The present work developed a three-equation local thermal non-equilibrium (LTNE) model to model the initial effect on the internal thermal conduction in the solid phase of porous medium. The two energy equations in the original LTNE model were extended with a third equation to govern the dimensionless thermal penetration depth for the internal thermal conduction in the solid phase, I, which can describe the initial effect. The model was validated by comparisons between an analytical solution of the I equation and exact solutions of thermal conduction problems and between the macro-scale and the pore-scale numerical simulations. The comparisons demonstrate that the three-equation LTNE model is more accurate than the two-equation LTNE model with a constant I during the initial period. An important similarity number, eta, is obtained for the enhanced geothermal system (EGS) heat extraction problem from the dimensionless three-equation LTNE model, which measures the internal thermal conduction effect in the solid phase of a porous medium. The initial effect in the EGS heat extraction can be neglected for eta < 0.5. Criteria for degeneration of the three-equation LTNE model are proposed. (C) 2017 Elsevier Ltd. All rights reserved.