Estimation of the inland extending length of the freshwater–saltwater interface in coastal unconfined aquifers

ABSTRACT

The inland extending length of the freshwater–saltwater interface toe is useful in studies of seawater intrusion in coastal areas. The submarine fresh groundwater discharge in coastal zones is affected not only by hydraulic conductivity and hydraulic gradient of the aquifer, but also by the position of the interface. Two observation wells at different distances from the coast are required to calculate the fresh groundwater flow rate in coastal unconfined aquifers. By considering that the submarine groundwater discharge is equal to the groundwater flow rate, the length of the interface toe extending inland can be estimated when the groundwater flow is at a steady-flow state. Aquifers with horizontal and sloping confined beds and without/with unique surface vertical infiltration are considered. Examples used to illustrate the application of these methods indicate that the inland extending lengths of the interface toe in aquifers with vertical surface infiltration are much shorter than those in aquifers without vertical surface infiltration, and the length of the interface in aquifers with a horizontal confining lower bed are smaller than those in aquifers with a confining lower bed sloping towards the sea. The extent of the interface on the northwestern coast near the city of Beihai in southern Guangxi, China, on 18 January 2013 was estimated as 471–478 m.

Editor M.C. Acreman Associate editor not assigned

KEYWORDS:

• coastal aquifer
• freshwater–saltwater interface
• seawater intrusion
• submarine fresh groundwater discharge
• unconfined aquifer

Introduction

Seawater intrusion has been the focus of a number of studies related to coastal hydrogeology (e.g. Bear 1979, Kashef 1986, Cheng and Ouazar 1999, Zhou et al. 2000, Rajmohan et al. 2009, Park et al. 2012, Trabelsi et al. 2012, Elewa et al. 2013). Understanding of the length of the freshwater–saltwater interface extending into coastal aquifers is important in the investigation of the groundwater environment in these areas, especially in the study of prevention of seawater intrusion in coastal zones. Under natural conditions fresh groundwater in coastal zones commonly discharges into the sea. The submarine fresh groundwater discharge in coastal zones is controlled not only by the hydraulic conductivity of the aquifer and the difference in hydraulic heads between the fresh groundwater and the sea, but also by the position of the freshwater–saltwater interface (Bear 1972, 1979, Rushton 1980, Zhou et al. 2010). Attention has been paid to mathematical models related to freshwater–saltwater interfaces by numerous researchers (Strack 1976, 1989, Van der Veer 1977, Reilly and Goodman 1985, Essaid 1986, Isaacs and Hunt 1986, Singh and Stammers 1989, Ledoux et al. 1990, Dagan and Zeitoun 1998, Person et al. 1998, Maas 2007, Xun et al. 2008, Zhou 2011, Lu et al. 2015). Submarine fresh groundwater discharge in both unconfined aquifers and confined aquifers was examined by Bear (1979), Cheng and Ouazar (1999) and Bear et al. (1999). Some researchers (e.g. Robinson et al. 2007, Tang et al. 2007) examined the location of the interface that is affected by the tide. Other researchers (e.g. Urish and McKenna 2004, Robinson et al. 2007) analysed groundwater discharge into the sea in a coastal zone that is affected by the tide. Studies of the freshwater–saltwater interface related to submarine groundwater discharge were made by a number of researchers (Kashef 1986, Izuka and Gingerich 1998, Kim et al. 2007, Maas 2007, Xun et al. 2008, Xun and Ying 2009, Zhou 2011). Knowledge of the length of the freshwater–saltwater interface toe extending inland is commonly necessary information in the investigation of seawater intrusion in coastal areas. Based on the Ghyben-Herzberg relation, Domenico and Schwartz (1990) presented a simple model to determine the length of saltwater wedge in a coastal unconfined aquifer where the location of the interface toe is known. Zhou (2014) described the methods for estimating the interface extending into coastal confined aquifers with horizontal and sloping confining beds where the location of the interface toe is not known in advance. The purpose of the this paper is to provide methods of estimating the length of the freshwater–saltwater interface extending into inland unconfined aquifers where the location of the interface toe is not known a priori. Unconfined aquifers with horizontal and sloping lower confining beds and without/with unique vertical surface infiltration are examined. Examples are also given to illustrate the application of the methods for estimating the inland extending length of the freshwater–saltwater interface in coastal unconfined aquifers. The extending distance of the interface in the northwestern coast of Beihai in southern Guangxi, China, is calculated as an actual example. These methods are simple and suitable for regional hydrogeological studies in coastal regions.

