1. Introduction
Analytical models and numerical simulation are important means to increase understanding and to enhance efficiency of hydraulic fracturing. The reasons for developing and using them are clearly explained, for instance, by Mack and Warpinski [1]. Thus, there is no need to dwell on them. Rather we focus on improving analytical and numeric methods used to the date. Our objective is to suggest new approaches for developing accurate, robust and stable simulators on the basis of recent analytical and computational findings [2-6].
The approaches discussed in the paper stem from the fact [2,3] that the conventional formulation of the hydraulic fracture problem (for example, see [7]), when neglecting the lag and fixing the position of the fracture front at a time step, is ill-posed. This feature has not been reported for more than three decades of studying hydraulic fractures because of two reasons. Numerical simulators, based on the conventional formulation (for example, see [7,8]), employ quite rough meshes, which themselves serve as specific ‘regularizators’. On the other hand, rare solutions of model problems either also employed rough meshes [9], or they were obtained by solving the initial value (Cauchy) problem [10-12] rather than the boundary value problem (a discussion of the difference may be found in references [3,4]). The disclosure of the mentioned fact has led to (i) explicit formulation of the speed equation (SE) in its general form To the authors’knowledge, Kemp [11] was the first who clearly distinguished the speed equation when revisiting the Nordgren’s problem. When introducing the SE, numbered (5) in his paper, Kemp wrote (p. 289): „Nowhere is (5) mentioned. (5) is called the Stefan condition and is always present in moving boundary problems“. Kemp used the SE to solve the Nordgren’s problem as an initial value rather than boundary value problem. This excluded solving ill-posed boundary value problem. Mack and Warpinski [1] have noticed the beneficial property of the velocity. On p. 6-21 of their fundamental work they wrote: „Detailed numerical simulations have shown that the velocity varies much more slowly than the flow rate“. These authors made the best of this property by using the velocity as the unknown variable; they actually employed the speed equations, as well, although not writing it explicitly.
Below we employ these options. Section 2 contains a concise review of the modified formulation. In Section 3, we illustrate its analytical advantages by simple solutions for the Perkins-Kern-Nordgren (PKN) [9,13] and Khristianovich-Geertsma-de Klerk (KGD) [14,15] models. In Section 4, we turn to computational advantages of the modified formulation, present new computational schemes and illustrate their efficiency by numerical results for the PKN model. Section 5 contains the extension of these schemes to the pseudo-three-dimensional (P3D) models [1], the importance of which grows nowadays because of their employment in simulators for hydraulic fractures in low permeable shales [16]. A brief summary concludes the paper (Section 6).
2. Modified formulation of hydraulic fracture problem
The modified equations [3,4] use as variables the velocity
where
It is obtained by integration of the Navier-Stokes equations for a flow of a viscous fluid in a narrow channel. Herein,
The equations (1) and (2) represent the modified system of PDE. Substitution of (2) into (1) gives the modified lubrication equation. We shall not write it down explicitly because in the general case, keeping the velocity, which is a smooth function near the fluid front, as unknown is more convenient than to replace it by
The elliptic operator requires only one
where q₀(x) is a known function at
Of special importance is that the limit value of the particle velocity
Herein,
Thus we have the
The speed equation (6) also yields important implications for numerical simulation of hydraulic fractures by finite differences. Indeed, when at a time step we have known both
The
where
The
The described modification concerns mostly with the
In terms of the modified opening, the
is solved under the BC of zero opening at points of the fracture contour. When neglecting the lag, this condition coincides with the condition (5) of zero flux at the fluid front.
The
where
3. Analytical solutions
In particular cases of 1D problems for the PKN [9,13] and KGD [14,15] models, one can solve an initial value rather than boundary value problem (for example, see [10-12]). Then the problem is well-posed and it does not require regularization. Still in these cases, the modified variables are of use to obtain simple analytical solutions of problems, which otherwise requite involved calculations. For a Newtonian fluid, the analytical solutions are given in [4] both for the PKN and KGD models. In a similar way, by employing the modified variables, we may solve these 1D problems for a non-Newtonian fluid.
