Temperature-related covariates were the most important predictors, with the mean diurnal range (BIO2) being shared by all the base-learners . The model was next used to predict the distribution of future biomes for the periods 2040–2060 and 2061–2080 under three climate change scenarios . Comparisons of predictions for the three epochs  show that increasing aridity and higher temperatures will likely result in significant shifts in natural vegetation in the tropical area  and around the Arctic Circle . Projected global maps at 1 km spatial resolution are provided as probability and hard classes maps for BIOME 6000 classes and as hard classes maps for the IUCN classes (six aggregated classes). Uncertainty maps (prediction error) are also provided and should be used for careful interpretation of the future projections.The global potential distribution of biomes  was modelled using 8,959 training points from the BIOME 6000 dataset and a stack of 72 environmental covariates representing terrain and the current climatic conditions based on historical long term averages (1979–2013). An ensemble machine learning model based on stacked regularization was used, with multinomial logistic regression as the meta-learner and spatial blocking (100 km) to deal with spatial autocorrelation of the training points. Results of spatial cross-validation for the BIOME 6000 classes show an overall accuracy of 0.67 and R Representative Concentration Pathways (RCPs)  , future s) e.g., , human ps) e.g., , future s) e.g., , future s) e.g., , and futs) e.g., . Even ths) e.g., : 1.Process-based mechanistic modeling: simulating evolution of the environment using biophysical process-based Earth System Models (ESM); 2.Data-based modeling: training predictive models using observations from the past and then extrapolating these models into the future;i.e. monitoring stations to fit spatiotemporal models l models . An adval models . In addil models . Yet, stl models , includil models  or usingl models .https://probablefutures.org/). e.g. extreme cold days). Their results indicate that prediction accuracy can be improved by extending the averaged climatic conditions with maps of climate extremes e.g. bioclimatic variables and similar.Predictions of future states of climate, land cover, terrestrial ecosystems, human population and similar have proven to be useful, with many of the datasets being frequently cited and used to communicate our possible futures  we first describe our predictive mapping framework based on using biome training points ; (2) we “Representative Concentration Pathways”  at 1 km spatial resolution, with the probabilities in each pixel summing to 100%. For each class we also provide model uncertainty maps. We excluded the continent of Antarctica, because of the presence of permanent ice areas and lack of training points. Also, other areas covered by water bodies, barren land and permanent ice according to ESA’s global land cover maps for the period 2000–2015  were exctaxa to PFTs, which were next ascribed to a specific biome following PFT-based biomes definitions. From the first version of the data set to its final publication by megabiome classification legend . While the megabiome system implies a necessary loss of information due to generalization, the original standardized classification system devised by We used the BIOME 6000 data set, compiled by A total of 72 spatially explicit and harmonized variables representing climatic, bioclimatic and topographic factors were used for modeling purposes. All the layers were resampled to a standard grid covering latitudes between 87.37°N and 62.0°S and reprojected to the coordinate reference system EPSG:4326 before the analysis. The original spatial resolution of the layers was used during the spatial overlay with the point dataset, while for the rest of the calculations all the layers were resampled to a spatial resolution of 30 arcseconds (approximately 1 km at the equator).W/m2) values by 2100 due to increase in greenhouse gasses (GHG) emissions. The new IPCC AR6  and with climate warming in response to carbon dioxide emissions that might be larger than supported by other evidence  was not included because of its excessive number of missing values in the layers of the future time periods. We also used monthly minimum, average and maximum temperature and monthly precipitation, for a total of 66 climatic and bioclimatic predictor variables. They can be downloaded from https://chelsa-climate.org/downloads/. We used six additional predictor variables representing topographic conditions, i.e., sine and cosine of aspect, slope, upslope curvature and downslope curvature. These covariates have a three arcsecond resolution (∼90 m at the equator) and they were derived from MERIT DEM  which then produces the final predictions. We used Random Forests (RF)  , generalsts (RF)  with Lassts (RF)  and gradsts (RF)  as composts (RF) : the outgression  as the mi.e. no difference between the first two classes, highest confusion possible) to 100 ; in short, high values would be measures of low uncertainty, while low values would indicate a high uncertainty. All the analysis were performed using R (version 4.1.1) (mlr package (Predictions are delivered as probability maps (0–100%) together with uncertainty maps: the standard deviation of the predicted values by the base learners serves as an indication of model uncertainty. The principle is that the higher the standard deviation, the more uncertain the model is regarding the probability to be assigned to the pixel . In contn 4.1.1)  and, spe package . For morλ for GLM  biome; •T7 - Intensive land-use biome.T7 - Intensive land-use biome” class was not considered. The complete translation scheme is available in i.e., low confusion) predictions. To assess change in potential biome class in fact, we calculated the difference in hard class between the potential biomes map of the current period and each of the future periods and RCP scenarios. We first reprojected all the IUCN classes and relative margin of victory maps to the Interrupted Goode Homolosine projection, which is an equal-area composite projection. We chose it specifically to provide an unbiased  estimate of the areas subjected to change. In the results we discuss change dynamics only for the aggregated IUCN classes and only for pixels having a margin of victory ≥ 50%; pixels with a margin of victory < 50% are not considered.Since the focus of this paper is on potential biomes, the “The hyperparameter tuning resulted in the following architecture for the ensemble model: •mtry 10, while the other hyperparameters were set to default;Random Forest: 452 trees, minimum node size 9,  •Gradient boosted trees: 20 boosting rounds, maximum depth per tree 5, learning rate 0.5, minimum loss reduction to split a leaf node 10, subsample ratio of the training instances 1, subsample ratio of columns when constructing each tree 0.5. The other hyperparameters were set to their defaults; •λ value 1.1 × 10−5;Generalized Linear Models with Lasso:  •Multinomial logistic regression: multinomial function to minimize the loss.tropical evergreen broadleaf forest” is the class with the greatest gain in predictive performances  number of points in the training data for this specific class. It is also the only class with negative gain in predictive performances: all the other classes go from weak (“5 km2) and the most in the scenario RCP 8.5 for the epoch 2061–2080 (5.0 × 106 km2). For epoch 2040–2060, the main changes shown by all three scenarios are as follows: areas belonging to the polar/alpine biome will transition to the temperate-boreal forest biome and areas from the tropical forest biome will transition to more drier biomes, like the savannas and grasslands biome the shrublands biome and, in some cases, the deserts and steppes biome.Examining the biome state transitions from the current conditions to the future epochs, we found that most locations remained stable. Filtering the transitional areas with the margin of victory across all scenarios and epochs using 50% as a safety threshold value considerably reduced the predicted transitional area: less than 1% of the Earth’s surface showed signs of change, with the least changes found in the scenario RCP 2.6 ; while our TPR values are consistently lower across all classes, TPR values reported by In this study we trained an ensemble machine learning model to classify the current and future potential distribution of biomes under different climate change scenarios. Our results show that it is possible to produce relatively accurate maps of natural vegetation using ensemble machine learning approaches and to reach consistent accuracy values even with a limited selection of predictor variables. Comparing our results with the previous work of ced data . The faci.e., less features) while maintaining consistent values of accuracy, and keeping the model simple enough to be able to transfer it to future epochs without introducing too many assumptions in the modeling framework; a similar approach to calculate future projections was used by In machine learning, increasing the size of the feature space is expected to provide more discriminating power , at the i.e., invasive species modeling, past vegetation reconstruction, future vegetation forecasting etc.)  . Howeverng etc.) . For futng etc.) ; startinng etc.) , hence wng etc.) , the samWe evaluated changes in potential biomes in two future epochs and across three different climatic scenarios: our results show that the distribution of the biomes on land in the future will mostly (≥99% land surface) remain the same. The limited geographic extent of the biome shifts under all three scenarios has probably to do with the chosen conservative threshold in margin of victory; despite that, the projections show specific emerging trends in biome shifts in precise locations of the globe that, while differing in size, are common across all the climatic scenarios.“boreal forests and woodland biomes” class mostly due to the fact that the low temperatures are a limiting factor for the presence of trees. The hypothesis that the thawing of permafrost would lead to the tree line advancing towards the North Pole finds more and more evidence, the last one provided by hot spots of change, from which the shift can then expand: the model used in this study does not take into account many factors, such as the feedback effects on the carbon cycle caused by the permafrost thaw : process-based models are known to be able to include and parametrize this factor, event though the mechanisms and limits of it are still not completely understood. 2 for every RCPs: so while our model may accurately capture a shift in the boreal region, due to the fact that the limiting factor for those biomes is temperature, it may capture a different type or extent of shifts in the tropical region, where neither temperature or precipitation are the limiting factors. On the other hand, however, hot spots of change in the tropical region, there is still high uncertainty for the future of biomes and their shifts there.Secondly, while the use of expert knowledge to select the predictor variables let us to reduce the complexity of the task, it may also have introduced biases or limitations to the model’s ability to accurately represent the full range of conditions present in different biomes; feedback loops (vegetation–climate interactions) or anthropogenic factors  were also not considered in the study. One of these effects which are difficult to incorporate in data-driven models is the fertilization effect of increasing concentration of COx for epoch 2061–2080 is not only a function of the predictor variables in the features space, but also of the realization of the phenomenon in previous states. The result, in the worst case scenario, is that the predicted values for one location may not be reliable over time: an example in our case would be a pixel labeled as “desert” for the current epoch, “forest” in the epoch 2040–2060 and “desert” once again in epoch 2061–2080, something that can be perfectly explained from a mathematical standpoint, but highly questionable from an ecological perspective.We also excluded from the study area those locations that are currently covered by permanent ice: glacial retreats due to climate change is a well known issue and the exclusion of these areas may also underestimate the potential changes in biomes in surrounding areas that may be influenced by the loss of glaciers. It is important to consider the inclusion of these areas in future studies to obtain a more comprehensive understanding of the potential impacts of climate change on the distribution of biomes. Another limitation is that the model cannot be considered spatiotemporal: while spatial relationships are taken into account during the modeling through the use of spatial partitioning, the temporal relationship is not considered since the model is trained on only one point in time. The model does not know that the three epochs analyzed in the study have an order in the temporal dimension: the phenomenon we want to predict at location Together with the heavily imbalanced dataset, this is another important reason to use the margin of victory to analyze the results. The inclusion of the margin of victory allows users to more accurately interpret the predicted maps and make more informed decisions based on the data. Without the margin of victory, the probability outputs could potentially be overinterpreted, leading to incorrect conclusions. The classification task examined in this study presented a total of 20 classes, with the model output per class constrained, for each pixel, to sum to 100%; the conservative threshold allowed us to focus our analysis only on those areas where the model found considerable differences in probability output between the dominant class and the remaining classes see . Even thIn this article we applied a methodological framework to predict current and future potential distribution of biomes under different climatic change scenarios using an ensemble machine learning approach. We focused our efforts on improving the caveats of previous work from “savannization”. In contrast, biomes located at higher latitudes, such as boreal forests, are expected to experience an expansion in their potential distribution in the future time periods at the expense of the polar biomes.Even though relatively small shifts in the distribution of biomes were projected under the RCP 2.6 and RCP 4.5 when compared to RCP 8.5, one of the significant findings of this study was the identification of areas where the change in climatological conditions could lead to a shift in the potential distribution of biomes regardless from which of the emission pathways analyzed will happen in the future. The biomes expected to shift the most are the tropical and subtropical biomes, particularly the tropical rainforests: these biomes are expected to experience a decrease in their potential distribution in the future time periods towards savanna and grassland biomes, a process called Further research is needed to better understand the factors that drive these shifts and the potential consequences for the distribution and diversity of plant and animal species, as well as for human communities. We hope that this study will contribute to the broader field of study by providing a framework that can be used to better understand the potential impacts of climate change on the distribution of biomes and their associated ecosystems, and by identifying areas where these impacts could be particularly significant. We recommend that this information is used by policy makers and land managers to make informed decisions about the management and conservation of these ecosystems, and to take action to mitigate the negative consequences of climate change.


