Mathematical analysis, finite element approximation and numerical solvers for the interaction of 3D reservoirs with 1D wells

Cerroni, Daniele; Laurino, Federica; Zunino, Paolo

doi:10.1007/s13137-019-0115-9

Cited by 31 publications

(28 citation statements)

References 29 publications

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“…Although the focus of the present work is mostly on the analysis and approximation of the proposed approach, we stress that it aims to build the mathematical foundations for tackling various applications involving 3D-1D mixed-dimensional PDEs, such as fluid-structure interaction of slender bodies [26], microcirculation and lymphatics [29,33], subsurface flow models with wells [8], and the electrical activity of neurons.…”

mentioning

confidence: 99%

Analysis and Approximation of Mixed-Dimensional PDEs on 3D-1D Domains Coupled with Lagrange Multipliers

Kuchta¹,

Laurino²,

Mardal³

et al. 2021

SIAM J. Numer. Anal.

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Coupled partial differential equations (PDEs) defined on domains with different dimensionality are usually called mixed-dimensional PDEs. We address mixed-dimensional PDEs on three-dimensional (3D) and one-dimensional (1D) domains, which gives rise to a 3D-1D coupled problem. Such a problem poses several challenges from the standpoint of existence of solutions and numerical approximation. For the coupling conditions across dimensions, we consider the combination of essential and natural conditions, which are basically the combination of Dirichlet and Neumann conditions. To ensure a meaningful formulation of such conditions, we use the Lagrange multiplier method suitably adapted to the mixed-dimensional case. The well-posedness of the resulting saddle-point problem is analyzed. Then, we address the numerical approximation of the problem in the framework of the finite element method. The discretization of the Lagrange multiplier space is the main challenge. Several options are proposed, analyzed, and compared, with the purpose of determining a good balance between the mathematical properties of the discrete problem and flexibility of implementation of the numerical scheme. The results are supported by evidence based on numerical experiments.

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confidence: 99%

Analysis and Approximation of Mixed-Dimensional PDEs on 3D-1D Domains Coupled with Lagrange Multipliers

Kuchta¹,

Laurino²,

Mardal³

et al. 2021

SIAM J. Numer. Anal.

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“…To model the fluid exchange with the vascular system, we adapt a coupling concept discussed in References 29‐31. Thereby the influence of the vascular system is incorporated by means of a source term in the corresponding Darcy equation.…”

Section: Mathematical Modellingmentioning

confidence: 99%

“…Contrary to existing 3D‐1D coupled flow models, 1,29,30,32,33 we project by means of the operator Π the 1D pressure onto the vessel walls, to compute the pressure differences on Γ k . In the aforementioned references, the 3D pressures are projected on the 1D vessels E k , by means of an average operator 33 (see Figure 3).…”

Section: Mathematical Modellingmentioning

confidence: 99%

“…The convection‐diffusion equation is enhanced by the Michaelis‐Menten law modelling the consumption of oxygen by the tissue cells (chapter 2 of Reference 1). To couple the PDE systems for flow and transport, we use a specific coupling concept presented in References 29‐31. The key ingredient of this concept is to couple the source terms of the PDEs for flow and transport.…”

Section: Introductionmentioning

confidence: 99%

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A 3D‐1D coupled blood flow and oxygen transport model to generate microvascular networks

Köppl¹,

Vidotto²,

Wohlmuth

2020

Numer Methods Biomed Eng

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In this work, we introduce an algorithmic approach to generate microvascular networks starting from larger vessels that can be reconstructed without noticeable segmentation errors. Contrary to larger vessels, the reconstruction of fine-scale components of microvascular networks shows significant segmentation errors, and an accurate mapping is time and cost intense. Thus there is a need for fast and reliable reconstruction algorithms yielding surrogate networks having similar stochastic properties as the original ones. The microvascular networks are constructed in a marching way by adding vessels to the outlets of the vascular tree from the previous step. To optimise the structure of the vascular trees, we use Murray's law to determine the radii of the vessels and bifurcation angles. In each step, we compute the local gradient of the partial pressure of oxygen and adapt the orientation of the new vessels to this gradient. At the same time, we use the partial pressure of oxygen to check whether the considered tissue block is supplied sufficiently with oxygen. Computing the partial pressure of oxygen, we use a 3D-1D coupled model for blood flow and oxygen transport. To decrease the complexity of a fully coupled 3D model, we reduce the blood vessel network to a 1D graph structure and use a bi-directional coupling with the tissue which is described by a 3D homogeneous porous medium. The resulting surrogate networks are analysed with respect to morphological and physiological aspects. K E Y W O R D S 3D-1D coupled flow models, blood flow simulations, dimensionally reduced models, flows in porous media, oxygen transport, vascular growth

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“…Чуть позже аналогичный подход был предложен Д. Писмэном [22,23] и затем развит многими авторами (см., например, [8,9,28,16]). Из недавних работ данного направления отметим [4,18]. Отсутствие необходимости сгущать сетку делает соответствующие вычислительные алгоритмы значительно более экономичными.…”

Section: Introductionunclassified

On wells modeling in filtration problems

Ivanov¹,

Kremer²,

Laevsky³

2019

Sib. Elektron. Mat. Izv.

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The work is devoted to one of the approaches of wells modeling within numerical oil reservoir simulation. The approach can be consider as fictitious domain method at mixed finite element approximation, which is used for non-stationary filtration processes of two phase fluid in Bukley-Leverett problem. The numerical results are compared with the results for the problem with usual Neumann conditions at the wells boundaries.

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Mathematical analysis, finite element approximation and numerical solvers for the interaction of 3D reservoirs with 1D wells

Cited by 31 publications

References 29 publications

Analysis and Approximation of Mixed-Dimensional PDEs on 3D-1D Domains Coupled with Lagrange Multipliers

Analysis and Approximation of Mixed-Dimensional PDEs on 3D-1D Domains Coupled with Lagrange Multipliers

A 3D‐1D coupled blood flow and oxygen transport model to generate microvascular networks

On wells modeling in filtration problems

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