Two-dimensional numerical investigations on the termination of bilinear flow in fractures

Abstract

Abstract. Bilinear flow occurs when fluid is drained from a permeable matrix by producing it through an enclosed fracture of finite conductivity intersecting a well along its axis. The terminology reflects the combination of two approximately linear flow regimes: one in the matrix with flow essentially perpendicular to the fracture, and one along the fracture itself associated with the non-negligible pressure drop in it. We investigated the characteristics, in particular the termination, of bilinear flow by numerical modeling allowing for an examination of the entire flow field without prescribing the flow geometry in the matrix. Fracture storage capacity was neglected relying on previous findings that bilinear flow is associated with a quasi-steady flow in the fracture. Numerical results were generalized by dimensionless presentation. Definition of a dimensionless time that, other than in previous approaches, does not use geometrical parameters of the fracture permitted identifying the dimensionless well pressure for the infinitely long fracture as the master curve for type curves of all fractures with finite length from the beginning of bilinear flow up to fully developed radial flow. In log-log scale the master curve's logarithmic derivative initially follows a 1/4-slope straight line (characteristic for bilinear flow) and gradually bends into a horizontal line (characteristic for radial flow) for long times. During the bilinear flow period, isobars normalized to well pressure propagate with the fourth and second root of time in fracture and matrix, respectively. The width-to-length ratio of the pressure field increases proportional to the fourth root of time during the bilinear period, and starts to deviate from this relation close to the deviation of well pressure and its derivative from their fourth-root-oftime relations. At this time, isobars are already significantly inclined with respect to the fracture. The type curves of finite fractures all deviate counterclockwise from the master curve instead of clockwise or counterclockwise from the 1/4-slope straight line as previously proposed. The counterclockwise deviation from the master curve was identified as the arrival of a normalized isobar reflected at the fracture tip 16 times earlier. Nevertheless, two distinct regimes were found in regard to pressure at the fracture tip when bilinear flow ends. For dimensionless fracture conductivities T D < 1, a significant pressure increase is not observed at the fracture tip until bilinear flow is succeeded by radial flow at a fixed dimensionless time. For T D > 10, the pressure at the fracture tip has reached substantial fractions of the associated change in well pressure when the flow field transforms towards intermittent formation linear flow at times that scale inversely with the fourth power of dimensionless fracture conductivity. Our results suggest that semi-log plots of normalized well pressure provide a means for the determination of hydraulic parameters of fracture and matrix after shorter test duration ...

Bilinear pressure diffusion and termination of bilinear flow in a vertically fractured well injecting at constant pressure

“…We define the master curve as the one that describes the behavior of an infinitely long fracture (red line in Taking into account all the aspects previously described, when type-curves start departing from the bilinear-fit-curve (Fig. 2), this indicates that the transition from bilinear flow regime to formation linear flow regime (cases with high ) or to pseudo-radial flow regime (cases with low ) begins (Ortiz R. et al, 2013).…”

“…Instead of using the conventional definition of dimensionless time = / 2 , we prefer to use a modified definition presented by Ortiz R. et al (2013):…”

“…In order to characterize the different isobars, the following definition is used (Ortiz R. et al, 2013):…”

“…it has only been investigated when injecting at constant flow rate in the well (Cinco-Ley and Samaniego-V., 1981;Weir, 1999). In this regard, new criteria to determine the end of bilinear flow, which are also used in this investigation, were introduced by Ortiz R. et al (2013).…”

On attenuation of seismic waves associated with flow in fractures

“…Storage capacity S F and transmissibility are accounted for as single macroscopic properties though pressure is calculated locally. For deformable fractures, the bulk storage capacity has two contributions [ Ortiz et al , , ], one deriving from the fluid compressibility and one from changes in the mean fracture aperture caused by pressure variations. Here we neglect details of the fracture deformation and express its contribution to the storage capacity by parameterization of the bulk storage capacity in multiples of the fluid compressibility, i.e., where denotes the bulk storage capacity of a rigid fracture and δ v 0 ( t 0 ), the maximum fracture aperture at time t 0 =0 s, approximates the mean aperture of the vertical fracture with deviations within approximately 20%.…”