Characterization of heterogeneous reservoirs such as multilayered or fractured systems is an important issue in different disciplines such as hydrology, petroleum and geothermal systems. One of the popular methods that can be used for this purpose is tracer tests. Better understanding of the mechanisms of mass transfer (convection–diffusion process) is essential for having a proper test interpretation. In this study, the solutions of different scenarios of tracer flow in a pair of high and low-permeable layered reservoirs including convection and diffusion mechanisms are discussed. Although analytical solutions generally provided exact solutions, they involve several assumptions and might be hard to use for complex problems. As a result, numerical methods are selected for the investigation of different scenarios and addressing cases that are beyond access of analytical methods. In this study, several scenarios of considering diffusion and convection in low and high permeable zones and effective parameters on tracer concentration are investigated. According to the results of this study, the higher the porosity ratio of low to high permeable layer, the more time is needed to get the final concentration value. Also, by increasing the value of the dispersivity coefficient, the time needed to increase the concentration decreases. In other words, the sharp increase in concentration for lower times is seen in higher dispersivity values. The concentration profile variation is affected by Peclet number. The difference among concentration profiles in different cases is considerable, especially in low Peclet numbers where the diffusion mechanism is dominant. This behavior is more common in low permeable mediums such as multilayered tight or shale reservoirs. Any prediction of reservoir performance in these types of reservoirs based on limited information may not be true in some cases and should only serve as a starting point for analyzing the mediums. Multilayered and fractured reservoirs are two common types of heterogeneous systems. In these reservoirs, the fluid flow will be done in two or more different environments. Fractured reservoirs are composed of two different medium, including a matrix system with high porosity and low permeability, as well as fracture network with low porosity and high permeability in which reservoir fluid may be flow between these two mediums. On the other hand multi-layer reservoirs can also include different environments with porosity and permeability distribution, where their fluid flow calculation may be somewhat similar to fractured reservoirs. Therefore, a better understanding of the mechanism of mass transfer (convection–diffusion process) in these reservoirs and also the identification of affected parameters to flow is an important topic in fluids flow phenomena. Besides the different methods of reservoir characterization, the tracer flow tests are the methods for the better investigation of the mass transfer process in heterogeneous reservoirs. Different investigations have been done on tracer flow in heterogeneous reservoirs such as naturally fractured and multilayer reservoirs7.Essentially reservoir heterogeneity is a variation in reservoir properties as a function of space and this challengeable factor exists in a lot of underground reservoirs. Therefore, identifying reservoir properties and also the mechanisms of fluid flow in heterogeneous systems is a vital goal of reservoir engineers for better characterization and reservoir management8. Yuen et al. presented a computer algorithm to determine the degree of heterogeneity in these reservoirs using Brigham equations9. Abbaszadeh-Dehghani and Brigham (1984) reported analytical expressions to analyze tracer flow in layered reservoirs using a nonlinear optimization technique10. Sato and Abbaszadeh used tracer pulse test response as a function of layer number to evaluate the maximum produced concentration from homogeneous and multilayered reservoirs11. Sometimes later Samaniego et al. presented new models to evaluate tracer flow behavior in heterogeneous reservoirs using dimensionless parameters12. Shook used a flow-storage capacity diagram to get an idea of flow geometry, swept volume, and saturation distributions in the layered reservoir13.To better evaluation of multilayered reservoirs as a type of heterogeneous system, Brigham and Smith proposed a tracer flow model in a water flooding process. This model was able to describe permeability heterogeneity in layered reservoirs14. Jason et al. show that changes in tracer concentration gradients can be good indicators of changes in porosity (or water saturation) between layers in multilayered reservoirs15. Shen et al. presented a new formulation to evaluate multilayered reservoir behavior regarding limited crossflow16. Bahamon and Mora described a method to evaluate water channeling and assess sweep efficiency improvements in a multilayered high water cut field using an inter-well tracer test to better perform the water flooding process17. Davidescu et al. used a tracer test in the polymer and water injection process to evaluate the sweep efficiency of a multilayered reservoir18.Suarsana et al. used the comparison of tracer test results and analysis of connectivity injector and producer to investigate pilot water flooding performance in a multilayered reservoir19. After that Raven et al. investigated the tracer flow in a fractured geothermal reservoir by introducing the Peclet dimensionless number to evaluate the time of tracer breakthrough in matrix fracture media20. Ramirez et al. investigated the response of the tracer test in matrix fracture medium and tried to estimate fracture opening and also matrix diffusion coefficient in different geometries21. Samaniego and Rodriguez used the tracer data in an observation well to estimate some reservoir parameters in a fractured reservoir using a trial method22. Qasem et al. showed that parameters such as porosity and permeability and their relationship significantly affect tracer results in fracture-matrix systems23. After that Samaniego et al. in 2005 presented a model to investigate the tracer flow in the homogenous and heterogeneous reservoir. They reported that some reservoir parameters such as block size and also dispersion coefficient can be estimated using tracer analysis12. Kocabas and Maier used analytical and numerical models to investigate the tracer flow in a fractured reservoir24. Haddad et al. used analytical methods to estimate the dispersion coefficient in different geometries of matrix-fracture media by solving the convection–diffusion equation25. Aymen et al. used the tracer response in analytical and numerical models to estimate water cut in fractured reservoirs26. Abbasi et al. used tracer data and discussed the analytical solution of the mass transfer equation to evaluate the shape factor in a fractured reservoir27. Jing et al. estimated the future performance of an ultralow-permeability fractured reservoir using tracer test and production data28. Kumar et al. reported that the combination of tracer data, completion data and also stimulation data in a machine-learning framework could show better dynamic behavior in a fracture network system29. Also a brief history of activities performed in the analysis of tracers is presented in Appendix C57.Some investigations also worked on tracer flow in the matrix-fracture mediums in fractured reservoirs as other types of heterogeneous systems. In 1986 a mathematical model was developed to estimate fracture width using tracer test results. The results showed good agreement in comparison with laboratory dataIn this study, first, an introduction to the general convection–diffusion equation is presented. Then, tracer flow in a two-layered system as a heterogeneous reservoir is evaluated in three scenarios. In the first scenario (Base Case), convection in higher permeable and diffusion from higher to lower permeable layers are dominant mechanisms. In the second case, the diffusion toward the lower permeable layer (L2) is also included in the evaluation. Finally, the convection mechanism toward L2 is added to the calculations. In this work, three different cases of mass transfer in tracer injection with different boundary conditions are investigated. Because only the analytical solution of the simplest case is available in the literature, the numerical methods are considered to solve mass transfer equations in these cases. Accordingly in the next section the results of analytical and numerical methods are compared for base simple case to check the validity of numerical methods. After that, the sensitivity analysis of affecting parameters in the convection–diffusion equation is done. Finally, a comparison among different cases of tracer flow in these layers is investigated.d and  molecular diffusion occurs from higher to lower concentration. Diffusion in porous media can be explained using Fick's first law. The mass of diffused fluid is related to the concentration gradient as below:If dilution of solute at the advancing edge of flow occurs, the mixing will be mechanical dispersion. In this condition, the mixing which occurs along the advancing edge of flow is longitudinal dispersion and the mixing in the direction normal to the flow path is called transverse dispersion.DL and DT are hydrodynamic dispersion coefficient parallel  and perpendicular (transverse) to the principal direction of flow.If the mechanical dispersion is described by Fick’s law for diffusion and it is assumed as a function of the average linear velocity, then the coefficient of mechanical dispersion will be calculated by introducing the dispersivity. Dispersivity denoted by α is a property of porous media. Dispersivity times the average linear velocity i.e. α.v will be called the coefficients of mechanical dispersion The combination of this parameter and molecular diffusion (D*) is called hydrodynamic dispersion coefficient which is represented by the following formulas;vi is uniform in space, Eq. (DL) will be used:DL—the longitudinal hydrodynamic dispersion (L2/T); DT—the transverse hydrodynamic dispersion (L2/T).For those domains where the average linear velocity ace, Eq.  for one-xis, Eq.  will be 59:On the other hand, the radial flow from a well in polar coordinate can be written as bellowIn this study, the above equations will be discretized and solved numerically to investigate the convection–diffusion mechanisms in heterogeneous reservoirs. These types of reservoirs may be presented by high and low-permeable systems such as Matrix-Fracture or multilayered media.Numerical modeling is a powerful tool to evaluate physical phenomena such as convection diffusion in heterogeneous reservoirs. Although the analytic solution of an equation or expression is the exact solution and guarantee that the resulting quantitative predictions are accurate, due to some complicated boundary conditions and limitations of analytical methods, numerical calculations are more useful. In the following sections of this study, numerical methods are used to evaluate physical phenomena such as convection diffusion in heterogeneous reservoirs. This study investigates the convection–diffusion mechanism of two-layered reservoirs with different degrees of contrast between layer's permeability. Different cases of flow in this reservoir are introduced to solve by numerical modeling.First of all, it is assumed that tracer flow is occurred in two-layered systems from the wellbore toward the higher permeable layer (L1) by convection mechanism. Also, diffusion in vertical direction is the dominant mechanism for mass transfer from the higher permeable layer to the lower permeable layer (L2). This case may be