3E-5 1.6E-6 6.0E-8 8.3E-9 4.1E-9 1.2E-9 1.1E-9 q 0.320 0.287 0.212 0.147 0.089 0.042 0.019 0.014 0.011 0.010 0.009 0.008 0.007 O8 K(q) 9.08 2.33 1.39 0.90 0.49 1.6E-1 2.6E-2 5.4E-3 1.1E-3 1.2E-4 2.3E-5 4.3E-6 1.4E-6 q 0.470 0.462 0.451 0.438 0.417 0.372 0.269 0.239 0.191 0.140 0.111 0.088 0.075 O9 K(q) 2.23 0.86 0.58 0.41 0.26 1.0E-1 2.1E-2 5.1E-3 1.1E-3 1.5E-4 3.2E-5 6.7E-6 2.3E-6 q 0.460 0.455 0.448 0.439 0.422 0.382 0.303 0.240 0.185 0.130 0.098 0.075 0.062 O10 K(q) 2.12 0.45 0.29 0.20 0.12 5.0E-2 1.2E-2 3.2E-3 8.4E-4 1.3E-4 3.3E-5 8.1E-6 3.1E-6 q 0.480 0.475 0.469 0.461 0.449 0.419 0.363 0.315 0.269 0.216 0.183 0.155 0.138 O11 K(q) 13.79 0.79 0.41 0.25 0.13 4.2E-2 7.9E-3 2.0E-3 4.8E-4 7.3E-5 1.7E-5 4.0E-6 1.5E-6 q 0.420 0.412 0.404 0.397 0.385 0.362 0.325 0.295 0.267 0.233 0.210 0.189 0.176 O12 K(q) 1.02 0.11 0.07 0.05 0.03 1.3E-2 3.8E-3 1.4E-3 4.6E-4 1.0E-4 3.4E-5 1.1E-5 5.1E-6 q 0.560 0.555 0.550 0.544 0.534 0.512 0.470 0.431 0.392 0.342 0.308 0.278 0.259 O13 K(q) 4.37 0.10 0.05 0.03 0.02 7.6E-3 2.0E-3 6.6E-4 2.2E-4 4.8E-5 1.5E-5 5.0E-6 2.3E-6 q 0.570 0.563 0.556 0.550 0.540 0.521 0.490 0.464 0.439 0.406 0.382 0.360 0.346 O14 K(q) 1.51 1.28 1.15 1.03 0.86 5.6E-1 1.8E-1 4.2E-2 6.3E-3 3.7E-4 4.1E-5 4.3E-6 9.4E-7 q 0.381 0.380 0.379 0.377 0.374 0.362 0.313 0.242 0.167 0.094 0.061 0.041 0.032 O15 K(q) 3.70 1.11 0.74 0.53 0.32 1.3E-1 2.1E-2 4.0E-3 6.3E-4 4.9E-5 6.9E-6 9.5E-7 2.5E-7 q 0.410 0.407 0.403 0.398 0.389 0.367 0.318 0.273 0.229 0.179 0.148 0.122 0.108

Source: Wösten et al., 2001.

Table A5.4. Data set of soil hydraulic functions described with the Van Genuchten model

 qres(cm3/cm3) qsat(cm3/cm3) Ksat(cm/d) a(cm-1) l(-) n(-) Topsoils Sand B1 0.02 0.43 23.41 0.0234 0.000 1.801 B2 0.02 0.42 12.52 0.0276 -1.060 1.491 B3 0.02 0.46 15.42 0.0144 -0.215 1.534 B4 0.02 0.46 29.22 0.0156 0.000 1.406 B5 0.01 0.36 52.91 0.0452 -0.359 1.933 Silt B7 0.00 0.40 14.07 0.0194 -0.802 1.250 B8 0.01 0.43 2.36 0.0099 -2.244 1.288 B9 0.00 0.43 1.54 0.0065 -2.161 1.325 Clay B10 0.01 0.43 0.70 0.0064 -3.884 1.210 B11 0.01 0.59 4.53 0.0195 -5.901 1.109 B12 0.01 0.54 5.37 0.0239 -5.681 1.094 Loam B13 0.01 0.42 12.98 0.0084 -1.497 1.441 B14 0.01 0.42 0.80 0.0051 0.000 1.305 Subsoils Sand O1 0.01 0.36 15.22 0.0224 0.000 2.286 O2 0.02 0.38 12.68 0.0213 0.168 1.951 O3 0.01 0.34 10.87 0.0170 0.000 1.717 O4 0.01 0.35 9.86 0.0155 0.000 1.525 O5 0.01 0.32 25.00 0.0521 0.000 2.374 O6 0.01 0.33 33.92 0.0162 - 1.330 1.311 O7 0.01 0.51 39.10 0.0123 -2.023 1.152 Silt O8 0.00 0.47 9.08 0.0136 -0.803 1.342 O9 0.00 0.46 2.23 0.0094 -1.382 1.400 O10 0.01 0.48 2.12 0.0097 -1.879 1.257 Clay O11 0.00 0.42 13.79 0.0191 -1.384 1.152 O12 0.01 0.56 1.02 0.0095 -4.295 1.158 O13 0.01 0.57 4.37 0.0194 -5.955 1.089 Loam O14 0.01 0.38 1.51 0.0030 -0.292 1.728 O15 0.01 0.41 3.70 0.0071 0.912 1.298

At the water table z = 0 and h = 0. To calculate the pressure head, profiles steps of h = -10 cm will be used. In the first step, h1 = 0, h2 = -10 cm and = -5 cm. The parameters found in Table A5.3 and A5.4 can be used to calculate with the model developed by Van Genuchten (Equation 4). Table A5.3 could also be used to estimate the values of K(q). However, this is less accurate as interpolation between two values of h is required, which is difficult due to the non-linear relation.

