Abstract
We present a novel numerical technique for computational models of thin tubular networks embedded in a bulk domain, for example a porous medium. These systems occur in the simulation of fluid flow in vascularized biological tissue, root water and nutrient uptake in soil, hydrological or petroleum wells in rock formations, or heat transport in micro-cooling devices. The key processes, such as heat and mass transfer, are often dominated by the exchange between the network system and the embedding domain. By explicitly resolving the interface between these domains with the computational mesh, we can accurately describe these processes. The network is efficiently described by a network of line segments. Coupling terms are evaluated by projection of the interface variables. The new method is naturally applicable for nonlinear and time-dependent problems and can therefore be used as a reference method in the development of novel implicit interface 1D-3D methods and in the design of verification benchmarks for embedded tubular network methods. Implicit interface models, not resolving the bulk-network interface explicitly have proven to be very efficient but have only been mathematically analyzed for linear elliptic problems so far. Using two application scenarios, fluid perfusion of vascularized tissue and root water uptake from soil, we investigate the effect of some common modeling assumptions of implicit interface methods numerically.
Highlights
There is a strong demand for efficient and accurate models describing flow and transport processes in porous media with embedded tubular network systems, such as vascularized biological tissue, plant root system growing in soil, hydrological, geothermal or petroleum wells in rock formations
Network and bulk partial differential equations (PDEs) are coupled by source terms that depend on state variables from both domains
Effect of neglecting vessel resistance to bulk flow we show with a numerical example comparing the explicit interface ps method with implicit interface methods that neglecting the resistance of the vessel to bulk flow introduces some error in the bulk pressure field and the computed exchange source term
Summary
There is a strong demand for efficient and accurate models describing flow and transport processes in porous media with embedded tubular network systems, such as vascularized biological tissue, plant root system growing in soil, hydrological, geothermal or petroleum wells in rock formations. Simple mixed-dimension methods have been used for more than two decades root water uptake simulations [13], the proper grid resolution required to accurately solve the model equations in dry soils is rarely considered in the literature [14, 15]. While convergence rates alone are inconclusive about the error at a given practical discretization length, the results in [19, 10, 12] indicate that in order to achieve sufficiently accurate numerical results, the discretization length in the embedding bulk domain has to be chosen in the order of the network tube radii or smaller This is in stark contrast to typical grid resolutions in root water uptake simulations, where soil cells are routinely chosen an order of magnitude larger than the root radius [14, 4]. We compare the new method with previously published methods for examples for numerical test cases of tissue perfusion and root water uptake in Sections 3.2 and 3.3
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