similar to a dual porosity system in fractured reservoirs. In this case (base case), diffusion exist in L1 but it is not the dominant mechanism.In the second case, the diffusion toward the lower permeable layer (L2) is also included in the evaluation. Finally, the convection mechanism toward L2 will be added to the calculations.hb/2, α, qo, h and D1 represent the porosity ratio of L2 to L1, thickness of layer L2, dispersivity, production rate, thickness of L1 and diffusion coefficient in L1 respectively. The derivation of this equation is presented in Appendix A.To solve the convection–diffusion equation, first of all, the dominant mechanism in L2 will be assumed to be the diffusion from L1 in the vertical direction using the following formula where subscript numbers 1 and 2 represent the higher and permeable layer properties respectively:1D, CD2, and tD represent dimensionless concentrations of L1 and L2 layers and dimensionless time respectively. Also, hR is the dimensionless ratio of L2 layer thickness to wellbore radius and ZD represents the dimensionless ratio of the vertical coordinate of the L2 layer to wellbore radius. The terms αD and D2_1D show dimensionless dispersivity and dimensionless diffusion coefficients of L1 to L2 respectively. On the other hand, to define the relation between convection and diffusion mechanisms, dimensionless Peclet number is introduced. Peclet number (Pe) is the ratio of advective velocity to the molecular diffusion coefficient. Fundamentally the value of this number is lower in regions of lower permeable than higher permeable layers.If the diffusion coefficient term is assumed to equal zero in high permeability medium which is not unusual, then the equations will be presented in the following short form:Pe < 0.3 presents the diffusion regime in which convection is not the case and diffusion is dominant.The regime 0.3 < Pe < 5 is the transition zone in which the effect of convection is important but the diffusion effect is strong.In the range 5 < Pe < 300 the convection dominates dispersion, but the effect of molecular diffusion cannot be neglected.5 the flow regime is the purely convective regime.When 300 < Pe < 105 dispersion is in the turbulent regime in which the Peclet number is no longer the only correlating parameter, as the Reynolds number should also be considered. In this condition or flow through porous media, this regime is not of interest.For Pe > 10In this study, the relation between the velocity of the high permeable layer (L1) and diffusion between high and low permeable layers (L2) will be presented with Pe2_1 as defined in Eq. , a, a18), aIn this section, the convection–diffusion equation for two layered defined systems will be presented in three following cases named I, II, and III. In all cases, it is assumed that the convection from L1 to L2 is ignored. Also, the diffusion in L1 wouldn’t include in calculations due to its low effect on mass transfer. It is considered that Eq.  will be high_k and hlow_k are the half length of L1 and L2 layers respectively. In this figure, the convection exists in L1 and the diffusion is considered to exist form L1 to L2. As is presented in Fig. low_k and Nhigh_k are the number of grids in vertical and radial directions for lower and the high permeable layers respectively. The total number of grids in high permeable layer equals Nhigh_k and the total number of grids in the low permeable layer equals NN-Nhigh_k. It is assumed that L1 and L2 are divided by 10 grids for low permeable layer and 20 grids for high permeable layer in Fig. The schematic of diffusion-convection flow in pours media, in this case, is shown in Fig. In this case, Eq.  will be g to Eq. , the iniIn this case, the diffusion toward the L2 is introduced as another existing mechanism of mass transfer in addition to diffusion from L1 to L2. The schematic of diffusion-convection flow, in this case, is shown in Fig. D = 1) is equal to one and its value in far distances from the injection point will be zero.In this condition the mass transfer equation in L2 can be shown in the following forms:In this case, it is considered that the convection also exists in L2 as a tracer flow mechanism in addition to before mentioned ones. In the discretization process, the boundary condition of tracer flow from the wellbore to its adjacent grid is different from other grids.em) will be introduced to describe the convection flow of the tracer from the wellbore toward the lower permeability layer (L2). This case is similar to the dual permeability mechanism in fractured reservoirs. The initial and boundary conditions are similar to the previous case. The schematic of flow in case III is illustrated in Fig. In this case, the rates of tracer flow from the wellbore toward the L1 and L2 will be different. Due to this difference, a new dimensionless Peclet number  is considered to equal five. The value of time difference (dt) is assumed to be equal to 0.5, the value of horizontal changes (dz) equals to 0.5 and the value of the radial change (dr) equals to 5 in numerical modeling. Therock and fluid properties and also the porosity ratio of the low permeable to the high permeable layer is assumed to be constant; therefore, the difference in porosity ratio in different grids is ignored. On the other hand the properties of the grids are assumed to be constant with time, except for the changes in concentration.The definition of the new Peclet number and also general convection–diffusion equation in L2 will be presented in the following forms:The dimensions of the grids are assumed unchanged in numerical calculations. The possible existence of diffusion mechanism from the low permeable layer to the high permeable layer has been neglected. The other assumed parameters are presented in Table r changes from 1 to N high_k, and the value of z + 1 will change from 1 to N low_k + 1 in grid numbering.On the other hand, other assumptions will be introduced individually in each case. The total number of discretization grids from the injection well to a production well is assumed to be 220 grids for L1 and L2 in which N low_k and N high_k are equal to 10 and 20 respectively according to Eq. . In our The detailed procedure to solve the convection–diffusion problem is presented in Appendix B for a twelve 12) grid system in which (N low_k + 1 × N high_k) equals 12 grids according to Fig.  grid sysIn this section, the result of convection–diffusion equations in the above-mentioned cases will be discussed. First of all, a comparison between analytical and numerical calculations will be evaluated to check the accuracy of numerical modeling. Then sensitivity analysis on affected parameters in the concentration profile and also the comparison between cases will be investigated in more details.25.Although analytical solutions are generally considered to be more rigorous than numerical methods due to providing exact solutions, sometimes these methods might not be able to handle complex problems with specific assumptions. As it mentioned in previous section, three different cases with different boundary condition will be discussed in this study in which only the first case is solved using analytical methods in literatureIt is obvious that in order to compare the numerical and analytical solutions, the values of 25:The analytical solution of a mass transfer equation is solved in base case Eqs.  and 30 wDue to complexity of assumptions in other cases, in this study, the numerical methods are selected to use for the investigation and evaluation of mass transfer equations in the three mentioned cases, but due to check the accuracy, the first case (Convection in L1—diffusion from L1 to L2) is solved by both the analytical and numerical methods. Figure Figure As it is shown in Fig. Also, this figure shows that the time to reach the final value of concentration in L1 will be increased in higher numbers of the discretized grid for all values of porosity ratio. As an example, it is shown that the concentration profile will yield faster to the final value in the numerical grid  in comparison to the numerical grid  because of the lower distance to the source of tracer injection.On the other hand, Fig. The value of the Peclet number is another important affected parameter in concentration–time profile evaluation. By increasing the value of the Peclet number, the concentration will be faster received to its final value. Also due to less effect of the convection mechanism in lower values of Peclet number, the time to reach the final concentration value is increased in low Peclet numbers. This behavior is depicted in Fig. The variation of concentration versus location at different times for the first layer also is shown in Fig. After discussing the concentration variation in L1, the profile of concentration time in L2 will be evaluated in the following figures. In this condition, only diffusion from L1 to L2 is included in calculations for this layer.As is shown, the shape of the concentration profile in grids near layer L1 Fig. a is diffAlthough the trend of concentration value approaches to unity in L2 like L1, because of the diffusion mechanism and also lower permeability values, the final concentration will be yielded at later times.In this case, the diffusion mechanism in L2 will be included in addition to diffusion from L1 to L2. Therefore two new boundary conditions will be used in numerical calculations of mass transfer for this layer. The concentration profile for the higher permeability layer (L1) is similar to the previous case.In this case, adding a new diffusion mechanism may not have so significant effect on mass transfer calculation. It is because, in higher values of the Peclet number, the convection mechanism is dominant. In this condition, the concentration–time profile wouldn’t be so different in comparison to case 1 as shown in Fig. To better understand the importance of the diffusion mechanism, especially in L2, a sensitivity analysis is done on Peclet number values. To compare the results, two grid numbers are selected and the concentration–time profile will be evaluated for different Peclet numbers in cases I and II. To do this, the grid number  will be selected. Because the effect of diffusion is so important in low Peclet number values, the values of 10, 100, and 1000 for this dimensionless number were considered for comparison.A comparison of the results of the concentration–time profile in Case I and Case II is shown in Fig. The above figure also shows that the effect of tracer diffusion in L2 is not considered when values of the Peclet number increase in the range of convection dominance.In this case, the convection mechanism in L2 is also included in addition to other defined mechanisms in previous cases. Although the concentration–time profile for L1 is not the case of change, due to including the convection mechanism in calculations of L2, the concentration profile will be affected by a considerable change in comparison to the other cases. The results of this case can simulate the convection–diffusion flow mechanisms in the real porous medium.According to Eq. , a new PThe diffusion mechanism is dominant and has a more visible effect in low permeable mediums such as single/multilayered tight or shale reservoirs. Also, diffusion mechanism is more considerable in lower values of Peclet number.As the porosity ratio of low-to-high permeability layer increases, more time is needed to reach the final concentration value. Also by increasing the dispersivity coefficient, the sharp and fast increase in concentration occurs in smaller times.Although the trend of concentration in the low and high permeable layers are similar to each other, because of diffusion mechanism and also permeability values, the maximum concentration in low permeable layer will yield at later times.The effect of including convection in the low permeable layer is considerable at initial locations near the tracer injection point, but the effect of diffusion from the higher to lower permeable layer will be more important and affect the results in farther locations.In the current study, a detailed numerical evaluation of the convection–diffusion equation in layered reservoirs is discussed using tracer response. The dispersivity coefficient, the effect of some related parameters to Peclet number, as well as changes in the concentration profile of the tracer, especially at low Peclet numbers, have also been investigated. On the other hand, the time of importance and dominance of Convection and Diffusion mechanisms in the low permeable layer is also one of the main subjects that was evaluated in this study. The main results of this paper are summarized as follows:Supplementary Information.


