In science and engineering, mathematical modeling serves as a tool to understand processes and systems acting as a testing bed for several hypotheses. The selection of a specific model, as well as its variables and parameters, depends on the nature of the system under analysis and the acceptable simplifying assumptions. Therefore, it must allow for a good fit between both the hypothesis and the available data. Opposite to other design approaches based on experimental data or/and complex models, this work presents a simpler numerical design method for efficiency maximization of an Hydraulic Jet Pump (HJP) for oilwell extraction process, considering its hydraulic and geometric parameters. The design process consists in setting and solving a constrained nonlinear optimization problem by taking into account the hydraulic model of the HJP in terms four design variables: throat area, nozzle area, injection flow, and injection pressure to the oilwell. The objective function of this case aims to maximize the HJP's efficiency avoiding to approach cavitation condition as well fulfilling technical constraints. A numerical technique, Differential Evolution Algorithm (DEA), has been implemented to solve the optimization problem. The proposed methodology leads to a solution set by considering only commercial geometries and feasible operating conditions for the HJP, which facilitates its practical implementation. A set of ten oilwells with land production data, operating in the southeaster of Mexico, is used to compare and validate several Jet pump designs, i. e., through comparison with actual oilwell's operation condition.
Keywords: Jet pump, optimization, efficiency, differential evolution
This section describes all the nomenclature used in this work.
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_s} = Suction area, [inFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle ^2} ].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_{th}} = Flow area of throat, [inFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle ^2} ].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle b=A_n/A_{th}}
Nozzle to throat area ratio, []
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle h_1} = Pump setting depth, [ft].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle H} =Dimensionless head recovery ratio, [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle H_v} = Jet velocity, [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle K_j} = Nozzle loss coefficient,[].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle K_d} = Diffuser loss coefficient, [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle K_s} = Suction loss coefficient, [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle K_t} = Throat loss coefficient, [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M} = Dimensionless flow ratio, [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M_c} =Cavitation limited flow ratio, [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle N} = Index of Losses by Friction, [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_1} = Pressure at the entrance of the nozzle, [Psi].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_2} =Pressure at the output of the throat, [Psi].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_3} = Pressure at the intake of the HJP, [Psi].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_s} = Injection pressure (Surface pressure), [Psi].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_f} = Average oilwell pressure, [Psi].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle PI} = Productivity index [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_v} =Vapor pressure, [Psi].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_{wf}} = Flowing bottom hole pressure, [Psi].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_1} =Injection flow, [Barrels/Day (BPD)].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_3} =Production flow, [BPD].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_2} =Sum of injection and production flows, [BPD].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_{max}} = Maximum oil flow from the oilwell, [BPD].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \sigma } = Cavitation index [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \eta } = HJP efficiency [].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma }
= Specific gravity [].
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= HJP Horsepower [hp].
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= Mutation factor [].
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= Crossover factor [].
The challenge of finding new technologies for production and/or improving existing ones in the oil and gas industry, is shaped because of the continuous increase in demand for and rapid depletion of this nonrenewable natural resource. Oil has been found naturally in large sedimentary basins at different depths ranging from 15 to 8,000 meters, occupying the empty space of porous and permeable rocks known as oilwell. As time goes by, the natural pressure of an oilwell decreases, until it can not longer produce naturally. This problem gives rise to use Artificial Lift Systems (ALS), as a feasible engineered system to exploit the oilwells. Indeed, is the fact that more than 95% of the world's oilwells use some ALS, starting from the oldest and simplest one such as mechanical pumping and pneumatic pumping, to the most technologies such as Electrical Submersible Pumping and hydraulic pumping.
In this work, we focus on the ALS of Hydraulic pumping with an HJP [1]. Hence, HJPs are of vast interest in petroleum engineering because of their design characteristics, and volumetric oil production they can handle. For the Hydraulic ALS, the HJP is introduced into the oilwell via the production pipe and it can be retrieved by simply reversing the working flow sense. In this application, the HJP has some disadvantages such as the cavitation phenomenon and typically low efficiency; nevertheless, both might be mitigated by an optimum design of the HJP and the selection of the best operating conditions for the ALS.
In general, the fundamental parts of this ALS are: a combustion engine that drives a hydraulic piston (or geared) pump; an horizontal threephase separator containing the mixture (oil, gas and sand) extracted from the oilwell; a control panel that controls the ALS and finally the HJP, installed at the bottom of the oilwell. Note that an HJP includes in a carrier that encloses the nozzle and throat (Figure 1).
