EFFECTS OF CLIMATE AND ANTECEDENT SOIL MOISTURE ON THE AREAL AVERAGE ABSTRACTION LOSSES

M. Fiorentino ¹, M. R. Margiotta ¹ and V. Iacobellis ²

¹ Dipartimento di Ingegneria e Fisica dell'Ambiente - Università della Basilicata

Contrada Macchia Romana, 85100, Italy

² Dipartimento di Ingegneria Civile e Ambientale - Politecnico di Bari

Via E. Orabona, 4, 70125, Bari, Italy

Tel.: +39-080-5963321; Fax: +39-080-5963414; E-mail: v.iacobellis@poliba.it

Abstract: The influence of climate on the flood generation process is nowadays a topic of striking impact and particular remark due to the necessity of improving the available techniques and procedures for risk assessments and land protection. In particular great efforts of the recent research are provided to support the flood frequency analysis exploiting the amount of information available by the observation of the frequency of precipitation, vegetation coverage, soil permeability, etc in order to improve the performances of models for flood prediction. The aim of this paper is to focus on links and significant relationships between infiltration models, valid at small scale, and average processes observed at the medium (hillslope) and large (basin) scales, by the light of climate and permeability of soil.

A significant interpretation to the observed relationships observed between total water losses F_A, climate and basin area A is provided. By means of results obtained on basins in Southern Italy, it is possible to conclude that in dry regions rain losses are mainly due to the initial abstraction phenomenon and the estimated values of F_A are quite  constant and invariant with basin area. Such findings are consistently matched to what one could expect from Hortonian behavior of soils. Conversely, in humid basins, the values F_A, show a strong relationship with climate represented by a climatic index. In this case, the basin’s behavior is consistent with Dunne’s model of runoff generation. Those results are supported by the theoretical framework provided by the classical analysis of infiltration in unsaturated porous media.

Keywords: flood, climate, soil moisture, water losses

1    INTRODUCTION

The influence of climate on the flood generation process is an investigation field of great remark aimed to improve the available techniques and procedures for risk assessment and land protection. The climate influences the flood regime not only through the precipitation patterns but also by means of antecedent soil moisture condition and its most probable state which depends on the frequency and intensity of precipitation, vegetation coverage, soil permeability, etc. The antecedent soil moisture is then responsible of boundary and initial conditions to be specified in the infiltration model used for the determination of water losses. At the basin scale, these last quantity has to be considered as a result of a number of overlapping processes related to the mechanisms of runoff generation, and to the variability in time and space of precipitation, sorptivity and permeability of soil. Yet, dealing with water losses at the basin scale is possible by referring to their average value in time and space.

In flood frequency analysis water losses processes are often taken into account in order to estimate position parameters like the mean annual flood. More rare is instead the case where these processes are invoked to infer variability characteristics of the flood distribution. However, we believe that more attention should be paid to this issue.

In this light, an important characteristic of the probability distribution of  floods is the mean annual number L of independent floods, which, when this distribution is schematized as a compound Poisson process (e.g. EV1, GEV, TCEV), represents the distribution parameter that more strongly controls the coefficient of variation C_v. For instance, in the case of the EV1 distribution, C_v depends on L only and is a decreasing function of it.

As shown in Iacobellis and Fiorentino (2000), in the case of floods generated by rain storms, L can be related to a water losses parameter f_A through the following formula:

In equation (1), L_p is the mean annual number of independent storms, E[×] is the expectation operator, and i_A,t  represents the average intensity of the maximum rainfall amount measured during the storm in a duration t, where t is the lag time of the basin of surface area A. Equation (1) is based on the simplified assumption that the peak discharge Q is related to i_A,t  by the following equation:

which may be considered sufficiently well suited for use in the frame of a theoretical model for deriving the flood distribution, although it is surely not particularly accurate at the event scale. In the above equation, x is a routing factor less than unity and f_A is the rainfall intensity threshold above which runoff is produced. Since equation (1) deals with rainfall in a duration t, this threshold can also be seen as a water volume. In fact, the quantity F_A = f_A t represents the rainfall depth that has to be reached in order to produce surface runoff. One should remark that the order of magnitude of the variability from an event to another of x and f_A is much smaller than that of i_A,t. Thus, it can be shown that equation (1) can be derived from equation (2) under the hypothesis that i_A,t  is a Weibull variate with shape parameter k. Incidentally, we may remind that k equals unity when the Weibull distribution reduces to the exponential one.

