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In this article, we examine the solution of the fuzzy linear vertical infiltration equation, which represents the water movement in porous media in that part which is called the vadose zone. This zone is very important for semi-arid areas, due to complex phenomena related to the moisture content in it. These phenomena concern the interchange of moisture content between the vadose zone and the atmosphere, groundwater and vegetation, transfer of moisture and vapor and retention of moisture. The equation describing the problem is a partial differential parabolic equation of second order. The calculation of water flow in the unsaturated zone requires the knowledge of the initial and boundary conditions as well as the various soil parameters. But these parameters are subject to different kinds of uncertainty due to human and machine imprecision. For that reason, fuzzy set theory was used here for facing imprecision or vagueness. As the problem concerns fuzzy differential equations, the generalized Hukuhara (gH) derivative was used for total derivatives, as well as the extension of this theory for partial derivatives. The results are the fuzzy moisture content, the fuzzy cumulative infiltration and the fuzzy infiltration rate versus time. These results allow researchers and engineers involved in Irrigation and Drainage Engineering to take into account the uncertainties involved in infiltration.

Vertical infiltration is a common physical phenomenon of water movement in porous media which is of great interest in many earth and plant sciences. Vertical soil-water flow plays an important role in understanding the phenomena of runoff, groundwater recharge, and transport of contaminants. Especially in the vadose zone, the soil moisture strongly influences the plants’ growing process. Historically, Buckingham [

∂ θ ∂ t = ∇ ⋅ ( K ∇ Φ ) (1)

where θ = the moisture content (cm^{3}/cm^{3}), K = the unsaturated hydraulic conductivity (cm/s) and Φ = the total potential (cm):

Φ = Ψ − z (2)

In Equation (2), Ψ = the pressure potential or capillary potential (cm) and z = the gravitational component (cm) and we adopt that z is taken positive downward. Introducing Equation (2) in (1) provides:

∂ θ ∂ t = ∇ ⋅ ( K ∇ Ψ ) − ∂ K ∂ z (3)

By introducing the diffusivity D (cm^{2}/s):

D = K ∂ Ψ ∂ θ , (4)

Equation (3) becomes:

∂ θ ∂ t = ∇ ⋅ ( D ∇ θ ) − ∂ K ∂ z (5)

and in the vertical dimension z:

∂ θ ∂ t = ∂ ∂ z ( D ∂ θ ∂ z ) − ∂ K ∂ z . (6)

The initial and boundary conditions are:

θ ( z , t ) | t = 0 = θ 0 , θ ( z , t ) | x = 0 = θ 1 , ∂ θ ( z , t ) ∂ z | z → ∞ , t > 0 = θ 0 . (7)

For θ 1 > θ 0 , Equation (6) with initial and boundaries conditions (7) describes the vertical infiltration of water if a constant moisture content at z = 0 is applied, as initially described by Philip [

Analytical solutions of the one-dimensional Equation (6) are available under several simplifications. Philip [_{1/2}. Parlange [

first integral ∫ θ θ S ∂ z ∂ t d θ of the transformed equation negligible compared to the

other terms. Subsequently, he developed an iterative method to solve the remainder equation. Philip [_{0}. In general, exact nonlinear solutions are derived for specific forms of the soil-water relationship [

The numerical methods—finite difference and finite element—presented in [

The calculation of water flow in the unsaturated zone requires the knowledge of the initial and boundary conditions as well as of the various soil parameters. Until today, these conditions and parameters were assumed well-defined, and this assumption is based principally in measurements. But they are subject to different kinds of uncertainty, due to human and machine imprecision. In many cases the uncertainties were considered in statistical terms as random variables with given mean values, variances and correlations. But these methods require the exact knowledge of mean values, variances and correlations, and often suffer from insufficient amount of accurate measurement data. For example, the accuracy of the linear distance between two points depends upon the precision of the location of the reference points. The precision rarely being perfect, dimensional limits would be imposed, often of the bilinear type ( x 0 − a i , x 0 − b i ) . This bilinear type set was assumed random and the probability theory was accepted valid. But randomness is an ideal tool only where a sufficiently long series of independent random experiments is available. In the cases where we have a small set of measurements, fuzzy set theory is ideal for formalizing incomplete information expressed in terms of fuzzy propositions with inherent vagueness. The same principle could clearly be extended to other applications, including non-geometric cases, such as chemical composition, machine registrations etc.