Figure 1. Schematic diagram of an ALS whit an HJP 
Early works on HJP design have considered either experimental or numerical approaches or a mix of both. Mallela [2], computed normalized geometries which leads to maximum efficiencies without considering the cavitation phenomenon. S. Mohan [3], reported a numerical analysis and an optimization of an HJP via multisurrogate model without considering cavitation. In Mohan's study, the area ratio, mixingtube length to diameter ratio and setback ratio, were varied during an optimization process. In the work of Saker [4], it was pointed out the importance and influence of factors such as cavitation phenomena, which impact the HJP performance. In a more recent paper [5], Xiao presented a numerical and experimental study on the HJP performance and inner flow details of annular Jet pumps under three area ratios. Xiao's results included cavitation phenomenon and indicate that by decreasing outlet pressure, the cavitation generated at the throat inlet renders unstable effects for the HJP. In 2013 [6], the Norwegian University of Science and Technology (NTNU) analyzed oilwells from the North Sea, by using the principles of the HJP, as those described in [1], to search for an optimal operational conditions of the HJP , i.e. pressures and flows. In a study closer to our approach for the HJP design, J. Fan. [7] developed a design oriented to improve the HJP efficiency by using an analytical approach that considers a computational fluid dynamics (CFD) model. Then, the influence of the pump's geometry on its performance and the CFD simulation results were used to build surrogate models of the pump's behavior. Finally, a global optimization was carried out by means of a genetic algorithm. Works surveyed so far require either extensive experimental data or CFD simulations to provide HJP performance improvement; this can be too costly and time consuming for the oilwell industry needs. For this reason, the creation of a numerical tool to determine the optimal geometric design and operational hydraulic conditions of an HJP is of paramount importance to satisfy industry requirements.
This work introduces a numerical design methodology to maximize the efficiency of an HJP, based on its hydraulic and geometrical models. According to this, the method is based on geometric and hydraulic models transcription into a constrained nonlinear optimization problem with four design variables: throat and nozzle areas, the input working fluid flow and the input pressure. The objective function of the optimization problem aims to render the maximum efficiency without reaching the cavitation condition for the HJP and fulfilling technical constraints. Because of the nonlinear characteristics and the number of the independent variables, a Differential Evolution Algorithm (DEA) is implemented to solve the optimization problem. As a strategy to calibrate the algorithm, only commercial geometries and real operating conditions for the HJP were used. Evidence on the feasibility of using this algorithm in real applications is demonstrated by comparisons of numerical results with ongoing oilwell production by using HJPs in the southeaster by the Mexican Company, Geolis/Nuvoil. It is important to highlight that feasible HJP geometries and operational conditions are be obtained without extensive simulations or experimentation.
This paper is structured as follows. Section 3 concerns the mathematical models considered, then the problem statement for this work is stated in Section 4. Section 5 of this paper, explains the solution methodology by using the DEA. In Section 6, numerical results are shown and analyzed, and finally Section 7 closes the paper with the conclusions and future work.
The fundamental of operation of an HJP of an ALS is based on the Venturi principle [8]. Basically, this principle consists of applying additional energy to the oil at the bottom of the oilwell, in order to force the fluid to flow to the surface. For this purpose, a driving or working fluid (typically water) is injected through the production pipe to the oilwell. At the return, the mixture between working fluid and oil, is forced to circulate through the annular space between the production and cladding pipes; this working principle is shown in Figure 2.
Figure 2. HJP schematic setup 
In order to improve the performance of a HJP, selection of the computational method gives rise to two options. One approach considers the use of pressures and flow ratios [9]. Although it is one of the most used, it is not adequate because in some cases relevant information is insufficient. In a second approach, experimental information such as adimensional coefficients, is used to perform the computations of the HJP efficiency [1].
In this work, we consider the hydraulic model of the HJP described in [1]. The efficiency of a HJP is defined as the ratio of the energy added to the production fluid in relation to the energy lost by the injection fluid. Therefore, we can compute the hydraulic efficiency of the pump as in equation (1), where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M}
is the input to net production flow ratio as expressed in equation (2), and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle H} represents the dimensionless head recovery ratio, as in equation (3)

(1) 

(2) 

(3) 
In equation (2), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_3}
represents the flow from the well in barrels per day (BPD) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_1} represents the flow injected from the surface in BPD. A third flow Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_2} is the sum of both flows, as shown in Figure 2. Note that the input flow Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_1} can be controlled via a hydraulic (piston or geared) pump on the surface.
