Modeling semi-steady state near-well flow performance for horizontal wells in anisotropic reservoirs

Abstract

This paper presents a novel methodology to model semi-steady state horizontal well flow performance in an anisotropic reservoir taking into account flow in the near-well region for an arbitrary well trajectory. It is based on an analytical productivity model describing coupled axial reservoir flow and radial well inflow. In order to apply this model in an anisotropic reservoir, the permeability field relative to the radial direction perpendicular to the well trajectory and the axial direction along the well trajectory must first be determined. A classical space transformation is used in concert with rotational transforms to obtain a virtual isotropic model. The transformation preserves the volumes and pressures. It is not a novel concept, but different from previous approaches in the sense that it is only applied in the near-well domain to formulate an equally isotropic media. As a result, the use of this virtual isotropic model requires the Dietz shape factor for an ellipse, transformed from the original cylindrical near-well domain. The Dietz shape factors are determined numerically in this research. The semi-steady state well/near-well model is implemented in a numerical simulator incorporating formation anisotropy and wellbore hydraulics. The specific productivity index along the well trajectory is generated using the virtual configuration. Numerical results for different anisotropy ratios and also incorporating frictional losses in the well are presented. Furthermore, the well/near-well model is applied in coupling with streamline reservoir model for a water flooding case. This appears to be the first coupling of a well hydraulics model and a streamline simulator. It presents the application of the well/near-well model in integrated reservoir simulation in an efficient and accurate manner. The results demonstrate that the coupling approach with a streamline reservoir model and the well/near-well is of great potential for advanced well simulation efficiently.

Appendix

The Dietz shape factor for ellipses with different major to minor axes ratios and with the well in the center is determined numerically in this Section. Previous research, [11, 26, 32], calculated the Dietz shape factor for triangular and rectangular configurations.

Consider a well producing a single-phase, slightly compressible fluid from the center of an elliptical reservoir, which is homogeneous and isotropic. The well is producing at a constant flow rate and the outer boundary of the reservoir is a no flow boundary. Using the above assumptions and boundary conditions, the conservation equation is

$$ \phi {c_{t}}\frac{{\partial p}}{{\partial t}} = \frac{K}{\mu }\frac{{{\partial^{2}}p}}{{\partial {x^{2}}}} + \frac{K}{\mu }\frac{{{\partial^{2}}p}}{{\partial {y^{2}}}}. $$

(24)

An implicit discretization [2] of Eq. (21) is

$$\begin{array}{@{}rcl@{}} \left( {1 + 2{T_{x}} + 2{T_{y}}} \right)p_{i,j}^{n + 1} &-& {T_{x}}p_{i + 1,j}^{n + 1} - {T_{x}}p_{i - 1,j}^{n + 1} - {T_{y}}p_{i,j + 1}^{n + 1}\\ &-& {T_{y}}p_{i,j - 1}^{n + 1} = p_{i,j}^{n} \end{array} $$

(25)

using a fixed point source well rate, where $T_{x}= \frac {K}{\phi \mu c_{t}} \frac {\Delta t}{\Delta x^{2}}$ and $T_{y}=\frac {K}{\phi \mu c_{t}} \frac {\Delta t}{\Delta y^{2}}$. The reservoir is defined by an ellipse with a unique major to minor axes ratio b/a. The equivalent wellbore radius (Peaceman, 1983) in the square grid blocks is given by

$$ \widetilde {{r_{w}}} = G{\left[ {{{\left( {\Delta x} \right)}^{2}} + {{\left( {\Delta y} \right)}^{2}}} \right]^{1/2}}, $$

(26)

where, $G=\frac {1}{4} e^{-\gamma } \approx 0.1404$. Matching the productivity equation, the Dietz shape factor can be determined by

$$ ln{C_{A}} = ln\frac{{4A}}{{\gamma {{\widetilde {{r_{w}}}}^{2}}}} - \frac{{4\pi Kh}}{{q\mu }}\left( {\bar p - {p_{w}}} \right). $$

(27)

A calculation example with b/a = 2 is given below using parameters from Table 3. The odd number of grid blocks is convenient in order to locate the well in the center. The pressure distribution after 30 hours is shown in Fig. 25. The red grid blocks are inactive, with no communication to the elliptic reservoir. The average reservoir pressure and the wellbore pressure vs. time are plotted in Fig. 26. The semi-steady state flow period can be determined by observing linearly decreasing pressure everywhere. It is observed that the flow is stabilized after approximately three hours, which means the flow after that is semi-steady state. After that point, the average reservoir pressure curve is parallel to the wellbore pressure, i.e., the difference between them is a constant. Substituting this constant pressure difference into Eq. 24, the Dietz shape factor of this major to minor axes ratio (b/a = 1/2) can be determined. Repeating the same simulation process for different major to minor axes ratios will finally results in the Dietz shape factor function for an elliptical reservoir. This function is plotted in Fig. 28.

Table 3 Basic parameters in ellipse reservoir simulator

Fig. 25

Pressure distribution in an elliptical reservoir

Fig. 26

Wellbore and average reservoir pressure profiles

Fig. 27

Pressure distribution in a rotated elliptic reservoir

Fig. 28

Dietz shape factor for ellipses with different major to minor radius ratio

The grid orientation effect is also tested in this research to confirm that it does not affect the results we achieved using a finite difference method. Reservoir simulators using finite difference methods are affected by grid orientation effects [6], i.e., fluid tends to flow in the direction of the grid axes rather than diagonal to them. To verify the previous results, the elliptic reservoir is rotated by π/4 from the original one. The shape factor calculation process is repeated again for the rotated elliptical reservoir. An example of pressure distribution for the rotated elliptical reservoir is shown in Fig. 27. The Dietz shape factor results, using both the original elliptic reservoir and the rotated one are plotted in Fig. 28, which indicates the grid orientation effect is insignificant.