Revisiting Pressure Transient Testing of Hydraulically Fractured Wells : A Single, Simple and Exact Analytical Solution Covering Bilinear, Linear, and Transition in Between Flow Regimes.
- Olivier Lietard | John Spivey (Phoenix Reservoir Software)
- Document ID
- Society of Petroleum Engineers
- SPE Hydraulic Fracturing Technology Conference, 24-26 January, The Woodlands, Texas, USA
- Publication Date
- Document Type
- Conference Paper
- 2011. Society of Petroleum Engineers
- 4.3.4 Scale, 5.8.6 Naturally Fractured Reservoir, 4.1.2 Separation and Treating, 5.6.3 Pressure Transient Testing, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 5.6.4 Drillstem/Well Testing
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Quite a lot has been published about pressure transient testing of hydraulically fractured wells since the pioneering work of Heber Cinco-Ley in 1976. A common approach of the problem since 1981 has been to solve the ordinary differential equations (ODEs) of rate and pressure versus time in the Laplace domain and to invert the Laplace solution back into the time domain. This method has led to exact asymptotic solutions for the so-called early time bilinear and formation linear flow regimes. Unfortunately, the inversion of the general Laplace solution was shown to be too complex. Therefore, the pressure and pressure derivative type curves of the transition period between the bilinear and linear regimes have always been numerically generated, most often by making use of Stehfest's inversion algorithm, sometimes by discretization methods in the time domain.
This paper shows that a single, simple and exact analytical solution of the ODEs can be obtained directly in the time domain, providing that the flow within the fracture is assumed to be incompressible (no fracture storage effects). This general solution has two asymptotes for early and late times that agree perfectly with the previously published solutions for bilinear and formation linear flow regimes.
The contribution of the present paper is that it provides exact pressure and pressure derivative type curves throughout the long (six log cycles) transition period between the bilinear and linear asymptotes. It therefore allows for rigorous definitions of transition times, which had been so far numerically approximated. In addition, the proposed solution handles without any difficulty the presence of skin at fracture walls (fracture face skin).
At the onset of production of a vertical well with a fully-penetrating, collinear hydraulic fracture, two flow regions are considered: the fracture itself, and the matrix rock in the vicinity of the fracture faces. Flow in each region is assumed to be linear, respectively along and across (orthogonal to) the fracture. In other words, the contribution of the reservoir past the tips of the fracture is neglected during this initial period of time, meaning that there is no deviation of the flow lines in the matrix rock (from linear to elliptical then to pseudo radial flows in an infinite acting reservoir). The initial treatment of the problem of rate/pressure responses as a function of time for this type of well led to quite complicated ODEs which could not be analytically solved in the time domain and required either discretizing in time and space to obtain a solution in the time domain (Cinco-Ley, 1978) or solving the ODEs in the Laplace domain, then using Stehfest's inversion algorithm (Cinco-Ley, 1981a; Wong, 1986) to obtain the time domain solution. As a consequence, all derived solutions were asymptotical and rigorous ones for the so-called early time bilinear and formation linear flow regimes, however remained numerical approximations in the transition period between the latter.
Actually, Heber Cinco-Ley himself indicated the way to simplify the ODEs so that they could be simply solved in the real space (1981a, page 1752, Eq. 11). He stated that the fracture linear flow period (during which compressibility effects are dominant) "occurs at a time too early to be of practical use??. The consequence is straightforward: the basic ODEs can be simplified by assuming purely viscous flow along the fracture (whereas compressible - i.e. transient - flow in the matrix is maintained, owing to extremely different orders of magnitude of fracture and matrix permeabilities). This is the very way used by the present paper, where ignoring the fracture porosity (as the governing factor of compressive effects) leads to much simpler ODEs, which can be solved in the time domain.
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