# Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space-Time Trefftz Method
## 6 Conclusions  
A novel meshless method for dealing with two-dimensional transient flows in heterogeneous layered porous media is presented in this article. The following key findings are given.  
To deal with transient flows in heterogeneous layered porous media, the continuity conditions at the interface of the subdomains are satisfied in terms of the domain decomposition method. Numerical solutions are approximated based on the superposition principle adopting the space-time basis functions of the diffusion equation. Utilizing the space-time collocation scheme, numerical approximations are solved with boundary and initial data assigned on the space-time boundaries, which combine spatial and temporal discretizations in the space-time domain. The transient flows through heterogeneous layered porous media in the space-time domain may therefore be solved without using the time-marching scheme, which is the novelty of this study.  
The results obtained show that an excellent agreement with the exact solution can be acquired. Additionally, the proposed approach may be relatively simple, because we only have to deal with the boundary data even with problems in heterogeneous layered porous media. Finally, compared with the conventional time-marching scheme, highly accurate solutions can be obtained and the error accumulation from the time-marching scheme can be avoided.  
**Author Contributions:** Conceptualization and method, C.-Y.K.; verification, L.-D.H.; writing--original manuscript, C.-Y.L. and C.-Y.K.; data collection, J.-E.X. and W.-P.H.; revising the manuscript, C.-Y.K. and C.-Y.L. All of the authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was supported by the Ministry of Science and Technology (MOST), Taiwan, Republic of China (MOST 109-2625-M-019-007).  
**Institutional Review Board Statement:** The study was conducted according to the guidelines of the Declaration of Helsinki, and approved by the Institutional Review Board.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The datasets generated during this research are available upon reasonable request.  
**Acknowledgments:** The authors sincerely thank MOST for providing financial support.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Nomenclature**  
\(r\) radius  
\(\hat{r}\) dimensionless radius  
\(t\) time  
\(\hat{t}\) dimensionless time  
\(R_{0}\) characteristic length  
\(\theta\) polar angle  
\(S_{S}\) volumetric specific storage  
i \(i\)th subdomain  
k \(i\) hydraulic conductivity  
\(k_{i}\) hydraulic conductivity of \(i\)th subdomain  
\(D\) hydraulic diffusivity  
\(D_{1}\) hydraulic diffusivity of subdomain \(1\)  
\(D_{i}\) hydraulic diffusivity of the \(i\)th subdomain  
\(\Omega_{t}\) space-time domain  
\(\Omega_{i}\) space-time domain of \(i\)th subdomain  
\(H_{0}\) initial total head  
\(H_{D}\) Dirichlet boundary data  
\(H_{N}\) [PERSON] boundary data  
\(A_{pq}\) vector of unknown coefficients  
T  basis functions  
T\({}_{i}\) basis functions of \(i\)th subdomain  
\(\Gamma_{12}\) sub-boundaries at the interface  
\(\Gamma_{22}\) sub-boundaries at the interface  
\(\mathbf{A_{w}}\) system matrix  
\(\mathbf{A_{\Omega_{i}}}\) matrix of the \(i\)th subdomain  
\(h\) total head  
\(\mathbf{A_{I}}\) interface matrix  
\(\mathbf{c_{w}}\) vectors of unknown coefficients  
\(\mathbf{c_{\Omega_{1}}}\) vectors of unknown coefficients  
\(\mathbf{c_{\Omega_{2}}}\) vectors of unknown coefficients  
\(\mathbf{B_{w}}\) vectors of boundary values  
\(\mathbf{B_{\Omega_{1}}}\) vectors of boundary values  
\(\mathbf{B_{\Omega_{2}}}\) vectors of boundary values  
\(\mathbf{B_{I}}\) vectors of interface  
\(0_{\Omega_{1}}\) zero matrices  
\(0_{\Omega_{2}}\) zero matrices  
\(0_{\Gamma_{12}}\) zero vectors  
\(0_{\Gamma_{22}}\) zero vectors  
\(m\) order of the basis functions