Today the fuzzy set theory provides methods for introducing imprecise information in a possibilistic sense. Zadeh [

In this paper, as is stated above, the case of linear vertical infiltration is studied, with imprecise boundaries conditions. The diffusivity is considered constant and the crisp problem is solved using the Laplace transform. For the fuzzy solution, the crisp solution is introduced first and then the problem is fuzzified. Then the problem is solved according to the theories presented in [

Equation (6) mentioned in the introduction, is called infiltration equation by Philip [

In the above equation, Philip [

D * = π S 2 4 ( θ S − θ r ) 2 , (8)

where S is the sorptivity [LT-1/2]. The linearized form of Equation (6) thus becomes:

∂ θ ∂ t = D * ∂ 2 θ ∂ z 2 − ∂ K ∂ z , (9)

with the same initial and boundary conditions:

θ ( x , t ) | t = 0 = θ r , θ ( x , t ) | x = 0 = θ S , ∂ θ ( x , t ) ∂ z | x → ∞ , t > 0 = θ r (10)

By writing the term ∂ K / ∂ z in the following form:

∂ K ∂ z = d K d θ ∂ θ ∂ z = k ∂ θ ∂ z , (11)

he has considered that k is constant by matching linear and non-linear values of infiltration rate ( lim t → ∞ v 0 ) and has obtained the value:

k = K S − K r θ S − θ r . (12)

In Equation (12), K_{s} = the hydraulic conductivity at saturation, K_{r} = the residual hydraulic conductivity, θ_{s} = the moisture content at saturation, and θ_{r} = the residual moisture content. He now poses in Equation (9):

Θ = θ − θ r , Θ 0 = θ S − θ r , d Θ = d θ (13)

and Equation (9) becomes:

∂ Θ ∂ t = D * ∂ 2 Θ ∂ x 2 − k ∂ Θ ∂ x (14)

with initial and boundaries conditions:

Θ ( x , t ) t = 0 = 0 , Θ ( x , t ) x = 0 , t > 0 = Θ 0 , ∂ Θ ( x , t ) ∂ x | x → ∞ = 0 (15)

The solution of this equation is ( [

Θ Θ 0 = 1 2 { erfc ( x − k t 2 D * t ) + e k x D * erfc ( x + k t 2 D * t ) } . (16)

The cumulative infiltration is:

I = K S t + 1 2 [ S t exp ( − K S 2 t π S 2 ) + 1 2 π S 2 K S erf ( K S S t π ) − K S t erfc ( K S S t π ) ] , (17)

while the infiltration rate is:

v 0 = K S 2 [ S K S t exp ( − K S 2 t π S 2 ) − erfc ( K S t S π ) ] + K S . (18)

Note: In order to facilitate the readers non-familiar with the fuzzy theory, we describe here some definitions concerning preliminaries of fuzzy theory and some definitions about the differentiability.

A fuzzy set U ˜ on a universe set X is a mapping U ˜ : X → [ 0 , 1 ] , assigning to each element x ∈ X a degree of membership 0 ≤ U ˜ ( x ) ≤ 1 . The membership function is also defined as μ U ˜ ( x ) with the properties:

1) μ U ˜ is upper semi continuous, 2) μ U ˜ ( x ) = 0 , outside of some interval [ c , d ] , 3) there are real numbers c ≤ a ≤ b ≤ d , such that μ U ˜ is increasing on [ c , a ] , decreasing on [ b , d ] and μ U ˜ ( x ) = 1 for each x ∈ [ a , b ] , 4) U ˜ is a convex fuzzy set (i.e. μ U ˜ ( λ x + ( 1 − λ ) x ) ≥ min { μ U ˜ ( λ x ) , μ U ˜ ( ( 1 − λ ) x ) } .

