A FEM for unsaturated transient seepage is established by using a quadrilateral isoparametric element, considering the fact that the main permeability does not coincide with the axis situation. It creates a function by using the element’s node hydraulic head and shape function instead of the real head in the Richard seepage control equation. With the help of the Galerkin weighted residual method, a FEM equation is given for analyzing 2-dimensional transient seepage problem. Further, based on the Jacobi matrix and Gauss numerical integral, it determines the elements of stiffness and capacitance matrices. This FEM equation considers not only the anisotropic of soil but also the uncoincidence between permeability and the axis. It is a common form of transient seepage. In the end, two examples illustrate the node accuracy of the quadrilateral element and the correctness of this FEM equation.

Transient seepage of unsaturated soils is a hot issue of research currently [

Lam et al. [

Under the condition of shape functions sharing the same power, approximate functions constructed by values on the nodes and shape functions are defective, while the quadrilateral element has no similar problem. In this regard, some scholars begin to use the rectangular element to calculate some complex engineering problems [

Further research [

At present, triangular element is still the simplest element form in the FEM of unsaturated soil’s transient seepage problem. The stiffness matrix and capacitance matrix of the triangular element can be used to code a calculation program. But the triangular element has relatively large errors when analyzing some symmetry problems. The quadrilateral element is symmetric, so it is more accurate than the triangular element in the condition of symmetrical flow. Quadrilateral element is also widely used in various seepage analysis software, but its stiffness matrix and capacitance matrix have not been reported yet. This makes it difficult for geotechnical engineers who need to code a calculation program to analyze seepage problems. In addition, soil anisotropy also needs to be reflected.

Based on this, this paper attempts to consider the phenomenon of the anisotropy of soils and main seepage coefficient not coinciding with the coordinate axis and adopt one-dimensional quadrilateral isoparametric element to research on establishing the calculating method of the finite element for transient seepage of unsaturated soils.

On this basis, by using the Galerkin weighted residual method, this paper establishes a transient seepage finite element calculation method of unsaturated soils. This method uses the one-dimensional quadrilateral isoparametric element and gives its stiffness matrix and capacitance matrix. This method also considers the anisotropy coefficient of soil and seepage is not consistent with the coordinate axis. Two examples are given: one is to illustrate the difference between triangular element and quadrilateral element and the other one gives a calculation of certain dam seepage to clarify the accuracy of the proposed method in seepage flow.

Assume that, in the plane problem, the seepage coefficients of soils are, respectively, known to be _{1} and _{2} in two mutually perpendicular directions _{1} and _{2}. They are functions of matric suction of unsaturated soils (_{a}−

Permeability not coinciding with the axis in an anisotropic unsaturated soil.

According to Darcy’s law,_{1} direction

Formula _{1} direction can be written as

Similarly, the flow velocity _{2} direction can be obtained as

At present, introducing formula (

On the other hand, the continuity condition of seepage of unsaturated soils contains_{a} −

Then, the control equation of transient seepage for anisotropic can be written as follows:_{1}) and the normal flow velocity on the Neumann boundary (Г_{2}) are known as

When the quadrilateral one-dimensional isoparametric element is adopted, as shown in Figure _{i}, _{i}) and (_{i}, _{i}) are global and local coordinate values of point

Diagram of coordinate transformation for a 4-node quadrilateral isoparametric element.

In the element, an approximate head function is constructed by means of the product of the node head and the shape function to approach the real head of any point on the nodes, such as

To replace formula (_{Ω}; formula (_{2}, denoted as _{Г2}. When shape functions of the quadrilateral element are used to multiply these margins, respectively, the integral sum of _{Ω} and _{Г2} in each integral domain should be zero, and this method is called Galerkin weighted residual method. According to the Galerkin method, the following mathematical expression can be obtained:

The method of subsection integration is adopted to the first item of formula (

In formula (

Then, formula (

For the isoparametric quadrilateral element, there are a total of 4 × 4 = 16 elements in the stiffness matrix, among which

To desire for the expression of stiffness matrix, first of all, it is necessary to convert the integral into the local coordinate system, which requires the assistance of the transition of Jacobi matrix.

The partial derivative of the global coordinate to the local coordinate can be written as

The coefficients in the formula are

All of them are only related to the node coordinates, which are constant for the local coordinates. Then, the determinant of the Jacobi matrix is given by

The coefficients among them are

Similarly, these coefficients are constant.