With respect to Equation (3), it presents has a dual definition. On the one hand, in this equation, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle H}
represents the dimensionless head recovery ratio for the pressures Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_1}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_2}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_3} of the HJP. On the other hand, the term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle N} represents experimental information considering friction loss coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle K_j}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle K_s} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle K_t} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle K_d}
and the geometry of the HJP, [1]. Given the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M}
, the adimensional coefficients, and once the nozzle area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_n}
and throat area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_{th}} are selected to compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle b}
, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle N}
is obtained using equation (4), see [1]

(4) 
Pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_1} , can be expressed by equation (5), as the sum of the pressure generated by the column of fluid at the production pipe and the surface pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_s}
given by an hydraulic pump

(5) 
Because of the damage problems associated with cavitation phenomenon (to be considered in the following section), it is also desirable to compute the intake pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_3}
while the pump is operating. This can be accomplished from the input flow Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_1} at the nozzle, according to equation (6). Note that the intake pressure depends on the input flow, the surface pressure, and the selection of the nozzle area. It is important to remark that, once Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle N} are computed, pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_2} is obtained from equation (3)

(6) 
The flow from the oilwell Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_3}
is one of the most challenging data to obtain. When the performance of oilwell is considered, it is often assumed that it can be estimated by the productivity index [9]. This concept is only applicable for oilwells producing under singlephase flow conditions, i.e. pressures above the oilwell fluid's bubblepoint pressure. For oilwell pressures less than the bubblepoint pressure, the oilwell fluid presents a twophase behavior, gas and liquid; then, other techniques must be applied to predict oilwell performance. In this work, we consider the case when the fluid exists as two phases, oil and gas. Thus, to estimate the production rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_3}
, a Vogel's performance relationship is considered [10]. In Vogel's relationship, to estimate the maximum oil production rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_{max}}
from the oilwell, it is required to measure the real oil production rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_3} and flowing bottomhole pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_{wf}} from a production test. Then, to obtain an measure (or estimate) of the average oilwell pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_f} at the time of the test, equation (7) is used

(7) 
The maximum oil production rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_{max}}
can be used to estimate the production rates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_{3}} for other flowing bottom hole pressures Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_wf^*} at the current average oilwell pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_f} as in equation (8). Moreover, in the following we set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_wf^*=P_3}

(8) 
By considering the hydraulic models presented in the last section, the efficiency of the pump depends mainly on four operational variables: the nozzle area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_{n}} , throat area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_{th}} , the input working flow ratio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_1}
and the surface pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_s}
. System pressures are computed with equations (5)(6), loss coefficients in equation (4) also affect the HJP efficiency, nevertheless they are considered as constants taken from the literature [1] (Table 3).
The objective function of the optimization problem, represents a mathematical model that allows to quantify the performance of the HJP. In this case, we consider to maximize the HJP efficiency, defined in equation (1). Although hydraulic pumping is an effective and reliable system, it presents some operational risks and disadvantages that are considered by means of the following constraints.
The first constraint is related to the cavitation problem in the HJP [5]. The high speed that is generated between the nozzle and throat, combined with the presence of gas and other solid particles causes cavitation, i.e. the sudden formation of small vapor bubbles into the liquid oil. This is a fundamental phenomenon to be considered because it might cause considerable efficiency decrease and severe damage to the nozzle and throat of the HJP. The mathematical expression of the cavitation is represented by the limit flow ratio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M_c}
[11], as in equation (9)

(9) 
To compute the Jet velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle H_v} , equation (10) is used

(10) 
Substituting (10) into equation (9), then equation (11) is obtained, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_v}
has been set to zero because low gasoil ratio (GOR) relationships for the studied oilwells are assumed. In the case that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M} is less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M_c}
, there is a low risk of cavitation in the HJP and efficiency might be increased

(11) 
A second set of constraints arise from geometrical characteristics of the HJP performance, specifically related to the commercial availability of nozzle and throat produced by manufacturers. Some technical guidelines are considered as follows. Both, nozzle and throat, use a strict progression of diameter and holes provided by the manufacturer, as depicted in Table 1. The progression establishes areas of ratio between nozzles and throats. In general, high flow volumes renders low lifting and vice versa. Very small area ratios are used in shallow wells, also the injection pressure is very low for these cases. The largest area ratios are installed for high lifting heads, but this is only applicable in specific cases, [12].
