# Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space-Time Trefftz Method
###### Abstract  
In this study, we developed a novel boundary-type meshless approach for dealing with two-dimensional transient flows in heterogeneous layered porous media. The novelty of the proposed method is that we derived the Trefftz space-time basis function for the two-dimensional diffusion equation in layered porous media in the space-time domain. The continuity conditions at the interface of the subdomains were satisfied in terms of the domain decomposition method. Numerical solutions were approximated based on the superposition principle utilizing the space-time basis functions of the governing equation. Using the space-time collocation scheme, the numerical solutions of the problem were solved with boundary and initial data assigned on the space-time boundaries, which combined spatial and temporal discretizations in the space-time manifold. Accordingly, the transient flows through the heterogeneous layered porous media in the space-time domain could be solved without using a time-marching scheme. Numerical examples and a convergence analysis were carried out to validate the accuracy and the stability of the method. The results illustrate that an excellent agreement with the analytical solution was obtained. Additionally, the proposed method was relatively simple because we only needed to deal with the boundary data, even for the problems in the heterogeneous layered porous media. Finally, when compared with the conventional time-marching scheme, highly accurate solutions were obtained and the error accumulation from the time-marching scheme was avoided. T +
Footnote †: journal: Applied Physics Letters  
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As the system of linear equations obtained from the collocation Trefftz method is ill-posed, the application of the collocation Trefftz method is less widespread [31]. Prior studies have demonstrated that applications of the collocation Trefftz method may be limited to linear and stationary problems. Recently, a study on modeling the subsurface flow problems with transient moving boundaries utilizing the Trefftz method was developed [12]. Moreover, the space-time collocation Trefftz method was developed to deal with the transient groundwater inverse problems [24]. However, the engineering applications of the Trefftz method with complete Trefftz functions for dealing with transient fluid flow through heterogeneous porous media have been less studied, which is what initiated this research. The pore network model and the self-organized percolation (SOP) model may perform well compared with computational fluid dynamics modeling and the lattice Boltzmann method [32; 33]. In contrast with other numerical methods, we developed a meshless method using the Trefftz space-time basis function. The proposed method is rooted in the Trefftz method, which is more like the semi-analytical method. Consequently, the proposed method is advantageous for dealing with problems with complex geometry because of its characteristics of simplicity and being meshfree. The proposed method may also be used to integrate regional-scale hydrological models [34]. In this study, a novel meshless approach for modeling transient flows in heterogeneous layered porous media is developed. In contrast to the conventional meshless method, the proposed approach is categorized into the boundary-type meshless method, such that the collocation points are placed only on the boundaries of the domain. Furthermore, we developed a boundary-type meshless method combining the conventional Trefftz method with the space-time collocation scheme. The proposed method is advantageous for dealing with engineering problems with a complex geometry because of its characteristics of simplicity and being meshfree. Based on the superposition principle, numerical approximations were approximated using the space-time basis functions of the governing equation. The numerical approximations of the problem were solved with boundary and initial data assigned on the space-time boundaries, which combined spatial and temporal discretizations in the space-time domain.