New Numerical Schemes for Near-Well Modeling Using Flexible Grid
- Y. Ding (Institut Français du Pétrole) | L. Jeannin (Institut Français du Pétrole)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2004
- Document Type
- Journal Paper
- 109 - 121
- 2004. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 1.8 Formation Damage, 1.11 Drilling Fluids and Materials, 5.6.4 Drillstem/Well Testing, 4.3.4 Scale
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Control-volume schemes are discussed for the modeling of vertical wells using flexible grids.
The main difficulty of well modeling in reservoir simulation is the problem of the difference in scale between the reservoir size (several kilometers) and the wellbore radius (several centimeters). In applications, although the wellbore boundary can be discretized using flexible grids, the gridblocks in the vicinity of the well are usually not small enough compared to the wellbore radius, and the gridblock sizes vary, usually geometrically, in the radial direction from the well. This kind of grid makes the commonly used linear approach (that is linear with respect to coordinates of physical space) inefficient for near-well flow modeling.
In this paper, new numerical schemes, based on logarithmic approach, are proposed to improve the linear approach for near-well modeling. In particular, a multipoint and a two-point flux approximation scheme are presented. These schemes can reduce the calculation errors for the dominant "singular" well flow and consequently give good results. The proposed approach can be used for any kind of flexible grid for near-well modeling.
Accurate well modeling is very important for flow simulations in reservoir engineering. The key point of well modeling is to perform accurate fluid flow simulations in the near-well region. The computational accuracy of well parameters such as the well flow rate or the wellbore pressure depends greatly on the near-well flow modeling.
This paper discusses 2D near-well flow modeling for a vertical well using a flexible grid with discretizations around the wellbore boundary. Using a flexible grid provides a good description for the near-well flow behavior, and the wellbore boundary can be discretized with small gridblocks. Moreover, the numerical productivity index (PI) (Peaceman1,2) is not required with this kind of gridblock. Theoretically, using numerical methods, based on the linear approach (that is, linear with respect to coordinates of physical space), can give accurate results with the flexible grid, provided that the gridblock sizes are small in the whole reservoir. In order to be as accurate as possible, it is necessary to have many small gridblocks in the near-well region. The gridblock sizes increase very smoothly toward the far-well region. To obtain a reasonable accuracy with the linear approach, the gridblocks should be smaller than the wellbore radius in the region near the well. This kind of grid implies a large number of gridblocks and usually small timesteps in simulations, which is not possible in field applications.
The main difficulty of well modeling in reservoir simulation is the problem of the difference in scale between the reservoir size (several kilometers) and the wellbore radius (several centimeters). In practice, the gridblock size far from the well is on the order of hundreds of meters, while the gridblock size around the wellbore is of the order of tenths of centimeters to several meters in the radial direction. This cannot be considered small compared to the wellbore radius. Usually, the gridblock size varies geometrically in the radial direction because of this problem of scale (Fig. 1), and the ratio between two neighboring gridblocks in the radial direction can be large. With this kind of flexible grid, the commonly used numerical schemes,3-7 based on the linear approach, are not always efficient for the modeling of radial well-flow behavior. Consequently, the numerical results on the wellbore boundary and in the near-well region might not be sufficiently accurate. Therefore, precautions should be taken when using this kind of gridblock. To improve accuracy of the flow calculation, a radial logarithmic-type approach can be applied. In this paper, we study new numerical schemes based on the logarithmic approach for near-well flow modeling with flexible grids.
Numerical schemes in the near-well region were discussed by Ding and Jeannin8 for the point sink/source modeling using quadrilateral gridblocks. In that paper, the well is considered as a point sink/source, and the wellbore boundary is not discretized. An analysis of flux truncation errors was given using solution splitting in the vicinity of the well. Based on the solution splitting, new numerical schemes based on a logarithmic approach, which can eliminate the errors for singular solution approximation, were proposed. These schemes showed the efficiency and the accuracy for point sink/source modeling with quadrilateral gridblocks. In this paper, we will follow the same analysis for well modeling, taking into account the existence of the wellbore boundary.
Similar to the point sink/source model, the near-well solution can also be split into a "singular" solution and a regular solution with the consideration of the wellbore boundary. The "singular" flow is the sum of a point-source "singular" flow, which corresponds to the solution of the point source in an infinite reservoir, and a boundary "singular" flow, which depends on variables on the wellbore boundary. It should be mentioned that, unlike point sink/source modeling, no real singularity appears in the existence of the wellbore boundary. The flux in the well vicinity is on the order of O(1/r), and r can be very small, which is similar to a singularity problem. However, r is greater than the wellbore radius rw, so the flux cannot tend to infinity. As discussed above, the gridblock sizes h are usually from several centimeters to several meters in the vicinity of the well, which cannot be considered small compared to the wellbore radius rw. With these kinds of gridblocks, it can be shown that the point-source "singular" flow is dominant in the near-well region, and the contribution of the boundary "singular" flow is small compared to the point-source "singular" flow. Based on this solution splitting, truncation errors can be analyzed for each splitting solution, and new numerical schemes are proposed based on logarithmic approach for near-well flow modeling with flexible gridblocks.
In this paper, a new multipoint and a new two-point flux approximation scheme are presented. These schemes can eliminate flux truncation errors for the calculation of the point-source "singular" solution, which is dominant in the well vicinity. Consequently, total flux truncation errors decrease in the near-well region. Compared to the commonly used linear approach, the new multipoint scheme gives an improvement with a factor of r in the error estimation, which is important for the near-well flow calculation. Although the two-point scheme might be less accurate for the regular flow and the boundary "singular" flow calculation because of the grid distortion, it remains a good approach in most cases, owing to the elimination of errors for the dominating point-source "singular" flow calculation. Moreover, the two-point scheme can be easily used and implemented in flow simulators.
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