Rheological Properties Of Fluid Research Articles

Influence of calcium chloride in coal on guar-based foam fracturing fluid rheological property: An experimental study

To investigate the influence of calcium chloride (CaCl2) in coal seams on the rheological property of a foam fracturing fluid, the guar powder was used as a foam stabilizer, and the anionic and cationic surfactants were used as foaming agents. The impact of CaCl2 concentrations on the foaming volume, half-life, viscosity, and surface tension of foam was investigated. The results show that CaCl2 has a more significant effect on the foaming amount and foam stability of the anionic surfactant-prepared foam. When the concentration of CaCl2 exceeds 0.25%, the viscosity of the base fluid shows a downward trend and gradually tends to a stable level below 10 mPa s. The foam half-life increases as the shape parameter an in the gamma function increases and decreases as the scale parameter b increases. The effect of CaCl2 on the bubble size of the anionic surfactant foam fracturing fluid is significantly greater than that of the cationic surfactant foam fracturing fluid. For coal reservoir foam fracturing, a foaming agent with better CaCl2 compatibility should be employed to reduce the influence of CaCl2 in the formation water on fracturing performance.

Rheological identification of jetted fluid using machine learning

The understanding of flowing properties of fluids and the knowledge of the related rheological properties are crucial from both a research and industrial point of view. To determine the complex rheological properties of fluids, many devices have thus been developed, the so-called rheometers. The main objective of the present paper is to identify the rheological properties of a fluid jetted using continuous inkjet (CIJ) printing process by comparing the morphology of the aforementioned jetted fluid to a dataset of known (rheologically speaking) fluid jet morphologies and properties of a fluid by the viscosity, the surface tension, and the density of fluids using large datasets and a CIJ printing process. When ejecting a fluid, the CIJ ejection process competes among several forces: inertial, viscous, surface tension, and elasticity, which affect the morphology of the resulting jet. Also, under certain conditions, the morphology of the jet is unique and directly related to the rheological properties of the fluid. We want to use this uniqueness to identify the fluid among a large dataset of known fluid jet morphologies to be compared with, to obtain its rheological properties. Using a large numerically generated dataset of Newtonian fluid jets, we show in this article that the identification of the viscosity using neural network is not only feasible but also proves to be very accurate with an average error of less than 1% for a large range of viscosities.

Design and use of an online drilling fluid pipe viscometer

The accurate and frequent measurement of the drilling fluid's rheological properties is essential for drilling engineering. To measure drilling fluid's rheological properties automatically and frequently, we design an online pipe viscometer. First, we introduce the measurement principle, calculation flow chart, and design scheme. The pulsation damper and back pressure valve are used to stabilize flowrate and pressure which can improve the accuracy of the pipe viscometer. Then the errors under various static hole sizes are measured, and the static hole with the smallest error is selected as the measurement pipe of the pipe viscometer. Finally, various drilling fluids are tested. Compared with the standard six-speed viscometer, the shear stress error of the six standard shear rates (1022, 511, 340, 170, 10, 5s−1) is within 1Pa. The designed pipe viscometer can frequently measure drilling fluid's rheological properties and meets the needs of on-site online testing.

Settling behavior of spherical particles in eccentric annulus filled with viscous inelastic fluid

Hole cleaning is a complex process as there are many variables affecting cuttings removal (e.g. drilling fluid type, density, flow rate and rheological properties, cuttings size, drill pipe rotation speed). With the increasing number of drilling small diameter wells (e.g. coiled tubing drilling applications, ultra-deep wells drilled for exploitations of unconventional oil and gas resources), the wall resistance of the micro annulus also emerges as one of the most critical factors affecting the cuttings accumulation in wellbore. The eccentricity of drill pipes commonly observed during the drilling process of ultra-deep well and coiled tubing well makes the wall resistance effect on the cuttings transport even more prominent. Understanding the wall resistance effect on the particle settling behavior in eccentric annuli is, therefore, crucial for hydraulic design of efficient cuttings transport operations in these wells. In this study, a total of 196 sets of particle settling experiments were carried out to investigate the particle settling behavior in eccentric annuli filled with power-law fluids. The test matrix included the eccentricity ranges of 0–0.80, the dimensionless diameter ranges of 0.13–0.75 and the particle Reynolds number ranges of 0.09–32.34. A high-speed camera was used to record the particle settling process and determine the influences of the eccentricity, the dimensionless diameter, the fluid rheological properties, and the solid particle characteristics on the wall factor and the particle settling velocity. The functional relationship among the dimensionless diameter, the particle Reynolds number, and the wall factor was determined by using the method of controlling variables. An eccentric annulus wall factor model with average relative error of 5.16% was established. Moreover, by introducing Archimedes number, an explicit model of particle settling velocity in the eccentric annulus with average relative error of 10.17% was established. A sample calculation of particle settling velocity was provided to show the application of the explicit model. Results of this study can be used as a guideline by field engineers to improve hydraulic design of cuttings transport operations in concentric and eccentric annuli.