Submarine fresh groundwater discharge in a coastal unconfined aquifer

To estimate the length of the freshwater–saltwater interface extending into inland unconfined aquifers, the submarine fresh groundwater discharge into the sea in the coastal zone must be described first.

Figure 1 shows a profile perpendicular to the coastline in a homogeneous and isotropic coastal unconfined aquifer with unique vertical infiltration on the land surface. It is assumed that fresh groundwater flow is in a steady state and discharges into the sea, and a seepage face near the coast does not exist. The flow rate in the salt-water zone is assumed too small in comparison with that in the freshwater zone and can be neglected. A possible capillary zone above the phreatic surface is also neglected (Strack 1976). An observation well is located at a distance L from the coastline. The water table at the observation well is denoted h_L. One needs to estimate the steady discharge of fresh groundwater into the sea and determine the location of the freshwater–saltwater interface. Bear (1979) examined the groundwater flow in the coastal unconfined aquifer and provided the solution for the steady discharge of fresh groundwater into the sea by placing the x axis along the aquifer base with the origin at the interface toe. Bear’s solution was applied in coastal unconfined aquifers by Qiu (1991) and others. For completeness, the solution for the steady discharge of fresh groundwater into the sea in a coastal unconfined aquifer with large thickness is deduced in the following paragraphs by placing the x axis along the mean sea level with the origin at the coastline. The coordinate system used is shown in Figure 1.

Figure 1. Schematic cross-section showing a coastal unconfined aquifer with unique vertical infiltration on the land surface (OW stands for observation well) (modified from Zhou et al. 2010). Horizontal arrows are used for q₀, q_x and q_L for simplification, but they are not necessarily horizontal.

All the hydraulic heads are referenced to the mean sea level. The fresh groundwater flow in the coastal zone is assumed to satisfy the Dupuit assumption. A flow section (approximately a vertical section) at distance x is selected. According to Darcy’s law and the principle of continuum of groundwater flow, the fresh groundwater flow rate of unit width through the vertical section at distance of x = 0, q₀ can be expressed as (Zhou et al. 2010):

(1)

where q₀ represents the fresh groundwater discharge into the sea, K is the hydraulic conductivity of the aquifer, h_f is the water table at the vertical section of distance x, M_f is the depth of the freshwater–saltwater interface below the mean sea level (or the thickness between the mean sea level and the interface) at the section of x, and w is the infiltration rate of unit area on the land surface. The term M_f in Equation (1) can be determined according to the Ghyben-Herzberg relationship, M_f = δh_f, where δ = ρ_f/(ρ_s − ρ_f), ρ_s and ρ_f are the densities of saltwater and freshwater, respectively. Thus, Equation (1) can be rewritten as:

(2)

or

(3)

Equation (2) is derived and integrated between h_f = 0 at x = 0 and h_f = h_L at x = L, giving:

(4)

Equation (3) is derived and integrated between M_f = 0 at x = 0 and M_f = M_L at x = L, giving:

(5)

Equation (3) can also be derived and integrated between M_f = 0 at x = 0 and M_f = M_f at x = x, giving the depth of the interface at any section of distance x (0 < x < L):

(6)

Equations (4) and (5) indicate that if L, h_L, M_L, K, w and δ are known, one can estimate the fresh groundwater discharge into the sea in the coastal zone. Equations (4)–(6) suggest that if the water table h_L or the depth of the interface M_L at the observation well of distance L are known, that is, only one observation well is required, the fresh groundwater discharge into the sea and the depth of the interface at any distance x (0 < x < L) in the coastal zone can be determined.

