Well Test Analysis of Horizontal Wells In Bounded Naturally Fractured Reservoirs
Authors: Ng M.G. , Aguilera R. 
Published: March 2010

Abstract
Using the source function approach and the method of images, we generated the pressure response of a horizontal well in a bounded homogeneous reservoir. Then we converted 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.

Introduction
Mathematical modelling of horizontal wells, though abundant in the literature, seldom pays much attention to computational efficiency. Only recently do a couple of papers 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 Vo(1), 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 approach(11) or Ozkan's approach(6).
Our main interests lie in naturally fractured reservoirs. Therefore we follow closely the schemes of Thompson et al.(2), although Carvalho and Rosa(3–4) present yet another way of generating pressure response of horizontal wells in naturally fractured reservoirs. Most of these solutions use source functions, Fourier Series, Boundary Element Methods(4) or even Fourier Transform Methods(5).

Theory
We define the following Dimensionless Pressure Drop:
Equation 1–13 (available in full papper)
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 Appendices, we give two subroutines to calculate the error function. These subroutines give good graphical accuracy. The above formulation is efficient computationally. The one thing to watch out for is the definition of the point of evaluation, (XD, YD, ZD) in space. Ozkan and Raghavan(6) 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.

Dual Porosity System
Once we obtain PD from the above formulation we use a scheme given in Houze, Home, and Ramey(7) to generate double porosity solutions. The approach involves the following steps:Find the Homogeneous SolutionLaplace TransformMultiply by sChange s to s F (s)Divide by sStehfest InversionDouble Porosity Solution
where s is the Laplace parameter
Equation 14–21 (available in full papper)

Model Verification
Figure 1 was generated using data from Table 1 of Reference (5). Stars are data of Reference (5) and the solid line is simulated with the equations presented in this paper. They agree very well [(See Figure 9 of Reference (5)].
Figure 2 was generated using data from Table 2 of Reference (5). In this case the reservoir is anisotropic. The comparison between the data presented in Reference (5) and the plot (Figure 2) developed using the equations presented in this paper is good.