How Much Does Filler Apparatus Influence Ease of Injection (and Hence, Potential Safety)?

To document the relative contributions of intrinsic filler fluid dynamics versus delivery systems for ease of injection-specifically, to measure extrusion force variability across different syringes and needles (with the characterization of intrinsic rheological fluid properties vs. delivery apparatus contributions to ease of injection). Six different fillers were tested: Belotero balance (Bel), Juvederm Voluma XC (Vol), Revanesse Versa (Rev), Restylane Lyft (Res), Radiesse (Rad), and Teosyal RHA3 (RHA). Extrusion force was measured in Newtons (N) for each by testing using the provided injection apparatus (needle + syringe), and also by standardizing all fillers to the same syringe and then varying needle sizes (30-ga, 27-ga, 25-ga/1.5-inch, 25-ga/2-inch, and 22-ga). Five trials were conducted for each scenario, with comparison via t -test (2-tailed, unpaired, assuming unequal variance). The following results were noted: (1) in order of least to highest extrusion force in box-provided syringe + needle at 0.2 ml volume, the following were noted: Vol < RHA = Bel (27-ga) < Bel (30-ga) < Rev < Res = Rad; (2) for each filler (except for Vol which was similar), the box-provided syringe involved greater extrusion force than the standardized syringe used in this study (each 1-cc, p < 0.05); (3) for 27-ga and 30-ga needles, after standardization of delivery syringe at 0.2 ml volume, a significant difference was noted (proportional to increasing resistance): Bel = Vol = RHA3 < Res < Rev < Rad (for needles of 30-ga [ p < 0.05] and 27-ga [ p < 0.01]); (4) for testing cannulas after standardization of syringes no reproducible order was noted with increasing resistance when using 25-ga/1.5-inch long, 25-ga/2-inch, and 22g/2-inch cannulae; and (5) confirming expectation (validating study technique), the extrusion force was significantly higher for smaller needles and longer needles. The delivery apparatus appeared to be the most influential contributor to filler injection extrusion force, with significant changes in ease of injection correlated to the filler's intrinsic rheological properties, such as viscosity (when standardized to the same syringe for needles tested). Knowledge of such data could influence the injector's ability to maximize patients' safety and clinical results.

Probabilistic assessment of equivalent fracture aperture constrained on quasi-real-time drilling mud loss data

We provide a rigorous workflow to quantify the effects of key sources of uncertainty associated with equivalent fracture aperture estimates w constrained through mud loss information acquired while drilling a well in a reservoir. A stochastic inverse modeling framework is employed to estimate the probability distribution of w. This choice is consistent with the quantity and quality of available data. The approach allows assessing the probability that values of w inferred from mud loss events exceed a given threshold. We rely on a streamlined analytical solution to model mud losses while drilling. We explicitly consider uncertainties associated with model parameters and forcing terms, including drilling fluid rheological properties and flow rates, pore fluid pressure, and dynamic drilling fluid pressure. A synthetic scenario is considered to provide a transparent reference setting against which our stochastic inverse modeling workflow can be appraised. The approach is then applied to a real-case scenario. The latter is associated with data monitored on a rig site. A direct comparison of the impact of data collected through two common techniques (respectively, relying on flow meter sensors or pump strokes) on the ensuing probability of w is provided. A detailed analysis of the uncertainty related to the level of data corruption is also performed, considering various levels of measurement errors. Results associated with the field setting suggest that the proposed workflow yields probability distribution of w that are compatible with interpretations relying on traditional analyses of image logs. Results stemming from direct and indirect flow data display similar shapes. This suggests the viability of the probabilistic inversion methodology to assist quasi-real-time identification of equivalent fracture apertures on the basis of routinely acquired information during drilling.