From Equation (6), one can notice that when x = 0, M_f = 0, that is, no outlet of fresh groundwater exists along the coastline, which is not the case in realistic coastal zones. This error is caused when the Ghyben-Herzberg relation is used to estimate the depth of the freshwater–saltwater interface. Therefore, Equation (6) is not suitable for estimating the depth of the interface near the coastline. Now it is assumed that the depth of the interface at the coastline (x = 0) is M₀ (Fig. 1), and Equation (3) is derived and integrated between M_f = M₀ at x = 0 and M_f = M_L at x = L, giving:

(7)

If M₀ = 0, Equation (7) reduces to Equation (5).

Thus, if M₀ is known, one can use Equation (7) to estimate q₀. However, M₀ is not determined easily in general cases. Henry (1959) suggested that M₀ can be calculated approximately using the following equation:

(8)

Equations (4) and (5) can be used to estimate the submarine fresh groundwater discharge into the sea in a coastal unconfined aquifer with unique vertical infiltration on the land surface. From Equations (4) and (5), one can easily obtain the fresh groundwater flow rate of unit width at any section of x (0 < x < L), q_x, and at the section of x = L, q_L, as follows (Zhou et al. 2010):

(9)

or

(10)

and

(11)

and

(12)

or

(13)

and

(14)

where q₀ > q_x > q_L. Equation (13) is the same as that derived by Bear (1979).

As a special case, if the unique vertical infiltration on the land surface does not exist, i.e., w = 0, Equation (4) reduces to (Zhou et al. 2010):

(15)

and Equation (5) becomes:

(16)

where q is the fresh groundwater flow rate of unit width through any section x (0 ≤ x ≤ L) or fresh groundwater discharge into the sea in a coastal unconfined aquifer without unique vertical surface infiltration. If M₀ is considered, Equation (7) reduces to:

(17)

From Equation (16) one can obtain:

(18)

If M_L in Equation (18) represents the vertical distance from the mean sea level to the horizontal lower confining bed of the coastal unconfined aquifer, L in Equation (18) is the distance of the interface toe extending into the unconfined aquifer (Bear 1979, Strack 1989, Zhou et al. 2010, Lu et al. 2015). In this case, only one observation well is required to determine the distance M_L. However, q and K must be known in advance in calculation of L according to Equation (18), and this is not easy when only one observation well is present. In the following sections, L is estimated when two observation wells are present in a coastal unconfined aquifer where q and K are not known in advance.

Estimation of the length of the freshwater–saltwater interface extending into coastal unconfined aquifers

A coastal unconfined aquifer with a horizontal lower confining bed without vertical infiltration on the land surface

Shown in Figure 2 is a profile showing a coastal unconfined aquifer with a horizontal lower confining bed without vertical infiltration on the land surface. Groundwater discharges into the sea in a homogeneous and isotropic unconfined aquifer. It is assumed that groundwater flow is in a steady state and a freshwater–saltwater interface exists in the coastal zone. The groundwater flow lines from a plane view are parallel to each other and perpendicular to the coastline. The freshwater–saltwater interface toe is at the distance of L from the coast, at which point the hydraulic head and the depth of the interface below the mean sea level are denoted as h_L and M_L, respectively. Two observation wells, Well 1 and Well 2, are drilled in the same profile, which are at the distances of L₁ and L₂ (L < L₂ < L₁) from the coast, respectively. The observation wells tap the lower confining bed at a depth of M₁ (= M₂ = M_L) below the mean sea level. Elevations of the hydraulic heads in Well 1, h₁, and Well 2, h₂, are measured. The fresh groundwater flow rate per unit width in the unconfined aquifer, q, is expressed as:

(19)

Figure 2. Schematic cross-section showing a coastal unconfined aquifer with a horizontal lower confining bed without vertical infiltration on the land surface (L < L₂ < L₁, M_L = M₂ = M₁).

where K is the hydraulic conductivity of the aquifer.

