A new method is proposed to estimate the compliance and conductivity of induced unpropped fractures as a function of the effective stress acting on the fracture from diagnostic-fracture-injection-test (DFIT) data. A hydraulic-fracture resistance to displacement and closure is described by its compliance (or stiffness). Fracture compliance is closely related to the elastic, failure, and hydraulic properties of the rock. Quantifying fracture compliance and fracture conductivity under in-situ conditions is crucial in many Earth-science and engineering applications but is very difficult to achieve. Even though laboratory experiments are used often to measure fracture compliance and conductivity, the measurement results are influenced strongly by how the fracture is created, the specific rock sample obtained, and the degree to which it is preserved. As such, the results may not be representative of field-scale fractures.

During the past 2 decades, the DFIT has evolved into a commonly used and reliable technique to obtain in-situ stresses, fluid-leakoff parameters, and formation permeability. The pressure-decline response across the entire duration of a DFIT reflects the process of fracture closure and reservoir-flow capacity. As such, it is possible to use these data to quantify changes in fracture conductivity as a function of stress. In this paper, we present a single, coherent mathematical framework to accomplish this. We show how each factor affects the pressure-decline response, and the effects of previously overlooked coupled mechanisms are examined and discussed. Synthetic and field-case studies are presented to illustrate the method. Most importantly, a new specialized plot (normalized system-stiffness plot) is proposed, which not only provides clear evidence of the existence of a residual fracture width as a fracture is closing during a DFIT, but also allows us to estimate fracture-compliance (or stiffness) evolution, and infer unpropped fracture conductivity using only DFIT pressure and time data alone. It is recommended that the normalized system-stiffness plot (NS plot) be used as a standard practice to complement the G-function or square-root-of-time plot and log-log plot because it provides very valuable information on fracture-closure behavior and the properties of fracture-surface roughness at a field-scale, information that cannot be obtained by any other means.