Effects of Different Fatty Acids as Surfactants on the Rheological Properties of Kerosene-Based Magnetic Fluids

Three types of surfactants (oleic acid, linoleic acid, and a mixture of oleic acid and linoleic acid) were coated on ferromagnetic particles, which were dispersed in kerosene to prepare magnetic fluids, to study the effect of different fatty acids as surfactants on the rheological properties of magnetic fluids. The particles were analyzed by XRD, TEM, FT-IR, and VSM. Furthermore, a rheometer was used to examine the rheological properties of kerosene-based magnetic fluids dispersed with various surfactants. The yield stress at different magnetic fields was calculated by fitting the Herschel–Bulkley model. The fitted curve and the observed values of mixed fatty acids are identical. The graphs of viscosity increase with the shear rate for each magnetic fluid were measured at constant magnetic field strengths. At constant shear rates, the curves of viscosity increase with magnetic field intensity were measured. In the absence of a magnetic field, the relative change in viscosity from 40°C to 0°C was observed. The rheological measurements of the mixed fatty acid-dispersed ferrofluid with a rising magnetic field at a constant shear rate are smoother than the single-fatty-acid-dispersed ferrofluid, indicating that it is more stable. As the temperature is dropped, the viscosity–temperature curve evidence that mixed fatty acids as surfactants can lower the proportion of magnetic fluid viscosity rise.

Incorporating steel-industry waste in water based drilling fluids for hydrogen sulfide scavenging

Drilling formations with hydrogen sulfide (H2S) pockets causes significant corrosion and operating issues for drilling equipment, as well as exposes working personnel to health and safety risks. Steelmaking, on the other hand, is an energy-intensive process that generates enormous amounts of waste, which is difficult to manage both economically and environmentally. Therefore, the objective of this study is to assess the possibility of using steel industry waste as an H2S scavenger in water-based muds. The H2S scavenging capacity was evaluated for the base mud and fluids containing commercialized SourScav® and steel-waste. The scavenging capacities were calculated using the H2S concentrations measured at the glass column's outlet after contact with the fluid sample. Also, the effects on drilling fluid rheological and filtration properties were investigated, and the corrosion rate was identified for the prepared fluids. According to the experimental results, utilizing steel-waste improved the base mud's H2S scavenging capability by 61%, compared to 50% with the SourScav at the required pH for drilling applications in sour environments. Incorporating steel-waste also raised the plastic viscosity and yield point by 16 and 9%, respectively, and improved the gelling structure with higher gel strengths. Furthermore, adding steel-waste reduced the filtrated volume by 25% with less potential formation damage and a slight reduction in composed filter cake thickness and weight, whereas the SourScav-composed filter cake characteristics were preferred, with weight and thickness reductions of 20 and 22%, respectively. Steel-waste and SourScav both had non-corrosive effects, with corrosion rates of less than 0.9E-5 lb/ft2. By exploiting and recycling steel industry waste, this study delivers an innovative and low-cost H2S scavenger that aids in waste management and, thus, the decrease of negative environmental and economic repercussions of these wastes.

Investigation of rheological properties of oil during the formation of wax deposits

The relevance of research lies in the need to assess the change in rheological properties of oil in the process of wax deposits formation. The existing correlation dependences make it possible to determine the viscosity of the oil at a known temperature and density, however, their application is possible only for oils of a specific region and a constant component composition. However, in the process of oil production at many fields, including in the Perm Krai, the problem of the formation of wax deposits arises, which directly affects the composition of the fluid. At the laboratory stand «Wax Flow Loop» a number of laboratory studies have been carried out in various thermal conditions. According to research data, when the oil temperature is above 25 °C during the formation of deposits, the rheological properties of oil improve. When the oil temperature is below 20 °C, the formation of deposits occurs in the volume of oil and the rheological properties deteriorate. The obtained experimental results allow us to consider the dynamics of changes in the rheological properties of oil in the process of the formation of wax deposits. The use of laboratory research data will make it possible to predict the change in the rheological properties of fluids during the formation of wax deposits under various conditions, as well as to increase the accuracy of modeling the process of production of the target fluid.

