A novel numerical model for groundwater flow in karst aquifers is presented. A discrete-continuum (hybrid) approach, in which a three-dimensional matrix flow is coupled with a one-dimensional conduit flow, was used. The laminar flow in the karst matrix is described by a variably saturated flow equation to account for important hydrodynamic effects in both the saturated and unsaturated zones. Turbulent conduit flow for both free surface and pressurized flow conditions was captured via the noninertia wave equation, whereas the coupling of two flow domains was established through an exchange term proportional to head differences. The novel numerical approach based on Fup basis functions and control-volume formulation enabled us to obtain smooth and locally conservative numerical solutions. Due to its similarity to the isogeometric analysis concept (IGA), we labeled it as control-volume isogeometric analysis (CV-IGA). Since realistic verification of the karst flow models is an extremely difficult task, the particular contribution of this work is the construction of a specially designed 3D physical model ( dimensions: 5.66 × 2.95 × 2.00 m) in order to verify the developed numerical model under controlled laboratory conditions. Heterogeneous porous material was used to simulate the karst matrix, and perforated pipes were used as karst conduits. The model was able to capture many flow characteristics, such as the interaction between the matrix and conduit, rainfall infiltration through the unsaturated zone, direct recharge through sinkholes, and both free surface and pressurized flow in conduits. Two different flow experiments are presented, and comparison with numerical results confirmed the validity of the developed karst flow model under complex laboratory conditions.

Summary A novel adaptive algorithm that is based on new hierarchical Fup (HF) basis functions and a control volume formulation is presented. Because of its similarity to the concept of isogeometric analysis (IGA), we refer to it as control volume isogeometric analysis (CV‐IGA). Among other interesting properties, the IGA introduced k‐refinement as advanced version of hp‐refinement, where every basis function of the nth order from one resolution level are replaced by a linear combination of more basis functions of the n+1th order at the next resolution level. However, k‐refinement can be performed only on whole domain, while local adaptive k‐refinement is not possible with classical B‐spline basis functions. HF basis functions (infinitely differentiable splines) satisfy partition of unity, and they are linearly independent and locally refinable. Their main feature is execution of the adaptive local hp‐refinement because any basis function of the nth order from one resolution level can be replaced by a linear combination of more basis functions of the n+1th order at the next resolution level providing spectral convergence order. The comparison between uniform vs hierarchical adaptive solutions is demonstrated, and it is shown that our adaptive algorithm returns the desired accuracy while strongly improving the efficiency and controlling the numerical error. In addition to the adaptive methodology, a stabilization procedure is applied for advection‐dominated problems whose numerical solutions “suffer” from spurious oscillations. Stabilization is added only on lower resolution levels, while higher resolution levels ensure an accurate solution and produce a higher convergence order. Since the focus of this article is on developing HF basis functions and adaptive CV‐IGA, verification is performed on the stationary one‐dimensional boundary value problems.

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