The relation between the shape function and the partial derivative of global and local coordinates is obtained by using Jacobi matrix:^{−1} represents the inverse matrix of Jacobi matrix. Formula (

The coefficients in the formula are

Similarly, the shape function on the partial derivative of

The coefficients in the formula are

The integral domain is transformed by Jacobi matrix

The results obtained by formulas (_{ij} can be obtained as follows:

The integrand _{Dij} (

In formula (_{s} and _{t} are weighting coefficients, _{s} and _{t} are integral points, and

After being introduced into formula (

Among them, the expression of the integrand _{Dij} is shown in formula (

The capacitance matrix is the same as the stiffness matrix, which also has 16 items. Among them, the calculation formula of the element at any position is

By using the Gauss integral method, each element in the capacitance matrix can be obtained. Considering that the form of the integrand is relatively simple in the capacitance matrix, the expression of its integral form can be presented by simplification.

For any element in the capacitance matrix, after the expression of formula (

If formula (

The coefficients in the formula are

Formula (

After local coordinates on nodes are introduced, the capacitance matrix can be simplified as

Among which matrices [_{5}], [_{6}], and [_{7}] are all constant matrices, they, respectively, are

Matrix [F] is the matrix reflecting Neumann boundary conditions, which represents equivalent nodal flow in the normal direction.

The above method can be used to calculate unsaturated seepage under various conditions. Because all of the existing calculating methods are not involved in the problem of the main seepage coefficient not coinciding with the coordinate axis, for the convenience of comparative analysis, so does validate calculation examples, which can be achieved by demanding Corner _{1} or _{2}) in a certain direction can be further demanded to be zero. The coefficients simplified by formula (

At present, based on the formula derivation in this paper, Fortran language self-compiled program is applied to calculate two different problems.

Due to the asymmetry of the triangle element, node errors will appear in the analysis of some symmetric problems, while the quadrilateral element can overcome these errors. Considering the one-dimensional unsaturated seepage problem on a horizontal soil column, the linear triangular and quadrilateral elements are used to conduct discretization, as shown in Figure

Discretization of one-dimensional seepage.

The length of the soil column is 1 m, and the boundary condition of the head is given on both sides: the suction head on the left side is maintained at zero, and the suction head on the right side is maintained at −10 m. Side lengths of the element are 0.1 m without exception. The quadrilateral element is composed of 10 elements and 22 nodes. The triangular element needs 20 elements and 22 nodes. In the process of seepage calculation, because the seepage coefficient

Permeability function for analyzing transient seepage [

The two values of upper and lower nodal heads in two different elements away from the same position on the left side are shown in Figure

Hydraulic head differences with distance.

In order to verify the correctness of the formula deriving and program compiling in this paper, the condition of two-dimensional seepage of earth-rock dam in literature [

Divided cross section of a dam for an isoparametric element.

In the calculation, in order to be consistent with the conditions in the literature, it is assumed that the seepage coefficient of the soil is isotropic, and the seepage coefficient function used in the analysis is still given out by Figure

The phreatic line differences with time (days).

In view of the fact that the main seepage coefficient not coinciding with the coordinate axis is not considered in the existing numerical calculation of the moisture field of unsaturated soils, this paper adopts the calculating method of the finite element for transient seepage of unsaturated soils. By using the shape function on the element node and the head value, an approximate function can be constructed to replace the head function in the Richard seepage control equation. Galerkin weighted residual method is used to construct finite element equations by making the integral of the error generated by the approximation in the element equal to zero. The “weak” form of weighted residual method is obtained by the further application of multivariate function integral method and Green formula transforming the equation form, based on which, the finite element form is presented to analyze the seepage problem by using the quadrilateral isoparametric element, and stiffness matrix and capacitance matrix are determined by the Gauss numerical integration method. This finite element method takes the anisotropy of soils, the main seepage coefficient not coinciding with the coordinate axis, and other cases into account, which is the general form of transient seepage of unsaturated soil analysis.

All data and models generated or used during the study are available in the submitted article. All data included in this study are available upon request by contact with the corresponding author.

The authors have no conflicts of interest to declare.

This study was supported by the Fund Program for the National Natural Science Foundation of China (no. 51078309), Key Research and Development Program of Shaanxi (Program no. 2018ZDCXL-SF-30-9), and Shaanxi Key Laboratory of Geotechnical and Underground Space Engineering, XAUAT. The authors would like to thank the team members of the key discipline of geotechnical engineering and the workers of the Geotechnical Engineering Laboratory.