Murree Clay Problems and Water-Based Drilling Mud Optimization: A Case Study from the Kohat Basin in Northwestern Pakistan

Drilling fluids with subtle filtration and rheological characteristics are essential to optimize the functioning of oil and gas well drills. The Early Miocene Murree Formation in the Kohat Basin of Pakistan is generally referred to as the Murree Clays. When mixed with water-base drilling mud, ultra-fine particles of the Murree Formation tend to eradicate default fluid rheological properties and result in wellbore instabilities during drilling in that area. To obtain the optimized mud to deliver the drilling operation efficiently, we aimed to mitigate the impact of Murree clay on the drilling mud. We analyzed the drilling mud to observe the effects of the mud additive on clays on the basis of the samples from the Murree Clays. On the basis of the experimental tests, we observed that the rheological properties of mud significantly improved in the presence of KCl. KCl prevented the smectite group swelling inclinations and reduced rheological values to 25%, 33.3%, 48.6%, and 65.2%. The plastic viscosity increased as the concentration of clays increased; however, there was a noticeable reduction in the yield point values with the introduction of KCl. The laboratory results showed that Mud + 4% Clay + 1% KCl proved to be the best mitigation while preserving the rheological and performance characteristics of the mud. Tests enabled the scope to increase the inhibition efficiency and optimize customization. Depending on the clay present in the Murree formation, drilling fluid optimization is proposed to reduce mud-related drilling problems in this area.

A method for evaluating time-resolved rheological functionalities of fluid foods.

We have developed an effective method for evaluating time‐resolved rheological functionalities of swallowed foods using ultrasonic spinning rheometry (USR). USR can obtain variations over time in the rheological properties of fluids despite the fluids being in heterogeneous and nonequilibrium conditions. In addition, USR can evaluate time variations of shear‐thinning property changing in a few seconds. Demonstrations were conducted with typical thickener solutions: starch, guar gum, and xanthan gum‐based solutions, with alpha‐amylase as a digestive enzyme. The flow curve of the starch‐based solutions lowered with time, and a few minutes after addition of the amylase, the viscosity dropped to one‐hundredth of the original value. In contrast, the guar gum‐ and xanthan gum‐based solutions maintained the original viscosities as generally known. Applying the power law fitting to series of these flow curves, the time variation of the shear‐thinning property is quantitatively characterized by the plots on typical K‐n space, where K and n are parameters in the model, consistency index and power law exponent. The qualitative characteristics of the thickeners are successfully quantified in the K‐n space, and this will be a practical tool for evaluating the time‐resolved rheological properties of swallowed foods.

Development of New Models to Determine the Rheological Parameters of Water-Based Drilling Fluid using Artificial Neural Networks

It is well known that drilling fluid is a key parameter for optimizing drilling operations, cleaning the hole, and managing the rig hydraulics and margins of surge and swab pressures. Although the experimental works represent valid and reliable results, they are expensive and time consuming. In contrast, continuous and regular determination of the rheological fluid properties can perform its essential functions during good construction. The aim of this study is to develop empirical models to estimate the drilling mud rheological properties of water-based fluids with less need for lab measurements. This study provides two predictive techniques, multiple regression analysis and artificial neural networks, to determine the rheological properties of water-based drilling fluid using other simple measurable properties. While mud density, marsh funnel, and solid% are key input parameters in this study, the output models are plastic viscosity, yield point, apparent viscosity and gel strength. The prediction methods have been applied on datasets taken from the final reports of two wells drilled in the Ahdeb oil field, eastern Iraq. To test the performance ability of the developed models, two error-based metrics (determination coefficient R2 and root mean square error have been used in this study. The current results support the evidence that MW, MF, and solid% are consistent indexes for the prediction of rheological mud properties. Both mud density and solid content have a relative-significant effect on increasing PV, YP, AV, and gel strength. The results also reveal that both MRA and ANN are conservative in estimating the fluid rheological properties, but ANN is more precise than MRA. Eight empirical mathematical models with high performance capacity have been developed in this study to determine the rheological fluid properties using simple and quick equipment such as mud balance and marsh funnel. This study presents cost-effective models to determine the rheological fluid properties for future well planning in Iraqi oil fields.

