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A novel boundary-type meshless method for modeling geofluid flow in heterogeneous geological media was developed. The numerical solutions of geofluid flow are approximated by a set of particular solutions of the subsurface flow equation which are expressed in terms of sources located outside the domain of the problem. This pioneering study is based on the collocation Trefftz method and provides a promising solution which integrates the T-Trefftz method and F-Trefftz method. To deal with the subsurface flow problems of heterogeneous geological media, the domain decomposition method was adopted so that flux conservation and the continuity of pressure potential at the interface between two consecutive layers can be considered in the numerical model. The validity of the model is established for a number of test problems. Application examples of subsurface flow problems with free surface in homogenous and layered heterogeneous geological media were also carried out. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. The results also reveal that it has great numerical stability for solving subsurface flow with nonlinear free surface in layered heterogeneous geological media even with large contrasts in the hydraulic conductivity.

Numerical approaches to the simulation of various subsurface flow phenomena using the mesh-based methods such as the finite difference method or the finite element method are well documented in the past [

Several meshless methods have been reported, such as the Trefftz method [

To solve subsurface flow problems with the layered soil in heterogeneous porous media, the domain decomposition method (DDM) [

The subsurface flow problem with a free surface is a nonlinear problem in which nonlinearities arise from the nonlinear boundary characteristics [

In this paper, we proposed a novel boundary-type meshless method. This pioneering study is based on the collocation Trefftz method and provides a promising solution which integrates the T-Trefftz method and F-Trefftz method for constructing its basis function using one of the particular solutions which satisfies the governing equation and allows many source points outside the domain of interest. To the best of the authors’ knowledge, the pioneering work has not been reported in previous studies and requires further research. Two important phenomena in subsurface flow modeling were explored in this study using the proposed method. We first adopted the domain decomposition method integrated with the proposed boundary-type meshless method to deal with the subsurface flow problems of heterogeneous geological media. The flux conservation and the continuity of pressure potential at the interface between two consecutive layers can be considered in the numerical model. Then, we attempted to utilize the proposed method to solve the geofluid flow with free surface in heterogeneous geological media.

The validity of the model is established for a number of test problems, including the investigation of the basis function using two possible particular solutions and the comparison of the numerical solutions using different particular solutions and the method of fundamental solutions. Application examples of subsurface flow problems with free surface were also carried out.

Consider a three-dimensional domain

The CTM belongs to the boundary-type meshless method which can be categorized into the T-Trefftz method and F-Trefftz method. The T-Trefftz method introduces the T-complete functions where the solutions can be expressed as a linear combination of the T-complete functions automatically satisfying governing equations. On the other hand, the F-Trefftz method constructs its basis function space by allowing many source points outside the domain of interest. The solutions are approximated by a set of fundamental solutions which are expressed in terms of sources located outside the domain of the problem. The T-Trefftz method and the F-Trefftz method both required the evaluation of a coefficient for each term in the series. The evaluation of coefficients may be obtained by solving the unknown coefficients in the linear combination of the solutions which are accomplished by collocation imposing the boundary condition at a finite number of points.

The CTM begins with the consideration of T-complete functions. For indirect Trefftz formulation, the approximated solution at the boundary collocation point can be written as a linear combination of the basis functions. For a simply connected domain or infinite domain with a cavity, as illustrated in Figures

Illustration of four different types of domain in the CTM.

A simply connected domain

An infinite domain with a cavity

A doubly connected domain

A multiply connected domain

For the doubly and multiply connected domains with genus greater than one, as illustrated in Figures

On the other hand, there is another type of the Trefftz method, namely, the F-Trefftz method, or the so-called method of the fundamental solutions (MFS) [

Illustration of four different types of domain in the MFS.

A simply connected domain

An infinite domain with a cavity

A doubly connected domain

A multiply connected domain

The unknown coefficients in the linear combination of the fundamental solutions which are accomplished by collocation imposing the boundary condition at a finite number of points can then be solved. Due to its singular free and meshless merits, the indirect type F-Trefftz method is commonly used. An approximation solution of the two-dimensional Laplace equation using the MFS can also be obtained as

For the conventional Trefftz method, the number of source points is only one. Theoretically, one may increase the accuracy by using a larger order of the basis functions [

In the following, we proposed a novel boundary-type meshless method. This pioneering study is based on the collocation Trefftz method and provides a promising solution which integrates the T-Trefftz method and F-Trefftz method for constructing its basis function using one of the particular solutions which satisfies the governing equation and allows many source points outside the domain of interest. Differing from the CTM and the MFS, the numerical solutions of the proposed method are approximated by a set of basis functions which are expressed in terms of source points located outside the domain. An approximation solution of the two-dimensional steady-state subsurface flow equation using the proposed method can be obtained as

In this example, we adopted two possible particular solutions of Laplace equation as the basis functions. They are

For a two-dimensional simply connected domain

The analytical solution can be found as

Figure

The collocation of boundary and source points.

