Bridging hydraulic diffusivity from aquifer to particle-size scale: a study on loess sediments from southwest Hungary

Figures & data

Fig. 1 Location of the Szigetvár area within the territory of Hungary, showing spatial distribution of topography and lithologic units, and delimitation of a sector where a number of drilled wells were monitored for hydraulic head.

Fig. 2 Seasonal variation of precipitation and river flow data, based on long-term records. The river flow data pertain to the Péli-víz Felsőpél watershed (30.5 km²), measured at a hydrometric station located at the outlet point of this catchment.

Fig. 3 Grain-size curves of Hungarian loess and interlayered palaeosols (Paks formation), drawn from the data in Table 1.

Fig. 4 Classification of Hungarian loess and interlayered palaeosols according to texture, based on the grain-size data depicted in Table 1. The textural classification used the USDA soil textural classification.

Table 1 Average grain-size distributions of loess and interlayered palaeosol samples, collected at Paks exposure by Nemecz et al. (2000).

Fraction	Grain size (μm)	Percentage
		Loess	Palaeosol
A	<5	27.4	40.2
B	5–10	5.0	5.5
C	10–20	8.5	8.7
D	20–45	38.1	37.3
E	45–80	18.5	6.8
F	80–160	1.6	0.9
G	160–315	0.5	0.3
H	315–800	0.1	0.0

Table 2 Hydraulic head records of drilled wells represented in Fig. 1. ID: drilled well identification code; X, Y and Z: planimetric and altimetric location coordinates; h: hydraulic head.

ID	Observation date	X (m)	Y (m)	Z (m)	Water depth (m)	h (m)
219w	8 May 1993	407030	342210	123.0	12.4	110.6
201w		399720	343590	125.0	5.9	119.1
202w		401840	343910	132.0	5.4	126.6
204w		406710	344070	168.0	25.0	143.0
203w		405360	344580	139.0	10.5	128.5
171w		409280	349260	138.0	0.6	137.4
170w		403880	349270	137.0	2.0	135.0
168w		398120	349530	164.0	1.6	162.4
154w		403710	349890	150.0	10.1	139.9
153w		402090	351430	204.0	55.3	148.7
136w		398110	353220	201.0	2.7	198.3
106w		401780	356710	223.0	25.2	197.8
221w	16 May 1993	409390	342140	124.0	1.2	122.8
185w		399890	345530	127.0	14.4	112.6
187w		406250	345920	142.5	2.2	140.3
186w		404050	346440	133.0	23.6	109.4
188w		407040	347930	129.0	22.2	106.8
155w		405940	351620	160.0	12.7	147.3
140w		409130	352560	132.0	1.6	130.4
121w		400310	354690	142.0	7.9	134.1
122w		401750	354710	212.0	1.6	210.4
124w		407860	355220	175.0	11.9	163.1
123w		406480	355470	138.0	2.8	135.2
105w		399960	356120	145.0	5.3	139.7
107w		406610	358150	217.0	30.5	186.5
092w		408030	358470	182.0	12.3	169.7

Table 3 Empirical formulas relating hydraulic conductivity with grain-size distribution, used in the calculation of microscale hydraulic diffusivities. Source: Odong (2007). Symbols are explained in the text.

Formula		Main application requisites
Name	Equation
Hazen		U < 5
		0.1 < de < 3 mm
Kozeny-Carman		de < 3 mm
		not applicable to clayey materials
Breyer		1 < U < 20
		0.06 < de < 0.6 mm
Slitcher		0.01 < de < 5 mm

Table 4 Representative values of specific yield (S_y), specific retention (S_r) and total porosity (n) of unconsolidated materials comprising a wide range of grain sizes. Source: Morris and Johnson (1967).

Fraction	Maximum grain size (mm)	Sy	Sr	n
Coarse gravel	32	23	5	28
Medium gravel	16	24	8	32
Fine gravel	8	25	9	34
Coarse sand	1	27	12	39
Medium sand	0.5	28	11	39
Fine sand	0.25	23	20	43
Silt	0.063	8	38	46
Clay	0.004	3	39	42

Table 5(a) Megascale hydraulic diffusivities (D), as calculated by the GRADIENTS model for various cell sizes (Δx), with indication of median slope (d) and curvature (τ) of hydraulic head surface (h).

Cell size, Δx (m)	Slope of h–d	Curvature of h–τ	Hydraulic diffusivity, D (m^2/s)
10	0.0056	0.00000146	16.5
25	0.0062	0.00000125	18.9
50	0.0051	0.00000147	13.8
100	0.0053	0.00000143	14.0
500	0.0052	0.00000153	15.8
750	0.0049	0.00000140	15.8
1250	0.0043	0.00000111	21
1500	0.0043	0.00000088	27

Table 5(b) Microscale hydraulic diffusivities (D), as calculated by equations (4) (unconfined condition) and (5c) (confined condition), based on hydraulic conductivities (K) calculated by the empirical formulas depicted in Table 3.

Formula	K × 10^-6 (m/s)	D (m^2/s)
		Unconfined	Confined
Hazen	4.0	0.0016	13.9
Kozeny-Carmen	4.4	0.0018	15.4
Breyer	3.6	0.0015	12.7
Slicher	1.4	0.0006	5.0

Fig. 5 Megascale and microscale hydraulic diffusivities plotted as a function of cell size (Δx). The megascale diffusivities were calculated by the GRADIENTS model; the microscale are based on grain-size data and were calculated using equations (4) and (5c), with hydraulic conductivity calculated by the empirical formulas shown in Table 3. The megascale D values could be fitted to a couple of trend lines, valid for different spatial resolutions; the microscale were arbitrarily assigned the unit cell size (Δx = 1 m).

Fig. 6 Spatial distribution of hydraulic head (grid h_8 May 1993), slope of hydraulic head (grid H_x), curvature of hydraulic head (grid H_xx) and hydraulic diffusivity (grid D), as calculated by the GRADIENTS model.

Fig. 7 Plot of hydraulic diffusivities as a function of concomitant storage coefficients, for unconfined, semi-confined and confined aquifers. The literature data were compiled by Pacheco (2013) and Onder (1994). The Szigetvár loess aquifer, with a median diffusivity of D = 17.9 m²/s, behaves as a confined aquifer. The associated storage coefficient is S = bμβS_y = 7.9 × 10^-6, with b = 27.5 m (average aquifer thickness, reported in the ‘Geology’ section), ρ = 999.1 kg/m³ (specific weight of water), β = 4.7 × 10^-9 m²/kg (compressibility of water), and S_y = 0.061 (specific yield, estimated from values in Table 4, based on a median grain size d₅₀ = 0.0405 mm, deduced from data in Table 1).

Fig. 8 Hydraulic head wave (A + B) of a hypothetical heterogeneous aquifer composed of two media, one with a low (A) and the other with a high (B) hydraulic diffusivity.

Fig. 9 (a) Comparison of hydraulic diffusivity estimates, as calculated by the GRADIENTS model and equation (10a). The plot of GRADIENTS model diffusivities is based on D values given in Table 5(a) as a function of Δx (L in the figure). The plot of equation (10a) diffusivities is based on the curvatures and slopes depicted in Table 5(a) as a function of Δx (L in the figure), assuming that τ₁ = τ₁₀ and d₁ = d₁₀. The values of ρ (specific weight of water) and β (compressibility of water) are those referred to in the text. (b) Relationship between the curvature (τ_L) and slope (d_L) of hydraulic head surface (h) and scale (L).

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