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Spring recession flows are analyzed from a Bayesian point of view. Two general equations are derived and it is shown that the classical formulas of recession flow are particular cases of both equations. It is shown that most of the recession equations reflect a non-Markovian process. That means that the groundwater storage exhibits a memory effect and that there is a nonlinear relationship between flow and storage. The Bayesian approach presented in this paper makes it possible to give a probabilistic meaning to recession flow equations derived according to a physical approach and can be an alternative to the study of complex reservoir for which the physical processes governing recession flow are unclear. Twelve spring recession flow series are analysed in order to validate the probabilistic approach presented in this paper and a conceptual model of storage-outflow is proposed.

The description of recession can have a variety of applications in hydrologic studies, including the evaluation of aquifer properties and the sustainability of ground-water discharge. Low flow characteristics have been increasingly utilized in recent years as the demand for water has increased. Information on low flow characteristics provides threshold values for different water-based activities and is required for water resource management issues such as water supply, irrigation, and water quality and quantity estimates. An understanding of the outflow process from groundwater or other delayed sources is also essential in studies of water budgets and catchment response.

The most commonly used method of modelling baseflow recession is to use a linear store. This method has a long history, and was first noted in the literature by Boussinesq in 1877 [

Although the linear model of recession can be a reasonable approximation, there are a number of processes that will affect recession-curve shape, some causing departures from linearity [

Aksoy, Bayazit and Wittenberg [

Increasing with Δt but decreasing with t, according to Bayes’s theorem:

the following probabilistic equations can be deduced:

F is the cumulative distribution function (CDF) of the recession flow Q(t).

Let’s consider that the conditional probability is only proportional to the time interval Δt. This condition is not sufficient because the flow interval, which increases with the time interval, also decreases with time.

If Δt tends toward dt, the solution of Equation (5) is Boussinesq’s equation [

Q_{0} is the initial flow at t = 0.

Let’s consider that the conditional probability is proportional to the time interval and inversely proportional to time. This case reflects the behaviour of the recession flow

with

At t = 0,

And

If x = 0 and b = 0, Equation (9) gives Maillet’s formula [

Let’s consider that x = 1 in Equation (7):

If Δt tends toward dt, the solution of Equation (10) is:

As

Equation (12) is similar to the equation derived by Wittenberg [

With

Equation (13) is related to a power-law reservoir which is suitable for springs, unconfined aquifer and soil moisture.

Equation (12) is similar to Boussinesq’s equation [

With k/a = 2. Equation (14) is suitable for shallow unconfined aquifer.

Equation (12) is similar to the equation of Griffiths and Clausen [

With k/a = 3. Equation (15) is suitable for surface depression storage such as lakes and wetlands.

Equation (12) is similar to:

With k/a =1 and

Equation (12) is similar to Coutagne’s equation [

With

Equation (12) is similar to the following equation related to underground caverns reservoir [

With k/a = ‒1,

Do these equations reflect a Markovian process or not? In other words, does the future flow depend on the former or not? The answer can be found by the following equation:

By combining Equations (6) and (19), it can be found that for Maillet’s equation:

For Equation (9):

For Equation (12):

The process is also not Markovian.

The non-Markovian character of Equations (9) and (13) means that the groundwater storage exhibits a memory effect.

Twelve spring recession flow series (

With

The computed results of Equations (12) and (23) have been compared with the measured data of the springs. The results are recapitulated in

Name | State | Duration (day) | Q_{0} (m^{3}/s) | Equation (12) | Equation (23) |
---|---|---|---|---|---|

Bennet spring site 06923500 | Missouri | 98 | 40.2286 | r^{2} = 0.97 a = 20.793 λ = 0.332 | r^{2} = 0.93 β = 1.233 λ = 0.1578 |

Big spring site 07067500 | Missouri | 97 | 25.893 | r^{2} = 0.99 a = 84.2017 λ = 0.116 | r^{2} = 0.97 β = 0.603 λ = 0.12 |

Chesapeak spring site 06918444 | Missouri | 19 | 0.3116 | r^{2} = 0.99 a = 4.1527 λ = 0.289 | r^{2} = 0.97 β = 0.566 λ = 0.275 |

Crnojevica | Bosnia Herzegovina | 14 | 94.7 | r^{2} = 0.993 a = 1.218 λ = 1.281 | r^{2} = 0.98 β = 1.346 λ = 0.386 |