The accuracy of all the aforementioned hypotheses is shown in Iacobellis and Fiorentino (2000) and Fiorentino and Iacobellis (2001), to whom the reader is kindly sent for details also regarding the results of application to real basins, presented hereafter.

In this paper the emphasis is put on the behavior that F_A is expected to show under different climate conditions. In particular, a theoretical scheme based upon the use of  Philip’s equation is developed to point out the role played by  the soil moisture at the arrival time of the storm. In addition, results obtained analyzing data from real data relative to basins located in Southern Italy are shown to be consistent with this scheme.

2    INFILTRATION AND AREAL WATER LOSSES

Infiltration is the movement of water into the soil matrix. In particular, in the unsaturated zones of soil, the vertical flow is described by the Richards’ equation which can be wrote in the form:

where D is the diffusion coefficient or diffusivity, K_z the effective permeability, q the volumetric water content and z the coordinate direction, positive upward, with origin at the earth’s surface.

Different analytical solution are achievable for particular assumptions, boundary conditions and initial conditions. An exact solution to the variable diffusivity problem of equation(3), for initial and boundary conditions:

was given by Philip (1957) for q_o correspondent to saturated surface, in the form of a series expansion whose first three terms give:

where the local infiltration process, f(t), depends on time t with coefficients, sorptivity S and gravitational term c, equal to:

where  is an effective diffusivity over the range of possible moisture values [q_o,q_i] (e.g. Eagleson, 1970).

Integrating equation (5) with respect to time, the cumulated infiltration volume F(t) can be found as equal to:

Then, in the equations (5) and (8) the infiltration process is assumed on condition of saturated terrain surface and sorptivity depending on the soil moisture at time zero. Therefore they do not take into account such phenomena as interception, depression storage, and initial moisturizing of vadose zone. In order to account for them, it is possible to introduce a third quantity, the volume of initial abstraction W. Let t_b be the time at which W reaches its maximum value W_b, thus, putting t’ = t – t_b, we obtain

Equations (6) and (7) can be presumed to hold at the basin scale if parameters S, c and W are assumed as the spatial averages of their respective local values.

In order to evaluate the significant rainfall abstraction, the unit runoff occurring on a basin with surface area A may be related, by equation (2), to the maximum rainfall in the time interval t equal to the basin lag time and to the time where this interval drops into the time history defined by equation (9). Let t_i be the initial time of this time window and t_f its final time (t_f – t_i = t). According to the relative timing of the storm with respect to the F’(t) history, we may distinguish three significant cases.

Case 1: t_f < t_b

This case (Figure 1a) is representative of arid and semi-arid climates where, because of long lasting periods with no rains and high temperatures, the soil is likely to be very dry at the time of the storm. Consequently, at that time, W is close to zero. Therefore, in these climates, the majority of water losses during storms is due to the initial abstraction so that, according to equation (9), we obtain

                                   (10)

              Fig.1a  Precipitation and infiltration                            Fig.1b  Precipitation and infiltration
                          rate versus time; case 1                                             rate versus time; case 3

Results found by Fiorentino and Iacobellis (2001) over semi-arid basins in Southern Italy are in a significant agreement with the considerations presented above. In fact, the F_A values obtained in the investigated region show a quite constant pattern as in Figure 2.

Fig.2  Total abstraction loss in semi-arid basins of Southern Italy

Case 2: t_i t_b, t_f > t_b

This case is representative of dry-sub-humid and sub-humid climates where the water losses in the time window t (defined as above) are usually due to sorptivity (S in equations (5) and (6)) and, eventually, to the initial abstraction. Here, the behavior of water losses is also controlled by the quantity t’_i = t_i – t_b, that is in turn strongly affected by the rainfall antecedent to the storm. In fact, the greater the rainfall the higher the t’_i and the less, for a given area A, is F_A. One should note that the average antecedent rainfall depends on the yearly average number of storms and on their mean duration. In addition, the characteristic value of t’_i is also affected by the average evaporation rate between storms. All this quantities depend on climate and on its average characteristic. Yet, since the majority of, or the entire, time interval responsible for the flood peak, falls on the rapidly declining limb of the f(t) function, f_A depends on the area A too.

