Polynomial treewidth forces a large grid-like-minor

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \emph{grid-like-minor of order} $\ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{\ell}$-minor. For example, the rows and columns of the $\ell\times\ell$ grid are a grid-like-minor of order $\ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $c\ell^4\sqrt{\log\ell}$ has a grid-like-minor of order $\ell$. As an application of this result, we prove that the cartesian product $G\square K_2$ contains a $K_{\ell}$-minor whenever $G$ has treewidth at least $c\ell^4\sqrt{\log\ell}$.Comment: v2: The bound in the main result has been improved by using the Lovasz Local Lemma. v3: minor improvements, v4: final section rewritte

Circumference and Pathwidth of Highly Connected Graphs

Birmele [J. Graph Theory, 2003] proved that every graph with circumference t has treewidth at most t-1. Under the additional assumption of 2-connectivity, such graphs have bounded pathwidth, which is a qualitatively stronger result. Birmele's theorem was extended by Birmele, Bondy and Reed [Combinatorica, 2007] who showed that every graph without k disjoint cycles of length at least t has bounded treewidth (as a function of k and t). Our main result states that, under the additional assumption of (k + 1)- connectivity, such graphs have bounded pathwidth. In fact, they have pathwidth O(t^3 + tk^2). Moreover, examples show that (k + 1)-connectivity is required for bounded pathwidth to hold. These results suggest the following general question: for which values of k and graphs H does every k-connected H-minor-free graph have bounded pathwidth? We discuss this question and provide a few observations.Comment: 11 pages, 4 figure

Variable interaction empirical relationships and machine learning provide complementary insight to experimental horizontal wellbore cleaning results

Long horizontal wellbore sections are now a key requirement of oil and gas drilling, particularly for tight reservoirs. However, such sections pose a unique set of borehole-cleaning challenges which are quite distinct from those associated with less inclined wellbores. Experimental studies provide essential insight into the downhole variables that influence borehole cleaning in horizontal sections, typically expressing their results in multivariate empirical relationships with dimensionless cuttings bed thickness/concentration (H%). This study demonstrates how complementary empirical H% relationships focused on pairs of influential variables can be obtained from published experimental data using interpolated trends and optimizers. It also applies five machine learning algorithms to a compiled multivariate (10-variable) interpolated dataset to illustrate how reliable H% predictions can be derived based on such information. Seven optimizer-derived empirical relationships are derived using pairs of influential variables which are capable of predicting H% with root mean squared errors of less than 1.8%. The extreme gradient boosting model provides the lowest H% prediction errors from the 10-variable dataset. The results suggest that in drilling situations where sufficient, locally-specific, information for multiple influential variables is available, machine learning methods are likely to be more effective and reliable at predicting H% than empirical relationships. On the other hand, in drilling conditions where information is only available for a limited number of influential variables, empirical relationships involving pairs of influential variables can provide valuable information to assist with drilling decisions.Document Type: Original articleCited as: Wood, D. A. Variable interaction empirical relationships and machine learning provide complementary insight to experimental horizontal wellbore cleaning results. Advances in Geo-Energy Research, 2023, 9(3): 172-184. https://doi.org/10.46690/ager.2023.09.0

Well-log attributes assist in the determination of reservoir formation tops in wells with sparse well-log data

The manual picking of reservoir formation boundaries using limited available well-log data in multiple wells across gas and oil reservoirs tends to be subjective and unreliable. The reasons for this are typically caused by the combined effects of spatial boundary complexity and limited well-log data availability. Formation boundary characterization and classification can be improved when treated as a binary classification task based on two or three recorded well logs assisted by their calculated derivative and volatility attributes assessed by machine learning. Two example wellbores penetrating a complex reservoir boundary, one with gamma-ray, compressional-sonic, and bulk-density logs recorded, the other with just gamma-ray and bulk-density logs recorded, are used to illustrate a more rigorous proposed methodology. By combining attribute calculation, optimized feature selection, multi-k-fold cross validation, confusion matrices, feature-influence analysis, and machine learning models it is possible to improve the classification of the formation boundary. With just gamma-ray and bulk-density recorded well logs plus selected attributes. K-nearest neighbour, support vector classification, and extreme gradient boosting machine learning models are able to achieve high binary classification accuracy: greater than 0.97 for training/validation in one well; and greater than 0.94 for testing in another well. extreme gradient boosting feature-influence analysis reveals the attributes that are the most important in the formation boundary predictions but these are likely to vary from reservoir to reservoir. The results of the study suggest that well-log attribute analysis, combined with machine learning has the potential to provide a more systematic formation boundary definition than relying only on a few recorded well-log curves.Cited as: Wood., D. A. Well-log attributes assist in the determination of reservoir formation tops in wells with sparse well-log data. Advances in Geo-Energy Research, 2023, 8(1): 45-60. https://doi.org/10.46690/ager.2023.04.0

