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We present and study a stabilized mixed finite element
method for single-phase compressible flow through porous media. This
method is based on a pressure projection stabilization method for multiple-dimensional incompressible flow problems by using the lowest equal-order
pair for velocity and pressure (i.e., the

The mixed finite element method is frequently used to obtain approximate solutions to more than one unknown. For example, the Stokes equations are often solved to obtain both pressure and velocity simultaneously. Accordingly, we need a finite element space for each unknown. These two spaces must be chosen carefully so that they satisfy an inf-sup stability condition for the mixed method to be stable. Examples of an appropriate choice for the mixed spaces for the Stokes equations include the

Although the equal-order pairs of mixed finite element spaces do not satisfy the inf-sup stability condition [

In this paper, we extend the pressure projection stabilization method to solving the single-phase compressible flow problem in porous media that is described by a second-order parabolic equation. We first define this stabilized method using the equal-order pair of mixed finite element spaces

The rest of this paper is organized as follows. In the next section, the basic notation, the differential equation, and its mixed formulation are stated. Then, in the third section, the stabilized mixed finite element method is shown. Error estimates for this stabilized method are derived in the fourth section, and a superconvergence result is proved in the fifth section. In the final section, numerical experiments are given to illustrate the theoretical results.

The dynamical problem we consider is single-phase compressible flow in a porous medium domain

In order to introduce a mixed formulation on

For convenience, we state the Gronwall Lemma that will be used later.

Let

In addition to the assumptions (

We divide the proof into three parts.

Taking

Taking

Taking

For

As noted earlier, this choice of the approximate spaces

Let

Under the assumption (

To derive error estimates for the mixed finite element solution

Under the assumption of Theorem

Let

Under the assumptions of Lemma

Subtracting (

Under the assumption of Lemma

Differentiating the term

Under the assumptions (

This theorem follows from Lemmas

Next, we will estimate

Under the assumptions (

Under the assumptions of Lemma

Differentiating (

Under the assumptions of Lemma

It follows from the inf-sup condition (

Under the assumptions (

This theorem follows from Lemmas

The result in Theorem

Set

Under the assumptions (

Setting

Numerical results are presented to check the theory developed in the previous sections. In all the experiments, the triangulations

The purpose of this example is to check the convergence rates for the solution

The numerical experiments have been performed by using the equal-order finite element pair

Detailed numerical results are shown in Tables

Errors and convergence rates for the velocity.

8 | 0.0626682 | 0.0953064 | ||

16 | 0.0175218 | 1.83858 | 0.0514547 | 0.889269 |

24 | 0.00819644 | 1.87377 | 0.0355113 | 0.914632 |

32 | 0.0047615 | 1.88798 | 0.027229 | 0.923136 |

40 | 0.00311895 | 1.89594 | 0.0221371 | 0.927761 |

Errors and convergence rates for the pressure.

8 | 0.0303817 | 0.0200016 | ||

16 | 0.00742562 | 2.03262 | 0.00464606 | 2.10603 |

24 | 0.00328428 | 2.01198 | 0.00203124 | 2.04055 |

32 | 0.00184548 | 2.00363 | 0.00113789 | 2.01428 |

40 | 0.00118171 | 1.9977 | 0.000728781 | 1.99673 |

A selected numerical pressure and velocity for this example at

Numerical results on a 32 × 32 uniform triangular mesh.

The pressure and velocity

The pointwise pressure error

Here, we consider the unsteady-state single-phase flow of oil taking place in a two-dimensional, homogeneous, isotropic, horizontal reservoir, with its property parameters given in Table

Parameters for a reservoir.

Item | Description | Unit | Value |
---|---|---|---|

Permeability | md | 88.7 | |

Oil viscosity | cp | 10 | |

Thickness | ft | 100 | |

Compressibility | 1/psi | 0.0002 | |

Porosity | fraction | 0.2 | |

Initial pressure | psia | 4,000 | |

Oil flow rate | STB/D | 400 | |

Radius of wellbore | ft | 0.25 | |

Oil FVF | RB/STB | 1 | |

Transmissibility conversion factor | — | 1.127 | |

Volume conversion factor | — | 5.614583 | |

Length in the | ft | 4,000 | |

Length in the | ft | 4,000 |

The basic differential equation describing this reservoir is

Conditions adopted in simulation.

Domain and boundary condition

Base triangles

In Figures

Numerical results (pressure contours) with respect to time.

Pressure at time

Pressure at time

Pressure at time

Numerical results at

A stabilized mixed finite element method for an unsteady-state single-phase flow problem in a porous medium has been developed and analyzed. An optimal error estimate in divergence norm for the velocity and suboptimal error estimates in the

This work is supported in part by the NSERC/AERI/iCORE/Foundation CMG Chair Funds in Reservoir Simulation.