Campylobacter and Neisseria species. Current methods for identifying the expression states of phase-variable genes involve taking a high number of single colonies. This approach is subject to bias, sampling effects and high workloads that reduce the ability to perform intermediary sampling. The use of high concentration colony sweeps provides a work around but reduces the resolution of combinatorial expression profiles (termed phasotypes). A parsimonious approach combining both single colony and sweep data was developed to overcome these limitations. The critical methodological advance is the use of an algorithm that utilises the experimental data from the two sample types and a parsimonious, iterative mathematical analysis that outputs the phasotype distribution with the highest likelihood of underpinning the experimental data sets. The advantages of this unified method are increased resolution and accuracy of gene expression state combinations as compared to conventional single colony sampling, reduced requirement for sampling large numbers of colonies leading to reduced costs, and a higher capacity for collecting samples and replicates.•Inputting of sweep and single colony data into an algorithm for a rapid determination of the combinatorial phase variation states (phasotypes) for repeat-mediated phase-variable bacterial genes•This method reduces the number of single colony samples required to produce accurate estimates of phasotypes•This method will reduce the costs of phasotype analyses and increase potential to analyse more time points or sample sites leading to an improved understanding of how phase variation contributes to bacterial host persistence and the ability to cause diseaseShort sequence repeat mediated phase variation results in diverse phenotype presentation in many bacteria including   Specifications tableMicrobial population dynamics are subject to change due to environmental pressures and stochastic events that result in bottlenecks. A bottleneck refers to a rapid reduction in population size either due to selection, by the presence of specific environmental pressures that act against certain phenotypes, or non-selectively through random elimination of portions of the population regardless of phenotype Phase variation (PV) is mediated by heritable genetic mechanisms that generate intra-clonal diversity in asexually reproducing bacteria and is one outcome of localised hypermutation within contingency loci. One mechanism of PV is mediated by slip strand mispairing of hypermutable DNA regions containing simple sequence repeats (SSR). Expansion or contraction of the number of repeats in the SSR due to slip strand mispairing affects phenotypic expression of the contingency locus ,5. ChangNeisseria meningitidis and Campylobacter jejuni). Current methods for detecting repeat tract changes include, but are not limited to, whole genome or gene sequencing across single colonies or pooled isolates [Experimental analyses of PV are important for improving our understanding of microbial adaptability and determinants of microbial survival and replication of a number of important bacterial pathogens  and fragment length calling, relative to a size standard ladder, from output files with a software package such as PeakScanner. The most abundant PCR product produces a signature peak reflecting the PCR product length and is termed the major peak. Several smaller peaks, termed minor peaks, may surround the major peak but only the latter is considered for determining the repeat number of the colony. This fragment is then compared to a control PCR product of known repeat number and expression state.Contrastingly, the sweep data set may contain a combination of multiple populations with varying fragment lengths due to differences in the repeat tract lengths. PCR amplification and GeneScan of this population produces a series of peaks whose heights/areas are proportional to the repeat numbers in the population. In order to estimate the expression state(s) for the sweep, all peaks are considered and sub-divided into repeat numbers that produce ON and OFF states. Following this division, the combined total for the peak heights (or areas) of the ON peaks is divided by the combined total for all peaks. For example, if the total peak height showing an ON state is 21,253 while the total peak height for the OFF state is 7,271, then the ON % for that sweep would be ∼75 % . To generate realistic populations, a simple switching algorithm was developed. For a population of 64 phasotypes, an individual phasotype was selected through weighted random selection acting on its proportion within the total population. The phasotype was then subject to a single gene switch (i.e. 000000 to 10000) and then re-added to the population. This produces a normal like distribution around a defined starting population. This process was repeated 20 times to produce 20 unique pseudo-populations with known proportions of each phasotype. The final phasotype distributions were utilised to calculate the ON % for each gene.To test whether sweep data can be utilised in combination with single colony analyses to accurately predict phasotype distributions, the algorithm was tested To mimic the single colony sampling method of experimental protocols, phasotypes were selected randomly from each of the 20 pseudo-populations using a random number generator and the proportion of each phasotype in the whole population. Multiple cumulative samples were generated that contained between 1 and 100 phasotypes. This method simulates picking single colonies from dilution plates of a large bacterial population with the number of samples being equivalent to the number of colonies. The next step was to utilise the algorithm to correct the single colony sampling data. The input data for the algorithm is the ON % for each gene, as calculated for the whole pseudo-population, and the proportions of each phasotype, as calculated from the 1 to 100 sample populations . This correction was performed for each of the sample sizes and all of the 20 pseudo-populations.In order to assess the effectiveness of the algorithm in making improved predictions about the phasotype distribution of a sampled population compared to the real distribution, the Jensen-Shannon divergence equation was selected to determine the similarity of two distributions. A normalised distribution of phasotypes was used to seed each run of the algorithm. This distribution can be assumed to be a probability distribution, where each phasotype is represented as a proportion of the population. Two identical distributions of phasotypes  will produce a Jensen-Shannon divergence of 0 whereas a maximum value of 1 shows the highest possible divergence between the two distributions (e.g. phasotype proportions of 0.5:0.5:0:0 versus 0:0:0.5:0.5). Two scenarios were selected to test the robustness of the sweep corrected algorithm, a highly diverse input population where each phasotype is selected randomly (as outlined above), and a low diversity input population were a single phasotype is selected. This latter approach results in the selected phasotype forming an increased fraction of the population (i.e. 50 % of the population). These scenarios represents two possible situations often encountered during PV testing: A) a population under neutral selection; or B) one under positive selection.in silico scenarios . Thus the sweep corrected data has a tendency to overestimate some rarer phasotypes while underestimating the major phasotype. Conversely, the single colony sampling method overestimates the dominant phasotypes and in general underestimates the rare phasotypes but with occasions where one or a few rare phasotypes are overestimated due to chance picking of a rare phasotype. Overall, both methods provide a very close match to the true distribution at 100 samples with the over and underestimated values not producing a significant variation in phasotype proportion. Importantly, when sample size is reduced to 20 colonies, the single colony method produces significant over and under estimates of rare phasotypes .As a further test, a bimodal example was examined wherein two phasotypes, 000000 and 111111, were chosen to each form ∼33 % of the total population with the remaining phasotypes being randomly distributed by application of the switching method described previously. This scenario represents a rare situation where a population is split between two major divergent phasotypes. In this case, sweep correction only marginally outcompetes the single colony sampling method with sample sizes of >1 . This fiin silico generated data, the sample size would need to identical to the total size of the population. More of less samples will always lead to some phasotypes being over-sampled. For example, if a population has 4 members across 4 phasotypes , a divergence score of 0 can only be reached with samples sizes of 4, 8, 12, etc. and these samples must have identical proportions for each phasotype (i.e. 3 single colonies for each phasotype with a sample size of 12). A sample size of 30 (non-divisible by 4) cannot produce a divergence of 0. In the case of the in silico data in In all cases, the Jensen Shannon divergence score did not reach 0 by 100 samples. This is in part due to the randomness of sampling. In order to achieve a divergence score of 0, for the Multiple SSR phase-variable genes are present in a diverse range of bacterial pathogens and commensals. Stochastic, independent switches in expression of these genes can give rise to high levels of variants, termed phasotypes, with combinatorial differences in expression of these genes. These phasotypes are potential determinants of important bacterial phenotypes and measures of the impact of population modulators such as bottlenecks. An accurate method for analysis of phasotype distributions is therefore critical for assessing changes in phase-variable populations during evolution both within and between hosts. Phasotype analyses have relied on sampling of small numbers of colonies with consequent sampling and resolution issues. The method outlined here describes a novel algorithmic approach to eliminating random selection bias or insufficient sampling and an overall improved resolution. Our tool combines both experimental sweep and single colony data to provide a more accurate picture of the phasotype distributions than currently achieved by the current approach of using low numbers of single colonies alone. The use of sweep data has been under utilised to date due to the risks associated with slippage in SSRs occurring during PCR reactions that create biases in the phasotype distributions. A PCR correction is applied to the sweep data prior to