Let X be a Banach space and U ˜ be a fuzzy set on X. We define the a-cuts of U ˜ as [ U ˜ ] α = { x ∈ R | U ˜ ( x ) ≥ α } , α ∈ ( 0 , 1 ] , and for α = 0 , we define the closure [ U ˜ ] 0 = { x ∈ R | U ˜ ( x ) > 0 } .

Let Ҡ(X) the family of all nonempty compact convex subsets of a Banach space. A fuzzy set U ˜ on X is called compact if [ U ˜ ] α ∈ Ҡ(X), ∀ α ∈ [ 0 , 1 ] . The space of all compact and convex fuzzy sets on X is denoted as Ƒ(X).

Let [ U ˜ ] ∈ RF . The α-cuts of U ˜ , are: [ U ˜ ] α = [ U α − , U α + ] . According to representation theorem of Negoita and Ralescu [

Let

1)

Note:

Let _{0}. We say that (Bede and Stefanini, [

· _{0} if (i)

· _{0} if (ii)

Let

The gH-differentiability implies g-differentiability, but the inverse is not true.

A fuzzy-valued function

_{0}, t_{0}) if:

_{0}, t_{0}) if:

Let

We write Equation (14), in its fuzzy form as follows:

with the new initial and boundary conditions:

where

We can find solutions to the fuzzy problem (Equation (19)) and the initial and boundary conditions (Equation (20)), utilizing the theory developed in [

(1,1) System (1,2) System

(1,3) System (1,4) System

(2,1) System (2,2) System

(2,3) System (2,4) System

We will hereby restrict ourselves to the solution of the (1,1) system, which is described in detail.

(1,1) system

In

the spread r is equal to 0.15.

1^{st} case

Boundary conditions

Initial condition

By setting

with boundary conditions:

The solution of Equation (26) becomes:

The first derivative w.r.t. z is:

The variable B(s) should be equal to 0, in order to satisfy the boundary condition (Equation (27)):

So, Equation (28) becomes:

For the first condition for z = 0, we have:

and Equation (30) becomes:

Applying now the inverse Laplace transform [

2^{nd} case

Boundary conditions

Initial condition

In Equation (34) we set

with boundary conditions:

Applying the same process as in case 1, we have:

Finally, the fuzzy solution is:

In Equation (39) the fuzzy number

Existence statement of Equations ((21), (22))

Initial condition

Equation (39) satisfies the initial condition:

due to:

Boundary conditions

he first boundary condition is:

because it is:

The second boundary condition is:

We apply now the “L’Hospital Rule”

Thus, it is proven that the initial and boundary conditions of Equations ((23) and (34)) are satisfied.

Fuzzy derivatives

First Derivative of

First Derivative of

Second Derivative of C versus z:

We have now to prove:

where:

In the right part of Equation (41) we apply the theorem 1 of Bede and Gal [

By substituting in Equation (41) the above expressions of

As proven above, Equation (39) satisfies Equations ((23) and (34)), or their equivalent fuzzy Equation (19), provided that functions

In order to investigate the positivity or negativity of the above functions, we set in

where:

As derived from

The Vadose zone thickness is considered approximately 15 m and the values

Sampl | Soil texture | θ_{r} (cm^{3}/cm^{3}) | θ_{r} (cm^{3}/cm^{3}) | α | n | m | K_{s} (cm/min) | D (cm^{2}/min) | k (cm/min) | D/k (cm) | η |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | Sand | 0.05 | 0.43 | 0.15 | 2.68 | 0.63 | 0.50 | 61.28 | 1.29 | 47.66 | 0.03 |

