Buildup Solution in a System With Multiple Wells Producing at Constant Wellbore Pressures
- Rodolfo Camacho-V. (Pemex E&P) | Agustin Galindo-N. (Pemex E&P) | Michael Prats (Michael Prats and Assocs. Inc.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- August 2002
- Document Type
- Journal Paper
- 278 - 283
- 2002. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 4.1.5 Processing Equipment, 5.8.6 Naturally Fractured Reservoir, 5.6.4 Drillstem/Well Testing, 4.1.2 Separation and Treating
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This paper presents a new analytical solution to obtain the shut-in wellbore pressure at a vertical well that had been producing at a constant wellbore hole pressure from a closed-boundary reservoir with multiple wells producing at constant, different wellbore pressures.
The analytical solution is a set of coupled simultaneous equations in the Laplace space (one for each well). These are obtained by using separation of variables, Green's functions, and the unitstep function with the methods of Rodriguez and Cinco-Ley1 and Camacho-V. et al.2 The equation for the shut-in well is nonlinear, preventing us from obtaining the formal solution of the set of equations either in real time or in Laplace space.
A number of approaches were used to make the solution tractable. These included expansion of Green's function for large times, a Taylor series expansion of the rates around the time of shut-in, and application of the concepts of Najurieta.3 None of the approaches gave satisfactory results.
There are, in the petroleum literature, several analytical solutions for wells producing a liquid at constant pressure from a homogeneous system.4-7 However, most of the work has been directed to the case of a single well. In 1993, Ref. 1 presented an analytical study for multiple-well systems, with each well starting to produce at the same time and continuing to produce at constant but different wellbore pressure from a rectangular closed reservoir. In Ref. 2, some of the assumptions made in Ref. 1 were relaxed, developing more general analytical solutions and using these in an optimization problem.
In this work, the buildup response is evaluated in a bounded reservoir, where all wells are producing at constant bottomhole pressure.
In some respects, this paper is a continuation of the lines of thought initiated in Refs. 1 and 2. However, the presentation here is intended to be self-contained while avoiding unnecessary repetition.
We consider a rectangular homogeneous reservoir with uniform thickness and closed boundaries. The reservoir is produced through nw wells at constant but different wellbore pressures. Fluids are slightly compressible and have constant compressibility and viscosity. Pressure is uniform at initial conditions. Flow in the reservoir is described by the following equation:1
for 0<xD<xeD, 0<yD<yeD, tD>0. The initial condition is pD (xD, yD, 0)=0, and closed outer boundary conditions are considered.
We consider that one well, well i=m, is shut in at time tD=tDm. At this well, the boundary condition2 is
where CDm=the wellbore storage constant and Sm=the mechanical skin factor.
At all other wells, i m, and the boundary condition for tD>0 is
The general solution of Eq. 1 in Laplace space, considering the external boundary conditions,1 is
where Gj(xD,yD;s)=G(xD, yD, xDj, yDj; s) is Green's function, defined in Refs. 1 and 2, and s is the Laplace variable.
The Laplace transform of Eq. 2, the boundary condition at well m, is
and the Laplace transform of the conditions at the other wells, for i m, is given by
At well i=m, substituting Eq. 5 into Eq. 4 yields
Similarly, substituting Eq. 6 into Eq. 4 yields the set of equations for the other wells, i m:
with Gij=G(xDi, yDi, xDj, yDj; s), and aij=-Si/2p if i=j and zero otherwise.
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