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Pseudo-steady state (or pseudo steady-state), is also referred to as "stabilized," or as "steady state in a bounded drainage area." This type of reservoir flow occurs much more frequently than steady-state flow or unsteady-state flow with an expanding drainage radius.

## Mechanism

Pseudo-steady state (PSS) flow occurs during the late time region when the outer boundaries of the reservoir are all no flow boundaries. (http://www.fekete.com/SAN/TheoryAndEquations/WellTestTheoryEquations/Pseudo-Steady_State_Flow.htm) This includes not only the case when the reservoir boundaries are sealing faults, but also when nearby producing wells cause no flow boundaries to arise. During the PSS flow regime, the reservoir behaves as a tank. The pressure throughout the reservoir decreases at the same, constant rate. PSS flow does not occur during build-up or falloff tests.

## Pseudo-steady state and the approximation technique

The approximation technique addresses the problem of two-dimensional flow of a slightly compressible fluid into a well bisecting a plane vertical fracture. The solution sought is subject to appropriate boundary and initial conditions. Here Ψ is the flow potential, P + ρgH(x̄,ӯ) with H(x̄,ӯ) being vertical height above a hydrostatic datum plane at the point (x̄,ȳ) and we have assumed a homogeneous reservoir of uniform thickness and rock properties which may however be anisotropic. Thus x̄,ȳ are artificial coordinates related to the real rectangular coordinates x,y by

values of the permeability tensor and the coordinate axes are oriented along the principal permeability axes of the formation. This approximation of the flow problem is obtained from Darcy's law, the equation of continuity for fluid of small, constant compressibility and constant viscosity as in Collins. Consistent units are assumed in all equations. Insight into the approximation technique we propose to use for solving the problem of a hydraulically fractured well is provided by examining the well-known "line-sink" solution of this partial differential equation.

where r̄ denotes x̄2 + ȳ2 and

is defined as the exponential integral function. This describes a well of "zero radius" at the origin in an infinite reservoir of thickness h producing at a constant volumetric rate, q, for t > 0. From this solution, we can show that the flow rate across the "circle" of radius r̄w is so for even moderately large values of t, this is an excellent representation for a well of radius r̄w with a constant rate q. Actually, for the anisotropic case, Kx ~ Ky , the "circle" of fixed radius r̄w is in fact an ellipse with ratio of axes proportional to .