The discharge of fresh groundwater into the sea is obtained as given in Equation (16). Thus, from Equations (16) and (19), the length of the interface toe from the coast, L, is obtained:

(20)

The discharge of fresh groundwater into the sea is also obtained as given in Equation (17) when the outlet of fresh groundwater is considered. Thus, from Equations (17) and (19), the length of the interface toe from the coast, L, is also obtained:

(21)

To illustrate the application of Equations (20) and (21), in Figure 2, let L₁ = 1500 m, L₂ = 1000 m, h₁ = 6 m, h₂ = 5 m, M₁ = M₂ = M_L = 50 m, and δ = ρ_f/(ρ_s − ρ_f) = 1.0/(1.025 – 1.0) = 40. According to Equation (20) we obtain:

According to Equation (19), we obtain:

(22)

where β₁ = 0.111 m. According to Equation (8), when the outlet of fresh groundwater is considered, M₀ is obtained by substituting Equation (22) into Equation (8):

Thus, according to Equation (21) we obtain:

On the northwestern coast of the city of Beihai (Fig. 3) in southern Guangxi, China, occur Quaternary and Neogene unconsolidated sediments (5–350 m in thickness). Overlying the basement rocks are Neogene sand with gravel and scattered lenses of clay or sandy clay (N₂sh). Overlying the N₂sh sediments are sand and gravel with the upper clay layer of the Lower Pleistocene Zhanjiang Group of the Quaternary age (Q₁z). The surficial sediments are the lower sand with gravel and upper sandy clay of the Middle Pleistocene Beihai Group of the Quaternary age (Q₂b). Along the coast occurs Quaternary Holocene sand (Q₄) with thickness of 0–5 m. One unconfined aquifer is roughly grouped in the Q₄, Q₂b and Q₁z sediments on the northwestern coast of the study area, considering the semi-perviousness and termination of the clay (Zhou et al. 2000). Under natural conditions, groundwater flows from south to north and discharges into the sea. In 2012 observation wells were drilled along the Yunnan Road and the thickness of the aquifer below the mean sea level is M_L = 54.14 m. Groundwater levels were measured in wells S2 and S3, which are L₂ = 675.0 m and L₁ = 937.50 m from the actual coast, respectively. Groundwater levels in wells S2 and S3 in the dry season (18 January 2013) are selected (i.e. h₂ = 1.770 m and h₁ = 2.138 m) for the calculation of the extending distance of the interface. It is assumed that fresh groundwater flow was at a steady state at the specific moment. According to Equation (20), the extending distance of the interface can be estimated as L = 477.568 m. When the outlet of fresh groundwater is considered, M₀ is obtained as 6.334 m based on q = 0.214K (according to Equation (19)) and Equation (8), and the extending distance of the interface can be estimated as L = 471.032 m.

Figure 3. Location of the observation wells on the northwestern coast of Beihai in Guangxi, China.

A coastal unconfined aquifer with a sloping lower confining bed and without vertical infiltration on the land surface

Shown in Figure 4 is a profile showing a coastal unconfined aquifer with a sloping lower confining bed without vertical infiltration on the land surface. Two observation wells, Well 1 and Well 2, tap the lower confining bed at depths of M₁ and M₂ below the mean sea level, respectively (M_L > M₂ > M₁). Other conditions are the same as those in Figure 2. The fresh groundwater flow rate per unit width in the unconfined aquifer, q, is expressed as:

(23)

Figure 4. Schematic cross-section showing a coastal unconfined aquifer with a sloping lower confining bed without vertical infiltration on the land surface (L < L₂ < L₁, M_L > M₂ > M₁). Horizontal arrow is used for q for simplification, but it is not necessarily horizontal.

In Figure 4, let L₁ = 1500 m, L₂ = 1000 m, h₁ = 6 m, h₂ = 5 m, M₁ = 40 m, M₂ = 45 m, and δ = ρ_f/(ρ_s − ρ_f) = 1.0/(1.025 – 1.0) = 40. According to Equation (23) we obtain:

(24)

where β₂ = 0.096 m.

As show in Figure 4, M_L can be calculated as follows:

(25)

(26)

For simplification, Equation (16) is thought to be suitable for the aquifer shown in Figure 4. Thus, from Equations (16), (26) and (24), we obtain:

(27)

From Equation (27) one can obtain:

According to Equation (8) when the outlet of fresh groundwater is considered, M₀ is obtained by substituting Equation (24) into Equation (8):

Thus, according to Equations (17), (26) and (24), we obtain:

(28)

From Equation (28) one can obtain:

According to Equation (26), one can also obtain M_L = 51.465 m when M₀ = 0 and L = 353.497 m, and M_L = 51.475 m when M₀ = 2.845 m and L = 352.615 m. On the coast, where L = 0, the vertical distance from the mean sea level to the lower confining bed is 55 m.

