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The objective of this study was to propose a mathematical regression model to estimate the exploitation flow rate of a water borehole from geophysical parameters in the midst of a fissured basement in the central-eastern part of C
ôte d’Ivoire. The data of the study are both hydrogeological and geophysical parameters from one hundred and eleven (111) data sheets of (111) water and geophysical boreholes. Two methods were used. The Normal Principal Component Analysis (NPCA) method applied to the data made it possible to select the explanatory variables (geophysical parameters) for borehole productivity. The multiple linear regression method subsequently made it possible to propose a mathematical model capable of estimating the exploitation rate from the geophysical parameters. The results indicate a very strong correlation (0.87) between longitudinal conductivity and flow rate, with flow rate and apparent resistivity negatively correlated. The multiple linear regression method highlighted two relevant explanatory variables, longitudinal conductivity and apparent resistivity. These two geophysical parameters contributed to a mathematical model in the form
*Q* = *C*_{1}*X*_{1} + *C*_{2}*X*_{2} + ... + *C*_{n}*X*_{n} + *C*_{0}. the real model obtained in this work is
*Q* = 0.82*Cl* - 0.12*Rho.app* + 2.5. The resulting model is efficient with a correlation of 86% in calibration and 95% in validation. A bias of 0.37 in calibration and 0.82 in validation is observed. Finally, the square root mean square error (RMSE) is 3.10 to 3.38 respectively in calibration and validation.

Access to water and sanitation is one of the major challenges of the 21st century. According to the WHO [^{3}, of which 35.5 billion m^{3} are renewed each year. Despite the availability of this resource, drinking water supply remains a concern. According to the National Drinking Water Office (ONEP) in 2018, the drinking water coverage rate was 61 percent in urban areas, 76 percent in rural areas and 13 percent in semi-urban areas. This disparity is said to be linked to a lack of a serious and in-depth study on the productivity of water drilling in areas of cracked basement. In general, the drinking water supply of populations in rural and semi-urban areas is provided by basement aquifers.

Indeed, basement aquifers are formed from fractures created in the bedrock by tectonic constraints and weathering [

It is therefore necessary to use complementary methods, in particular non-destructive geophysical methods [

The general objective of our research is to propose a mathematical model capable of estimating the flow rate of a water borehole based on geophysical parameters in a cracked basement environment.

The study area is located in the central-eastern part of Côte d’Ivoire between longitudes 3˚40' and 4˚55' West and latitudes 6˚20' and 8˚10' North and includes nine departments. It includes the N’zi, Iffou and Moronou regions (

The relief of the study area is dominated by a series of hills that rise to 300, 400 and even 500 m [

The study area has a transitional equatorial climate. It is particularly hot and relatively dry with two rainy seasons from April to July and October to November and two dry seasons from December to March and August to September. Winds are generally light. Only the harmattan is important especially in the northern part of the region, but its influence decreases rapidly towards the south. However, with climate change, this regime is now disrupted.

The search for water in areas where the basement is crystalline or crystallo-phyllized is limited to areas where the rock has undergone the phenomenon of fracturing. The fractured environment is heterogeneous and its hydraulic characteristics are determined by the geometry and density of the cracks. The hydraulic continuity of this hydrogeological system depends on the interconnection of the cracks [

The data used in this research was obtained from the technical files of 111 boreholes drilled in 2010 within the framework of the (West African Monetary Economic Union) UEMOA projects in the N’Zi, Iffou and Moronou Regions and within the framework of the 2013 Presidential Emergency Programme (PPU) in the Iffou Region (

Two hydrogeological parameters, i.e. the operating flow (Q) and the specific flow (Qs), and four geophysical parameters, such as longitudinal conductance (Cl), transverse resistance (Rt), apparent resistivity (Rho.app) and simulated alteration thickness (Eas), were analysed. The method used to determine the specific flow rate (Qs) is based on the following simple Equation (1).

Q s = Q d s (1)

where Qs is the Specific flow rate in m^{3}/h/m, Qd is the flow rate of the last pumping stage in m^{3}/h and s, the drawdown at the end of the 4 hours of pumping.