input into the sweep correction algorithm leading to a further enhancement in the accuracy of this approach.in-silico modelling of various scenarios showed that sweep correction of the distribution of phasotypes leads to more accurate determination of the phasotype population distribution than colony samples alone  and OFF (0) state stochastically, reversibly and independently. As such, the probability of a single gene switch is significantly greater than the probability of 2 or more gene switches for a given phasotype.The probability (p) of n gene switches occurring simultaneously for a total of N genes within a phasotype is:For the simplest case of a single bacterium, and hence a single phasotype, this means that only a limited number of phasotypes are likely to be occupied by the majority of bacteria within the population. Additionally, the phasotype with the greatest number of gene switches, as compared to the starting phasotype, will comprise the lowest proportion of the population. This mutational pattern applies limitations to the potential mutational space occupied by a specific expression pattern for individual genes and hence provides a framework for predicting the phasotypes. This framework is the basis for the method elaborated herein. It should be noted that growth of a population through ‘enough’ generations, and in the absence of any selective or non-selective bottlenecks, will eventually lead to generation of a population containing all possible phasotypes. This latter occurrence is likely to be a rare observation in natural populations.Our approach is to combine the minimum parsimony described above with ON % data, generated through a GeneScan analysis of sweep data, and phasotype distributions calculated from a small number of single colonies. This process involves application of a series of iterative steps starting with data from single colonies to seed the system and generation of a new phasotype distribution for each iteration of the algorithm. The phasotype data for N genes is normalised based on the total number of phasotypes within the distribution such that the total population will sum to 1. The input data are, therefore, the ON % for each PV gene in a phasotype derived from analyses of sweep data and the proportions of each phasotype derived from the single colony analysis.The assumed starting point for this method is multiplex PCR and GeneScan data for multiple phase-variable genes as performed on a sweep and multiple single colonies derived from a specific bacterial population. The sweep and single colonies should be obtained from the same serial dilution of this bacterial population. The sweep should be obtained from a plate where there is either confluent growth or 1000+ colonies and the single colonies from plates where there are isolated single colonies. To compute the sweep adjusted phasotypes for this population, the following information must be provided, the number of genes, the ON % for each gene as determined from the sweep data see  and the Step 1Determine ON % from sweep data1.Import the autosequencer file for the sweep data into the PeakScanner software (or other appropriate software) and analyse with the relevant size standard for the samples (e.g. GS600LIZ).2.Examine the peaks for the sample and identify the major and minor peaks for each repeat tract.3.Output the areas for the major and minor peaks.4.Assign repeat numbers to major and minor peaks relative to a control of known repeat number.5.Assign ON and OFF states to peaks based on known associations between the repeat number and in frame translation of the gene sequence.Step 2Analyse single colony data for proportions of slippage1.Analyse the GeneScan data for multiple single colonies as described in 2.For each gene, determine the areas of all major and minor peaks.3.Calculate the total area for all the combined minor peaks and divide by the total area of all peaks to obtain the % area for the minor peaks.4.For each gene, determine the % area of the minor peaks across 10 or more single colonies for samples where the major peak has the same number of repeats.Step 3Correct ON % for slippage1.Calculate the average % area for the minor peaks from multiple single colonies (see Step 2) as a measure of the degree of slippage occurring during the PCR reactions (note that the correction is applied utilizing peaks of the same repeat number and for the same gene).2.Subtract the % area for the minor peaks obtained from single colonies from the % area for the minor peaks in the sweep data.3.Add the % area for the minor peaks obtained from single colonies to the area for the major peaks in the sweep data.4.The output data has now been subject to sweep correction and can be utilized as input for Step 4Input data into the algorithm1.Open program.2.Input phasotypes of single colonies.3.Input ON % values for each gene as derived from sweep data.4.Indicate number of iterations.5.Run algorithm.The algorithm is run in Python v3.9.5 and a suitable computer is required for utilizing this program. The program can produce data for phasotypes of 2 or more genes. The expression states of each gene are coded as 1 for ON and 0 for OFF.Step 5Run algorithm1.The distribution of phasotypes derived from the experimental single colony analyses is converted into a series of ON % for each gene in the phasotype.2.Divergence between the values for ON % for each gene in the system derived from the sweep data versus the single colony phasotype distribution is calculated and the largest variance in ON % between the sweep and colony is used to identify the gene requiring the largest correction.3.All phasotypes present in the distribution are considered for potential changes. This process leads to identification and selection of those phasotypes whose ON or OFF states could be switched to reduce the divergence in the ON % of the identified gene.4.The percent change required to align the ON % for a specific gene from the phasotype data to the sweep data is proportionally distributed between all selected phasotypes. This process simultaneously decreases the relative representation of the most divergent phasotypes in the population while increasing those phasotypes most closely aligned with the observed expression state.5.This new population then becomes the new phasotype distribution. The algorithm goes through multiple cycles adjusting each gene individually until either there is convergence - where the difference in gene ON % is the same between the new phasotype distribution and sweep data - or a user defined number of cycles is completed.6.The final output file contains the corrected gene ON % for the population and estimates of the proportions for each possible phasotype within the population.Each iteration of the algorithm goes through the following steps:-For each cycle of the algorithm only a single PV gene state is considered for each phasotype in the system, therefore producing the minimum number of state changes required to mirror the percentage of gene ON % between the phasotype and sweep data.Step 6Perform check of distributions of the phasotype output of the algorithm1.Utilise the single colony data to derive the phasotypes .6.Where such differences are observed, examine the single colony data for evidence of a bimodal distribution of two phasotypes that have non-overlapping combinations of expression states.7.Flag these samples as potentially requiring further investigation (e.g. sampling of more single colonies) or low confidence.Serial PassageC. jejuni strain NCTC 11168ca was grown for 24 h in Brucella broth (6 mL). After this initial incubation, a sample of 600 µL was transferred to a new liquid culture (6 mL: i.e. a 1 in 10 dilution) and regrown for another 24 h. This cycle was repeated across five days in order to increase diversity within the bacterial population. After the last cycle, a serial 10-fold dilution was performed in triplicate with Brucella broth as diluent. A 10 uL aliquot from each dilution was spread with a sterile cotton swab onto a blood-agar plate and incubated for 2–3 days at 42 °C under microaerophilic conditions:- 83 % N2, 10 % CO2 and 7 % O2 in a VAIN cabinet . Multiple single colonies (n = 287) were picked and individually re-suspended in deionised water in non-skirted PCR plates. Bacterial sweeps were taken from 10°, 101 and 102 dilution plates, re-suspended in deionised water and aliquoted into non-skirted PCR plates. The single colonies and sweep samples were then denatured by heating to 94 °C for 5 min in a PCR machine.GeneScan of multiple phase variable genesA total of 6 phase-variable genes were selected and the repeat tracts of these genes were amplified from all single colony and sweep samples using multiplex PCR reactions, as described by Lango-Scholey et al. Following confirmation of the expected PCR products, an A-tailing reaction was performed on the PCR products as previously described The ‘sweep’ GeneScan analyses were utilised for determining the relative proportions of repeat numbers that correlate with the ON and OFF states by determining the areas under the major and minor peaks observed in the ThermoFisher dashboard. The ON % was determined by dividing the area of the peak representing the ON number of repeats by the area of all combined peaks and multiplying by 100. Finally, a correction for PCR slippage was applied to the sweep gene ON percentage. Slippage in the repeat tracts occurs during the PCR reactions and can alter the estimated size of the major and minor peaks. The slippage correction was calculated utilising the PCR data from the single colony GeneScan data as the majority of single colonies will contain a low number of variants due to derivation from a single cell and to the low number of generations. For each colony, the ratio of the area of the minor to the major peak was selected from the data in the PSAnalyse output file, and an estimate of the average level of slippage across multiple colonies was generated for each gene. A value for slippage rate (as a % from the main peak) was estimated by taking the average major to minor peak ratio and dividing it across the total number of single colonies to gain an average combined −1 and +1 repeat peak error rate per colony for each gene (ranging from 0.3 to 17 %). This was then applied to the sweep data by correcting the values for the major and minor peaks to give a new ON %. For example, if all single colonies for a given gene show a secondary peak of 10 % of the height or area of the major peak, then this is an indication that slippage has occurred within the single colony PCR reactions and will also have occurred in the sweep PCR reactions. Therefore, before estimating the ON % for a given sweep set, 10 % of the major peak data should be subtracted from the minor peak data and added to the major peak data  peak means that the true values for these major and minor peaks are 7700 and 1300, respectively).C. jejuni population in triplicate. The ON % states for six genes were calculated from either the phasotypes of the 288 single colony analyses or from the sweep peak ratios. Three sweep duplicates were combined for each sweep data set while all colonies were combined to build the gene ON % values. The values for the ON % for each of the colony datapoints were corrected for slippage against the single colony values , sweep corrected sampling showed a large reduction in the divergence score with the sweep corrected divergence score for a sample size of 10 colonies being 2-fold lower than for the uncorrected score. Across the corrected sampled populations the divergence scores are similar across all samples sizes ranging from a peak of 0.324 with 10 colonies to the lowest value of 0.14 with 100 colonies