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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


  • Aftab A, Ali M, Arif M, Panhwar S, Saady NM, Al-Khdheeawi EA, Mahmoud O, Ismail AR, Keshavarz A, Iglauer S (2020) Influence of tailor-made TiO2/API bentonite nanocomposite on drilling mud performance: Towards enhanced drilling operations. Appl Clay Sci 199:105862

    Article  CAS  Google Scholar 

  • Aftab A, Ismail AR, Ibupoto ZH (2017) Enhancing the rheological properties and shale inhibition behavior of water-based mud using nanosilica, multi-walled carbon nanotube, and graphene nanoplatelet. Egypt J Petrol 26(2):291–299

    Article  Google Scholar 

  • Aftab A, Ismail AR, Ibupoto ZH, Akeiber H, Malghani MGK (2017) Nanoparticles based drilling muds a solution to drill elevated temperature wells: a review. Renew Sustain Energy Rev 76:1301–1313

    Article  CAS  Google Scholar 

  • Ahmad HM, Kamal MS, Al-Harthi MA (2018) High molecular weight copolymers as rheology modifier and fluid loss additive for water-based drilling fluids. J Mol Liq 252:133–143

    Article  CAS  Google Scholar 

  • Hafiz Mudaser Ahmad TI, Muhammad Shahzad Kamal, Mamdouh A. Al-Harthi (2020) Influence of hydrophobically modified polymer and titania nanoparticles on shale hydration and swelling properties. Energy & Fuels

  • Ahmed EM (2015) Hydrogel: preparation, characterization, and applications: a review. J Adv Res 6(2):105–121

    Article  CAS  PubMed  Google Scholar 

  • Al-Ansari APC, Abahussain A, Abuhamed AM, Pino R, El Bialy M, Mohamed H, Lopez C (2017) Reservoir drill-in fluid minimizes fluid invasion and mitigates differential stuck pipe with improved production test results. In SPE Middle East Oil Gas Show Conf., SPE, Editor. 2017: Manama, Kingdom of Bahrain

  • Ali SI, Lalji SM, Haneef J et al (2021) Asphaltene precipitation modeling in dead crude oils using scaling equations and non-scaling models: comparative study. J Petrol Explor Prod Technol 11:3599–3614

    Article  CAS  Google Scholar 

  • Alsaba M, AlMarshad A, Abbas A et al (2020) Laboratory evaluation to assess the effectiveness of inhibitive nano-water-based drilling fluids for Zubair shale formation. J Petrol Explor Prod Technol 10:419–428

    Article  CAS  Google Scholar 

  • Azam M, Arshad A, Aslam M et al (2017) A control chart for monitoring the process mean using successive sampling over two occasions. Arab J Sci Eng 42:2915–2926

    Article  CAS  Google Scholar 

  • Balaban R, Borges VEM (2015) Design of experiments to evaluate clay swelling inhibition by diferent combinations of organic compounds and inorganic salts for application in water base drilling fuids. Appl Clay Sci 105–106:124–130

    Article  Google Scholar 

  • Bruschi ML (2015) 5-Mathematical models of drug release. Strategies to Modify the Drug Release from Pharmaceutical Systems. 2015: Woodhead Publishing

  • Christiansen C (1991) From oil-based mud to water-based mud. In: paper presented at the SPE Health, Safety and Environment in Oil a.

  • Caccavo D (2019) An overview on the mathematical modeling of hydrogels’ behavior for drug delivery systems. Int J Pharm 560:175–190

    Article  CAS  PubMed  Google Scholar 

  • Daud F, Khan GN, Ibrahim M (2012) Remaining hydrocarbon potential in Pakistan a statistical review.

  • Elkatatny S (2019) Enhancing the rheological properties of water-based drilling fluid using micronized starch. Arab J Sci Eng 44:5433–5442

    Article  CAS  Google Scholar 

  • Fu Y, Weiyuan JK (2010) Drug release kinetics and transport mechanisms of non-degradable and degradable polymeric delivery systems. Expert Opin Drug Deliv 7(4):429–444

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Hashmi S, Nadeem S, Awan Z, Ghani AA (2019) Synthesis, applications and swelling properties of poly (Sodium Acrylate-Coacrylamide) based superabsorbent hydrogels. J Chem Soc Pak 41(4):668–678

    CAS  Google Scholar 

  • Higuchi T (1961) Rate of release of medicaments from ointments bases containing drugs in suspension. J Pharm Sci 50(10):874–875

    Article  CAS  PubMed  Google Scholar 

  • Higuchi WI (1962) Analysis of data on the medicament release from ointments. J Pharm Sci 51:802–804

    Article  CAS  PubMed  Google Scholar 

  • Kabiri K, Hashemi HOS, Zohuriaan-Mehr M (2003) Synthesis of fast-swelling superabsorbent hydrogels: effect of crosslinker type and concentration on porosity and absorption rate. Eur Polym J 39(7):1341–1348

    Article  CAS  Google Scholar 

  • Khan MA, Haneef J, Lalji SM et al (2021) Experimental study and modeling of water-based fluid imbibition process in Middle and Lower Indus Basin Formations of Pakistan. J Petrol Explor Prod Technol 11:425–438