2 | Loam | 0.08 | 0.43 | 0.04 | 1.56 | 0.36 | 0.02 | 4.75 | 0.05 | 96.6 | 0.06 |

3 | Silt | 0.03 | 0.46 | 0.02 | 1.37 | 0.27 | 0.00 | 1.47 | 0.01 | 148.9 | 0.1 |

4 | Silty loam | 0.07 | 0.45 | 0.02 | 1.41 | 0.29 | 0.01 | 2.58 | 0.02 | 131.6 | 0.09 |

5 | Clay loam | 0.10 | 0.41 | 0.02 | 1.31 | 0.24 | 0.00 | 1.43 | 0.01 | 104.62 | 0.07 |

6 | Sandy loam | 0.07 | 0.41 | 0.08 | 1.89 | 0.47 | 0.07 | 14.09 | 0.21 | 65.68 | 0.04 |

7 | Loamy sand | 0.13 | 0.40 | 0.04 | 2.55 | 0.61 | 2.70 | 1866.05 | 10.0 | 186.65 | 0.12 |

8 | Sandy loam | 0.11 | 0.35 | 0.05 | 4.60 | 0.78 | 1.04 | 722.01 | 4.4 | 163.92 | 0.11 |

9 | Sandy loam | 0.05 | 0.42 | 0.05 | 3.45 | 0.71 | 2.60 | 1188.85 | 7.02 | 169.29 | 0.11 |

of η are derived from:

It is derived as a conclusion that functions

Fuzzy Infiltration rate and fuzzy cumulative Infiltration

The fuzzy infiltration rate is:

and the fuzzy cumulative infiltration is:

The Fuzzy model was applied to 3 selected soils. These soils are shown in

For the first case the following are valid:

where

In

For the second case the following are valid:

where

In

illustrated in real times t = 5, 10, 30 and 60 min at z = 70cm. In

For the third case the following are valid:

where

In

Important Remark

As pointed out in introduction the linearized equation of Philip does not give accurate detailed description of flow profiles. However, this linear equation yields useful estimates of integral properties of cumulative infiltration I and of infiltration rate v_{0}. In order to evaluate this property, we used the Valiantzas model [

^{−5}.

The Bede and Stefanini [

Vertical infiltration linear equation regarding the water movement in vadose zone has a fuzzy solution with a function

The fuzziness of soil water movement diminishes as flow moves up in vertical direction. The fuzziness of cumulative infiltration rises vs time and the fuzziness of infiltration rate diminishes vs time.

Since there exist no previous treatments of the problem of vertical infiltration with Fuzzy Logic, comparison of the results of the present work is only possible between crisp solutions and the fuzzy solution. The difference between crisp solution (CS) and fuzzy solution (FS) appearing in Figures 4-15, remains constant

irrigation systems to be calculated with crisp solution, without considering the fuzziness of the problem. It is important for engineers and researchers to take into account uncertainties of such magnitude in order to proceed to decision making. More specifically, in problems of Irrigation and Drainage Engineering, the related design of irrigation and drainage networks can be more accurate if the possible lower and higher limits of the water-front are known beforehand.

As is pointed in the remark, the linearized solution estimates well the phenomena of cumulative Infiltration and Infiltration rate for real soils.

Conceptualization, C.T.; methodology, C.T. and G.P.; validation C.T. and K.P.; writing, C.T., G.P. and K.P.; review and editing, G.A. and C.E.; supervision, C.T. and C.E.

The authors declare no conflicts of interest regarding the publication of this paper.

Tzimopoulos, C., Papaevangelou, G., Papadopoulos, K., Evangelides, C. and Arampatzis, G. (2020) Fuzzy Analytical Solution to Vertical Infiltration. Journal of Software Engineering and Applications, 13, 41-66. https://doi.org/10.4236/jsea.2020.134004