Using the source function approach and the method of images, we have generated the pressure response of a horizontal well in a bounded homogeneous reservoir. Then we have convened the homogeneous solution to the case of a dual-porosity system by means of Laplace transform. Furthermore we introduce skin and wellbore storage in the solution. Use of the technique is illustrated with the case of a horizontal well in a naturally fractured reservoir.
Mathematical modelling of horizontal wells though abundant in the literature, seldom pay much attention to computational efficiency. Only recently a couple of paper appear to deal with the situation. 1–2.They use the method of images for early time and Fourier series for late time. Ohaeri and Vo1, using Laplace Transform, handle wellbore storage effects, skin and phase redistribution. Thompson et al, 2 on the other hand, deal with double porosity reservoirs, also through Laplace transform. They appear to be both faster than Daviau's approach11 or Ozkan's approach6.
Our main interests lie in naturally fractured reservoirs. Therefore we follow closely schemes of Thompson et al2, although Carvalho and Rosa3–4, present yet another way of generating pressure response of horizontal wells in aturally fractured reservoirs. Most of these solutions usesource functions, Fourier Series, Boundary Element Methods4 or even Fourier Transform method5.
Equations (Available in full paper)
Thompson et al. 2 indicated that they were not successful in finding an approximate (polynomial) expression which reproduces the error function with sufficient accuracy. In the Appendix, we give two subroutines to calculate the error function. These subroutines give quite good graphical accuracy.
The above formulation is efficient computationally. Theone thing to watch out for is the definition of the point of evaluation, (XD, YD, ZD) in space. Ozkan and Raghavan6 give a very detailed discussion on this point.
Since we are using a line source, and it is not possible to compute pressure drop on the source, one has to decide on a point away from the well-axis, to account for the radius of the actual well.
Once we obtain PD from the above formulation we use a scheme given in Houze, Home and Ramey7 to generate double porosity solutions. The approach involves the following steps:
Find the Homogeneous Solution
Multiply by s
Change s to s F (s)
Divide by s
Double Porosity Solution where s is the Laplace parameter