In the present paper 3D analytical simulator for modeling horizontal well (HW) and slanting well (SW) performance is presented. Hydrodynamic characteristics of the development process are treated in terms of Newton's and heat potentials, and the developed simulators provide the ability for modeling both stationary and non-stationary states. The computational experiments allowed to analyze the impact of the geometry of the system «reservoir + HW » on performance indexes and to infer new flow rate formulas for stationary and non-stationary drainage.

The proposed method is based on iterative algorithms and domain decomposition principle reducing the drainage problem in domains with complicated structure to problems in standard (simple) domains, such as ball, cylinder, etc. The developed numerical methods and flow rate formulas have been validated and proofed to be highly precise.


The fluids drainage in the neighborhood of one or several HW is still less fully studied than in the case of vertical wells (VW). This default may be accounted for by the essentially 3D nature of the hydrocarbons influx toward the well in a reservoir limited by its top, bottom, and the no flow barrier.

The main problem consists in determining the dependency of the well's productivity on the reservoir geometry and on the wells interference.

A number of analytical formulas and charts [1–8] linking production rate and differential pressure in relation to geometric and hydrodynamic parameters of the system «reservoir + HW » are available. Analytical formulas are mostly obtained by reduction of the authentic 3D model to 2D flow. Precise solution of essentially 3D drainage problems is, very complicated and can be constructed only for special cases (see e.g., [9–10]).

Another efficient approach consists in development of computational models to simulate drainage processes in 3D reservoirs [11–13]. In the present paper an'explicit method [14,16] is presented to solve the problem of one-phase incompressible elastic fluid inflow toward a single HW or a group of HW in a 3D reservoir limited by top, bottom, and no-flow barrier.

The wellbore is modeled as a superposition of Source Functions (SF) such that in each inner reservoir point these functions satisfy Laplace equation in case of stationary drainage or heat equation in the non-stationary case; the boundary conditions are satisfied on the reservoir top, bottom, and on the no-flow barrier. The solution is sought in terms of discrete density distribution subject to one of the following conditions: the pressure on the wellbore or the flow rate at discrete points of the well is given.

In view of the maximum principle (see [15]) it is easy to check that for the approximate solution the deviation from the precise solution in each inner point of the reservoir doesn't exceed the deviation from the given value on the boundary of the well. Thus, the main issue consists in construction of Green's function possessing the outlined properties.

Development of a highly precise algorithm for construction of this function in the stationary case and of a parabolic analogue of Schwarz's algorithm [16], utilizing standard Green's function and introducing image well in the non-stationary case, is the main achievement of the presented work.

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