A coastal unconfined aquifer with horizontal lower confining bed and unique vertical infiltration on the land surface

Shown in Figure 5 is a profile showing a coastal unconfined aquifer with a horizontal lower confining bed and unique vertical infiltration on the land surface. Other conditions are the same as those in Figure 2 except the presence of vertical surface infiltration. The fresh groundwater flow rate per unit width through the vertical section of distance L₂ in the unconfined aquifer, q₂, is expressed as follows:

(29)

Figure 5. Schematic cross-section showing a coastal unconfined aquifer with a horizontal lower confining bed and unique vertical infiltration on the land surface (L < L₂ < L₁, M_L = M₂ = M₁, q₀ > q_L > q₂ > q₁).

In Figure 5, let L₁ = 1500 m, L₂ = 1000 m, h₁ = 6 m, h₂ = 5 m, M₁ = M₂ = M_L = 50 m, K = 30 m/d, w = 0.01 m/d, and δ = ρ_f/(ρ_s − ρ_f) = 1.0/(1.025 – 1.0) = 40. According to Equation (29) we obtain:

As shown in Figure 5, q_L can be calculated as follows:

(30)

where γ₁ = 0.01 m/d. According to Equations (13) and (30) we obtain:

(31)

From Equation (31), one can obtain:

and q_L = 15.211 m²/d when L = 61.914 m.

As shown in Figure 5, q₀ can be calculated as follows:

According to Equation (8) when the outlet of fresh groundwater is considered, M₀ is obtained:

Thus, according to Equations (14) and (30) we obtain:

(32)

From Equation (32), one can obtain:

We can also obtain q_L = 15.273 m²/d when M₀ = 15.64 m and L = 55.746 m, and q₁ = q₂ − w(L₁ − L₂) = 0.83 m²/d.

A coastal unconfined aquifer with a sloping lower confining bed and unique vertical infiltration on the land surface

Shown in Figure 6 is a profile showing a coastal unconfined aquifer with a sloping lower confining bed and with unique vertical infiltration on the land surface. Other conditions are the same as those in Figure 2 except the existing vertical surface infiltration. The fresh groundwater flow rate per unit width through the vertical section of distance L₂ in the aquifer, q₂, is expressed as follows:

(33)

Figure 6. Schematic cross-section showing a coastal unconfined aquifer with a sloping lower confining bed and unique vertical infiltration on the land surface (L < L₂ < L₁, M_L > M₂ > M₁, q₀ > q_L > q₂ > q₁). Horizontal arrows are used for q₀, q_L, q₂ and q₁ for simplification, but they are not necessarily horizontal.

In Figure 6, let L₁ = 1500 m, L₂ = 1000 m, h₁ = 6 m, h₂ = 5 m, M₁ = 40 m, M₂ = 45 m, K = 30 m/d, w = 0.01 m/d, and δ = ρ_f/(ρ_s − ρ_f) = 1.0/(1.025 – 1.0) = 40. According to Equation (33) we obtain:

As shown in Figure 6, q_L can be calculated as follows:

(34)

where γ₂ = 0.01 m/d.

M_L can be calculated according to Equation (25):

According to Equations (13) and (34), we obtain:

(35)

From Equation (35), one can obtain:

and q_L = 14.660 m²/d and M_L = 54.280 m when L = 71.952 m.

q₀ can be calculated as follows:

According to Equation (8) when the outlet of fresh groundwater is considered, M₀ is obtained:

According to Equations (14) and (34), we obtain:

(36)

From Equation (36), one can obtain:

One can also obtain q_L = 14.714 m²/d and M_L = 54.334 m when M₀ = 15.195 m and L = 66.570 m, and q₁ = q₂ − w(L₁ − L₂) = 0.38 m²/d.

