Abstract

We examine the use of natural boundary conditions and conditions of the Sommerfeld type for finite element simulations of convective transport in viscous incompressible flows. We show that natural boundary conditions are superior in the sense that they always provide a correct boundary condition, as opposed to the Sommerfeld‐type conditions, which can lead to a singular formulation and a great loss of accuracy. For the Navier–Stokes equations, the natural boundary conditions must be combined with a simple method to eliminate perturbations on the pressure at the open boundary, which is the source of most errors.

Abstract

The finite point method (FPM) is a gridless numerical procedure based on the combination of weighted least square interpolations on a cloud of points with point collocation for evaluating the approximation integrals. In the paper, details of a procedure for stabilizing the numerical solution for advective-diffusive transport and fluid flow problems using the FPM are given. The method is based on a consistent introduction of the stabilizing terms in the governing differential equations. One example showing the applicability of the FPM is given.

Abstract

We propose a novel approach for flow simulations in fractured porous media based on an equi-dimensional representation of the fractures and a standard continuous finite element (FE) method. We employ an adaptive mesh refinement strategy to automatically adjust an initial regular mesh to any fracture distribution with a desired accuracy. The proposed approach is easily implementable in any FE software, does not involve the coupling of different discretizations, and provides symmetric and positive definite stiffness matrices. We provide a systematic validation of our method and we show that it can be used to simulate fluid flow for fracture networks of realistic complexity.

Abstract

The efficient extraction of geothermal energy from fractured hot dry rock reservoirs requires accurate prediction of subsurface thermohydrodynamic processes. In this study, a novel simplified mid-plane fracture model was developed and validated as an approach that bridges the computational efficiency of two-dimensional approximations with the physical accuracy of fully three-dimensional simulations. Three distinct fracture representations were systematically compared: full 3D models, simplified 2D plane fractures, and the proposed rough mid-plane fractures. Discrepancies in flow dynamics and heat transfer predictions were quantified using coupled steady-state and transient numerical simulations. The influence of “X”-type and “V”-type fracture intersections on thermo-hydrodynamic processes was further examined. The results show that the rough mid-plane model accurately captures permeability and flow channeling effects inherent in realistic 3D fracture geometries, outperforming traditional plane approximations. Flow fields and velocity distributions are substantially modified by surface roughness, directly governing heat exchange efficiency. Flow and temperature fields are shown to be dramatically redistributed by variations in fracture aperture and intersection geometry, with heat extraction significantly enhanced by optimal aperture and injection rate combinations. Notably, “X”-type intersections are found to exhibit 40%–60% greater effective heat conduction areas than “V”-type configurations, highlighting their preferential heat transfer characteristics. These findings provide critical insights for the optimization of HDR reservoir performance, contributing to improved strategies for efficient geothermal energy extraction.OPEN ACCESS Received: 14/09/2025 Accepted: 21/10/2025 Published: 29/05/2026