For h = -5 cm, K(q) = 3.33 (cm/d)

With Equation 6, the corresponding z2 value for a flux of 1 cm/d (10 mm/d) can be calculated:

TABLE A5.5. Tabulated z-values (cm) for a silty soil for stationary capillary fluxes

 h (cm) (cm) (cm/d) z-values (cm) q = 10 mm/d q = 7.5 mm/d q = 5.0 mm/d q = 2.5 mm/d q = 1.0 mm/d q = 0 mm/d 0 0 0 0 0 0 0 -10 -5 3.3 7.7 8.2 8.7 9.3 9.7 10.0 -20 -15 1.8 14.1 15.2 16.5 18.1 19.2 20.0 -30 -25 1.1 19.4 21.2 23.4 26.2 28.4 30.0 -40 -35 0.8 23.8 26.3 29.5 33.8 37.2 40.0 -50 -45 0.6 27.4 30.6 34.8 40.8 45.7 50.0 -60 -55 0.4 30.4 34.2 39.5 47.1 53.8 60.0 -70 -65 0.3 32.9 37.3 43.5 52.8 61.5 70.0 -80 -75 0.3 35.0 39.9 46.9 58.0 68.8 80.0 -90 -85 0.2 36.8 42.2 49.9 62.6 75.6 90.0 -100 -95 0.2 38.3 44.1 52.6 66.7 82.0 100.0 -110 -105 0.1 39.6 45.7 54.8 70.4 88.0 110.0 -120 -115 0.1 40.7 47.1 56.8 73.8 93.5 120.0 -130 -125 0.1 41.6 48.4 58.6 76.8 98.7 130.0 -140 -135 0.1 42.5 49.5 60.1 79.5 103.5 140.0 -150 -145 0.1 43.2 50.4 61.5 81.9 107.9 150.0 -160 -155 0.1 43.9 51.3 62.7 84.1 112.0 160.0 -170 -165 0.1 44.4 52.1 63.8 86.0 115.8 170.0 -180 -175 0.1 45.0 52.7 64.8 87.8 119.4 180.0 -190 -185 0.0 45.4 53.3 65.7 89.5 122.7 190.0 -200 -195 0.0 45.8 53.9 66.5 90.9 125.7 200.0 -210 -205 0.0 46.2 54.4 67.2 92.3 128.5 210.0 -220 -215 0.0 46.6 54.8 67.9 93.5 131.1 220.0 -230 -225 0.0 46.9 55.3 68.5 94.7 133.6 230.0 -240 -235 0.0 47.2 55.6 69.1 95.7 135.8 240.0 -250 -245 0.0 47.4 56.0 69.6 96.7 137.9 250.0 -260 -255 0.0 47.7 56.3 70.0 97.6 139.9 260.0 -270 -265 0.0 47.9 56.6 70.5 98.4 141.7 270.0 -280 -275 0.0 48.1 56.9 70.9 99.2 143.5 280.0 -290 -285 0.0 48.3 57.1 71.2 99.9 145.1 290.0 -300 -295 0.0 48.4 57.3 71.6 100.6 146.6 300.0 -310 -305 0.0 48.6 57.5 71.9 101.2 148.0 310.0 -320 -315 0.0 48.8 57.8 72.2 101.8 149.3 320.0 -330 -325 0.0 48.9 57.9 72.5 102.3 150.6 330.0 -340 -335 0.0 49.0 58.1 72.7 102.8 151.8 340.0 -350 -345 0.0 49.2 58.3 73.0 103.3 152.9 350.0 -360 -355 0.0 49.3 58.4 73.2 103.7 153.9 360.0 -370 -365 0.0 49.4 58.6 73.4 104.2 154.9 370.0 -380 -375 0.0 49.5 58.7 73.6 104.6 155.9 380.0 -390 -385 0.0 49.6 58.8 73.8 104.9 156.8 390.0 -400 -395 0.0 49.7 59.0 74.0 105.3 157.6 400.0

In the second step, h1 = -10, h2 = -20 cm and = -15 cm. For = -15 cm, = 1.76 (cm/d). The corresponding z2 value for a flux of 1 cm/d (10 mm/d) is:

In this manner, all the other values can be calculated (Table A5.5). Figure A5.1 presents the pressure head profiles as calculated.

Figure A5.1. Pressure head profiles for a silty soil for stationary capillary fluxes