The relationships between hydrogeological and geophysical parameters in the study of aquifers are established through the work [

F = ρ ρ w (2)

F = ϕ − m (3)

where, F is the formation factor, ρ is the rock resistivity in ohm∙m, ρ w is the resistivity of formation water in ohm∙m, ϕ is the total porosity of the formation, m is the cementing coefficient of the formation. The cementing factor depends on the permeability of the rock and therefore on the fracturing density for magmatic rocks. The knowledge of these hydrogeophysical relationships has led to the establishment of the following expressions [

C l = ∑ i = 1 n b i ρ i (4)

R t = ∑ i = 1 n b i ⋅ ρ i (5)

where Cl is the longitudinal conductance, b_{i} are the layers thicknesses, ρ_{i} are the layers resistivity and Rt the transverse resistance.

Hydrogeological and geophysical parameters from the analysis of drilling data and geophysical studies were subjected to a standardized principal component analysis (SCA). The NPAC, which is a statistical tool, defines the main factors whose correlation with the variables allows an explanation of the phenomena involved. In order to assume that the phenomenon involved is sufficiently expressed, the cumulative sum of the contributions of the main factors must be about 70% [

In addition, multiple linear regression was used to predict the values of a dependent variable from explanatory or independent variables [

Y = C 1 X 1 + C 2 X 2 + ⋯ + C n X n + C 0 (6)

where Y is the variable explained, X_{i} is the explanatory variable, C_{0} is the constant, C_{i} ( 1 ≤ i ≤ N ) the weighting coefficients of the explanatory variable X_{i}.

Indeed, Y is a vector of observed values of water drilling flow, X_{i} is a matrix of independent or explanatory variables, C_{i} is a vector of parameters or regression coefficients to be estimated, and C_{0} is a vector of residuals or random disturbances. Linear regression estimates the vector C_{i} as a least squares solution [

C i = ( X i T X i ) − 1 ( X i T Y ) (7)

with X^{T} the transpose of X.

Multiple regression is a variant of the simple regression method that can help deal with collinearity by iteratively choosing the variables with the highest explanatory value. An ascending multiple regression starts with no variable, or a subset of the available variables, and adds the most significant variable (the one with the lowest p-value, combined with estimated F-statistics) at each step of the model. A stepwise downward regression starts with all available variables and removes the least significant variable at each step. This is the second form of multiple regression that was chosen for this study.

One of the most relevant steps in the development of a model is the estimation of its parameters [

1) Evaluation of model performance

The performance of the models and their robustness were evaluated using numerical and graphical criteria. The analysis of the simulation results focuses on the performance of the models in the calibration and validation phases. Indeed, according to [

R = ∑ i ( Q i − Q ¯ ) 2 × ( Q ′ i − Q ¯ ′ ) ∑ i ( Q i − Q ¯ ) 2 × ∑ i ( Q ′ i − Q ¯ ′ ) 2 (8)

with, Q i are the measured flow rate, Q ′ i are the simulated flow rate, Q ¯ i are the average of measured flows and Q ¯ ′ i the average of simulated flows.

The relation is said to be perfect if R = 1, very strong if 0.8 ≤ R < 1, strong if R is between 0.5 and 0.8, medium if R is between 0.2 and 0.5, weak if R is between 0 and 0.2 and nil if R = 0. The square root mean square error (RMSE) is used as the measure of the overall performance of the model. The model is well optimized if the RMSE value is close to zero, which tends towards a perfect forecast. Its mathematical formulation is given by the following relationship Equation (9).

∑ i n ( Q i − Q ′ i ) 2 n (9)

with Q_{i} the Observed flow, Q ′ i are the simulated flow, n is the sample size.

Bias is a criterion for highlighting the difference between two quantities. It must be minimized (the optimum is the null value). It then gives the relative error between the observed values and those simulated during the analyses. When the biased mean “B” tends towards zero, the model results are unbiased, i.e. the two values are close and therefore the model is efficient. This parameter is defined by the following relationship Equation (10).

B = 1 n ∑ n = 1 i | Q i − Q ′ i | (10)

where Q_{i} are the observed flow rate, Q ′ i are the simulated flow.

In addition to the numerical evaluation, a graphical analysis comparing the observed flows to the simulated flows was carried out in order to assess the quality of the modelling carried out. In practice, if the simulation was perfect, i.e., if each of the values simulated by the model was equal to the observed value, the resulting scatterplot would be aligned and merged with the line of equation y = x. However, since the modeling is not perfect, the qualitative assessment of the performance of the different models consisted in assessing the dispersion of the scatter of the scatterplot around the first diagonal.

