The effect of the partial penetration of an infinite conductivity fracture on the transient pressure behavior of a vertically fractured well is investigated.
Analysis of results shows that the pressure behavior of a well intersected by a partially-penetrating infinite conductivity vertical fracture can be divided into three flow periods:
the early time flow period which is characterized by a formation linear flow as in the case of a fully-penetrating infinite-conductivity vertical fracture,
the infinite-acting flow period and
the pseudoradial flow period which develops after the effects of the vertical boundaries of the reservoir are felt in the pressure behavior of the well.
A log-log graph of log(hf/h)pWD versus log t shows a slope of one half during the early time, flow period of a well with an infinite-conductivity period of a well with an infinite-conductivity partially penetrating fracture. The time for the end of the partially penetrating fracture. The time for the end of the early time flow period is directly related to the square of the dimensionless height of the fracture, hfd which is defined as the ratio between the height of the fracture and its half length. This time becomes so small for small values of hfD that the linear formation flow period will not show up for practical purposes.
The solutions during the infinite-acting flow period are, for the different values of penetration period are, for the different values of penetration ratio and a given value of characterized by a single envelope curve in the log-log graph previously mentioned.
The time for the start of the pseudoradial flow period shows no dependence on the penetration ratio and period shows no dependence on the penetration ratio and on the dimensionless fracture height except for fractures with dimensionless heights larger than about 1 and penetration ratios smaller than 1/5.
Even though solutions were mainly obtained for the case of a fracture located at the middle of the formation, the effect of offsetting the fracture was investigated for some particular cases. As expected, the solutions are the same up to the end of the infinite acting flow period which slightly varies as a function of the fracture location.
The solutions are presented in the form of type curves suitable for transient pressure analysis procedures. procedures
Most of the work done on the behavior of fractured wells have considered a totally-penetrating fracture and only a few studies have dealt with the effects of a partially-penetrating fracture. partially-penetrating fracture. Tinsley et al. used an electrolytic model to deter mine the effect of the height of the fracture on the production of a well intersected by a finite- production of a well intersected by a finite- conductivity vertical fracture under steady-state conditions.
Cinco derived an analytical solution for the unsteady-state distribution created by a well intersected by a partially-penetrating uniform-flux inclined fracture in an infinite slab reservoir. An expression for the pseudo-skin factor was derived as well, based on a late time approximation to the solution. No further analysis was however presented.
Raghavan et al. investigated the effect of the partial penetration of a vertical fracture on the partial penetration of a vertical fracture on the transient pressure behavior of a fractured well. A uniform flux across the fracture was assumed, for which case an analytical solution was possible. An approximate solution for the transient pressure behavior well with a partially-penetrating infinite- conductivity fracture was obtained by evaluating the uniform flux solution at a point in the fracture which was assumed to yield the infinite-conductivity solution. This approximation was an extension of the relations between the infinite and uniform-flux solutions previously found by Gringarten et al., in the case of a previously found by Gringarten et al., in the case of a totally-penetrating vertical fracture, and by Muskat in the case of steady-state flow towards a partially-penetrating well. The solution to the partially- partially-penetrating well. The solution to the partially- penetrating infinite-conductivity problem would have required penetrating infinite-conductivity problem would have required however of a semi-analytical type of solution.
Mao studied the productivity of vertically fractured wells using a potentiometric model. Finite conductivity and partial penetration were considered.
It is the purpose of this paper to formulate and to solve properly the unsteady-state flow problem to a well with an infinite-conductivity partially- penetrating fracture and to present the results in a suitable penetrating fracture and to present the results in a suitable form for fracture design and evaluation procedures.