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• ψ -Caputo derivative can be efficiently used to model water transport. • Particle swarm optimization algorithm was used to identify parameters. • Sigmoid functional parameter best described experimental data. The paper studies a ψ -Caputo space-time-fractional moisture transport model stated in terms of water head. We consider a functional parameter of ψ -Caputo derivatives in power and sigmoid forms and use particle swarm optimization algorithm to identify model's parameters. Input data for the identification were acquired on the irrigated field in the South of Ukraine within one irrigation cycle. Direct problems were solved using a finite-difference scheme. In these conditions, application of a fractional model with the classic Caputo derivative allowed more than ~ 2-times lowering of average absolute error compared to the integer-order model. The usage of functional parameter in power form allowed up to 19% accuracy enhancement compared to the classic Caputo derivative while the usage of sigmoid function up to 2.6-times lowers the error. Such behaviour remains when predicting soil moisture dynamics within the following irrigation cycles. [ABSTRACT FROM AUTHOR]