Block-centered Finite Difference Method Research Articles

A block-centered finite difference method for reduced fracture model in Karst aquifer system

A block-centered finite difference scheme is given for the approximation of reduced coupled model in the fractured media aquifer system. The fluid flow of aquifer system is governed by Darcy’s law in both fracture and surrounding porous media. The degrees of freedom are given separately on both sides of fracture to capture the discontinuity of pressure and velocity. Optimal error estimates in discrete L2 norms are obtained. Numerical experiments are provided to verify the second-order accuracy of presented method. It is demonstrated whether the fracture acts as a fast pathway or geological barrier is totally determined by the value of its permeability tensor.

Block-Centered Finite Difference Method for Simulating Compressible Wormhole Propagation

In this paper, the block-centered finite difference method is introduced and analyzed to solve the compressible wormhole propagation. The coupled analysis approach to deal with the fully coupling relation of multivariables is employed. By this, stability analysis and error estimates for the pressure, velocity, porosity, concentration and its flux in different discrete norms are established rigorously and carefully on non-uniform grids. Finally, some numerical experiments are presented to verify the theoretical analysis and effectiveness of the given scheme.

Characteristic block-centered finite difference method for simulating incompressible wormhole propagation

In this paper, the characteristic block-centered finite difference method is introduced and analyzed to solve the incompressible wormhole propagation. Error estimates for the pressure, velocity, porosity, concentration and its flux in different discrete norms are established rigorously and carefully on non-uniform grids. Finally, some numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

A Block-Centered Finite Difference Method for Slightly Compressible Darcy–Forchheimer Flow in Porous Media

A block-centered finite difference method is introduced to solve an initial and boundary value problem for a nonlinear parabolic equation to model the slightly compressible flow in porous media, in which the velocity–pressure relation is described by Darcy–Forchheimer’s Law. The method can be thought as the lowest order Raviart–Thomas mixed element method with proper quadrature formulation. By using the method the velocity and pressure can be approximated simultaneously. We established the second-order error estimates for pressure and velocity in proper discrete norms on non-uniform rectangular grid. No time-step restriction is needed for the error estimates. The numerical experiments using the scheme show that the convergence rates of the method are in agreement with the theoretical analysis.

A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation

In this article, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear time-fractional parabolic equation. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Stability results are proven rigorously. Error estimates are established on non-uniform rectangular grid which show that the discrete $$L^{\infty }(L^2)$$ and $$L^2(H^1)$$ errors are $$O(\triangle t^{2-\alpha }+h^2+H^3)$$ . Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.

A block-centered finite difference method for general non-Darcy flow in porous media

A block-centered finite difference scheme is introduced to solve the nonlinear equation of non-Darcy model, in which it can maintain local mass conservation and the velocity and pressure can be approximated simultaneously. The second-order error estimates in discrete $l^2$ norm for both pressure and velocity are established on a nonuniform rectangular grid. Numerical experiments using the scheme show the consistency of the convergence rates of our method with the theoretical analysis.

A parallel CGS block-centered finite difference method for a nonlinear time-fractional parabolic equation

In this article, a parallel Conjugate Gradient Squared (CGS) block-centered finite difference scheme is introduced and analyzed to cast about the numerical solution of a nonlinear time-fractional parabolic equation with the Neumann condition and a nonlinear reaction term. The unconditionally stable result, which just depends on initial value and source item, is derived. Some a priori estimates of discrete L2-norm with optimal order of convergence O(Δt2−α+h2+k2) with pressure and velocity are established on both uniform and non-uniform rectangular grids, where Δt is the time step, h and k are maximal mesh sizes of x and y-directional grids. In our model, because the simulation is iterated over a series of time steps, it is most beneficial if all the calculation units in a time step can be run separately. In CGS algorithm, adding OpenMP instructions to the circulation calculations in an iterative step can realize parallel computing. To examine the efficiency and accuracy of the proposed method, numerical experiments using the schemes are studied. The results clearly show the benefit of using the proposed approach in terms of execution time reduction and speedup with respect to the sequential running in a single thread.

