Derivative Analysis Reveals Which Marble Properties Truly Drive Heat Flow

Marble has decorated buildings for millennia, but in an era of geothermal energy, underground heat storage, and nuclear waste repositories, the stone’s thermal behavior has become a serious engineering question. A new study published in Results in Engineering tackles that question with an unusually honest piece of methodology: instead of simply predicting thermal conductivity from rock properties with a machine-learning model, researchers led by Khawar Jalil of the National University of Sciences and Technology, working with colleagues in Pakistan, China, and Ireland, asked how much each measurable property of marble actually changes the way heat moves through it — and whether those relationships survive rigorous, stone-by-stone testing. Their answer reshapes what engineers should look for when they need to know how a marble will behave under thermal load.

The team assembled a dataset of 390 specimens spanning 26 commercially recognized marble varieties from Pakistan, 15 specimens per variety. Every property was measured according to standardized ASTM protocols: density, porosity, moisture content, water absorption, uniaxial compressive strength, P-wave velocity, rebound number, thermal diffusivity, volumetric heat capacity, and, critically, thermal conductivity itself, tested in a dedicated thermal laboratory. The use of commercial classification rather than strict petrographic nomenclature reflects how the dimension-stone industry actually sorts its product, and the breadth of varieties gives the dataset genuine geological diversity rather than the narrow sameness that plagues many single-quarry studies.

The methodological core is deceptively simple. For each property, the researchers fitted low-order polynomial models linking it to thermal conductivity, then computed the first and second derivatives of those fitted curves. The first derivative measures how strongly thermal conductivity responds to a small change in the property — the local sensitivity. The second derivative measures curvature: whether that sensitivity intensifies, weakens, or stays constant across the property’s range. This is the kind of mechanistic information that a correlation coefficient simply cannot supply, and it is what most predictive studies, obsessed with accuracy, have left on the table.

What makes the study notable, however, is how skeptically it treats its own results. Rather than validating models on randomly split specimens, the team used grouped cross-validation in which entire marble varieties were withheld from training, preventing information leakage between stones from the same source. A one-standard-error criterion pushed model selection toward the simplest adequate model. And 3,000 cluster-bootstrap resamples — each drawing whole varieties with replacement, never splitting specimens of one variety apart — tested whether the estimated sensitivities held their sign and magnitude under resampling. The pipeline was fully scripted in Python, with no manual adjustment of results, making every table and figure reproducible from the raw data.

The headline finding is that simplicity won everywhere. For all nine predictors, the one-standard-error criterion selected a first-degree polynomial, meaning thermal conductivity changes at an essentially constant rate with each property across the observed domain, with no statistically supported curvature. The bootstrap confirmed this: the linear model was re-selected in 99 to 100 percent of 500 resampling iterations per predictor, and leave-one-variety-out tests across all 26 varieties left the conclusions intact. When the researchers compared polynomial fits against flexible cubic splines, spline second-derivative stability dropped to a range of roughly 0.73 to 0.87 — independent corroboration that the data contain no reproducible nonlinearity, not just an artifact of the polynomial form.

Perhaps the most striking lesson comes from a comparison the authors ran against their own earlier, looser analysis. Under the original specimen-level validation and minimum-error model selection, several properties appeared to require second- and third-degree polynomials, producing seductive apparent nonlinearities. Once whole-variety grouping, parsimonious selection, and cluster bootstrapping were imposed, those nonlinearities vanished entirely. In other words, the curvature was a methodological artifact, not a physical phenomenon — a cautionary tale for anyone who has ever published a wiggly fitted curve and called it science.

Not surprisingly, thermal diffusivity and volumetric heat capacity showed the strongest associations with conductivity, explaining roughly 91 and 92 percent of variance respectively in unseen varieties. But the authors are refreshingly blunt about why: these three properties are bound together by the fundamental heat-transfer relation k = α·ρCp, so the connection is partly definitional. They treat these two as boundary reference variables — benchmarks validating the dataset’s thermophysical consistency — rather than pretending they constitute independent predictive discoveries.

Among the genuinely independent properties, a clear hierarchy emerges that tells a coherent physical story. Uniaxial compressive strength (grouped out-of-fold R² of 0.596), density (0.593), rebound number (0.581), and P-wave velocity (0.571) all showed positive sensitivities of comparable magnitude. Dense, strong, elastically competent marble conducts heat better because its mineral grains maintain continuous contact, offering unbroken pathways for conduction, whereas microcracks and structural discontinuities interrupt the flow. Porosity, moisture content, and water absorption all showed negative sensitivities (out-of-fold R² between roughly 0.52 and 0.55), consistent with their role as proxies for void space and degraded structural continuity. The narrow spread among the independent predictors indicates that no single mechanical parameter dominates; thermal conductivity reflects the overall integrity of the mineral framework.

The comparison with conventional statistics proved equally instructive. Univariate Pearson correlations and standardized multivariate regression coefficients disagreed sharply for density, rebound number, and P-wave velocity — density, for instance, shows a correlation of 0.779 with thermal conductivity but a regression coefficient of essentially zero once other variables are accounted for. Variance inflation factors between 5.33 and 8.54 confirmed moderate-to-considerable collinearity, meaning these mechanical indicators largely measure the same underlying microstructural quality. The authors deliberately chose the univariate derivative framework to isolate each property’s interpretable functional contribution while avoiding exactly this ambiguity, and they note that in a multivariate setting much of the shared sensitivity would likely redistribute toward uniaxial compressive strength, whose coefficient proved most stable across measures.

For engineers, the practical payoff is direct: routinely measured properties — density, strength, rebound hardness, and ultrasonic velocity — can serve as reliable proxies for thermal performance when direct conductivity measurement is impractical, while porosity and moisture metrics flag stones likely to underperform thermally. The moisture result is particularly nuanced, since water’s conductivity sits between that of air-filled pores and the mineral grains, so saturation produces two partially offsetting effects. Beyond marble, the authors argue the framework — derivative-based sensitivity coupled with grouped validation and cluster bootstrapping — transfers readily to granite, limestone, sandstone, and even concrete, and could extend to multivariate, spatial, and machine-learning settings. The deeper message may resonate even further: how a model responds matters as much as how well it predicts, and rigorous validation is the only way to tell genuine physics from an artifact of your own methodology.

Subject of Research: Derivative-based sensitivity analysis of thermal conductivity in marble using polynomial modelling, grouped cross-validation, and bootstrap robustness testing

Article Title: Thermal conductivity in marble: A derivative-based approach to sensitivity and functional response

Article References: Jalil, K., Feroze, T., Khan, N. M., Raja, I. B., Xin, H., & Genc, B. (2026). Thermal conductivity in marble: A derivative-based approach to sensitivity and functional response. Results in Engineering, 32, Article 113450. https://doi.org/10.1016/j.rineng.2026.113450

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113450

Keywords: thermal conductivity, marble, derivative analysis, rock physics, grouped cross-validation, bootstrap resampling, polynomial regression, sensitivity analysis, geothermal energy, dimension stone, machine learning interpretability, microstructure