This paper treats the propagation of hydraulic fractures of limited vertical extent and elliptic cross-section with the effect of fluid loss included. Numerical and asymptotic approximate solutions in dimensionless form give the fracture length and width at any value of time or any set of physical parameters. The insight provided by The dimensionless parameters. The insight provided by The dimensionless results and approximate solutions should be useful in the design of fracture treatments.
The theory and practice of hydraulic fracturing has been reviewed by Howard and Fast. Therefore, we confine our discussion of previous investigations to those pertinent to the present study of the propagation of vertical fractures. propagation of vertical fractures. An important theoretical result is Carter's formula for the area of a fracture of constant width formed by injection at constant rate with fluid lost to the formation. For a vertical fracture of constant height, Carter's formula gives fracture length as a function of time. In general, Carter's assumption of constant width is not realistic. However, at large values of time the effect of this assumption becomes insignificant since the effect of fluid loss dominates. The width of a vertical fracture was first investigated by Khristianovic and Zheltov under the assumption that the width does not vary in the vertical direction. Thus, a state of plane strain prevails in horizontal planes and the width can be prevails in horizontal planes and the width can be determined as the solution of a plane elasticity problem. An approximate solution is found in Ref. 3 problem. An approximate solution is found in Ref. 3 upon neglect of fluid loss, fracture volume change, and pressure variation along the fracture. The fracture length is determined by the condition of finite stress at the fracture tip. Baron et al. and Geertsma and de Klerk have included the effect of fluid loss in the approach of Ref. 3. Geertsma and de Klerk give simple approximate formulas for fracture length and width. A different approach to the determination of fracture width was taken by Perkins and Kern. They considered a vertically limited fracture under the assumption of plane strain in vertical planes perpendicular to the fracture plane. The perpendicular to the fracture plane. The cross-section of the fracture is found to be elliptical, and the maximum width decreases along the fracture according to a simple formula that contains the fracture length. In the derivation of this formula, fluid loss and fracture volume change are neglected in the continuity equation and the fracture length is not determined. In a subsequent application, a "reasonable" fracture length was assumed. Carter's formula for length and the width formula of Perkins and Kern are both cited by Howard and Fast, and combined use of the two formulas is believed to be common practice. The present theoretical investigation is concerned with vertically limited fractures of the type studied by Perkins and Kern. However, we include the effects of fluid loss and fracture volume change in the continuity equation. Consequently, fracture length is determined as part of the solution. General results for the variation of fracture width and length with time are obtained in dimensionless form by a numerical method. In addition, asymptotic solutions are derived for large and small values of time. The small-time solution is also the exact solution for the case of no fluid loss to the formation. For large values of time our asymptotic formula for fracture length is identical with Carter's formula at large time. Our large-time formula for fracture width differs from the formula of Perkins and Kern by a numerical factor that varies along the fracture. In comparison with our formula, this formulas overestimates the width by 12 percent at the well and 24 percent at the midlength of the fracture. At early times Perkins and Kern's formulas for width in terms of length is again a fair approximation to our result. However, our formula for length differs from Carter's formula, which is not applicable since the neglected width variation is important at early times. The results for the width of a vertically limited fracture as obtained here and in Ref. 6 differ from the results for vertically constant fractures.