Malibert Monts du Pardailhan (Hérault) | France | 7 | 1.23 | r^{2} = 0.985 a = 0.686 λ = 0.2869 | r^{2} = 0.972 β = 0.1658 λ = 0.5793 |

Poussarou Monts du Pardailhan (Hérault) | France | 8 | 1.7 | r^{2} = 0.9987 a = 0.48 λ = 0.8441 | r^{2} = 0.999 β = 0.394 λ = 0.591 |

Mill spring site 03494500 | Tennessee | 207 | 0.538 | r^{2} = 0.996 a = 0.03835 λ = 1.03 | r^{2} = 0.988 β = 0.149 λ = 0.511 |

Annie spring site 11503000 | Oregon | 54 | 0.1388 | r^{2} = 0.99 a = 0.1338 λ = 0.496 | r^{2} = 0.995 β = 0.0935 λ = 0.621 |

Big spring fish hatchery site 05411950 | Iowa | 30 | 1.218 | r^{2} = 0.914 a = 2137.88 λ = 0.124 | r^{2} = 0.906 β = 0.977 λ = 0.1 |

Silver spring site 02239500 | Florida | 127 | 12.861 | r^{2} = 0.984 a = 0.01685 λ = 0.44 | r^{2} = 0.97 β = 0.0106 λ = 0.808 |

Sulphur spring run site 02306000 | Florida | 14 | 0.991 | r^{2} = 0.965 a = 1.67 λ = 0.127 | r^{2} = 0.97 β = 0.1554 λ = 0.37 |

Fay spring site 01616075 | Virginia | 23 | 0.0793 | r^{2} = 0.993 a = 1.264 λ = 0.2153 | r^{2} = 0.991 β = 0.2313 λ = 0.373 |

is illustrated by

For each spring, the volume of water released during the recession duration can be computed by:

Equation (24) is a discrete sum. T is the duration of the recession. The volume released can also be computed in a continuous point of view by:

The Euler gamma function satisfies:

The incomplete gamma function satisfies:

Equation (26) has a limiting value when the duration t is important (mathematically, when the duration t approaches infinity):

Equation (25) also has a limiting value when the duration is important and when λ > 1:

On the other hand, if λ < 1, Equation (25) does not have a limiting value when the duration t is important:

That means if Equation (25) with λ < 1 is used for computing the released volume by the spring and if the recession duration is important, the error made on the calculation of this volume can be large. In this case, Equation (26) is more suitable than Equation (25) although it is a little less powerful to simulate the recession flow.

In a general way, Equation (12) is a little more powerful than the Equation (23) to simulate the recession flows. However, for the calculation of released volume, in the case of a long period of recession, the Equation (26) will have to be used because if the parameter λ of the Equation (12) is less than one, the integral of Equation (12) (Equation (25)), mathematically speaking, does not have limiting value if λ < 1.

It was shown in 1.3 that Equation (13) was similar to the Equation (14) derived by Wittenberg [

He suggested a value of b = 0.5 for average conditions even if the true value of b is not exactly met. The assumption of b = 0.5 would be more physically based and better fitting for the majority of river basins than the linear reservoir.

Equation (14) is related to the following power-law reservoir:

It is obvious that if λ < 1, Equation (13) can not be related to the power-law reservoir described by Equation (30) because b should be negative.

The volume released is computed by Equation (25) which can be rewritten as:

If S_{0} is the initial volume of groundwater in the aquifer, then, at time t, the groundwater storage is:

A general storage-outflow relation can be expressed by:

With

The conceptual model expressed by Equation (33) explains the recession flows of the springs for which the exponent l of Equation (13) is less than one.

A simple analysis of a classical recession curve leads to conclude that this natural phenomenon is generally not a Markovian process. That means that a future value of the recession flow depends on the former and that ground- water reservoir exhibits a memory effect. The Bayesian approach makes it possible to give a probabilistic mean- ing to recession flow equations derived according to a physical approach.

The probabilistic approach can lead to derive general equations of recession flow which can be transposed to complex reservoir for which the physical approach could be difficult to use.

The logical continuation of this work would be used to determine the contributions which the probabilistic approach could have for river recession flow which is more complicate to model than spring recession flow because the recession curves are related to overland flow then to subsurface flow and, finally, to groundwater flow. For advances in recession analysis, the probabilistic approach can be an alternative when the physical processes governing recession flow are unclear.