The proposed work provides a new definition of the pressure-derivative function [i.e., the ß-derivative function, ?pßd(t)], which is defined as:
(?pd(t) is the "Bourdet" well testing derivative)
This formulation is based on the "power-law" concept (i.e., the derivative of the logarithm of pressure drop with respect to the logarithm of time) --- this is not a trivial definition, but rather a definition that provides a unique characterization of "power-law" flow regimes.
The "power-law" flow regimes uniquely defined by the ?pßd(t) function are: [i.e., a constant ?pßd(t) behavior]
Case ?pßd(t) Wellbore storage domination:1 Reservoir boundaries:--- Closed reservoir (circle, rectangle, etc.).--- 2-Parallel faults (large time).--- 3-Perpendicular faults (large time). 11/21/2 lFractured wells:--- Infinite conductivity vertical fracture.--- Finite conductivity vertical fracture. 1/21/4 Horizontal wells:--- Formation linear flow. 1/2
In addition, the ?pßd(t) function provides unique characteristic responses for cases of dual porosity (naturally-fractured) reservoirs.
The ?pßd(t) function represents a new application of the traditional pressure derivative function, the "power-law" differentiation method (i.e., computing the dln(?p)/dln(t) derivative) provides an accurate and consistent mechanism for computing the primary pressure derivative (i.e., the Cartesian derivative, d?p/dt) as well as the "Bourdet" well testing derivative [i.e., the "semilog" derivative, ?pd(t)=d?p/dln(t)]. The Cartesian and semilog derivatives can be extracted directly from the power-law derivative (and vice-versa) using the definition given above.
The following objectives are proposed for this work:To develop the analytical solutions in dimensionless form as well as graphical presentations (type curves) of the ß-derivative functions for the following cases:–Wellbore storage domination.–Reservoir boundaries (homogeneous reservoirs).–Unfractured wells (homogeneous and dual porosity reservoirs).–Fractured wells (homogeneous and dual porosity reservoirs).–Horizontal wells (homogeneous reservoirs).
To demonstrate the new ß-derivative functions using type curves applied to field data cases using pressure drawdown/buildup and injection/falloff test data.
The well testing pressure derivative function,1?pd(t), is known to be a powerful mechanism for interpreting well test behavior --- it is, in fact, perhaps the most significant single development in the history of well test analysis. The ?pd(t) function as de-fined by Bourdet et al.[i.e., ?pd(t)=d?p/dln(t)] provides a constant value for the case of a well producing at a constant rate in an infinite-acting homogeneous reservoir. That is, ?pd(t) = constant during infinite-acting radial flow behavior.
This single observation has made the Bourdet derivative, ?pd(t), the most used diagnostic in pressure transient analysis --- but what about cases where the reservoir model is not infinite-acting radial flow? Of what value then is the ?pd(t) function?
The answer is somewhat complicated in light of the fact that the Bourdet derivative function has almost certainly been generated for every reservoir model in existence. Reservoir engineers have come to use the characteristic shapes in the Bourdet derivative for the diagnosis and analysis of wellbore storage, boundary effects, fractured wells, horizontal wells, and heterogeneous reservoirs. For this work we prepare the ß-derivative for all of those cases --- but for heterogeneous reservoirs, we only consider the case of a dual porosity reservoir with pseudosteady-state interporosity flow.