2) Assessment of model robustness

One of the most widely used techniques to assess the robustness of a model is the double-sampling technique [

R ′ = 100 × | R validation − R calage | (11)

Six parameters have been presented in the parameter characteristics

1) Analysis and Interpretation of Own Values

The analysis of

Paramètres | Number | Minimum | Maximum | average | Ecart-type | Cv (%) |
---|---|---|---|---|---|---|

Apparent resistivity (rho.app) | 111 | 42 | 2050 | 435.80 | 459.7 | 105.48 |

Longitudinal conductance (Cl) | 111 | 0.000806 | 3.51 | 0.53 | 0.80 | 150.94 |

Transversal resistance (Rt) | 111 | 762.35 | 199,694,050 | 2,861,528 | 20,937,715 | 731.69 |

Calculated weathering thickness (Ep.cal) | 111 | 4.22 | 100 | 38.73 | 20.13 | 51.97 |

Flow rate (Q) | 111 | 0.38 | 43.20 | 5.43 | 6.67 | 122.83 |

Specific flow (Q/s) | 111 | 0.02 | 4.09 | 0.45 | 0.72 | 160 |

Factors | Eigenvalue % | Variance Cumulative % | Eigenvalue Cumulative | Cumulative Variance |
---|---|---|---|---|

Factors 1 | 2.93 | 48.83 | 2.93 | 48.83 |

Factors 2 | 1.17 | 14.85 | 4.10 | 68.37 |

Factors 3 | 0.89 | 11.56 | 4.99 | 83.22 |

2) Factor Plan F1 - F2 and F1 - F3

3) Analysis of the correlation matrix between the variables

Analysis of the correlation matrix in

Variables | Qal | Q/s | T | Rho.app | Ea.cal | Cl | Rt |
---|---|---|---|---|---|---|---|

Qal | 1.00 | 0.85 | 0.70 | −0.45 | −0.04 | 0.87 | −0.05 |

Q/s | 0.85 | 1.00 | 0.66 | −0.34 | −0.03 | 0.79 | −0.04 |

T | 0.70 | 0.66 | 1.00 | −0.30 | 0.01 | 0.53 | −0.04 |

Rho.app | −0.45 | −0.34 | −0.30 | 1.00 | −0.11 | −0.41 | 0.05 |

Ea.cal | −0.04 | −0.03 | 0.01 | −0.11 | 1.00 | 0.02 | 0.16 |

Cl | 0.87 | 0.79 | 0.53 | −0.41 | 0.02 | 1.00 | −0.05 |

Rt | −0.05 | −0.04 | −0.04 | 0.05 | 0.16 | −0.05 | 1.00 |

1) Design of multiple linear regression models

The models were designed using the backward or backward elimination regression method. The results are presented in ^{2} = 0.78). This step shows us that longitudinal conductivity has an important weight (82%) in the prediction of borehole flow rates. Next comes the apparent resistivity, which influences the productivity estimate by 12%. However, transverse resistance and calculated alteration thickness have no significant influence (0% and 4% respectively). Next, step 1 gives us 3 parameters, namely longitudinal conductance, apparent resistivity and calculated thickness of alteration. The transverse resistivity being very insignificant (0.0004) in the prediction of productivity was rejected in step 2. The weights of the variables remain the same in this step as well as the multiple R and the R. The standard error varies slightly at the C_{0} constant from step 0 to step 1. Finally, step 2 includes two explanatory variables which are the apparent resistivity with an influence weight of (−0.12) or (12%) and the longitudinal conductivity with a much larger influence weight of (0.83) or (83%). These values show that the longitudinal conductivity can help predict the flow rate of a borehole. Also, the R^{2} of 78% means that the apparent resistivity and the longitudinal conductivity are likely to predict the productivity of a borehole. The set of standard errors is less than 1%. These results express a good relationship between the flow rate (explained variable) and the explanatory variables (apparent resistivity and longitudinal conductance).

Step | Explanatory variable | Coefficient (C_{i}) | R | R² | Error-type |
---|---|---|---|---|---|

0 | Constante (C_{0}) | 2.88 | 0.88 | 0.78 | 0.70 |

Rho.app | −0.120 | 0.0007 | |||

Ep.cal | −0.040 | 0.009 | |||

Cl | 0.82 | 0.43 | |||

Rt | 0.0004 | 0.000 | |||

1 | Constante (C_{0}) | 2.88 | 0.88 | 0.78 | 0.69 |

Rho.app | −0.120 | 0.0007 | |||

Ep.cal | −0.040 | 0.009 | |||

Cl | 0.82 | 0.42 | |||

2 | Constante (C_{0}) | 2.50 | 0.88 | 0.78 | 0.55 |

Rho.app | −0.12 | 0.0007 | |||

Cl | 0.83 | 0.42 |

2) Results of multiple linear regression models

Following the analysis of

- Model 1 with the parameters longitudinal conductance (Cl), apparent resistivity (Rho.app) and calculated alteration thickness (Ep.cal). It is expressed in the form of the following equation.