whereas the scores for the uncorrected samples range from 0.81 to 0.11.Commonly a range of between 18 and 30 The mean divergence score for the uncorrected population was approximately 0.3 as compared to 0.2 for the corrected population. The Welch's T Test returned a P value of <0.0035 indicating a significant change in the mean for the population from the sweep corrected distribution as compared to the uncorrected population for a sample size of 20.All the comparisons described above were predicated on the assumption that ON % values derived from sweeps had a high accuracy relative to the actual population. However, the sweep PCRs may have inaccuracies due to the quality of the PCR, difficulties in determining slippage rates, peak height/size cut-off effects in the PeakScanner software or a bias in the sweep sampling. These effects may result in major differences between the sweep and actual populations in gene ON % values. The role of these potential error rates on estimating population phasotype proportions was assessed by simulating generic error rates. Twenty sets of 16 single colonies were randomly sampled from the in vivo data set. These samples were compared to the known phasotype distribution using the Jensen Shannon divergence. An additional 7 single colonies were randomly selected across 20 replicates and corrected using the sweep derived ON % states. These samples were also compared to the known phasotype distribution. Finally, the gene ON % values, as derived from the sweep data, were randomly altered by a fixed percentage error change  that ranged from 1 to 100 % . For errn = 288) of single colonies and correcting for PCR slippage using single colony data, we have shown that the ON % of a gene can be estimated accurately for a population through taking sweeps as shown in Both the new model and the current single colony sampling strategies were assessed for accuracy through a PV analysis of experimental data of 6 phase-variable genes. By combining known sweep data for a set of genes with a large volume (The application of the algorithm across a range of sample sizes was compared to uncorrected distributions. As shown in No ethical issues identifiedJonathan Holmes: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Visualization. Lickson Munjoma: Validation, Investigation, Data curation, Writing – original draft. Christopher D. Bayliss: Conceptualization, Writing – original draft, Writing – review & editing, Supervision.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


The surface roughness (Ra and Rz lower than 1 µm and 6 µm respectively), cylindricity (0.045 mm), roundness (0.025 mm), perpendicularity of the hole axis (0.025 mm), diameters and position of the individual holes were achieved for a drill with point angle 138.32°and clearance angle 6.92 respectively. The increase of the drill point angle by 6° resulted in the decrease in the feed force of more than 150 N. In addition, an increase of the clearance angle by 1° resulted with a decrease in the feed force of 70 N. The results of the experiment showed that with the correct geometry of the tool the effective machining without using internal cooling can be realised.Drilling into the solid material is one of the basic technological operations, which creates a cylindrical hole in an appropriate time with required quality. Drilling operation demands a favourable removal of chips from the cutting area because a creation of an undesirable shape of chips can impart a lower quality of the drilled hole corresponding with the generation of excess heat due to the intense contact of the chip with drill. The solution for a proper machining is a suitable modification of the drill geometry i.e., point and clearance angles as presented in current study. The tested drills are made of M35 high-speed steel characterized by a very thin core at the point of the drill. An interesting feature of the drills is the use of cutting speed higher than 30 m min Stainless steels are used in a wide range of industries due to their exceptional properties in the field of resistance to the external environment and elevated temperature. This steel can be used as surgical instruments in medicine, components for the exhaust system of cars or large reservoirs and pipelines in the petrochemical industry.1 examined strain hardening during drilling of austenitic stainless steel. With the help of the cut breaker, it was found that the highest intensity of deformation hardening occurred in the vicinity of the axis of the drill close to the point of the cross cutting edge. The result of strain hardening was that the microhardness of the material increased from 300 to 600 HV 50. Austenitic stainless steel is characterized by up to four times lower thermal conductivity compared to low-carbon steel, which causes an intense increase in temperature at the edge of the cutting tool2. The highest heat occurs on the periphery of the tool in the area of the outer point due to the increasing cutting speed outwards to the drill perimeter3. According to the simulation based on the experiment in the study3 performed on AISI 1045 carbon steel, almost 50% of the generated heat transferred into the tool. The intensity of heat generation can be reduced by decreasing the area of the inefficiently machining core of the drill, as in the case of the CZ005 tool from this study, or by using the appropriate clearance angle of the drill4. Effective cooling of the tool edge is another option for eliminating the generated heat. In the study4, the effect of internal cooling with the pressure of 80 bar towards to the cutting edge was analysed experimentally and through the simulation. It was found that, despite the high speed of liquid supply through the cooling channels, inefficient cooling of the tool edge occurred due to an undesirable fluid turbulence and friction on the drill. According to the results of a study5 conducted on austenitic stainless steel, an increase in the value of the point angle in the range of 82°–118° led to the increase in the temperature at the cutting edge. Under certain conditions, there may be a reduction in cutting forces due to the so-called softening effect6 leading to an improvement of surface roughness. The effect of the cutting speed on the roughness of the machined surface depends on the temperature at which it occurs to prevent the formation of growth in the given material, the values of the cutting speed used and the machined material. In the study7, the lower surface roughness of austenitic stainless steel (about 20–50%) was observed when the cutting speed increased from 18 to 30 m min−1. On the contrary, in the study8 when drilling low-alloy steel, there was an increase in roughness more than 50% when the cutting speed increased from 32 to 51 m min−1. The softening effect featured a major impact in the experiment used by Huang et al.9. They analysed the influence of individual parameters on the magnitude of the feed force when drilling thin plates made of composite material reinforced with silicon carbides. Increasing the cutting speed from 50 m min−1 to the double value featured a different effect on the amount of feed force at different stages of drilling. Before entrance of the entire tool tip, there was a slight increase in feed force due to the increase of cutting speed. In case of drilling workpiece with a thickness higher than 1 mm, when the cutting speed increased, the temperature increased and, but due to the softening effect, the feed force decreased by more than 15%.When drilling stainless steel, several limitations can act on the tool. These limitations include low thermal conductivity, high ductility, susceptibility to strain hardening and high abrasiveness of the machined material. Dolinšek11. The same trend was observed in the current study. On the other hand, the increase of the feed force when the point angle increased was presented by the other researchers13. When the point angle increased from 82° to 118° in the study5 and 110° to 118° in14 the intense deformation of the material at the point of the cross edge caused the higher temperature. Karpat6 performed a development of the special drills with a double edge aimed for machining composite materials. The drills featured the diameter of 6.4 mm, while the length of each drill varied as well as the length of the cutting edge, which enabled to change the dependence of the feed force on time. The drill with a lower point angle exhibited lower feed force compared to the drill with the higher point angle. They found out that the best results were achieved when the drill had the longest cutting edge with a point angle of 120°6. The lower feed force can decrease the delamination of the composite layers, or the formation of burrs when drilling ductile metal materials. Murad et al.15 presented the paper evaluating an effect of point angle when drilling of the titanium Ti6Al4V alloy. Due to the temperature increase in the hole surface, the hole featured the hardening layer. The higher point angle caused an increase in the thickness of the hardened layer of the hole from 0.05 to 0.1 mm. The increase in layer thickness was also observed when tool wear increased.The difficult conditions of drilling austenitic stainless steel described in the previous paragraph require the correct selection of individual geometric parameters of the tool, otherwise the rapid wear is observed. The tool wear is accompanied with the lower quality of the hole corresponding with the increase of the temperature and the high toughness of the material. From the complex geometry of the helical drill, the influence of the point angle on the cutting process is most often discussed in the studies. The first parameter analysed is the feed force. The dependence of the size of the point angle on the feed force varies in each study. It was found that when the point angle increased, the feed force slightly decreased3. At the same time, the long chip can block the access of the cutting fluid into the cutting area4. When the conventional drilling of tough materials, it is not possible to prevent the formation of long chips. The solution can be ultrasonic drilling technology, where vibrations at the frequency of ultrasound are introduced into the classic drilling process. Zhu et al.16 focused on drilling the 0.5 mm diameter holes in nickel superalloy with the ultrasound technology. The vibration added to the drilling process broke the chip into small pieces and the high-density energy input disrupts the material under the tool edge resulting in lower feed force.When machining tough stainless steel, long helical chips, which causes abrasive effect in the drill groove and in the drilled hole are formed. Due to the friction between the chip and the tool, an excess heat is generated, and the surface of the drilled hole may be damaged11, they observed an increase in surface roughness when the value of the point angle increased, but contrary in studies15 the authors presented a decrease in surface roughness when the point angle increased. In the study16, authors found that the surface roughness decreased about 20–30% when implementing ultrasonic drilling technology.The influence of the point angle setting on the achieved surface roughness of the drilled hole varies in individual studies. In the case of the studies17 examined an effect of material structure on the quality of drilled holes when drilling low-alloy chromium-molybdenum steel subjected to different heat treatments. The monitored hole quality parameters were circularity, cylindricality, deviation of the diameter from the nominal value and roughness of the machined surface. It was found that the quality of drilled holes depends on the hardness of the