The accuracy of the maximum absolute error versus

Using

The accuracy of the maximum absolute error versus the number of source points.

Figure

The condition number versus the number of source points.

Similar to the previous example, we verified the accuracy of the proposed method with the consideration of a complex star-like boundary. For a two-dimensional simply connected domain

The analytical solution can be found as

The collocation of boundary and source points.

The accuracy of the maximum absolute error versus

The accuracy of the maximum absolute error versus the number of source points.

Figure

The condition number versus the number of source points.

The first application under investigation is a free surface seepage problem of a rectangular dam as depicted in Figure

Subsurface flow with free surface through a rectangular dam.

The subsurface flow with a free surface is a nonlinear problem in which nonlinearities arise from the nonlinear boundary characteristics. Such nonlinearities are handled in the numerical modeling using iterative schemes. Typically, the methods, such as the Picard method or Newton’s method, are iterative in that they approach the solution through a series of steps. In this study, the Picard method is adopted.

There are 16 boundary collocation nodes uniformly distributed in the initial guess of the moving boundary with the spacing of 1 m as shown in Figure

Result comparison of the computed free surface for a rectangular dam.

The previous examples have demonstrated that the proposed method can be used to deal with the subsurface flow with a free surface. Since the appearance of layered soil in heterogeneous geological media is much more common than homogeneous soil in nature, we further adopted the proposed method to deal with the subsurface flow problems of layered heterogeneous geological media using the DDM.

This example under investigation is a rectangular dam in layered soil as depicted in Figure

The collocation of boundary, source points (a) and configuration of boundary condition (b).

In this study, we adopted the nonoverlapping method to deal with the subsurface flow problems of layered soil profiles. The problems on the subdomains are independent, which makes the DDM suitable for describing the layered soil in heterogeneous porous media.

For the modeling of the layered soil, we split the domain into smaller subdomains in which subdomains were intersected only on the interface between soil layers, as shown in Figure

For the first subdomain, there are a total of 250 boundary collocation nodes where 50 boundary collocation nodes are uniformly distributed in the initial guess of the moving boundary. For the second subdomain, there are also a total of 250 boundary collocation nodes where 50 boundary collocation nodes are uniformly distributed in the initial guess of the moving boundary.

Figure

Comparison of free surface for a rectangular dam in layered heterogeneous geological media.

Because the basis function,

The boundary collocation points of three-dimensional subsurface flow problem.

The analytical solution of the problem is given as

The accuracy of the maximum absolute error versus

Figure

The accuracy of the maximum absolute error versus the number of source points.

This study has proposed a novel boundary-type meshless method for modeling geofluid flow in heterogeneous geological media. The numerical solutions of geofluid flow are approximated by a set of particular solutions of the subsurface flow equation which are expressed in terms of sources located outside the domain of the problem. To deal with the subsurface flow problems of heterogeneous geological media, the domain decomposition method was adopted. The validity of the model is established for a number of test problems. Application examples of subsurface flow problems with free surface were also carried out. The fundamental concepts and the construct of the proposed method are addressed in detail. The findings are addressed as follows.

In this study, a pioneering study is based on the collocation Trefftz method and provides a promising solution which integrates the T-Trefftz method and F-Trefftz method for constructing its basis function using one of the negative particular solutions which satisfies the governing equation and allows many source points outside the domain of interest. The proposed method uses the same concept of the source points in the MFS, but the fundamental solutions can be replaced by the negative Trefftz functions. It may release one of the limitations of the MFS in which the fundamental solutions may be difficult to find.

It is well known that the system of linear equations obtained from the Trefftz method may also become an ill-posed system with the higher order of the terms. In this study, the proposed method integrates the collocation Trefftz method and the MFS which approximates the numerical solutions by superpositioning of the negative particular solutions as basis functions expressed in terms of many source points. As a result, only two Trefftz terms were adopted because many source points are allowed for approximating the solution. Meanwhile, the ill-posedness from adopting the higher order terms for the solution with only one source point in the collocation Trefftz method can be mitigated. In addition, results from the validation examples demonstrate that the proposed method may obtain better accuracy than the MFS.

The validity of the model is established for a number of test problems, including the investigation of the basis function using two possible particular solutions and the comparison of the numerical solutions using different particular solutions and the method of fundamental solutions. Application examples of subsurface flow problems with free surface were also carried out. Numerical results demonstrate that the proposed method is highly accurate and computationally efficient. This pioneering study demonstrates that the proposed boundary-type meshless method may be the first successful attempt for solving the subsurface flow with nonlinear free surface in layered heterogeneous geological media which has not been reported in previous studies. Moreover, the application example depicted that the proposed method can be easily applied to the three-dimensional problems.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This study was partially supported by the Ministry of Science and Technology, Taiwan. The authors thank the Ministry of Science and Technology for the generous financial support.