Discussion

In the above calculation of distance L in the coastal unconfined aquifers, it is worth noting that the hydraulic conductivity K and the groundwater flow rate q do not need to be known in advance in the aquifers of Figures 2 and 4 with horizontal or sloping lower confined bed and without vertical surface infiltration, when the two observation wells are located far from the coast. However, the hydraulic conductivity K, the infiltration rate of unit area on the land surface w and the groundwater flow rate q must be known in advance in the calculation of L in the coastal unconfined aquifers of Figures 5 and 6 with horizontal or sloping lower confined bed and with vertical surface infiltration, even though two observation wells are present.

In order to examine the factors affecting the value of L, the above methods are used to calculate the distance L of the coastal confined aquifers of the same K for different cases in Figures 2, 4, 5 and 6, considering the absence or presence of vertical surface infiltration, changes in hydraulic gradient, changes in the infiltration rate of unit area on the land surface if the vertical surface infiltration is present, and the outlet of fresh groundwater at the coast M₀. The values of ρ_f = 1.0 g/cm³, ρ_s = 1.025 g/cm³, L₁ = 1500 m and L₂ = 1000 m are kept constant for each case. The results are listed in Table 1 for a rough comparison.

Table 1. Calculation results for the different cases in Figures 2, 4, 5 and 6.

Figure	Case	M1 (m)	M2 (m)	ML (m)	M0 (m)	h1 (m)	h2 (m)	K (m/d)	w (m/d)	L (m)	q1 (m^2/d)	q2 (m^2/d)	qL (m^2/d)	q0 (m^2/d)
Figure 2	1			50		6	5			288.570
	2			50	3.290	6	5			287.320
	3			50		6.5	5.25			229.306
	4			50	4.140	6.5	5.25			227.734
Figure 4	5	40	45	51.465		6	5			353.494
	6	40	45	51.475	2.854	6	5			352.547
	7	40	45	52.122		6.5	5.25			287.814
	8	40	45	52.134	3.585	6.5	5.25			286.588
Figure 5	9			50		6	5	30	0.005	103.079	2.080	4.580	9.065	9.580
	10			50	9.465	6	5	30	0.005	99.284	2.080	4.580	9.084	9.580
	11			50		6.5	5.25	30	0.005	94.161	2.941	5.441	9.970	10.441
	12			50	10.315	6.5	5.25	30	0.005	90.063	2.940	5.441	9.990	10.441
	13			50		6	5	30	0.01	61.914	0.830	5.830	15.211	15.830
	14			50	15.640	6	5	30	0.01	55.746	0.830	5.830	15.273	15.830
	15			50		6.5	5.25	30	0.01	58.602	1.691	6.691	16.105	16.691
	16			50	16.490	6.5	5.25	30	0.01	52.125	1.691	6.691	16.169	16.691
Figure 6	17	40	45	53.819		6	5	30	0.005	118.121	1.630	4.130	8.539	9.130
	18	40	45	53.850	9.021	6	5	30	0.005	115.033	1.630	4.130	8.555	9.130
	19	40	45	53.900		6.5	5.25	30	0.005	109.986	2.378	4.878	9.328	9.878
	20	40	45	53.934	9.760	6.5	5.25	30	0.005	106.606	2.378	4.878	9.345	9.878
	21	40	45	54.280		6	5	30	0.01	71.952	0.380	5.380	14.660	15.380
	22	40	45	54.334	15.195	6	5	30	0.01	66.570	0.380	5.380	14.714	15.380
	23	40	45	54.311		6.5	5.25	30	0.01	68.831	1.128	6.128	15.440	16.128
	24	40	45	54.368	15.935	6.5	5.25	30	0.01	63.159	1.128	6.128	15.497	16.128

Note: for each case ρ_f = 1.0 g/cm³, ρ_s = 1.025 g/cm³, L₁ = 1500 m, and L₂ = 1000 m.