The working fluid (water) and the production fluid (oil and gas), must go through the throat area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_{th}} . The suction area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle (A_s)}
is the separation between the nozzle and throat; this area is where the oilwell fluid enters and it also raises a constraint. Moreover, the relationship Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle b} is also constrained by maximum and minimum values due to practical implementations. Considering this, the geometric constrains described above can be mathematically stated as follows:
must be smaller than the throat area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_{th}}
. This fact implies the existence of the suction area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_s=A_{th}A_n} .
, must be bounded as follows: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_{s_{min}} < A_s < A_{s_{max}} } .
.
In addition, equation (3) and Vogel relationship given by equation (8), renders a set of constraints for the systems pressures in order to get feasible (positive) values for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle H}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_3}
, this is:
.
.
After the above considerations, we propose to set the design of a HJP as solving the following constrained nonlinear optimization problem:

subject to the hydraulic pump model functions (2)(8), the cavitation constraints (9)(11) and the following geometrical and hydraulic constraints:

The solution to this problem offers an optimum configuration of the variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_n} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_{th}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_1}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_s}
There exist several theoretical and numerical techniques, to solve nonlinear constrained optimization problems. In this work, we consider a numerical approach by using a Differential Evolution Algorithm (DEA), [13]. A DEA is a numerical method for the determination of the global minimum or maximum, for highly nonlinear problems. It can handle an optimization problem with or without constraints, based on a process of natural selection that imitates biological evolution, [14]. The DEA repeatedly updates a set of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P}
initial vector designs Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle [\mathbf{x}_{1,0}, \mathbf{x}_{2,0},..., \mathbf{x}_{P,0}]} called population, in order to reach a final set of vector designs Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle [\mathbf{x}_{1,f}, \mathbf{x}_{2,f},..., \mathbf{x}_{P,f}]}
, for which an objective function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle f(\mathbf{x}_{i,f})}
is minimized or maximized for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle i=1,2,..,P}
, as shown in Figure 3.
Figure 3. Flow diagram for the DEA 
The design vector is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{x}=[x_1, x_2,...,x_{D}]^T} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle D}
is the number of design variables; and the objective function is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle f(\mathbf{x})}
. It is important to mention that the design vector might belong to a set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \Omega =\{ \mathbf{x} : \mathbf{x}_{min}\leq \mathbf{x}\leq \mathbf{x}_{max} \} } . In the following, we assume the search for a minimum of the objective function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle f} , while for searching maximum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle f}
must be considered. Each population, is a set of vector designs Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{x}_{i,g}}
, where subindex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle i}
represents the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle ith} vector design for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle gth} generation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle i=1,2,...,P}
.
The initial population for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle g=0} , can be randomly obtained for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle j_{th}} component of the vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{x}_{i,0}}
as: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle x_{i,j,0}=rand_j(0,1)*(x_{j,U}x_{j,L})+x_{j,L}} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle x_{j,L}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle x_{j,U}} represents, respectively, lower and upper bounds for each variable of the vector design and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle rand_j(0,1)} is a random number between 1 and 0.
At each generation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle g} , the DEA algorithm randomly selects individual solutions from the current population and uses them as parents to produce the offspring of the next generation. This is known as mutation and it states that the vector design for the next generation can be computed as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{v}_{i,g+1}= \mathbf{x}_{r_1,g}+F(\mathbf{x}_{r_2,g}\mathbf{x}_{r_3,g})} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle r_1, r_2}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle r_3} are integers randomly selected from the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \{ 1,2,3,...P\} }
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle F}
is a mutation factor. To guarantee that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle v_{i,g+1} \in \Omega } if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle v_{i,j,g+1}\leq x_{j,min}} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle v_{i,j,g+1}=2x_{j,min}v_{i,j,g+1}} else if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle v_{i,j,g+1}\geq x_{j,max}} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle v_{i,j,g+1}=2x_{j,max}v_{i,j,g+1}}
.