A two-grid block-centered finite difference method for nonlinear non-Fickian flow model

In this paper, a two-grid block-centered finite difference scheme is introduced and analyzed to solve the nonlinear parabolic integro-differential equation arising in modeling non-Fickian flow in porous media. This method is considered where the nonlinear problem is solved only on a coarse grid of size H and a linear problem is solved on a fine grid of size h. Error estimates are established on non-uniform rectangular grid which show that the discrete L∞(L2) and L2(H1) errors are O(▵t+h2+H3). Finally, some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.

A block-centered finite difference method for Darcy-Forchheimer model with variable Forchheimer number

A block-centered finite difference scheme is introduced to solve the nonlinear Darcy–Forchheimer equation with variable Forchheimer number, in which the velocity and pressure can be approximated simultaneously. For variable Forchheimer number the second-order error estimates for both pressure and velocity are established on nonuniform rectangular grid. An iteration process is given to solve the nonlinear system. Numerical experiments using the scheme show the consistency of the convergence rates of the presented methods with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1603–1622, 2015

A Two-Grid Block-Centered Finite Difference Method For Darcy--Forchheimer Flow in Porous Media

A two-grid block-centered finite difference method is proposed for solving the two-dimensional Darcy--Forchheimer model describing non-Darcy flow in porous media. To construct the two-grid method we modify the original nonlinear elliptic operator of Darcy--Forchheimer flow to a twice continuously differentiable one by introducing a small and positive parameter $\varepsilon$. By using the two-grid method, solving a nonlinear equation on a fine grid is reduced to solving a nonlinear equation on a coarse grid together with solving a linear equation on a fine grid. Optimal order error estimates for pressure and velocity in discrete $L^2$ norms are obtained. Some numerical examples are given to show the accuracy and efficiency of the presented method.

Characteristic block-centered finite difference method for Sobolev equation

Characteristic block-centered finite difference method for Sobolev equation

Comparison of Two Mathematical Models for 3D Groundwater Flow: Block-Centered Heads and Edge-Based Stream Functions

Traditionally, groundwater flow models, as well as oil reservoir models, are based on the block-centered finite difference method. Well-known models based on this approach are MODFLOW (groundwater) and ECLIPSE (oil and gas). Such models are well proven and robust; their underlying principles are well understood by hydrologists and petroleum reservoir engineers. Nevertheless, the desire to improve the block-centered finite difference paradigm has always been alive, for instance, to be able to apply deformed grid blocks, or to model anisotropy that is not aligned along the coordinate axes. This article introduces the edge-based stream function as a potential alternative to the paradigmatic model, not only to mitigate the above mentioned limitations, but especially for its promise to inverse modeling. Computer programs have been developed for the discrete analog equations of the stream function method and the conventional method. The two methods are tested by using synthetic forward modeling problems of uniform and radial flow. The theoretical formulation and the numerical results show that the two methods are algebraically equivalent and yield the same flux output. However, for rectangular grid blocks and anisotropy aligned along the coordinate axes, the block-centered method is shown to be computationally more efficient than the edge-based stream function method. The major advantage of the stream function method is that it is linear in the resistivities, proving it an ideal candidate for direct inverse modeling. Moreover, any arbitrary specification of stream functions yields a solution that satisfies the mass balance.

A Direct Inverse Model to Determine Permeability Fields from Pressure and Flow Rate Measurements

The determination of the permeability field from pressure and flow rate measurements in wells is a key problem in reservoir engineering. This paper presents a Double Constraint method for inverse modeling that is an example of direct inverse modeling. The method is used with a standard block-centered finite difference method. With an a priori grid block permeability field as input, two forward runs are made: the first is constrained with the measured pressures; the second is constrained with the measured flow rates. We calculate the pressures in the grid block centers from the first run, while from the second run we calculate the fluxes through the faces between the grid blocks. Substitution of these pressures and fluxes into Darcy's law then yields the transmissibilities at the faces and hence the permeabilities in the grid blocks. In this way the "hard" data (measured pressures and flow rates) are always honored while the "soft", geological data can be incorporated at the discretion of the geologist. Using a synthetic example, we demonstrate the method and compare the results with another method: Ensemble Kalman Filtering. The two methods agree within the scope of their applicability. The Double Constraint method focuses initially on determining spatial distributions of the permeability field for single-phase, steady state flow. For history matching an extension is required to non-steady state, two-phase flow conditions, which is already possible with EnKF. We are currently investigating the possibility of combining the two methods, whereby the strengths of the two methods could be fully exploited. © International Association for Mathematical Geology 2008.

