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A new finite difference scheme, the development of the finite difference heterogeneous multiscale method (FDHMM), is constructed for simulating saturated water flow in random porous media. In the discretization framework of FDHMM, we follow some ideas from the multiscale finite element method and construct basic microscopic elliptic models. Tests on a variety of numerical experiments show that, in the case that only about a half of the information of the whole microstructure is used, the constructed scheme gives better accuracy at a much lower computational time than FDHMM for the problem of aquifer response to sudden change in reservoir level and gives comparable accuracy at a much lower computational time than FDHMM for the weak drawdown problem.

Natural porous media exhibit a significant spatial variability in most attributes of hydrogeological interest. For instance, it is quite typical for hydraulic conductivity to vary orders of magnitude over distances [

Multiscale solution methods are currently under active investigation for the simulation of subsurface flow in heterogeneous formations [

Most of existing multiscale methods have been limited to the finite element method [

Here, we propose a new scheme of FDHMM for simulating not only the steady saturated flow problem but also the transient saturated flow problem in geostatistical random porous media. The constructed scheme employs an idea presented by Ming and Zhang [

Our method is also analogous to the classical upscaling method, where the upscaled hydraulic conductivities are precomputed [

This paper is organized as follows. We firstly describe the flow problem and introduce the principle and the algorithm of the constructed scheme in detail. Numerical examples to illustrate the performance of the constructed scheme are arranged in Section

The transient saturated flow through a heterogeneous porous medium is governed by the parabolic partial differential equation

The discretization in this study is the mesh-centered finite difference. To simplify the presentation of the constructed scheme, we assume that the solution domain

Notice that a macroscopic model is known to exist according to the homogenization theory, and the idea of the constructed scheme is to evolve a macroscopic model for the flux form of (

To solve (

Illustration of the macroscopic and microscopic computational domains.

The control volumes at the coarse node

To estimate the macroscopic flux, we need to solve a set of local microscale problems in the control volumes. Actually, the saturated hydraulic conductivity tensor

In every control volume

Set the head of the basic elliptic problem with no dimensional change. Let

After solving the basic elliptic problems, we estimate basic macroscopic fluxes based on the solutions of the basic elliptic problems. In

Based on the estimation of the basic macroscopic fluxes, we will estimate macroscopic fluxes. Let

We first solve a microscopic elliptic problem with the Dirichlet-Neumann boundary condition at every control volume

For the control volume

Like the assumption in [

By applying the above assumption, we derive the macroscopic flux

Let

Then, we solve the following equation at the coarse mesh by using MGD9V [

Thus, the algorithm is completed. The solution procedure at a time step is illustrated in Figure

Flow chart of the new scheme at a time step.

The algorithm described above is easy to extend to the steady flow problem in heterogeneous porous media. Under the condition of the steady flow, the left-hand side of (

The locally hydraulic conductivity (

All porous media in nature are heterogeneous. The heterogeneity in this study comes from the hydraulic conductivity. As the standard deviation of logarithmic hydraulic conductivity increases, the heterogeneity increases. The random conductivity field is generated by the Turning Band method [

The algorithm has been implemented in a FORTRAN code. Because it is difficult to construct interesting multiscale problem with an exact solution, people often compare the coarse scale solution obtained by the multiscale method with a computed reference solution obtained on the fine scale. We have employed the conventional finite difference method with multigrid over a fine mesh to solve the original equation and refer to this solution as the “exact” solution.