Aging Effects on the Rheological Properties of Novel Magnesium Bromide Hexahydrate-Based Completion Fluids for Oil and Gas Reservoirs

Aging Effects on the Rheological Properties of Novel Magnesium Bromide Hexahydrate-Based Completion Fluids for Oil and Gas Reservoirs

Data-driven closure model for the drag coefficient of the creeping flow past a translating sphere in a shear-thinning viscoelastic fluid

A drag force closure model for particle-laden viscoelastic fluid flows is the key to describing the ensemble-averaged behavior of the mixture. The effects of fluid rheological properties on the flow dynamics of a spherical particle in viscoelastic fluids in the creeping flow regime are parameterized using the Giesekus rheological model. Direct numerical simulations are performed within a large range of Deborah number(0−10).The drag force of a sphere in unbounded Giesekus fluids decreases monotonically with the increase of Deborah number. The negative wake may occur when the viscosity ratio is larger than 0.6 in Giesekus fluids but is absent in Oldroyd-B fluids in all conditions. An explicit closure model for the drag coefficient of a sphere in Giesekus fluids is established using the backpropagation artificial neural network. The drag closure model provides a method for explicitly formulating the momentum exchange model for dilute suspensions of solid particles in shear-thinning viscoelastic fluids.

A simple yet efficient approach for electrokinetic mixing of viscoelastic fluids in a straight microchannel

Many complex fluids such as emulsions, suspensions, biofluids, etc., are routinely encountered in many micro and nanoscale systems. These fluids exhibit non-Newtonian viscoelastic behaviour instead of showing simple Newtonian one. It is often needed to mix such viscoelastic fluids in small-scale micro-systems for further processing and analysis which is often achieved by the application of an external electric field and/or using the electroosmotic flow phenomena. This study proposes a very simple yet efficient strategy to mix such viscoelastic fluids based on extensive numerical simulations. Our proposed setup consists of a straight microchannel with small patches of constant wall zeta potential, which are present on both the top and bottom walls of the microchannel. This heterogeneous zeta potential on the microchannel wall generates local electro-elastic instability and electro-elastic turbulence once the Weissenberg number exceeds a critical value. These instabilities and turbulence, driven by the interaction between the elastic stresses and the streamline curvature present in the system, ultimately lead to a chaotic and unstable flow field, thereby facilitating the mixing of such viscoelastic fluids. In particular, based on our proposed approach, we show how one can use the rheological properties of fluids and associated fluid-mechanical phenomena for their efficient mixing even in a straight microchannel.

Effect of Synthetic Quadripolymer on Rheological and Filtration Properties of Bentonite-Free Drilling Fluid at High Temperature

High temperature would dramatically worsen rheological behaviors and increase filtration loss volumes of drilling fluids. Synthetic polymers with high temperature stability have attracted more and more attention. In this paper, a novel quadripolymer was synthesized using 2-acrylamido-2-methylpropanesulfonic acid (AMPS), acrylamide (AM), sodium styrene sulfonate (SSS), and dimethyl diallyl ammonium chloride (DMDAAC). Firstly, the molecular structure was studied by Fourier transform–infrared spectroscope (FT-IR) and nuclear magnetic resonance (1H-NMR) analysis. It was shown that the synthetic polymer contained all the designed functional groups. Moreover, the effect of temperature and the quadripolymer concentration on the rheological behavior and filtration loss of the bentonite-free drilling fluid were investigated. It was experimentally established that when the adding amount of the quadripolymer was 0.9 wt%, the prepared drilling fluid systems exhibited relatively stable viscosities, and the filtration losses could be controlled effectively after hot rolling aged within 180 °C. Further, it was confirmed that the bentonite-free drilling fluid containing the synthesized quadripolymer had good reservoir protection performance. In conclusion, the synthetic quadripolymer is a promising rheology modifier and a filtrate reducer for the development of the bentonite-free drilling fluid at high temperature.

Emulsion-based drilling fluids: Rheological properties preservation facing changes on the temperature, pressure and dispersed phase

The biodiesel transesterification reaction generates crude glycerol as a by-product, which has been excessively generated due to the growing production of this biofuel. The possibility of allocating this low value-added waste to applications would solve several problems, especially regarding the need to dispose of this material. In this sense, the use of crude glycerol in olefin-based inverse emulsion fluids is something new in the literature. However, its performance in drilling fluids must ensure that the fluid's rheological properties are achieved. For this purpose, the plastic viscosity, yield stress and thixotropy of olefin-based fluids were studied, varying the temperature, pressure and the composition of the dispersed phase (sodium chloride brine (NaCl), crude glycerol and pure glycerol). A 23 factorial design was also carried out to assess whether the dispersed phase type, pressure and temperature variables affect the rheological properties of these fluids. The results showed that crude glycerol maintains the rheological properties of the conventional NaCl brine fluid even with changes in temperature and pressure, enabling the use of this industry waste for this type of application.