For instance, for t’_i = 0 we obtain

                           (11)

According to consolidated geomorphological theories that assume:

                               (12)

equation (11) leads to the individuation of a dependence of F_A on basin area.

On the other hand, for a generic t’_i, we get

                      (13)

which shows that F_A also depends on the characteristic values t’_i.

As already stated, t’_i is dependent on climate and it is possible to observe that the individuation of the dominant factor between t and t’_i seems to be sensitive to the way t’_i is correlated to climate.

In this case results provided by Fiorentino and Iacobellis (2001, see Figure 3) for climate from dry sub-humid to humid show F_A estimates strongly correlated to climate represented  by the climatic index (Thornthwaite, 1948):

                                    (14)

with h mean annual rainfall depth and Ep mean annual potential evapotranspiration.

Fig. 3  Total abstraction loss in dry-sub-humid and sub-humid basins of Southern Italy

Case 3: t_i >> t_b

This case, schematically depicted in Figure 1b is representative of humid and hyper-humid climates. Here, in fact, because of the very high number of rainy days, the rainfall responsible for the flood peak is likely to occur when the infiltration rate into the soil is close to the gravitational capacity. Therefore, neglecting the first term at the right-hand side of equation (8) and, following equation (12) we deduce

                                 (15)

Looking at equations (10), (12), (13) and(15), one could suggest to account for different processes of the global hydrologic loss, by means of a global expression:

                      (16)

where a₁, a₂ and a₃ are coefficients mainly depending on the spatial averages of initial abstraction, characteristic sorptivity and gravitational infiltration rate. In equation (16), a_1, a_2, and a₃ are weights, ranging from 0 to 1, whose relative value is a function of climate. In particular, the first of them tend to prevail in arid and semi-arid climate, while the last one is expected to be mainly active in hyper-humid climates. Thus, when the first term prevails, equation (16) reduces to equation (10), on a constant value. In the case of prevalence of the other terms, related to the infiltration rate through unsaturated soils, one would expect a stronger influence of climate, but also a scaling relationship with basin area particularly conditioned by the gravitational infiltration rate.

3    CONCLUSION

As a main results of this investigation, a significant and climatically consistent interpretation to the observed relationships observed between water losses, climate and basin area is provided. According to the presented arguments, it is possible to observe that in dry regions rain losses are mainly due to the initial abstraction phenomenon and the estimated value of areal averages of total water losses are constant over basins of different area. Such findings are consistently matched to what one could expect from Hortonian behavior of soils. Conversely, in humid basins, the small values of F_A, together with the relationship between total abstraction rate and climatic index, highlight a strong control of climate. Also in this case we underline the role of the gravitational infiltration rate while the basin’s behavior is consistent with Dunne’s model of runoff generation.

Acknowledgements

This work was supported by funds granted by the Consiglio Nazionale delle Ricerche-Gruppo Nazionale per la Difesa dalle Catastrofi Idrogeologiche and by the ECAPI Research Program, funded by MURST, Italian Government.

References

Eagleson, P.S., Dynamic Hydrology, McGraw-Hill Book Company, 1970.

Fiorentino, M., and V. Iacobellis, New insights about the climatic and geologic control on the probability distribution of floods”, accepted by Water Resources Research, in press, 2001.

Iacobellis, V., and M. Fiorentino, Derived distribution of floods based on the concept of partial area coverage with a climatic appeal, Water Resour. Res., 36(2), 469-482, 2000.

Philip, J.R., The theory of infiltration - Sorptivity and algebraic infiltration equation, Soil Sci., 84, 257-264, 1957.

Thornthwaite, C. W., An approach toward a rational classification of climate, Am. Geograph. Rev., 38, 1, 55-94, 1948.