Saturated-Unsaturated 3D Groundwater Model. II: Verification and Application

Verification and application results are presented for a new saturated-unsaturated 3D groundwater flow model (SU3D) developing in Dogan and Motz, which can be used to calculate the pressure distribution over the entire groundwater flow domain in response to rainfall and evapotranspiration. SU3D solves the nonlinear 3D, modified mixed form of the Richards equation continuously throughout the groundwater flow domain, including both the unsaturated and saturated zones. The block-centered finite-difference method, a modified Picard iteration scheme, and the preconditioned conjugate gradient method are used to solve the governing partial differential equations in the new model. SU3D can simulate evaporation from land surface and transpiration from the root zone. Potential evapotranspiration is partitioned into potential evaporation and potential transpiration as a function of the leaf area index; actual evaporation and actual transpiration are then calculated individually. As shown in this paper, SU3D has been verified by reproducing the results of five analytical and numerical studies and a 3D unconfined-aquifer pumping test from the published literature.

An improved approach for assigning pumping rates to heterogeneous aquifer models

A common assumption of ground water models formulated using a block-centered finite-difference method is that a well is located at the center of a cell regardless of its actual location. Due to this assumption, errors are introduced in the spatial distribution of simulated heads. This paper presents an alternative approach for assigning the pumping rates of wells that are located off cell centers. This approach consists of assigning the pumping rate not only to the cell in which the well is located but also to adjacent cells, taking into account the length of the well screen, the hydraulic conductivity, and the distance from the well to the center of its cell. The advantage of this alternative approach over the conventional one is illustrated with a test problem of a synthetic aquifer. Statistical measures of error indicate a much better model fit when pumping rates of wells are distributed over several cells.

The Edge-Based Face Element Method for 3D-Stream Function and Flux Calculations in Porous Media Flow

We present a velocity-oriented discrete analog of the partial differential equations governing porous media flow: the edge-based face element method. Conventional finite element techniques calculate pressures in the nodes of the grid. However, such methods do not satisfy the requirement of flux continuity at the faces. In contrast, the edge-based method calculates vector potentials along the edges, leading to continuity of fluxes. The method is algebraically equivalent with the popular block-centered finite difference method and with the mixed-hybrid finite element method, but is algorithmically different and has the same robustness as the more conventional node-based velocity-oriented method. The numerical examples are computed analytically and may, therefore, be considered as an 'heuristic proof' of the theory and its practical applicability for reservoir engineering and geohydrology. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.

Monte Carlo study of conservative transport in heterogeneous dual-porosity media

In this study, a Monte Carlo simulation method is applied to study groundwater flow and solute transport in heterogeneous, dual-porosity media. Both the hydraulic conductivity and the interregional mass diffusion rate are assumed to be spatial random variables, and their random distributions are generated through a Fast Fourier Transform (FFT) technique. A block-centered finite difference (FD) method is used to solve the flow equation. Based on the generated flow fields, a random walk particle-tracking algorithm is invoked to study the solute transport. The mass diffusion between the mobile and immobile water regions is simulated by a two-state, homogeneous, continuous-time Markov chain. The Monte Carlo simulation results are compared to those obtained through the first-order, Eulerian perturbation method. It is shown from the comparison that the first-order analytical method is robust for predicting mean concentration in mild heterogeneous dual-porosity media. However, large deviations are observed between the analytical and Monte Carlo results for predicting transport in moderately-highly heterogeneous media. The Monte Carlo method is also used to study the variance of the solute flux through a control plane.