Summary
In reservoir simulation, linear approximations generally are used for well modeling. However, these types of approximations can be inaccurate for fluid-flow calculation in the vicinity of wells, leading to incorrect well-performance predictions. To overcome such problems, a new well representation1 has been proposed that uses a "logarithmic" type of approximation for vertical wells. In this paper, we show how the new well model can be implemented easily in existing simulators through the conventional productivity index (PI). We discuss the relationship between wellbore pressure, wellblock pressure, and flow rate in more detail, especially for the definition of wellblock pressure. We present an extension of the new approach to off-center wells and to flexible grids. Through this extension, the equivalence of various gridding techniques for the well model is emphasized. The key element is the accurate calculation of flow components in the vicinity of wells.
Introduction
The well model plays an important role in reservoir simulation because the precision of calculation in well-production rate or bottomhole pressure is directly related to this well model. The main difficulty of well modeling is the problem of singularity because of the difference in scale between the small wellbore diameter (less than 0.3 m) and the large wellblock grid dimensions used in the simulation (from tens to hundreds of meters), and to the radial nature of the flow around the well (i.e., nonlinear but logarithmic variation of the pressure away from the well). Thus, the wellblock pressure calculated by standard finite-difference methods is not the wellbore pressure. Peaceman2,3 first demonstrated that wellblock pressure calculated by finite difference in a uniform grid corresponds to the pressure at an equivalent wellblock radius, r0, related to gridblock dimensions. Assuming a radial flow around the well, he demons-trated that this radius could be used to relate the wellblock pressure to the wellbore pressure. However, there are problems with this approach in many practical reservoir simulation studies:For routinely used nonuniform Cartesian grids,4 there is no easy means to determine an r0 value.In three-dimensional (3D) cases with non-fully-penetrating wells, the basic radial flow assumption does not apply,5 whereas vertical flow effects must be included.6Off-center wells are not correctly treated.7,8Treatment of the well model is much more complicated with non Cartesian or flexible grids.9–11
The aim of this paper is to show that the new well representation1 proposed in a previous paper can handle these problems accurately.
Wellblock Pressure Calculation
A previous paper1 presented a new approach particularly well-suited to nonuniform grids for the modeling of vertical wells in numerical simulation. The principle of this new approach, which is based on a finite-volume method, is to calculate new interblock distances that improve the modeling of flow in the vicinity of wells. Because the new approach was originally presented for two-dimensional (2D)-XY problems, it was shown that for such problems the wellbore pressure could be calculated without both the intermediate computation of the wellblock pressure and introduction of an equivalent wellblock radius. However, for at least two reasons, it is convenient to keep this standard method commonly used in numerical models, which consists of relating the wellbore pressure and wellblock pressure through the use of a numerical PI and equivalent wellblock radius. One reason is practical. To implement the new approach more easily into standard numerical models, it is better to keep their internal structure unchanged. The other reason is dictated by the necessity of having a wellblock pressure in particular 3D simulation studies. When a well partially penetrates the reservoir or when there is communication between different layers, there is a vertical flow component in the vicinity of the well that necessitates that the wellblock pressure be calculated.
How should the new approach be implemented in standard reservoir simulators- In these simulators, a numerical PI is used in the well model to relate the wellbore pressure, pw, to the wellblock pressure, p0. Usually, this PI is written as
where r0 is the equivalent wellblock radius at which the pressure is equal to p0.
Within the new well representation,1 to obtain a pressure p0 corresponding to a radius r0, it is sufficient to use equivalent wellblock transmissibilities relating p0 to the pressures of adjacent blocks through equivalent interblock distances, Leq, i (Fig. 1:
where ?x0, ?y0 are the wellblock dimensions. For instance, in the x+ direction, Leq,1 is written
where ?1+2 arctg (?y0 /?x0) is the angle formed by the right wellblock interface seen from the well.
Because wellblock transmissibilities in standard models are conventionally expressed by
the new approach can be implemented easily in standard models multiplying the conventional wellblock transmissibilities by constant factors. For instance, in the x+ direction, this factor is
By use of equivalent transmissibilities, the calculated wellblock pressure, p0, should correspond to the equivalent wellblock radius, r0, which is involved in transmissibility calculations (Eq. 3). Then, the wellblock pressure can be related to the wellbore pressure with the conventional PI (Eq. 1).
Modelling of multi-lateral well geometries for geothermal applications
