A mathematical model is presented to evaluate pressure response of a horizontal well in bounded homogeneous and naturals fractured reservoirs. The model also used to understand the pressure behavior of a horizontal well in an infinite acting, one, two or three sealing faults and closed rectangular Furthermore we introduce skin and wellbore storage in the numerical solution. Use of the technique is illustrated with the case of a horizontal well in a naturally fractured reservoir.
As a horizontal well technology is becoming more suitable in developing naturally fractured reservoirs, tight reservoirs, heavy oil reservoirs and fine reservoirs. The pressure transient of horizontal well has become a very interesting topic. Many papers on well testing have been presented since Ramey's1 and Gringarten's2 review in 1982 and 1984, The earlier articles are about reservoir performance and productivity of horizontal well.4,5 In recent years, the paper deals with pressure distribution of horizontal well for homogenous and naturally fractured reservoirs6–8. Most recently, a number of papers have cared for the interpretation of horizontal well test data and horizontal well-layered reservoir and multi-lateral well.3,13,14
Unlike vertical wells, reservoirs have to be seen as three-dimensional formations. Hence, the pressure behavior of a horizontal well is more sophisticated than that of a vertical well. Transient behavior is an important factor in understanding the horizontal well performance.
Although Mathematical modeling of horizontal wells is abundant in the literature, seldom papers pay attention to boundaries including rectangular, channel, parallel, one fault and computational method. Only recently several papers appear to discuss boundaries case3,10,11,13, R. Aguilera and M.C. Ng3 using the method of images, handle wellbore storage and skin in pressure drawdowns and buildup of vertical well in bounded rectangular naturally fractured reservoirs. Tompson et al13, on the other hand, deals with double porosity reservoirs for horizontal well, also through Laplace transform. Ozkan's approach10–12 is very good method to handle complex boundaries, This paper uses this method to generate pressure response of horizontal Well under different boundaries, including infinite, single faulted, parallel, channel, and closed rectangular homogeneous and naturally fractured reservoirs.
Let us consider a horizontal well completed in an anisotropy medium, which is infinite in the × and y directions. kx, ky and kz denote the perrneabilities of the formation in the principle directions. Although solutions have been presented for the three-dimensional anisotropy medium, we will deal with the solution for isotropic media (kh = kx = ky and kv = kz).
We consider the flow of a single-phase, slightly-compressible fluid of constant viscosity and a horizontal well has length L in a reservoir of height h. The upper and bottom boundaries (z=O and zc=h) are assumed to be impermeable. The well is assumed to be parallel to the top and bottom boundary, and gravity effects are considered to be negligible. The origin of the coordinate system, as shown in Figure 1, is the center of the well. Initially, the pressure is uniform throughout the reservoir.
For convenience, we use results in dimensionless form. Dimensionless pressure and dimensionless time are defined as:
