This paper provides a framework for adding after-closure fracturing-pressure analysis to the pre-treatment calibration-testing sequence that defines fracture geometry and fluid loss characteristics. The after-closure period contains the reservoir pseudo-linear flow period that is the focus of this paper and the pseudo-radial flow period that has been previously addressed in a comprehensive manner. Considerations beyond linear-flow include the transition from linear-flow to radial-flow that permits extracting the fracture length; the synergy and validation provided by the various phases of a fracture calibration sequence; and application examples for a large range of reservoir parameters and conditions. The examples include a summary of the operational considerations and derived benefits obtained by extensive use of the analysis offshore Trinidad during a frac-and-pack campaign, and the paper concludes with a detailed analysis of the pumping, closing, and after-closure periods for a calibration ("minifrac") treatment during this campaign. A companion paper provides a detailed analytical-framework for the after-closure period. Reservoir linear-flow provides the remaining and missing link for the fracturing-pressure chain-of-events. This chain gives a continuum of increasing information about the fracture geometry, fracturing fluid, and reservoir with feedback to validate or question prior information. The proposed timeline of events (and information) begins with a small-volume injection (for closure pressure) and shut-in (for reservoir transmissibility and initial pressure); pumping the fracture calibration treatment (for fracture geometry characteristic); the shut-in closure-decline (for total fluid-loss coefficient and fracture length to validate geometry); immediately after closure (for separating the various fluid loss mechanisms and validating closure pressure); after-closure linear-flow (for spurt-loss and to validate fracture length); and in the case of high-permeability, transitional flow (for validating various parameter-combinations) and radial-flow (for validating reservoir transmissibility and initial pressure). The ensemble of calibrated and validated information provides all the prerequisite fracture and reservoir information for achieving an on-site economics-optimized design of the proppant treatment.


Figure 1 shows a typical history of the fracturing pressure from the beginning of pumping until the reservoir disturbance from the fracture decays back to the initial reservoir pressure. Of particular importance for this paper is the last period of the pressure response, or the after-closure response noted on the figure as "transient reservoir pressure near the wellbore." A calibration test is generally performed without proppant and, therefore, retains negligible conductivity when it closes. The after-closure pressure behavior is independent of the physical properties governing fracture propagation and depends only on the previous spatial and temporal history of the fluid loss, the fracture length, and the reservoir parameters. The "late-time" behavior becomes pseudo-radial flow and provides reservoir transmissibility (kh/) and initial pressure in a manner similar to more traditional methods for a well test. The after-fracture-closure application of radial-flow has been comprehensively covered in two companion papers. The first paper by Gu et al. focused on application aspects, and the second paper by Abousleiman et al. focused on theoretical aspects. The latter paper also considered approximations for the "early-time" pseudo-linear flow regime.

This paper and a companion provide a similar division of focus for after-closure linear-flow. The primary role for linear-flow is to define spurt, loss and validate information available from other parts of a calibration sequence. The following sections provide illustrative examples, a cursory review of the related literature, the role of numerical simulation, an outline for incorporating after-closure and its synergy with other phases of calibration testing, and conclude with a detailed example of a combined analysis for the pumping, closure, and after-closure periods of a calibration treatment.

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