Nonlinear mixed-dimension model for embedded tubular networks with application to root water uptake

We present a numerical scheme for the solution of nonlinear mixed-dimensional PDEs describing coupled processes in embedded tubular network system in exchange with a bulk domain. Such problems arise in various biological and technical applications such as in the modeling of root-water uptake, heat exchangers, or geothermal wells. The nonlinearity appears in form of solution-dependent parameters such as pressure-dependent permeability or temperature-dependent thermal conductivity. We derive and analyse a numerical scheme based on distributing the bulk-network coupling source term by a smoothing kernel with local support. By the use of local analytical solutions, interface unknowns and fluxes at the bulk-network interface can be accurately reconstructed from coarsely resolved numerical solutions in the bulk domain. Numerical examples give confidence in the robustness of the method and show the results in comparison to previously published methods. The new method outperforms these existing methods in accuracy and efficiency. In a root water uptake scenario, we accurately estimate the transpiration rate using only a few thousand 3D mesh cells and a structured cube grid whereas other state-of-the-art numerical schemes require millions of cells and local grid refinement to reach comparable accuracy.

[1]  Oliver Sander,et al.  The Dune FoamGrid implementation for surface and network grids , 2015, ArXiv.

[2]  H. Vosteen,et al.  Temperature dependence of the relationship of thermal diffusivity versus thermal conductivity for crystalline rocks , 2008 .

[3]  Olaf Kolditz,et al.  Simulation of heat extraction from crystalline rocks: The influence of coupled processes on differential reservoir cooling , 2006 .

[4]  Jan Vanderborght,et al.  Use of a Three‐Dimensional Detailed Modeling Approach for Predicting Root Water Uptake , 2008 .

[5]  Y. Mualem.  A New Model for Predicting the Hydraulic Conductivity , 1976 .

[6]  L. Rosenhead.  Conduction of Heat in Solids , 1947, Nature.

[7]  M. Mori,et al.  The double-exponential transformation in numerical analysis , 2001 .

[8]  D. W. Peaceman.  Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability , 1983 .

[9]  Ingeborg G. Gjerde,et al.  Splitting method for elliptic equations with line sources , 2018, ESAIM: Mathematical Modelling and Numerical Analysis.

[10]  Loïc Pagès,et al.  MODELLING OF THE HYDRAULIC ARCHITECTURE OF ROOT SYSTEMS : AN INTEGRATED APPROACH TO WATER ABSORPTION : MODEL DESCRIPTION , 1998 .

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  Laura Cattaneo,et al.  Computational models for fluid exchange between microcirculation and tissue interstitium , 2014, Networks Heterog. Media.

[13]  Ralf Kornhuber,et al.  Fast and Robust Numerical Solution of the Richards Equation in Homogeneous Soil , 2011, SIAM J. Numer. Anal..

[14]  K. Friedrichs.  On the differentiability of the solutions of linear elliptic differential equations , 1953 .

[15]  Timo Koch,et al.  Projection-based resolved interface mixed-dimension method for embedded tubular network systems , 2021, Comput. Math. Appl..

[16]  Carlo D'Angelo,et al.  Finite Element Approximation of Elliptic Problems with Dirac Measure Terms in Weighted Spaces: Applications to One- and Three-dimensional Coupled Problems , 2012, SIAM J. Numer. Anal..

[17]  K. Friedrichs.  The identity of weak and strong extensions of differential operators , 1944 .

[18]  Matthias Ehrhardt,et al.  On Dirac delta sequences and their generating functions , 2012, Appl. Math. Lett..

[19]  Raducanu Razvan,et al.  MATHEMATICAL MODELS and METHODS in APPLIED SCIENCES , 2012 .

[20]  Rainer Helmig,et al.  A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model , 2019, J. Comput. Phys..

[21]  Martin Beck,et al.  DuMux 3 - an open-source simulator for solving flow and transport problems in porous media with a focus on model coupling , 2019, Computers & Mathematics with Applications.

[22]  T. Secomb,et al.  A Green's function method for analysis of oxygen delivery to tissue by microvascular networks. , 1989, Mathematical biosciences.

[23]  Jan Vanderborght,et al.  Call for Participation: Collaborative Benchmarking of Functional-Structural Root Architecture Models. The Case of Root Water Uptake , 2019, bioRxiv.

[24]  Jan Vanderborght,et al.  Call for Participation: Collaborative Benchmarking of Functional-Structural Root Architecture Models. The Case of Root Water Uptake , 2020, Frontiers in Plant Science.

[25]  Paolo Zunino,et al.  Mathematical analysis, finite element approximation and numerical solvers for the interaction of 3D reservoirs with 1D wells , 2019, GEM - International Journal on Geomathematics.

[26]  R. Helmig.  Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems , 2011 .

[27]  Günter Zimmermann,et al.  3D numerical modeling of hydrothermal processes during the lifetime of a deep geothermal reservoir , 2010 .

[28]  Ernst Steudle,et al.  THE COHESION-TENSION MECHANISM AND THE ACQUISITION OF WATER BY PLANT ROOTS. , 2001, Annual review of plant physiology and plant molecular biology.

[29]  Melvin T. Tyree,et al.  The Cohesion-Tension theory of sap ascent: current controversies , 1997 .

[30]  Van Genuchten,et al.  A closed-form equation for predicting the hydraulic conductivity of unsaturated soils , 1980 .

[31]  W. R. Gardner.  DYNAMIC ASPECTS OF WATER AVAILABILITY TO PLANTS , 1960 .

[32]  C. Dangelo,et al.  Multiscale modelling of metabolism and transport phenomena in living tissues , 2007 .

[33]  Paolo Zunino,et al.  Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction , 2019, ESAIM: Mathematical Modelling and Numerical Analysis.

[34]  N. Beudez,et al.  Influence of Three Root Spatial Arrangement on Soil Water Flow and Uptake. Results from an Explicit and an Equivalent, Upscaled, Model , 2013 .

[35]  Rainer Helmig,et al.  A New Simulation Framework for Soil–Root Interaction, Evaporation, Root Growth, and Solute Transport , 2018, Vadose Zone Journal.

[36]  Arnold Neumaier,et al.  Mathematical Modeling of the Dynamics of Macroscopic Structural Transformations in Self-Propagating High-Temperature Synthesis , 2004 .

[37]  Jan Vanderborght,et al.  Continuum multiscale model of root water and nutrient uptake from soil with explicit consideration of the 3D root architecture and the rhizosphere gradients , 2018, Plant and Soil.

[38]  Barbara Wohlmuth,et al.  Mathematical modelling, analysis and numerical approximation of second order elliptic problems with inclusions , 2018 .

[39]  Timo Koch.  Mixed-dimension models for flow and transport processes in porous media with embedded tubular network systems , 2020 .

[40]  Patrick Jenny,et al.  Modeling tissue perfusion in terms of 1d-3d embedded mixed-dimension coupled problems with distributed sources , 2020, J. Comput. Phys..

[41]  Bernd Körfgen,et al.  Effect of Local Soil Hydraulic Conductivity Drop Using a Three‐Dimensional Root Water Uptake Model , 2008 .

[42]  Lianhai Wu,et al.  Modelling root–soil interactions using three–dimensional models of root growth, architecture and function , 2013, Plant and Soil.