Abstract
Summary
A mathematical model was developed to study the transient pressure behavior in a well with an infinite-conductivity horizontal drainhole in an infinite slab reservoir. The physical model includes a fluid of small and constant compressibility flowing through an infinitely large anisotropic reservoir with upper and lower impermeable boundaries. The analytical solution is obtained by applying the concepts of instantaneous sources and Green's functions.
We discuss the uniform-flux model, a special case of the infinite-conductivity model, and present a simple way to use it to calculate the pressure at the wellbore face. We suggest that the pressure for the infinite-conductivity case can be evaluated with the uniform flux model at a fixed point along the wellbore with an error of less than 1 percent for combinations of the various parameters that may be encountered in real situations. We show to what extent the accuracy of the model will be affected by neglecting gravitational effects.
Introduction
A convenient model to represent the pressure behavior in a horizontal drainhole is one that assumes no pressure drop in its interior during fluid flow. This means that pressure is uniform along the wellbore face, and the well is said to have infinite conductivity.
In this work, equations that enable us to determine the wellbore pressure using such a model are developed. In practice, it is not feasible to evaluate the wellbore pressure directly from the infinite-conductivity model because of the computational work required. To perform these calculations, an equivalent uniform-flux model is proposed that produces the same pressure response that would be obtained with the more realistic model.
The uniform-flux case assumes that the flow rate, instead of the pressure, is uniformly distributed along the horizontal portion of the well.
The comparison between the short- and long-term approximations for both models enables us to calculate the point along the wellbore face where the uniform-flux and the infinite-conductivity cases produce the same answer for pressure. This value can be used with the uniform-flux model to calculate wellbore pressure.
Recent works on well-test analysis for horizontal wells have not dealt in detail with whether or not the uniform-flux model can be used at a fixed point along the wellbore to obtain the infinite-conductivity response. Goode and Thambynayagam emphasized the case where the well is in the center of the formation; Daviau et al. assumed that the uniform-flux solution with a dimensionless distance of 0.7 was a good approximation to simulate the infinite-conductivity well behavior, but no development for that was presented; and Clonts and Ramey evaluated the uniform-flux solution at a dimensionless distance of 0.732 by analogy with -the infinite-conductivity fracture solution. The main objective of this study is to investigate the effects of the assumptions made in Refs. 1 through 3.
Gravitational forces also can be extremely important for the horizontal boreholes. We explicitly state to what extent the accuracy of our model will be affected by neglecting gravitational effects.
Physical Model
Fig. 1 is a schematic of a horizontal well. The following hypotheses are made.The reservoir is horizontal, homogeneous, anisotropic, and has constant horizontal, and vertical, permeabilities. It has thickness h, constant porosity, constant effective formation compressibility, and infinite lateral extension.The production occurs through a well of radius, represented in the model by a line-source located at a distance, from the lower boundary, the length of which is equal to that of the horizontal portion of the well, 2L.A single-phase fluid, of small and constant compressibility c, constant viscosity, and FVF B, flows from the reservoir to the well at a constant rate at surface and q at bottomhole conditions.The reservoir is bounded by upper and lower impermeable formations, and the pressure remains constant and equal to the initial value at an infinite distance from the well.Small pressure gradients are assumed, and wellbore-storage effects are not included to simplify the solution.
We initially neglected gravity forces, but gravitational effects are considered in Appendix B.
Infinite-Conductivity Mathematical Model
Although the well is represented in the model by a line-source, flowing pressures will be calculated at the wellbore face, which is at a distance from the horizontal-well axis.
Pressure is uniform along the wellbore face, constituting the infinite-conductivity model. This condition is satisfied by dividing half of the well length, L, into segments, as shown in Fig. 2. Each one of these segments produces uniform flux per unit length equal to, where n equals 1,2 ... . The values of are determined by equating the pressure drops at the center of the segments, which give () equations. The last equation is obtained from the condition of constant total rate. Using the Cartesian system of coordinates (x, y, z) we have the following equations:
(1)
and (2)
where the pressure drop is evaluated at the wellbore face at a distance from the lower boundary.
The solution for the pressure drop is determined by applying the concepts of Green's functions and the product solution method (Newman's method) with source functions presented by Gringarten and Ramey. The Green's function for our problem is given by the intersection of three basic Green's functions corresponding toan infinite vertical plane in the × direction located in an infinite reservoir,an infinite vertical plane in the y direction placed in an infinite system, andan infinite horizontal plane located in a horizontal slab reservoir with upper and lower impermeable boundaries.
This product generates a Green's function equivalent to an instantaneous point source in a slab reservoir. Its integration yields the desired instantaneous source, the line-source that represents the horizontal well.
SPEFE
P. 559^
Publisher
Society of Petroleum Engineers (SPE)
Subject
Process Chemistry and Technology