To complement the mutation strategy, in DEA it is defined a recombination process that ensures each design vector copied from two different vectors, crosses with a mutant vector. This is, DEA generates a trial crossed vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{u}_{i,g}}
with the following definitions for its Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle jth} component: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle u_{i,j,g}=v_{i,j,g}} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle rand_{j} (0,1) \leq CR} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle j=j_{rand}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle u_{i,j,g}=x_{i,j,g}} otherwise, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle CR} is a userdefined crossover value that controls the fraction of parameter values that are copied from the mutant and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle j_{rand}} is a randomly chosen index from the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \{ 1,2,...,D\} }
.
Finally, the selection is performed by comparing each trial vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{u}_{i,g}}
with the target vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{x}_{i,g}} as follows: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{x}_{i,g+1}=\mathbf{u}_{i,g}} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle f(\mathbf{u}_{i,g})\leq f(\mathbf{x}_{i,g}) } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{u}_{i,g}} is feasible, or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{x}_{i,g+1}=\mathbf{x}_{i,g}} otherwise. The algorithm continues checking feasible individuals using the constraint handling mechanism proposed in [15], for each vector of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle gth} generation, until some criterion is fulfilled, e.g. a maximum number of generations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle G_{max}} is overcome or other.
After Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle G_{max}}
successive generations, the population evolves towards a set of optimal solutions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{x}_{i,G_{max}}=\mathbf{x}_{i,opt}} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle i=1,2...,N_P}
, which in this case render feasible and optimal designs of the HJP and its operational conditions.
Continuing with the solution to the optimization problem, it is necessary to establish the dimensions that will have nozzle and throat according to standard sizes and relationships offered by manufacturers. Figure 4 depicts in red squares commercial nozzle and throat.
Figure 4. Throat and nozzle for an HJP. 
There are many manufacturers of HJPs in the world. In Table 1, the nozzle and throat areas (in squared inches) are classified for three different manufacturers: Kobe, National and Guiberson. In this paper we select the last one, since Geolis/Nuvoil company uses Guiberson Jet pumps.
KOBE  NATIONAL  GUIBERSON  
Nozzle  Throat  Nozzle  Throat  Nozzle  Throat  
1  0.0024  1  0.0060  1  0.0024  1  0.0064  DD  0.0016  000  0.0044 
2  0.0031  2  0.0077  2  0.0031  2  0.0081  CC  0.0028  00  0.0071 
3  0.0040  3  0.0100  3  0.0039  3  0.0104  BB  0.0038  0  0.0104 
4  0.0052  4  0.0129  4  0.0050  4  0.0131  A  0.0055  1  0.0143 
5  0.0067  5  0.0167  5  0.0064  5  0.0167  A+  0.0075  2  0.0189 
6  0.0086  6  0.0215  6  0.0081  6  0.0212  B  0.0095  3  0.0241 
7  0.0111  7  0.0278  7  0.0103  7  0.0271  B+  0.0109  4  0.0314 
8  0.0144  8  0.0359  8  0.0131  8  0.0346  C  0.0123  5  0.038 
9  0.0186  9  0.0464  9  0.0167  9  0.0441  C+  0.0149  6  0.0452 
10  0.0240  10  0.0599  10  0.0212  10  0.0562  D  0.0177  7  0.0531 
11  0.0310  11  0.0774  11  0.0271  11  0.0715  E  0.0241  8  0.0661 
12  0.0400  12  0.1000  12  0.0346  12  0.0910  F  0.0314  9  0.0804 
13  0.0517  13  0.1292  13  0.0441  13  0.1159  G  0.0452  10  0.0962 
14  0.0668  14  0.1668  14  0.0562  14  0.1476  H  0.0661  11  0.1195 
15  0.0863  15  0.2154  15  0.0715  15  0.1879  I  0.0855  12  0.1452 
16  0.1114  16  0.2783  16  0.0910  16  0.2392  J  0.1257  13  0.1772 
17  0.1439  17  0.6594  17  0.1159  17  0.3146  K  0.1590  14  0.2165 
18  0.1858  18  0.4642  18  0.1476  18  0.3878  L  0.1963  15  0.2606 
19  0.0240  19  0.5995  19  0.1879  19  0.4938  M  0.2463  16  0.3127 
20  0.3100  20  0.7743  20  0.2392  20  0.6287  N  0.3117  17  0.375 
21  1.000  P  0.3848  18  0.4513  
22  1.2916  19  0.5424  
23  1.6681  20  0.6518  
24  2.1544 
Eddie Smart [16], proposed a throat and nozzle combinations in terms of the parameters Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle b}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_s}
, that are implementable in practical applications. This represents a set of constraints as explained in Section 4. The feasible relationships between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle b}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_s} are depicted in Table 2.