material. When the hardness of the drilled material increased from 34 to 67 HRC (by applied heat treatment), there was an improvement of each individual parameter in the range 30–50%, but conversely, when the hardness of the material decreased from 34 to 24 HRC , the decrease in hole quality of more than 15% was detected17. Improving the ability to drill into soft material can be achieved by implementing ultrasonic drilling technology. In the experiment (carried out by16) there was the decrease in the deviation from the nominal diameter from 25 to 5 µm and the conicity of the hole more than 30%.Accurate drilling of the tool is a prerequisite to produce a high-quality hole, when drilling off-axis, the diameter of the entrance part of the hole increases and the geometric tolerances of the hole deteriorate. The material with lower hardness deteriorates the stabilization of the drill in the drilled hole. Üllen et al.The main aim of this paper is to study an effect of the different drill geometry  on the quality of drilled holes. The quality of drilled holes was assessed by the surface roughness, cylindricity, roundness, perpendicularity of the hole axis, the diameters and position of the individual holes and their mutual comparison. The cutting forces were measured during the drilling operation to verify the effect of different drill geometry. The influence of the tested material surface hardness was evaluated. The novelty of present work is a special design of tested drills, which feature a thin core corresponding with the possibility to set the higher cutting speed with respect to the final hole quality. The thin core utilises to produce a deeper helical groove, which provides the better access for the grinding wheel enabling higher productivity of HSS drills.The material for drilling was the AISI 304 austenitic stainless steel. The material was in the form of rolled plates with dimensions of 206 × 120 × 20 mm, which were used for the experiment. The drilling operations were carried out into the solid material (without the centering of the hole). The microhardness was measured on both sites of the tested material, which featured the significant difference on the bottom and top site. The microhardness  was observed on the cross-section of the material, in which the microhardness exhibited the range 200–400 HV 0.1  manufactured by the company NÁSTROJE CZ, s.r.o. were used. The drills were made of M35 high-speed steel with different point angle and clearance angle on the main cutting edge . Total of 14 holes see Fig.  were dricutting forces (1st and 14th hole),roughness (1st and 14th hole),geometric tolerances—circularity, cylindricality, and perpendicularity of the axis of the hole to the surface of the plate ,—diameter ,pitch .The drilled holes were analysed considering the following factors:A KISTLER 9272 dynamometer  with accessories was used for the purpose of the experiment to measure the cutting forces. The dynamometer was clamped to the worktable of the machining centre using T-slot screws, a vice was attached to the dynamometer for clamping the drilled plate. Cutting forces were measured only when drilling into the side corresponding with the higher observed microhardness (400 HV). The cutting forces were affected only by the change in the geometry of the drill, the other parameters of the process were constant during the experiment.The surface roughness of the holes was measured using a Taylor Hobson Talysurf CLI 1000 roughness device. According to the ČSN EN ISO 4288 standard, the indicative value of the RSm parameter was first measured and the corresponding basic and evaluated length was chosen based on the measured values. After levelling the surface to a horizontal position, the profile of the entire length of the hole was measured. To obtain the roughness profile, it was necessary to perform thresholding, remove the shape of the measured profile using a 5th-order polynomial, and filter the measured data using a Gaussian filter. The evaluated lengths were extracted from the obtained roughness profile see Fig. , and theMeasurements of geometric tolerances i.e., hole diameters and pitches were performed on the Leitz Reference Xi coordinate measuring device. The circularity, the perpendicularity of the hole axis to the surface and the spacing of the holes were measured in two places along the length of the hole in places 3 mm from both sides of the material see Fig. . RoundneThe evaluation of the cylindricity and diameter of the drilled holes took place in three places along the length of the holes  and feed per rotation f = 0.2 mm (corresponding feed rate vf = 157 mm min−1). Quaker Cool 7350BFF cutting fluid with a concentration of 5% (recommended by the producer of the drills) was used during drilling.The cutting conditions were set to the indicated values and remained constant throughout the experiment i.e., cutting speed vThe following chapter presents the obtained results based on observing the influence of the point angle and the clearance angle on the quality of the drilled holes. Using the simulation in the Helitronic tool studio program, the effect of both mentioned angles on the size of the angle of the cross edge of the drill was determined see Fig. .Figure 8The feed force and torque were measured for individual drills used in the experiment. The Figs. Based on the acquired data, it can be stated that point and clearance angles influenced the applied feed force . When the clearance angle increased by less than 1° in drills A and B, the feed force decreased more than 70 N. The example of the influence of the size of the point angle can be observed in drills A and E, where the point angle increased by 6°, the feed force decreased more than 150 N see Fig. . An incrThe surface roughness of the drilled holes was evaluated based on the parameters Ra and Rz, the measurement was carried out according to the methodology described in chapter 2.3.1. Using the lower values of both observed angles resulted in surface roughness lower than 1 µm considering three compared drills i.e., A, B and C. The dependence of surface roughness on the point and clearance angle is depicted in Fig. The observed dependence of surface roughness on the point and clearance angles, showed the same trend in case of Rz parameter, therefore the display of the measured values of Rz was not performed. The surface roughness parameter Rz was evaluated in the range 5.7–11.7 µm. The influence of the angles on the surface roughness is based on the premise to maintain a smooth machining process without introducing additional vibrations and uneven force stress. The drill geometry with the lower clearance angle shows the sharper cutting wedge and makes it easier for the tool to cut into the workpiece material. Choosing a lower point angle influences the ability to stabilize the drill in the hole due to more favourable force action. When both angles described above are changed, the emerging phenomena interact with each other. In the case of choosing a lower value of the angle of the point and the clearance angle  to 0.1 for drill E see Fig. . The besIn terms of the accuracy of the dimension of the drilled holes, the best results were found for drills C and D, with a point angle of 138° and the clearance angle of 7° see Fig. . The devThe correlation between the measured values of the deviation of the pitch and the diameter from the nominal value for all compared drills except tool G was found. Drilling out of the correct position affects the accuracy of the hole dimension due to the creation of a radial force. In the entrance part of the hole, this effect manifests itself more intensively, because there is no stabilization of the drill bit in the hole .The best results in terms of geometric tolerances of roundness (0.025 mm), cylindricity (0.045 mm) and perpendicularity of the hole axis (0.025), deviation from the nominal value of the spacing (0.06 mm) and deviation from the nominal value of the diameter of the hole (0.03 mm) was achieved with the C drill.For drilled holes, the inlet diameter of the hole was higher than the outlet diameter due to unstable drilling. The lowest difference between the inlet and outlet diameter of the hole was found for drill C, when the value of the difference was only 0.0005. For holes created by other drills, the values of this parameter were up to 0.047 mm.Reducing the hardness of the material on the surface of the drilled plate from 400 HV0.1 to the half value resulted in the deterioration of the geometric accuracy of the drilled holes by up to 220%, depending on the individual parameters.The range of recommended values for the point angle is 138°–142°, the clearance angle 7°–10°.−1) in comparison to standard produced HSS drill tool.The performed research helped to obtain a new design of the drill tool, which is cheaper in price (compared to the sintered carbide tool) and can be used for higher cutting speed (32 m minDrills marked CZ005 made of M35 high-speed steel with different geometry were subjected to cutting tests when drilling into solid material. The cutting forces, surface roughness, geometric tolerances, spacing and diameter deviations were compared to the nominal values of the drilled holes. Based on the analysis of the measured results, the following conclusions were drawn:


The factors influencing the incidence of COVID‐19, including the impact of the vaccination programs, have been studied in the literature. Most studies focus on one or two factors, without considering their interactions, which is not enough to assess a vaccination program in a statistically robust manner. We examine the impact of the U.S. vaccination program on the SARS‐CoV‐2 positivity rate while simultaneously considering a large number of factors involved in the spread of the virus and the feedbacks among them. We consider the effects of the following sets of factors: socioeconomic factors, public policy factors, environmental factors, and non‐observable factors. A time series Error Correction Model (ECM) was used to estimate the impact of the vaccination program at the national level on the positivity rate. Additionally, state‐level ECMs with panel data were combined with machine learning techniques to assess the impact of the program and identify relevant factors to build the best‐fitting models. We find that the vaccination program reduced the virus positivity rate. However, the program was partially undermined by a feedback loop in which increased vaccination led to increased mobility. Although some external factors reduced the positivity rate, the emergence of new variants increased the positivity rate. The positivity rate was associated with several forces acting simultaneously in opposite directions such as the number of vaccine doses administered and mobility. The existence of complex interactions, between the factors studied, implies that there is a need to combine different public policies to strengthen the impact of the vaccination program. Mobility and vaccination are key factors in reducing COVID‐19 positivity rate, they both have a feedback effect on each otherSocioeconomic factors, public policy factors, environmental factors, and non‐observable factors and their interactions affect positivity rateA multivariate model shows complex relationships between vaccination and positivity rate Previous studies in the literature include fewer factors in their studies. Therefore, we believe the importance of our study is that it can be used to learn the importance of direct, indirect, and feedback effects when designing a response to future pandemics caused by Coronaviruses or air‐borne viruses. First, a cointegrated (Simultaneous Equations [SE]) system, at the national level, was estimated