    Article  Google Scholar 

  • Korsmeyer RW, Doelker RGE, Buri P, Peppa NA (1983) Mechanisms of solute release from porous hydrophilic polymers. Int J Pharm 15(1):25–35

    Article  CAS  Google Scholar 

  • Lal M (1999) Shale Stability: Drilling Fluid Interaction and Shale Strength.", in Paper presented at the SPE Asia Pacific Oil and Gas Conference and Exhibition, SPE, Editor. 1999: Jakarta, Indonesia

  • Lalji SM, Ali SI, Awan ZUH et al (2021a) A novel technique for the modeling of shale swelling behavior in water-based drilling fluids. J Petrol Explor Prod Technol 11:3421–3435

    Article  CAS  Google Scholar 

  • Lalji SM, Khan MA, Haneef J et al (2023) Nano-particles adapted drilling fluids for the swelling inhibition for the Northern region clay formation of Pakistan. Appl Nanosci 13:503–517

    Article  CAS  Google Scholar 

  • Lalji SM, Ali SI, Ahmed R et al. (2021b) Comparative performance analysis of different swelling kinetic models for the evaluation of shale swelling. J Petrol Explor Prod Technol

  • Lalji SM, Ali SI, Ahmed R et al (2022) Influence of graphene oxide on salt-polymer mud rheology and Pakistan shale swelling inhibition behavior. Arab J Geosci 15:612

    Article  CAS  Google Scholar 

  • Luo ZP, Wang L, Yu P, Chen Z (2017) Influence of an ionic liquid on rheological and filtration properties of water-based drilling fluids at high temperatures. Appl Clay Sci 136:96–102

    Article  CAS  Google Scholar 

  • Vigata M, Meinert C, Hutmacher DW, Bock N (2020) Hydrogels as drug delivery systems: a review of current characterization and evaluation techniques. Pharmaceutics 12:1188

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Umar M, Friis H, Khan AS, Kassi AM, Kasi AK (2011) The effects of diagenesis on the reservoir characters in sandstones of the Late Cretaceous Pab Formation, Kirthar Fold Belt, southern Pakistan. J Asian Earth Sci 40(2):622–635

    Article  Google Scholar 

  • Nisar UB, Rizwan M, Khan MR, Farooq M, Qureshi SH, Ahmed KA (2019) Identification of sealing potential through fault seal analysis: a case study of Badin area, Lower Indus Basin. Pakistan. Geofísica Internaciona 58(2):139–150

    Google Scholar 

  • Peppas NA, Bures P, Leobandung W, Ichikawa H (2000) Hydrogels in pharmaceutical formulations. Eur J Pharm Biopharm 50:27–46

    Article  CAS  PubMed  Google Scholar 

  • Peppas NA, Sahlin JJ (1989) A simple equation for the description of solute release III. Coupling of diffusion and relaxation. Int J Pharm 57(2):169–217

    Article  CAS  Google Scholar 

  • Sadeghalvaad MS (2015) The effect of the TiO2/polyacrylamide nanocomposite on water-based drilling fluid properties. Powder Technol 272:113–119

    Article  CAS  Google Scholar 

  • Sánchez FG (2007) Water diffusion through compacted clays analyzed by neutron scattering and tracer experiments. 2007, Der Universität Bern

  • Saud Hashmi SN, Awan Z, Rehman A, Ghani AA (2019) Synthesis, applications and swelling properties of poly (sodium acrylate-coacrylamide) based superabsorbent hydrogels. J Chem Soc Pak 41:04

    Google Scholar 

  • SurathAebtarm NB (2011) An empirical evaluation of attribute control charts for monitoring defects. Expert Syst Appl 38(6):7869–7880

    Article  Google Scholar 

  • Unagolla JM (2018) Drug transport mechanisms and in vitro release kinetics of vancomycin encapsulated chitosan-alginate polyelectrolyte microparticles as a controlled drug delivery system. Eur J Pharm Sci 114:199–2009

    Article  CAS  PubMed  Google Scholar 

  • Wahid NY, MAM, Hanafi NH (2015) Optimum nanosilica concentration in synthetic based mud (SBM) for high temperature high pressure well. In: SPE/IATMI Asia Pacific Oil and Gas Conference and Exhibition, SPE, Editor. 2015: Nusa Dua, Bali, Indonesia

  • Wang GD (2018) Rheological properties of KCl/polymer type drilling fluids containing particulate loss prevention material. Appl Rheol 28(3):35727

    Google Scholar 

  • Yan X, Tong Z, Chen Y, Mo Y, Feng H, Li P, Qu X, Jin S (2017) Bioresponsive materials for drug delivery based on carboxymethyl chitosan/poly(γ-glutamic acid) composite microparticles. Mar Drugs 15:5

    Article  Google Scholar 

  • Yu C, Yun-fei L, Huan-lin T, Hui-min T (2010) Study of carboxymethyl chitosan based polyampholyte superabsorbent polymer I: optimization of synthesis conditions and pH sensitive property study of carboxymethyl chitosan-g-poly (acrylic acid-co-dimethyl-diallylammonium chloride) superabsorbent polymer. Carbohydr Polym 81(2):365–371

    Article  Google Scholar 

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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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