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:

K_s = 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

	q_res (cm³/cm³)	q_sat (cm³/cm³)	K_sat (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