PRESSURE DROPS CREATED BY THE COMBINED PRESENCE OF VERTICAL AND HORIZONTAL WELLS IN A RESERVOIR

 

INTRODUCTION

 

This project aims to calculate the pressure drops created in bounded reservoirs due to the combined presence of vertical and horizontal wells. The boundaries will be modeled using the “method of images” while the combined pressure drop will be calculated using the superposition principle.

 

Knowledge of the pressure drop created by the wells (vertical, horizontal, fractured, etc) at any desired location in a reservoir is important for the following reasons:

(a)    It permits the calculation of the sizes and shapes of drainage boundaries during primary production.

(b)   It permits the calculation of average pressures within the drainage boundaries for the wells in a reservoir. These average pressures are needed in material balance calculations, as well as in productivity and inflow performance calculations.

(c)    It permits the delineation of fluid migration paths (streamlines) during secondary and enhanced recovery operations.  These streamlines help to determine sweep efficiencies and travel times of hydrocarbons, injected fluids, and pollutants to producing wells.

 

Horizontal wells can be treated as vertical wells lying on their sides. However, the three dimensional drainage shape of a vertical well (right circular cylinder) is different from that of a horizontal well which is an ellipsoid. Thus, the pressure at any location due to a vertical well in an infinite medium depends only on the distance of the point from the center of the well. That is, all equidistant points will have the same pressure. For horizontal wells however, the pressure at any point in an infinite medium depends not only on the distance between the point and the two ends of the well denoted as “a” and “b”. Thus, all points for which ½(a+b) is constant lie on an ellipsoid of revolution around the horizontal well. The pressure is constant for all such points.

For bounded reservoirs containing both vertical and horizontal wells, the pressure drop at any location is dependent on the distance of the location from the center of the vertical well, the distance from the ends of the horizontal wells, the orientation of the point from the horizontal well axes, as well as the distance of the point from the external boundaries. The external boundaries will be modeled using the “method of images” and the combined pressure drop calculated using the superposition principle.

 

Consider a three dimensional reservoir containing a horizontal well as shown in the top diagram below.  It is desired to calculate the pressure at any arbitrary observation point in the 3-D space. The reservoir is bounded at the top and bottom by impermeable boundaries. These boundaries are handled by the introduction of image wells.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


An example of a single image well is shown below.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


The following assumptions are made:

1.      The reservoir is homogeneous, isotropic and infinite in the lateral directions but bounded at the top and bottom.

2.      A slightly compressible single phase fluid is flowing under Steady state conditions.

3.      The horizontal well of length L has uniform flux of strength (Q/L).

4.      Existing vertical wells are fully penetrating wells.

5.      Both the vertical and horizontal wells can be treated as line sources or sinks

 

The figure below depicts a horizontal well of length AB and an observation point located at C such that AC and BC are the distances of the observation point from the two ends of the well.

 

Observation point

 

C
 

B
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


The pressure at any observation point in a 3-D reservoir due to a horizontal well flowing at a rate of Q/L uniformly along its length is given (1) as:

 

    …………………………………………………..(1)

 

Where

  and  are the angles (in radians) that the perpendicular from the observation point to the horizontal well (line D) makes with the lines AC and BC.

For a fully penetrating vertical well, the length of the well is the reservoir thickness (h) and the drainage volume is a right circular cylinder. Therefore the pressure at any fixed radius is the same at any point in the z-direction. For this reason, the 3-D problem can be reduced to a 2-D problem whose solution is well known as:

    …………………………………………………(2)

 

Where r is the distance from the observation point to the center of the well.

 

Example calculation:

 

 

 

 

Modeling of the upper and lower boundaries of the reservoir

 

Use image wells to model the upper and lower boundaries (assume both are no-flow boundaries). The major modifications will involve the number and placements of the image vertical and horizontal wells. This will affect only their distances from the observation point. Thus, the only new modifications will be:

(a)    For the vertical wells, the 2-D distances of the wells from the observation point.

(b)   For the horizontal wells, the 3-D distances from the observation point to the ends of the horizontal wells.

 

Start with the case of one vertical well and one horizontal well. Then consider the case of  a 5-spot pattern with one horizontal well in the center surrounded by 4 vertical wells at the 4 corners of a square pattern.