Characterization of the mud displacement in an enlarged wellbore: An integrated rock-fluid model

Cement-mud displacement plays a crucial role in the sealability of cement sheaths. Irregular geometric features of a wellbore due to washout can have a negative impact on mud and cement mobilization. An unstable interface between two fluids always leads to mud channeling, interfluid mixing, and cement contamination, degrading the cement quality. Many factors, such as mechanical and rheological properties of fluids, annulus geometry, flow pattern, and flow rate, significantly influence the displacement efficiency. This study investigates the characterization of the mud displacement in an irregular horizontal well using a 3D computational fluid dynamics (CFD) model. Mud is displaced in an enlarged wellbore by geopolymer and neat class G cement. The wellbore geometry is developed based on the caliper log data from an unconventional shale well in the Tuscaloosa Marine Shale (TMS) lithology. The effects of pump rate, density difference, and mud contamination are evaluated by numerical simulations. The results present those residual muds mainly exist in the upper annulus of the enlarged section. Geopolymer has a better sealing performance and can resist more water-based mud (WBM) contaminations than neat class G cement. The scenario with a low mud-cement density difference and high cement injection rate results in a high cement volume fraction, mitigating the gas migration.

Study on Attapulgite as Drilling Fluid Clay Additive in Persian Gulf Seawater

Drilling fluids are a vital part of every successful well construction operation. Water based fluids are used commonly due to better environmental compatibility, lower cost and easier preparation. In offshore drilling, seawater can be used as the basis of water based fluids. Salinity of seawater restricts application of some additives. For example, bentonite settles in saline environments. In this study, a synthetic water is prepared based on Persian Gulf seawater. Bentonite, pre-hydrated bentonite and attapulgite suspensions were developed based on fresh water and prepared synthetic water. Rheological and filtration properties of fluids were tested to check their performance in synthetic seawater. Results of filtration measurements showed a thick mud cake and high filtration volume in pre-hydrated bentonite fluids. In the case of attapulgite, filtration volume of suspensions in synthetic water increased comparing to suspensions in fresh water. However, filtration properties were acceptable. Study on rheological properties revealed that Herschel-Bulkley model can predict rheological properties with a good accuracy. This is the case for suspensions in both fresh and seawaters. Also it was seen that all suspension had a flow behavior index less than 1, showing their shear thinning character. By increasing clay concentrations, higher consistency index, yield stress and gel strength values were reported. At higher clay concentration a stronger three-dimensional network of clay particles in aqueous environment and consequently a stronger gel structure were formed. Overall, it can be concluded that attapulgite can be used in the saline environment of Persian Gulf seawater.

Physical-mathematical model applied in the study and simulation of dissolution of sodium chloride particles in brines in upward flow in tubes

Drilling trough salt layers has been a challenge for the oil drilling industry. In this way, water-based muds are a friendly environment, although they tend to dissolve generated salt particles along with the wall of the well. Particularly, the particle dissolution may cause changes in the fluid rheology and physicochemical properties. Other problems related to this are the enlarging of the borehole, the excessive torque, the accumulation of salt gravels in the bottom hole, the well controlling issues, and some other operational situations. Thus, the controlling and prediction of the salt dissolution process in drilling fluids are of great interest. The main objective of this work was to study the dissolution of halite (NaCl) particles in brines at different concentrations on upward flow in tubes that simulate the borehole flow. An experimental apparatus was built, capable of reproducing some drilling conditions as the upward fluid flow where the data of NaCl concentration as function of position and time were obtained. A mass conservation mathematical model based on the continuum mechanic was proposed in order to simulate the process. The model was composed of two differential partial equations generated by the conservation law for the mass of salt in two phases: a fluid phase of salt dissolved in water and a solid phase of salt particles. The dissolution data were used to estimate the global mass transfer coefficient in different operational conditions and to validate the mathematical model. The total mean deviations were lower than 5 % for the proposed and validated model.