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Statistical approach used to evaluate the performances of different diffusion correlations in modeling Pakistan shale swelling characteristics

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One of the leading problems in the oil and gas sector is the swelling of shale formation after the adsorption of water during drilling activity. This mechanism has some intense effects on wellbore integrity. Experimentally, the swelling characteristics of a shale formation can be best evaluated by the linear dynamic swell-meter (LDSM). In this study, modeling the diffusion of water in the shale formation is done using the three diffusion models proposed by Higuchi model, Peppas model, and Peppas and Sahlin model, and the best model is selected. During this article, swelling characteristics of Khadro formation obtained from the Lower Indus Basin of Pakistan were examined using three above-mentioned models. Experimentally, the swelling behavior of Khadro formation was investigated in salt polymer and silicate mud systems. For the first time, the effect of changing the mud type on the performances of the models is evaluated. The experimental swelling curves obtained from LDSM are divided into two separate segments that indicate transient and steady state zones, respectively. During the transient state, water molecules diffuse into the nano-platelets of shale formation. For this region, the Higuchi model is best in modeling the swelling results with near to 100% accuracy. However, the model performance substantially overestimates the swelling rate during the steady state condition. Moreover, maximum relative error of 120% and MAE > 50% in both muds is inadequacies associated with this model. On the contrary, the remaining two models exhibit a completely different behavior in the steady state region. Both the models estimate the swelling percentages with MAE values below 5%. Additionally, the performance of the models substantially improves as the quantity of the water in the formulation of the mud system reduces. The silicate mud system having low volume of water exhibits three times reduction in MAE values in Peppas model and Peppas and Sahlin model  respectively as compared with salt polymer mud system.

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Hydrogel-based systems


Linear dynamic swell-meter


Water-based drilling fluids


Oil-based drilling fluids

\({M}_{t}(t)\) :

Quantity of drug that is been released at each time period

\({M}_{\infty }\) :

Total quantity of drug release at the final time period


Time dependent function

\(k\) :

Characteristics constant

\(n\) :

Adjustable fitting models


Mean absolute error


Control charts


Upper control limit


Lower control limit


Central limit

H+ :

Hydrogen ions


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Lalji, S.M., Ali, S.I., Khan, M.A. et al. Statistical approach used to evaluate the performances of different diffusion correlations in modeling Pakistan shale swelling characteristics. Chem. Pap. 77, 4705–4717 (2023).

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