Predicting brittleness indices of prospective shale formations from sparse well-log suites assisted by derivative and volatility attributes

A technique is proposed that calculates derivative and volatility attributes from just a few well log curves to assist in brittleness index predictions from sparse well-log datasets with machine learning methods. Six well-log attributes are calculated for selected recorded well logs: the first derivative, the moving average of the first derivative, the second derivative, the logarithm of the instantaneous volatility, the standard deviation of volatility, and the moving average of volatility. These attributes make it possible to extrapolate brittleness index calibrations from the few cored and comprehensively logged wells to surrounding wells in which only minimal well-log suites are recorded. Data from two cored wells penetrating the lower Barnett Shale with distinct lithology and five well logs recorded are used to demonstrate the technique. Based on multi-K-fold cross validation analysis, the data matching K-nearest neighbour machine learning model provides the most accurate brittleness index predictions, closely followed by tree-ensemble models. For this dataset, recorded data from three well logs plus calculated attributes matches the brittleness index prediction accuracy that is achieved by the five recorded logs. Moreover, any one of the logs plus their calculated attributes yields better brittleness index prediction performance than that achieved by a combination of just those three recorded well logs. Analysis of the Gini indices of the tree-ensemble models reveals the relative influences of the recorded logs and their attributes on the brittleness index prediction solutions. Such information is used to perform feature selection to optimize the well-log attributes involved to generate reliable brittleness index predictions.Cited as: Wood, D. A. Predicting brittleness indices of prospective shale formations from sparse well-log suites assisted by derivative and volatility attributes. Advances in Geo-Energy Research, 2022, 6(4): 334-346. https://doi.org/10.46690/ager.2022.04.0

Machine Learning and Pattern Analysis Identify Distinctive Influences from Long-term Weekly Net Ecosystem Exchange at Four Deciduous Woodland Locations

A methodology integrating correlation, regression (MLR), machine learning (ML), and pattern analysis of long-term weekly net ecosystem exchange (NEE) datasets are applied to four deciduous broadleaf forest (DBF) sites forming part of the AmeriFlux (FLUXNET2015) database. Such analysis effectively characterizes and distinguishes those DBF sites for which long-term NEE patterns can be accurately predicted using the recorded environmental variables, from those sites cannot be so delineated. Comparisons of twelve NEE prediction models (5 MLR; 7 ML), using multi-fold cross-validation analysis, reveal that support vector regression generates the most accurate and reliable predictions for each site considered, based on fits involving between 16 and 24 available environmental variables. SVR can accurately predict NEE for datasets for DBF sites US-MMS and US-MOz, but fail to reliably do so for sites CACbo and MX-Tes. For the latter two sites the predicted versus recorded NEE weekly data follow a Y ≠ X pattern and are characterized by rapid fluctuations between low and high NEE values across leaf-on seasonal periods. Variable influences on NEE, determined by their importance to MLR and ML model solutions, identify distinctive sets of the most and least influential variables for each site studied. Such information is valuable for monitoring and modelling the likely impacts of changing climate on the ability of these sites to serve as long-term carbon sinks. The periodically oscillating NEE weekly patterns distinguished for sites CA-Cbo and MX-Tes are not readily explained in terms of the currently recorded environmental variables. More detailed analysis of the biological processes at work in the forest understory and soil at these sites are recommended to determine additional suitable variables to measure that might better explain such fluctuations

