Structuring an Ontology for Mathematical Modelling Wastewater Flow through Porous Media

Single or multi-layer infiltration rate models describe water penetration into soil porous zone at a variety of comparable complexity levels. Commonly, those models are indicating a route leading from general to more specific case depending on the mathematical structure and the simulated ongoing phenomenon. For the purpose of wastewater flow quantification through porous media, an algorithmic procedure was developed which includes certain interconnected activity stages and decision nodes. Furthermore, a schematic cross shaped representation of the infiltration rate models’ ontology is presented within a route, leading from general to more specific models and evaluation criteria are introduced to determine highest ranking and thus the best model amid those placed on the horizontal axis of the cross shaped scheme, via a simple Multi-Criteria Analysis Decision Making.


Introduction
This Complexity is the main characteristic of mathematical models that simulate wastewater flow through porous media either under laboratory/industrial conditions or in the Nature, where soil pollution takes place. By increasing complexity, we can 1) improve the explainability of the dependent variable, and 2) achieve parameters identification at deeper phenomenological levels; on the other hand, applicability difficulties and information acquisition/processing/ storing/retrieval cost also increase. Since such issues imply a tradeoff, we can determine the optimal value of a complexity index, including the number of independent Variants Parameters, Coefficients (VPCs) and the phenomenological knowledge depth, representing numerical and structural complexity, respectively.
Numerous models are not independent to each other but interconnected as "members" of the same category or "rings" in the same chain, representing a derivation path. Given in such a mode they build up an Ontology structure in terms of modern Informatics and Logic/Methodology of Science.
Ontologies entails concepts suitably bound by well-defined relationships in specific scientific fields which are applied in artificial intelligence in order to provide to all users an interaction framework with numerous application systems i.e. communication models between (KB) users and machines [1]. Such structures appear to form an entity with concepts, relations and controlled vocabulary, while logical operators and taxonomy/partonomy functions are used for processing the relevant knowledge.
Modeling application methodology can be improved by utilizing ontological structures for all stages of model building. This paper explores mechanisms by setting up algorithmic procedure as the path to build up an Ontology of mathematical models to approach wastewater flow quantification through porous media. Thus, it is manageable ontological entities to be incorporated into Knowledge Bases (KBs) in knowledge engineering over infiltration modelling discipline that combined with artificial intelligence and expert systems, information can be easily retrieved and applied in a variety of applications [2] [3] [4].

Methodology
For structuring an Ontology of mathematical models to quantify wastewater flow through porous media an algorithmic procedure was designed/developed which includes the following 30 activity stages and 8 decision nodes while the interconnection is shown in Figure 1  9. Determination of n criteria to be used for choosing the best mathematical model from the set of alternatives selected in stage 4.
10. Assignment of grades onto the elements of the multi-criteria n × m preference matrix (by using experts' opinion) in fuzzy version to count for uncertainty. 11. Performance of multi-criteria analysis.
12. Experimental validation of the proposed mathematical model. 13. Scale-up to the required level and estimation of the tolerance limits expected to constrain applicability.
14. Sensitivity-robustness analysis by changing the values assigned to the elements of the criteria vector and the preference matrix.  B. Is at least one of them dimensionally inhomogeneous? C. Is it valid to preset confidence intervals at the required significance level? D. Are the numerical results of sensitivity -robustness analysis within the range corresponding to the tolerance limits? E. Is it considered to be necessary to minimize p further without losing the initial information (i.e. decreasing the information granularity level) required to describe adequately this phenomenon?
F. Is splitting of at least one primary dimension indispensable in order to enhance explainability-predictability?
G. Is SEE within the pre-determined range?
H. Is there at least one complete set remained unexamined so far? Considering e.g. water penetration into soil porous zone, numerous single or multi-layer infiltration rate models can be employed to approach water infiltration development at comparable complexity levels.
Infiltration rate models are indicating a route leading from general to more specific ones according to the mathematical structure and simulated operation. Hereunder, certain infiltration models are presented which simulate infiltration phenomena through porous media.
Green-Ampt (1911) (3.1 in Figure 2) Simple model for one-dimensional vertical infiltration through ponded homogeneous soil of uniform antecedent moisture content. The water flow in the saturation zone was caused by constant soil water suction at the wetting front and gravity of soil water. Considering that the depth of the ponding is negligible [8]: where K S is considered to be the saturated hydraulic conductivity,   Simple 2-stage model for infiltration prediction before and after surface ponding initial Uniform moisture content assumption under rainfall with a constant intensity [11].
Case 1: infiltration prior to runoff and the time to the beginning of runoff, with I ≥ K s Case 2: infiltration after runoff begins where: K is the capillary conductivity θ dependent, [LT −1 ] k r is the relative conductivity (dimensionless) Salvucci & Entekhabi (1994) (3.5 in Figure 2) Model applicable for homogeneous soils, uniform antecedent water content, non-zero or constant ponding depth [12]. Green-Ampt model extension, infiltration process into layered soils with a decreasing hydraulic conductivity from the soil surface moving downwards. Constant ponding depth. Infiltration rate estimation at which the wetting front reaches some location in the nth layer [13]. into a multi-layered soil during unsteady rain. When surface ponding is occurring from the beginning of a rain event and is continuing, while the wetting front penetrates the mth soil layer [15]: where: Extended form of the original Green-Ampt model for infiltration into layered, non-uniform soils [16]. The infiltration models depicted on horizontal axis of schematic representation in Figure 2 shall undergo evaluation by means of a simplified Multi-Criteria Decision Making (MCDM) methodology [3]. Six criteria determine the final score accumulation and thereof the final ranking (see Table 1). The final ranking of the available infiltration models leads to the optimal final selection as the 25 th activity stage described in Chapter 2. Each criterion has its own gravity (weight) (see 2 nd column of Table 2). Criterion of higher weight value entails more influential impact on the final ranking. The overall value estimation is in fact the best model performance in a tradeoff evaluation. During the (MCDM) procedure an evaluation of each model over a criterion is carried out and a grade within 0 -10 climax was given. The higher the grade the better conformity of the model to the criterion direction. The selected criteria are given below in Table 1.
The scores presented in Table 2 form a 6 × 5 evaluation matrix ( k l × J ). where: 1, 2, , 6 k =  1, 2, ,5 l =  , k l a represents the score achieved each model to fulfill the corresponding criterion and is acquired by author's empirical knowledge after scrutinizing all relative publications. The values of J are already normalized since the evaluation range was set out to be between 0 -10.