by using time series data  and a wide set of other relevant factors at the national and state levels, determined from outside the model . The definition of the system of equations is described in detail in Supporting Information yit (the positivity rate for state i in week t) with respect to xit1 (the number of doses administered to people) as defined in Equation The assessment of the total effect of the U.S. vaccination program on positivity rate was done by taking the total derivative of α>βζ. This may happen if the effect of the vaccination program on the positivity rate (α) is greater than the effect of a higher population mobility on the positivity rate (β) multiplied by the effect of the vaccination program on population mobility (ζ). Equation Equation 2.2The parameters of the three long‐run equations, at the national level, were estimated with a time series data cointegrated SE system  model was estimated by using two different estimators that allowed the authors to consider different forms of heterogeneity in its parameters across states  restricted the long‐run parameters of the long‐run effects to be equal across states but allowed the short‐term effects of the model to be state‐specific. Also, the Mean Group estimator (MG) was used because it allowed the model to have completely heterogenous long‐run and short‐run parameters. The Hausman test was used to choose the appropriate estimator (PMG or MG) for our data. The empirical approach is explained in detail in Supporting Information Due to the possibility of including a wider number of factors in a panel Error Correction Model (ECM), two versions of the estimated models are proposed: (a) a comprehensive panel ECMs with all the available factors and (b) a more parsimonious panel ECMs with a reduced number of relevant factors, which are selected by using machine learning techniques such as the Lasso, Ridge, and Elastic Net regression models.3For the cointegrated SE model, weekly time series from the first week of January 2021 to the third week of January 2022 were used. Specifically, the model included US positivity rate  concentrations at the state level , during the same time span. This database included the level, first, second, and third lags of the first differences of the variables. Furthermore, the level and lags of the first difference of the percentage of fully vaccinated people over time was included as a factor. Moreover, some dichotomous variables were built using information from the CDC such as: (a) variables indicating the timeline of the emergence of COVID‐19 new variants CDCc,  such as vel EPA, , publishvel EPA, , which avel EPA, . All datvel EPA, .44.1Unit root tests performed on the data showed that the 7‐day average percent positivity rate, the number of doses of vaccines administered, a population mobility index, the average temperature and average relative humidity are first‐difference stationary. A Vector Autoregression (VAR) model with these variables and five lags was estimated. A Cointegration Rank test performed to the VAR model indicated the existence of three cointegrating vectors among our five variables, at the 95% confidence level. This is fully consistent with the hypothesis on the existence of three long‐run equations. Then, we reparameterized the VAR model as a system of three cointegrated SE: (a) the SARS‐CoV‐2 positivity rate, (b) the population mobility, and (c) the number of vaccine doses administered to people  was used to estimate the long‐run parameters and also to control for a wider set of observable factors by including them in the short‐run structure of the models. Moreover, we were able to control for all the unobservable factors by using state‐specific fixed effects. We included in our model the first, second, and third lags of the first differences of the factors associated with the positivity rate, population mobility, and the number of doses administered.Furthermore, the three Panel EC models were estimated with the PMG, and the MG estimators and valid estimates were chosen with the Hausman test. The first, second, and third lags of the first differences of the factors associated with the positivity rate (population mobility and the number of doses administered) were included in the model.The results of our models are shown in Table In column 1 of panel A, the estimated panel ECM for the positivity rate is reported. The positivity rate is negatively associated with the number of doses administered to people (−3.59), Thereby confirming that the U.S. vaccination program successfully reduced the positivity rate at the national level in the long run. On the other hand, population mobility increased the positivity rate (1.94) in the long‐run equation. It was also found that a higher relative humidity (−1.10) contributed to reduce the positivity rate. In contrast to the times series model, the average temperature was not found to be statistically significant in the long‐run structure of the positivity rate ECM model.In column 1 of panel B, it is shown that the number of vaccines (0.280) increased the population mobility in the long run. In contrast with the time series cointegrated SE system, there was not statistically significant long‐run relationship between population mobility and the positivity rate. However, this relationship did exist, with the same sign, in the very short term (−0.07)  in the short run. The panel ECMs also allowed us to control for unobserved effects that do not vary over time using state‐level fixed effects estimates. Specifically, we were able to control for unobservable preferences for the vaccine ‐at the state level‐ in the equation for the number of doses given to people.5Table According to the variance decomposition analysis, most of the variations in positivity rate are explained by its own shocks after 2 weeks. However, other factors started to play an important role after 3 weeks. After 7 weeks, 62.77% of the variations in the positivity rate were explained by internal shocks and 10.89% by the shocks to the number of doses administered to people. The combination of population mobility, the number of doses administered to people, and the climatic factors explained 37.21% of the variation in the forecast error for the positivity rate after 7 weeks, which was in line with the previous empirical findings (Dave et al., Figure Figure Vaccination had a positive effect on mobility, therefore, mobility increased when the vaccination rate increased. The increase in vaccination rates was a reason for relaxing mobility restrictions. Thus, mobility increased, and positivity increased in the short‐term. However, one of the main effects of full vaccination is the reduction of hospitalization or deaths caused by the SARS‐CoV‐2 virus and allowing to re‐start economic and social activities.The results displayed in Table S3 in Supporting Information All effects presented in Figure High temperatures (Sasikumar et al., 6–109 virus/ml for the SARS‐CoV‐2 virus (Oswin et al., Mobility restrictions help to reduce contagions; however, small droplets of the virus can remain airborne for long periods of time, and they can travel considerable distances. Therefore, when vaccinated or unvaccinated people inhale these droplets, they can acquire the infection. It is estimated that droplets in the air can contain 10The results of the short‐term model presented in Table S3 in Supporting Information Given the significant effect of mobility on positivity rate, the model included other variables affecting mobility such as weekly unemployment insurance claims. In this case, the results show that unemployment is not a significant factor affecting mobility which means that other factors such as stay‐home orders or work from home have a higher influence on mobility. A marginal effect of unemployment on mobility was found with a 2‐week delay.The results of this study suggest that policies implemented to contain the pandemic should consider the interactions between the factors here described and should focus on mobility control. Thus, policy makers should pay attention to trends in population mobility in order to avoid a higher number of infections, at least until total herd immunity is achieved. In order to reduce transmission when the transmission dynamics of SARS‐CoV‐2 is high, social distancing, mask wearing mandates, and complete vaccination schemes need to be implemented. These are lessons that can be learn in case a new pandemic from a coronavirus or from an air‐borne virus arises in the future.The limitation of this study is that only data from 2021 was used once the original variant passed. Furthermore, in 2021 some mobility restrictions were already lifted so our study only includes the effect of those remaining restrictions. Also, some effects here described are valid only for USA given its social and economic characteristics. Future work could include data from different parts of the world or include information about vaccine brand or maker.6A multivariate framework was built to assess the effect of the U.S. vaccination program on the positivity rate of the SARS‐CoV‐2 virus, which was formalized as a set of three SE. Its empirical validity was tested by estimating the parameters of such equations in the long‐run structure of a cointegrated SE system. The impulse response functions and variance decomposition analysis for the positivity rate were also estimated in this study.The use of a cointegrated SE system allowed the authors to consider a wider set of factors associated with the positivity rate and the feedbacks among them. This modeling approach is useful to obtain an unbiased estimate of the effect of the vaccination program and a better understanding of the role played by different factors on the positivity rate. However, the model here presented has the limitation that the results are valid only through the period January 2020–January 2021. As the pandemic advanced, the virus mutated and the attitudes toward restrictions changed so, these changes would have to be considered in future models.The results suggest that a higher number of doses administered to people reduces the positivity rate. However, the effect of the U.S. vaccination program seems to be partially offset by the combined effect of higher population mobility and dry weather. Also, statistically significant feedback was found between the vaccination program and population mobility, which may partly reduce the effect of the vaccination program. These findings suggest that population mobility, geographical and meteorological factors, and their interactions, must be considered to obtain a proper estimate of the effect of vaccination program in any statistical model. Such factors and their interactions must also be considered when designing public policies to cope with potential new waves of the SARS‐Cov‐2 virus.Policy makers should pay particular attention to the trends in population mobility in order to avoid a higher number of infections. Specifically, policy makers and government officials might usefully adjust mobility restrictions according to trends in population mobility. While it is difficult for the western world to apply such stringent mandates as exemplified by China's zero COVID tolerance policy, especially in the regards to controlling human mobility, policy makers should be aware that with increased mobility, infection rises and be prepared to implement tried and true measures. These include social distancing and mask wearing when the transmission dynamics of SARS‐CoV‐2 is high. Finally, complete vaccination schemes should be promoted to reduce transmission.The authors declare no conflicts of interest relevant to this study.Supporting Information S1Click here for additional data file.