Over the past two decades, Diagnostic Fracture Injection Tests (DFIT) or Injection-Fall-off Fracture Calibration Tests have evolved into a commonly used and reliable technique to evaluate reservoir properties, fracturing parameters and obtain in-situ stresses. Since the introduction of DFIT analysis based on G-function and its derivative, this method has become standard practice for quantifying minimum in-situ stress and leak-off coefficient. However, the pressure decline model that underlies the G-function plot makes two distinct and important assumptions: (1) leak-off is not pressure-dependent and, (2) fracture stiffness (or compliance) is assumed to be constant during fracture closure. Fracture closure is a gradual process that starts when asperities at the fracture tip first come into contact with each other. As pressure declines due to leak-off, more and more of the fracture wall comes into contact. It is important to model this process quantitatively to obtain good estimates of in-situ stresses and leak-off. In this paper, we present a model that accounts for changes in fracture stiffness/compliance as the fracture closes, with leak-off that is dependent on fracture pressure. The model is, therefore, capable of analyzing DFIT data from the end of pumping to days or even weeks after shut-in.

We first review Nolte's original G-function model and examine the assumptions inherent in the model. We then present a new global pressure transient model for pressure decline after shut-in which not only preserves the physics of unsteady-state reservoir flow behavior, elastic fracture mechanics and material balance, but also incorporates the gradual changes of fracture stiffness/compliance due to the contact of rough fracture walls during closure. This global model allows us to analyze the whole spectrum of DFIT data by bridging before closure and after closure data seamlessly. Analysis of synthetic cases, along with field data are presented to demonstrate how the coupled effects of fracture geometry, fracture surface asperities, formation properties, pore pressure and wellbore storage can impact fracturing pressure decline and the estimation of minimum in-situ stress.

It is shown that so-called "normal leak-off" behavior that is modeled using Carter leak-off is an oversimplification that leads to significant errors in the interpretation of the data. All the before closure analysis conducted under a "normal leak-off" assumption should be reexamined cautiously. Most importantly, this article reveals that previous methods of estimating minimum in-situ stress often lead to significant over or underestimates, because of their failure to account for changes in fracture stiffness/compliance correctly as the fracture closes progressively from the edge to the center. Based on our modeling and simulation results, we propose a much more accurate and reliable method to estimate the minimum in-situ stress, fracture pressure dependent leak-off rate and evaluate the compliance of the un-propped fracture.