A General Theory of Minifracturing
Authors: Poulsen D.K. 
Published: December 2007

Abstract
This paper presents a general equation applicable to all types of minifracturing tests: pump-in/shut-in, pump-in/flowback, microfracturing, and equilibrium tests. Using this equation as a basis, the paper demonstrates how to develop analysis equations for all of these tests by inserting the proper conditions and assumptions. New analysis equations and methods are developed for pump-in/shut-in and pumping/flowback tests.

Background and Introduction
Minifracturing takes several forms: the pump-in/shut-in test, the pump-in/flowback test, the microfracturing test, and the recently introduced equilibrium test.
The pump-in/shut-in test was introduced as a means for determining fluid-loss parameters in situ. Subsequent investigations have provided analysis techniques that attempt to account for phenomena that the original analysis ignored. These include considerations such as pressure-dependent fluid loss, fracturing-fluid compressibility, and formation inhomogeneity. As of yet, no analysis technique has been presented that considers the factors simultaneously. In addition, methods developed to consider fluid compressibility ignore the compressibility of the fluid in tubular goods in communication with the fracture. Also, although methods for considering compressibility and those for considering pressure-dependent fluid loss provide improved means for matching pressure data, they do not adequately address the determination of closure time, fluid efficiency, or fluid-loss coefficient from the match parameters.
Pump-in/flowback tests were designed to determine minimum principal in-situ stress of a formation, also known as fracture closure pressure. By accelerating the removal of fluid from the fracture through flowback, these tests allow fracture closure to be observed after a much briefer period than if the well had been simply shut in after the fracturing treatment. Initially, methods for analyzing these tests were developed intuitively. Subsequent theoretical and model studies have provided more firmly based techniques for analyzing these tests.
The microfrac test, which follows, on a much smaller scale, the procedure of a pump-in/shut-in test, was developed as another way of determining minimum principal stress.
The most recently developed minifrac test is the equilibrium minifracturing test, designed to determine most of the same parameters as a pump-in/shut-in test, but be independent of many of the factors that can complicate the analysis of a pump-in/shut-in test.
This paper demonstrates that it is possible to write a general equation from which analyses for all of these tests can be readily developed. In addition to most currently available analyses, new analyses that consider additional factors or new combinations of factors can be developed, as will be demonstrated by example in this work. Several new analysis equations and techniques are presented here, particularly for pump-in/shut-in and pump-in/flowback tests. Although the analyses presented here rely on particular, generally accepted relationships describing fracture growth and fluid loss, the derivations are presented in such a way that, should other relationships be preferred, they may be substituted in and relevant analyses developed.

Theory
In general for a hydraulic fracture, the rate of change in system (fracture plus connected tubular goods) volume equals the sum of the rates of change in volume due to (1) expansion (or compression) of the fluids present, (2) fluid loss, (3) injection (or flowback), and (4) reaction; P. 177^