Q = 0.82 C l − 0.12 R h o . a p p − 0.04 E p . c a l + 2.88 (12)

- Model 2 with the parameters longitudinal conductance (Cl) and apparent resistivity (Rho.app). It is expressed in the form of the following equation

Q = 0.82 C l − 0.12 R h o . a p p + 2.50 (13)

1) Results of the quality of the models developed through performance

Three evaluation criteria were selected to verify the performance of the model. Only Model 2 was subjected to the evaluation criterion after the top-down elimination regression test.

2) Results of the quality of the models developed using robustness

The robustness of the model was tested by the double sample technique. This technique allowed us to determine the robustness criterion R’, which is 9%. This value allowed us to test the robustness of the developed model because R’ is less than 10%. These results confirm those obtained in

3) Graphical validation of the developed model

The developed model was tested graphically by a correlation study betweenthe observed and simulated flow in calibration and validation. ^{3}/h. Above 12 m^{3}/h, we observe a dispersion of values around the equation line. However, this dispersion is important in

The study of the prediction of drilling productivity from geophysical parameters was carried out by analyzing six parameters, including four geophysical parameters

Evaluation Criteria | Calibration performance | Validation performance |
---|---|---|

Model 2 | Model 2 | |

R | 0.86 | 0.95 |

Biais (B) | 0.37 | 0.82 |

RMSE | 3.10 | 3.38 |

(longitudinal conductance, transverse resistance, apparent resistivity and calculated alteration thickness) and two hydrogeological parameters (flow, specific flow) that influence productivity. Indeed, authors such as [

Prediction of water drilling productivity based on surface geophysical parameters was done in a methodical manner. First, the choice of the hydrogeological parameters governing borehole productivity was made. Thus, two parameters, namely the flow rate and the specific flow rate, were selected as explanatory variables. But the flow rate being the parameter observable after a hydrogeological drilling was retained as the explained parameter. Also, four geophysical parameters such as longitudinal conductance, transverse resistance, calculated alteration thickness and apparent resistivity were analyzed and retained as explanatory parameters. Similarly, the study of coefficients of variation gave values greater than 100% except for the non-calculated alteration thickness, which is 51.97%. In view of these results, we can say that the parameters studied are highly variable in the research area.

Finally, a model for estimating drilling productivity was established based on surface geophysical parameters. This model allowed us to understand that the longitudinal conductance with a weighting coefficient of 0.86 and the apparent resistivity with a weighting coefficient of (−0.12) can be used to simulate a drilling rate after a geophysical study. To validate this model, we performed the performance test by determining the correlation R to show the link between the observed and simulated flows. The values of R = 0.86 in calibration and 0.95 in validation obtained show that there is a very strong correlation between the simulated and observed flows. The values of Bias B (0.37 in calibration and 0.82 in validation) tend towards zero, they also confirm the performance of the developed model. Another confirmation, that of the robustness test gave an R’ value of 9%. This also confirms the efficiency of the developed model. After this study, we retain that the developed model can predict the productivity of a water well after a geophysical study.

A.K.S.K., O.M., O.I., and I.S., designed the study, developed the methodology and wrote the manuscript. A.K.S.K., collected the data and performed the analysis; O.M., O.I., and I.S., supervised the data analysis; O.M., contributed to paper write up.

This paper was extracted from a Doctoral research study undertaken at University of Nanguy-Abrogoua (UNA), in Ivory Cost. My sincere acknowledgements go to the Ministry of Superior Education and Scientific Research (MSESR) for providing the scholarship. Finally thanks to JWARP for its collaboration and the publication of this article.

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

Kouadio, A.K.S., Moussa, O., Ouattara, I. and Savane, I. (2020) Use of a Multiple Regression Model in the Estimation of Water Borehole Flows in the Middle of Cracked Basement in Côte d’Ivoire. Journal of Water Resource and Protection, 12, 527-544. https://doi.org/10.4236/jwarp.2020.127032