It is found that the vertical surface infiltration has a significant effect on the inland extending distance of the interface L. The values of L for cases 1–8 without vertical surface infiltration are much larger than those of cases 9–24 with vertical surface infiltration. The values of L for cases 9–12 in Figure 5 and cases 17–20 in Figure 6 with lower infiltration rate of 0.005 m/d are also larger than those of cases 13–16 in Figure 4 and cases 21–24 in Figure 6 with higher infiltration rate of 0.01 m/d. Thus, an increase in the groundwater flow rate (which increases towards the sea from q₁ to q₀ as shown in Table 1) through vertical surface infiltration may effectively control the inland extension of the interface.

In Figures 2, 4, 5 and 6, the hydraulic gradient has an important bearing on the inland extending length of the interface L. The values of L in cases 1, 2, 5, 6, 9, 10, 13, 14, 17, 18, 21 and 22 with lower hydraulic gradient of 0.002 are larger than those of the corresponding cases 3, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23 and 24 with greater hydraulic gradient of 0.0025. Thus, increase in the hydraulic gradient in the aquifer may also control the inland extension of the interface. In particular, Equation (20) or (21) indicates that L is inversely proportional to the hydraulic gradient of the coastal unconfined aquifer when the vertical surface infiltration is absent.

The values of L in the cases of Figures 4 and 6 with sloping lower confining bed are larger than those of the cases of Figures 2 and 5 with horizontal lower confining bed. The reason for this is that the groundwater discharge into the sea in the unconfined aquifer with a sloping lower confining bed is less than that in the aquifer with a horizontal lower confining bed, leading to a longer inland extending distance of the interface. The angle between the sloping surface of the confining bed and the horizontal plane is not larger than 30°, as suggested by Chapman (1980).

In addition, the values of L in the cases of Figures 2, 4, 5 and 6 where the outlet of fresh groundwater at the coast M₀ is considered are a bit smaller than those of the corresponding cases of Figures 2, 4, 5 and 6 where the outlet of fresh groundwater at the coast M₀ is not considered. The difference in the values of L under these two conditions is not so obvious as expected.

Conclusion

The inland extending length of the interface toe in coastal unconfined aquifers can be estimated through examining the submarine fresh groundwater discharge in the coastal zones and the fresh groundwater flow rate in the coastal unconfined aquifers, which are assumed to be homogeneous and isotropic. Methods for estimating the length of the interface toe in coastal unconfined aquifers with or without vertical infiltration on the land surface and with a horizontal or sloping lower confining bed require two observation wells at different distances far from the coastline along a profile perpendicular to the coast. As an actual example, the extending distance of the interface on the northwestern coast of Beihai in southern Guangxi, China, in the dry season on 18 January 2013 is estimated as 471–478 m.

The subsurface fresh groundwater discharge into the sea is affected by the location of the freshwater–saltwater interface in coastal unconfined aquifers. The subsurface fresh groundwater discharge also plays an important role in the occurrence of the freshwater–saltwater interface. When vertical surface infiltration exists in the coastal unconfined aquifers, the inland extending lengths of the interface toe are much shorter than those in the coastal unconfined aquifers without vertical surface infiltration. Increase in hydraulic gradient in the unconfined aquifer can also lead to the decrease in the inland extending lengths of the interface toe. When the lower confining bed of a coastal unconfined aquifer slopes towards the sea (the slope angle is not larger than 30°), the fresh groundwater flow rate may decrease compared to that in the coastal unconfined aquifer with horizontal lower confining bed, causing the length of the interface to increase. The submarine fresh groundwater discharge in coastal zones is the key factor affecting the inland extending length of the interface toe. Thus, maintaining or increasing the subsurface fresh groundwater discharge into the sea from the inland unconfined aquifers in different ways, including an increase in groundwater level or hydraulic gradient and an increase in vertical surface infiltration, is vital for the prevention of seawater intrusion in coastal unconfined aquifers.

Although the methods described in this paper involve some assumptions, and thus have certain limits, they offer simple ways of estimating the distances of the interface extending into the coastal unconfined aquifers where only two observation wells are needed. They are relatively easily operated and are suitable in preliminary investigations of regional hydro-geology of coastal areas. More real case studies of the methods will be the subject of future studies.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was cooperatively supported by the Project of Development Field of High Priority of the Specialized Research Fund for the Doctoral Program of Higher Education of China (20110022130002) and the Natural Science Foundation of China (41172227).