One-dimensional Linear Flow With Constant Terminal Pressures
Authors: Prats M. , Camacho-Velazquez R. , Rodriguez F. 
Published: March 2010

Abstract
A one-dimensional linear flow system of slight compressibility and of uniform properties is considered. At each boundary the pressure is constant, below the initial value, and generally different. At any time, the flow rate distribution within the system is found analytically in terms of theta functions, and the position of the drainage front is given in terms of an incomplete elliptic function. Except for equal terminal pressures, no proportionality exists between production rates and drainage areas, probably because the time to obtain boundary dominated flow is larger than the time at which one of the rates becomes zero. Pressure buildups following shutting in one of the two ends were obtained using the rate distribution during production and the unit step-function. Results provide insight into the significance of parameters obtained from well tests in the presence of open nearby wells.

Introduction
For wells producing at constant rates from a finite and homogeneous reservoir, the volume drained by each well is proportional to its production rate once pseudo-steady-state has been reached. This paper explores the relationship between drainage volumes and rates (or other parameters) for wells producing at constant bottom-hole pressure, for which no pseudo-steady-state can exist. Under the customary idealized conditions of homogeneity and uniformity of properties and single fluid flow, the drainage volumes have two unknowns: shape and size. The one-dimensional linear system chosen for analyses reduces the unknowns to one: the size of the drainage volume. In plan view, the shapes of the drainage volumes are rectangles.
Rodriguez and Cinco-Ley(1) and Camacho et al.(2) have recently considered multi-well, multi-dimensional systems producing at constant bottom- hole pressure. Their results give rise to the investigation considered here. We are unaware of any prior investigation on this subject.

Model
The system considered is linear in x, of length L, and having a unit cross-sectional area normal to the x-direction. There are two plane sinks (wells), one at × = 0, the other at × = L, in a formation containing a single fluid having a small and constant compressibility c, and a mobility λ. The system has a uniform and homogeneous hydraulic diffusivity h, and is initially at a uniform pressure

Differential Equation
For the dimensionless variables defined in the Nomenclature, the differential equation and the initial and boundary conditions are
Equation 1–4 (available in full paper)
Sometimes it is advantageous to use a dimensionless time t defined by
Equation 5 (available in full paper)

Solution
The solution to the differential Equation (1) subject to the initial and boundary conditions given by Equations (2) – (4) is given by Carslaw and Jaeger(3) to be
Equation 6 (available in full paper)
The pressure may then be manipulated to obtain additional results of interest. The flow rate anywhere in the reservoir, obtained from the spatial derivative of Equation (6) and the definitions of the theta functions, is
Equation 7 (available in full paper)
Other results are concisely defined in terms of a dimensionless parameter m, related to the dimensionless time by
Equation 8 (available in full paper)