Geometry  B0  B1  B2  B3  B4  B5  B6  
b  0.9135  0.6643  0.5026  0.3942  0.3025  0.2500  0.2102  
As  0.0009  0.0048  0.0094  0.0146  0.0219  0.0285  0.0357  
Geometry  C1  C2  C3  C4  C5  C6  C7  
b  0.8601  0.6508  0.5104  0.3917  0.3237  0.2721  0.2316  
As  0.0020  0.0066  0.0118  0.0191  0.0257  0.0329  0.0408  
Geometry  D3  D4  D5  D6  D7  D8  D9  
b  0.7344  0.5637  0.4658  0.3916  0.3333  0.2678  0.2201  
As  0.0064  0.0137  0.0203  0.0275  0.0354  0.0484  0.0627  
Geometry  E4  E5  E6  E7  E8  E9  E10  E11 
b  0.7675  0.6342  0.5332  0.4539  0.3646  0.2998  0.2505  0.2017 
As  0.0073  0.0139  0.0211  0.0290  0.0420  0.0563  0.0721  0.0954 
In this work the vector design is set as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \mathbf{x}=[x_1,x_2,x_3,x_4]^T=[A_{th},A_n,P_s,q_1]^T}
and the parameters of the DEA are set to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle F=0.7} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle CR=0.8}
. The algorithm was implemented in MatlabFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle ^\mbox{©}} .
It is important to remark that in order to implement the considerations presented in Tables 2 and 1, indexed lists are used in the DEA, i.e. any list item can be identified by a sequential integer number that identifies its position. In order to keep operational conditions feasible, as indicated by the technical datasheet of the surface pumps, the range of values for the input flow and input pressure were set to minimum and maximum values as: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_{1_{min}}=600 [BPD]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle q_{1_{max}}=1500 [BPD]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_{s_{min}}=900 [Psi]}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_{s_{max}}=2500 [Psi]}
. Table 3 depicts testing parameters experimentally obtained, [9].
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): \sigma  1.35 [] 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): K_j  0.15 [] 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): K_d  0.1 [] 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): K_s  0.0 [] 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): K_t  0.28 [] 
In order to validate our design proposal, it has been evaluated a set of ten oilwells from the asset Aceite Terciario del Golfo (ATGMexico), an important oilwell zone in Mexico. The values of the operating oilwell conditions that have been used to perform this analysis have been provided by the Mexican Company Geolis/Nuvoil (Table 4). In this Table, it can be observed low efficiencies and high power consumption for most oilwells currently operating. It is important to mention that the efficiency and horsepower consumption, were actually computed with models considered in this paper, based in the operational parameters of the Geolis/Nuvoil Company.
Table 4. Field data of ten wells form ATGMexico. (HP= Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): 1.7 \cdot 10^{5} q_1 \cdot P_s
)Well  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_f  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_{wf}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): q_{max}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): h_1  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): G_w  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_s  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): q_1  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): A_n  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): A_{th}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): \eta
[%] 
HP 
1  3427  2530  373  5248  0.3854  1067  1336  0.0177  0.0452  20.0665  24.2337 
2  2700  2182  329  5112  0.3550  995  888  0.0123  0.0380  21.4368  15.0205 
3  2700  2182  333  7119  0.3498  1564  974  0.0123  0.0380  18.6433  25.8967 
4  2000  1470  129  4917  0.355  924  1141  0.0123  0.0314  4.0318  17.9700 
5  1830  1600  250  4331  0.3732  2133  1158  0.0177  0.0452  6.8621  41.9902 
6  2567  1539  105  7890  0.3113  1422  868  0.0123  0.0314  7.9871  20.9830 
7  1860  1737  250  4934  0.4205  1010  921  0.0123  0.0241  22.8988  15.8136 
8  3000  2035  190  4889  0.3585  1209  1425  0.0241  0.0804  7.1149  29.2880 
9  2450  2030  300  5243  0.3710  995  772  0.0123  0.0314  22.6438  13.0584 
10  3193  3000  260  6562  0.4317  1280  1184  0.0177  0.0804  6.6017  25.7638 
For the sake of clarity, the sixth oilwell in Table 4 is selected as a benchmark to show our design proposal. Geolis/Nuvoil company disposed of a pressuretemperature sensor of the brand Pioneer Petrotech Services Inc., into the oilwell. The sensor was operating during one month getting the average value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_{wf}=1355 [PSI]} , while the predicted with equation (6) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P_3=P_{wf}=1177 [Psi]}
getting a error of 13.14 %
After the application of our approach, the first set of results are depicted in Table 6.1, considering the initial population of 20 individuals and varying the number of maximum generations from 50 to 500. As it can be observed with low generations, convergence is not achieved and the optimal efficiency rounds 9% to 11% In these cases the cavitation indicator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M}
is quite lower than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M_c} and power consumption rounds 40 [HP]. Then, with higher number of generations the efficiency is improved up to 16.88 %, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M} closer to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M_c}
. Note also that in this case the power consumption is close to 12 [HP] with a clear energy saving.