As a measure of the error, we take the relative

In all test examples, the study domain

We impose the Dirichlet-Neumann boundary condition for the test steady flow problem. The left and right sides of boundary are Dirichlet boundaries. Head on the left is 20 m, and that on the right side is 10 m. The top and bottom sides are impermeable boundaries. To start the computation using the new scheme, we need to choose the size of the control volume

Four conductivity fields with isotropic correlation microstructure are first applied. Correlation lengths of these conductivity fields are

A realization of the random saturated conductivity fields of different standard deviations under

Relative (a)

Exact solution and the coarse solution of the new scheme in section

Convergence should be a necessary condition for the new scheme as a good numerical method. Here, we only consider the conductivity field with

Relative (a)

Next, we turn to consider three conductivity fields with anisotropic correlation microstructure. Fixing

Relative (a)

We design this transient test example based on the example in [^{−1} and 10 m, respectively. To generate the random hydraulic conductivity field, we assume that

At first, accuracies and efficiencies of the constructed scheme and FDHMM are compared. Let the size of the control volume

Relative (a)

Exact solution (top), the coarse solutions of the new scheme (middle), and FDHMM (bottom) at times

The results were obtained on a computer running Windows XP with 2.66 GHz processor, 2 megabytes of cache, and 512 megabytes of RAM. For this test example, memory requirements using the conventional finite difference method, the constructed scheme, and FDHMM are about 27.7, 4.3, and 4.3 megabytes, respectively; CPU times using the three methods are about 12.1 min, 0.1 min, and 7.2 min, respectively. Compared with the computational cost of the conventional finite difference method, in our test example, the present new can save about

Next, we discuss the effects of different cell sizes on the accuracy of the constructed scheme. In a coarse

Relative (a)

In this section, we first consider the steady flow problem with well drawdown in heterogeneous porous media. Similar to the examples discussed in [^{3}/min, 0.24 m^{3}/min, 0.36 m^{3}/min, and 0.48 m^{3}/min, respectively. The aquifer is 10 m thick. We also choose

Four conductivity fields with ^{3}/min, when ^{3}/min. There are larger errors of the results of the constructed scheme near point (500 m, 500 m), which are caused by the pumping well at this point. This is likely because heads near the well vary nonlinearly with distance to the well, which cannot be well described by the constructed scheme. On the other hand, the problem of the well singularity may be related to the chosen scale. If we choose a coarse

Relative (a)

Exact solution and the coarse solution of the new scheme in section ^{3}(min)^{−1}).

Exact solution and the coarse solution of the new scheme at the coarse ^{3}(min)^{−1}).

Next, we consider the transient well drawdown problem in heterogeneous porous media. Boundaries of the study area are Dirichlet types. Heads on four sides are all 10 m. Initial pressure head is also 10 m everywhere in the aquifer. The specific storage coefficient is ^{−1} and the aquifer is 10 m thick. There is a pumping well at the point (500 m, 500 m). The well has the constant flow rate of 0.24 m^{3}/min and is pumped for 1600 min in the problem. The time step is 1 min for every method. This test example is analogous to the examples used in [

Similar to Section

Relative (a)

Exact solution and two coarse solutions at times (a)

Computational costs of the three methods in this test example are similar to those in the test example discussed in Section

Relative (a)

A new scheme of the finite difference heterogeneous multiscale method, which puts more emphasis on the interaction between the macro- and microscale behaviors, has been presented for solving saturated water flow problems in random porous media. The macroscopic iteration formulas of steady and transient flow problems have been explicitly deduced. By solving basic microscopic elliptic problems and estimating basic macroscopic fluxes, it is subtly brought to the large scale for microscale information of the medium property and useful information about the gradients of the solutions of basic microscopic elliptic models. For the transient flow problem, different from that FDHMM needs the macroscopic and microscopic evolution at every time step, the constructed scheme implements the microscopic evolution at the preprocessing step and only needs the macroscopic evolution at every time step, which offers substantial saving in the computational cost. The constructed scheme saves about

This study is limited to two-dimensional saturated flow through heterogeneous porous media. We also plan to extend this scheme to solve unsaturated water flow problems with heterogeneity which would be more difficult to simulate.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Natural Science Foundation of China (51039007), the Natural Science Foundation of Hunan Province (13JJ3120), and the Construct Program of the Key Discipline in Hunan Province.