The natural gas sector needs to be mindful of its sustainability credentials

 Despite substantial growth in demand for natural gas, building of infrastructure and prevailing low wholesale prices, some energy-industry stakeholders are now questioning the long-term sustainability of the natural gas industry. This is due, in part, to it being a fossil fuel with associated greenhouse gas emissions, and its very poor long-term historical track record regarding gas flaring and fugitive methane and other light-hydrocarbon gas emissions. However, major environmental concerns have also arisen regarding the development of unconventional natural gas resources using hydraulic fracture stimulation, its large environmental and community impacts, water usage and potential contamination and induced seismic activity leading to increased surface impacts. There are however a number of technological opportunities identified and available for deployment for a number of years that could enable the industry to improve its sustainability credentials. Seriously developing these opportunities could convince public opinion that the natural gas sector should be part of long-term plans to develop and maintain a near-zero emissions energy sector. Most of the identified opportunities are obvious, such as eliminating flaring, improving production efficiency by gaining a better understanding of sub-surface reservoirs and fluid movements, and reducing its surface footprints, carbon capture sequestration and utilization, and more collaboration with the renewables sector to build hybrid power and energy storage plants. It is imperative for the natural gas sector’s long-term future that it fully embraces these opportunities and makes them visibly contribute to more eco-friendly energy supply-chain developments.Cited as: Wood, D.A. The natural gas sector needs to be mindful of its sustainability credentials. Advances in Geo-Energy Research, 2020, 4(3): 229-232, doi: 10.46690/ager.2020.03.0

Gamma-ray log derivative and volatility attributes assist facies characterization in clastic sedimentary sequences for formulaic and machine learning analysis

Rate of change, second derivative and volatility of gamma-ray (GR) well-log curves provide useful indicators with which to characterize lithofacies in clastic sedimentary sequences. Rolling averages of these variables, as they change with depth, are also able to distinguish certain lithofacies features. These attributes make it possible to accurately distinguish key facies by using only gamma-ray data, both with formulaic calculations and employing machine-learning (ML) algorithms. This is useful in the many wellbores for which only basic logging suites are available. As well as enhancing lithofacies classification more generally using well-log variables, these GR attributes can be used to forecast facies in real time based on logging-while-drilling data. The application is demonstrated with simple formula using synthetic GR logs featuring common clastic lithofacies and their transitions. Seven widely used ML methods are each trained and validated with a synthetic GR curve (1450 data points) displaying six distinct facies. The ability of the ML model to distinguish those facies using seven GR attributes is compared and further tested with an independent GR data set (800 data points). The random forest algorithm outperforms the other ML models in this facies prediction task, achieving a mean absolute error of 0.25 (on a facies class range of 1 to 6) for the independent testing dataset. The results highlight the benefit of this technique in providing reliable facies analysis based only on GR data. Random forest, support vector classification and eXtreme Gradient Boost are the ML models that provide the most reliable facies classification from the GR attributes defined. Annotated confusion matrices assist in revealing the details of facies class prediction accuracy and precision achieved by the ML and models and classification formulas.Cited as: Wood, D.A. Gamma-ray log derivative and volatility attributes assist facies characterization in clastic sedimentary sequences for formulaic and machine learning analysis. Advances in Geo-Energy Research, 2022, 6(1): 69-85. https://doi.org/10.46690/ager.2022.01.0

Machine Learning and Regression Analysis Reveal Different Patterns of Influence on Net Ecosystem Exchange at Two Conifer Woodland Sites

Variations in net ecosystem exchange (NEE) of carbon dioxide, and the variables influencing it, at woodland sites over multiple years determine the long term performance of those sites as carbon sinks. In this study, weekly-averaged data from two AmeriFlux sites in North America of evergreen woodland, in different climatic zones and with distinct tree and understory species, are evaluated using four multi-linear regression (MLR) and seven machine learning (ML) models. The site data extend over multiple years and conform to the FLUXNET2015 pre-processing pipeline. Twenty influencing variables are considered for site CA-LP1 and sixteen for site US-Mpj. Rigorous k-fold cross validation analysis verifies that all eleven models assessed generate reproducible NEE predictions to varying degrees of accuracy. At both sites, the best performing ML models (support vector regression (SVR), extreme gradient boosting (XGB) and multi-layer perceptron (MLP)) substantially outperform the MLR models in terms of their NEE prediction performance. The ML models also generate predicted versus measured NEE distributions that approximate cross-plot trends passing through the origin, confirming that they more realistically capture the actual NEE trend. MLR and ML models assign some level of importance to all influential variables measured but their degree of influence varies between the two sites. For the best performing SVR models, at site CA-LP1, variables air temperature, shortwave radiation outgoing, net radiation, longwave radiation outgoing, shortwave radiation incoming and vapor pressure deficit have the most influence on NEE predictions. At site US-Mpj, variables vapor pressure deficit, shortwave radiation incoming, longwave radiation incoming, air temperature, photosynthetic photon flux density incoming, shortwave radiation outgoing and precipitation exert the most influence on the model solutions. Sensible heat exerts very low influence at both sites. The methodology applied successfully determines the relative importance of influential variables in determining weekly NEE trends at both conifer woodland sites studied