Table 5. Numerical results for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P=20
and different number of maximum generationsGenerations  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): A_n  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): A_{th}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): q_{1}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_s  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): HP  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_1  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_2  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_3  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): \eta
[%] 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): M  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): M_c  Time [s] 
50  0.0177  0.0241  1436  1511  36.9  3967  2410  398  8.9962  0.0696  0.1071  17.0523 
100  0.0123  0.0189  1097  2047  38.2  4503  2560  188  11.4677  0.0940  0.1018  25.7528 
150  0.0095  0.0143  706  911  10.9  3367  1948  372  15.7920  0.1422  0.1573  35.1284 
200  0.0095  0.0143  704  905  10.8  3361  1948  379  15.8050  0.1424  0.1590  42.2058 
250  0.0109  0.0189  839  1029  14.7  3485  1967  275  13.5557  0.1216  0.1923  51.4344 
300  0.0075  0.0104  624  2039  21.6  4495  2731  744  16.7122  0.1484  0.1484  64.4303 
350  0.0095  0.0143  704  900  10.8  3356  1943  375  15.8147  0.1425  0.1582  29.9908 
400  0.0095  0.0104  656  2432  27.1  4888  2944  736  16.0420  0.1413  0.1413  25.9958 
450  0.0075  0.0143  601  1181  12.1  3637  1877  152  16.8812  0.1722  0.1722  82.6422 
500  0.0095  0.0143  705  905  10.8  3361  1945  374  15.8027  0.1424  0.1580  48.8717 
Table 6 presents numerical results by considering a fixed number of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle G_{max}=250}
generations and varying the number of initial population.
As it can be observed, with low initial population and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle G_{max}}
reasonably high, convergence is achieved and the optimal efficiency rounds 15.8% In this cases the cavitation indicator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M} is quite less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M_c} and power consumption rounds 11 [HP], thus saving energy. Nevertheless, in this case with higher number of initial population the efficiency is improved up to 17.23 %, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M} equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M_c} and more important, the power consumption is close to 19 [HP]. Here, the geometries selected for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_n} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle A_{th}} are smaller than the selected for the case when the efficiency rounds 16.8 % Moreover, the input pressure must be increased twice in order to render higher efficiency, which explains the 19 [HP] of power consumption.
Table 6. Numerical results for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): G_{max}=250
and different initial populationPopulation  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): A_n  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): A_{th}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): q_{1}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_s  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): HP  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_1  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_2  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_3  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): \eta
[%] 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): M  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): M_c  Time [s] 
50  0.0095  0.0143  704  900  10.8  3356  1944  376  15.8147  0.1425  0.1583  74.04058 
100  0.0095  0.0143  704  900  10.8  3356  1943  375  15.8149  0.1425  0.1583  131.5920 
150  0.0075  0.0104  600  1770  18.1  4226  2586  750  17.2324  0.1539  0.1539  181.0629 
200  0.0095  0.0143  704  900  10.8  3356  1943  375  15.8149  0.1425  0.1583  243.6755 
250  0.0095  0.0143  704  900  10.8  3356  1943  375  15.8149  0.1425  0.1584  177.0426 
300  0.0095  0.0143  704  900  10.8  3356  1943  375  15.8149  0.1425  0.1582  258.3238 
350  0.0075  0.0104  608  1867  19.3  4323  2641  754  17.0253  0.1517  0.1525  469.1144 
400  0.0075  0.0143  600  1178  12.0  3634  1881  163  16.8895  0.1723  0.1780  423.9130 
450  0.0095  0.0143  704  900  10.8  3356  1944  376  15.8149  0.1425  0.1584  320.7046 
500  0.0075  0.0104  608  1868  19.3  4324  2644  759  17.0113  0.1515  0.1531  474.2681 
The rest of the wells were analyzed with the proposed DEA by considering Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle G_{max}=500}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle P=100}
. Numerical results are depicted in Table 7. Note that power consumption is between 9 and 19 [HP], remarkably lower than the presented in Table 4. Moreover, the efficiencies are incremented in all cases as shown in Figures 5 and 6. Worst DEA and Best DEA means the worst and best result using the Differential Evolution Algorithm, respectively; while real means the result for the implemented HJP. Note that in most cases Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M=M_c} , thus putting in risk of cavitation in the HJP. Nonetheless this result can be improved by considering a safety factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle \beta < 1}
such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): {\textstyle M < \beta M_c}
.
Table 7. Results obtained with the DEA algorithm for ten wells form ATGzone Mexico
Well  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): A_n  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): A_{th}  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): q_1  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_s  HP  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_1  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_2  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): P_3  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): \eta
[%] 
M  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com:8081/localhost/v1/":): M_c  Time [s] 
1  0.0109  0.0189  600  1015  10.35  3037  2078  1394  34.9052  0.4889  0.4889  203.7145 
2  0.0109  0.0189  609  900  9.32  2715  1747  1022  32.7827  0.4375  0.4375  172.1843 
3  0.0095  0.0189  662  900  10.14  3390  1839  753  31.0225  0.4434  0.4434  205.3670 
4  0.0095  0.0143  605  900  9.26  2646  1561  444  20.1180  0.1953  0.1953  113.0932 
5  0.0109  0.0189  654  900  10.01  2516  1450  562  27.4801  0.3298  0.3298  215.3736 
6  0.0075  0.0104  608  1868  19.3  4324  2644  759  17.0113  0.1515  0.1531  115.9245 
7  0.0095  0.0189  652  900  9.97  2975  1532  422  25.9546  0.3504  0.3504  237.1783 
8  0.0095  0.0143  600  1376  14.03  3129  2024  965  25.8551  0.2721  0.2721  177.4631 
9  0.0075  0.0143  600  1506  15.36  4338  2361  867  29.0353  0.3842  0.3842  233.4054 
10  0.0095  0.0189  607  900  9.29  2845  1542  630  30.9925  0.4426  0.4426  252.6758 
Figure 5. Comparison of implemented and computed efficiency for oilwells 1 to 5 from ATGzone Mexico 
Figure 6. Comparison of implemented and computed efficiency for oilwells 6 to 10 from ATGzone Mexico 
For each oilwell, although belonging to the same productive zone, has different and unique characteristics; for that reason it should always be performed the analysis for each well. The efficiency of an HJP is linked to the implemented geometry. Nevertheless, the analysis must consider the production capacities of each oilwell, as well as the resources available in the installation of the ALS. The algorithm presented in this work is able to define the optimal geometry and the operational conditions that renders the maximum efficiency of the HJP for a given oilwell, based on its characteristics and its production capacities. By using this design methodology it will be easier for the engineers to select the HJP geometries to install besides knowing the efficiency that it can develop during operation while its conditions remain stable.
From the numerical results it can be concluded that efficiency of the operation of the studied wells can be improved up to 24% for a given implementation of the HJP. In addition, it was observed that similar efficiencies can be achieved with different amount of power required to implement the HJP artificial lift system. This fact encourage us to continue research on the optimization of the HJP in terms of the design of other sections such as the inlet holes for the working fluid and suction for the well fluids.
The authors would like to thank the support provided by Geolis/Nuvoil for sponsoring, training and advice provided. Their support served extensively for the development and validation of this research article, driving development within the energetic sector in the country.
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Published on 08/02/19
Accepted on 20/09/18
Submitted on 13/06/18
Volume 35, Issue 1, 2019
DOI: 10.23967/j.rimni.2018.11.002
Licence: CC BYNCSA license