Consider a fluid with the viscosity law of power-type
where
The geometrical scheme of the PKN model is given in Figure 1. In this model the plane-strain conditions occur in cross-sections parallel to the fracture front. Then the elasticity equation (10) takes the form [9]:
where
Physically significant speed of the front is neither zero, nor infinite. According to (14) and the SE (7) it is possible only when the function
Following [4], introduce the normalized variables
Consider the case when the influx is prescribed by the power dependence in time:
where
with
We may account for leak-off by assuming that the term
We prescribe the functions
where
with the starting values
Starting from
The value
In the case of a
4. Increasing efficiency of numerical simulations
As mentioned, in the general case, using the SE (7) opens options for tracing the fracture in the 3D space by level set, fast marching and other methods of the theory of propagating surfaces [17]. As to the authors’ knowledge, to this date, the only paper, in which the level set method has been applied to the hydraulic fractures, is that by Peirce and Detournay [20]. Since these authors used the conventional formulation not including the SE, special technique, called by them “implicit level set method”, was suggested and successfully used. Still, direct employing of the SE looks superior in simplicity and possibility to use the standard well-established technique.
From now on, we focus on other computational advantages of the modified formulation which evidently appear when considering the PKN model. Recall that this model is basic for a wide class of simulators employing the P3D models (for example, see [1]). Since the Nordgren’s problem is 1D, it is convenient to use the spatial coordinate
where
In the case when
In the considered problem the dependence (28) between the unknowns is explicit. Therefore, we may substitute (28) into (27). This yields the PDE of the first order in time and of the second order in ς:
where
From (34) and (35) it follows that after spatial discretization of these equations and the BC (30), (31), we obtain a well-posed system of ordinary differential equations (ODE) of the first order in time to be solved under the initial conditions (29) and (33). Actually, the problem does not require regularization. Still, as noted in [6], not to have too stiff system of ODE, it is reasonable to employ the
where
Solving the system of ODE, resulting from spatial discretization of (34), (35), under the initial conditions (30), (33) may be performed by well-developed methods, like the Runge-Kutta method. Standard solvers are of immediate use. Emphasize that this option has appeared only due to employing the
As an example, we use this scheme to extend the Nordgren’s numerical results [9] on the dependence of the fracture length on time for a Newtonian fluid and Carter’s leak-off. To this end, the normalizing length
Herein,
The results are summarized in Figure 2. The thick solid line presents the Nordgren’s curve (Fig. 2 of reference [9]), obtained in the time range [0.01, 5.0]. The author did not comment on the accuracy of his calculations. The accuracy, as stated in [3], is certainly below 1% which is also evident from the fact that the Nordgren’s curve intersects the asymptotic dotted line, corresponding to small time, and the asymptotic dashed line, corresponding to large time.
The results obtained by employing the computational scheme described are presented in Figure 2 by the thin solid line. It can be seen that the solution starting from the small time asymptotics tends to that corresponding to the large time one. The calculations are performed by using Previously, Kovalyshen and Detournay [21] also have solved the same problem in the range of time wider than that covered by Nordgren. The authors performed calculations starting from the small-time asymptote as an initial condition at τ =10-8 and presented numerical results in the range 10-5 < τ < 500. Their results are shown by circles in Figure 2. They are indistinguishable from our numerical solution. The authors do not discuss the accuracy, stability and robustness of their calculations. From our results, to which the relative error does not exceed 10-3, we may conclude that the relative error of the results, given for the fracture length in Table 1 of the paper [21], does not exceed 1% in the whole range of the calculations.
A similarly efficient computational scheme consists of solving the PDE (27), (28) with
5. Extension to P3D models
In the previous section we have stated that the modified formulation provides prerequisites for efficient solving the Nordgren’s problem. The latter, being the basis for the P3D models, we may extend the efficient schemes to these models.
The detailed description of the P3D models is given by Mack and Warpinski [1]. Thus it is sufficient to list only those their features, which distinguish them from the PKN model (Fig. 1) and which are significant for the extension.
(i) In P3D models, the in-situ rock stresses are different in various layers while the fluid pressure is assumed constant in any vertical cross-section. This implies that, in contrast with the PKN model, the net-pressure is now not constant in a vertical cross-section. Thus the P3D models employ the pressure itself rather than the net-pressure. Alternatively, one may employ the difference of the fluid pressure with a
(ii) The linear dependence (13) between the net-pressure
where for sufficiently small
(iii) The dependence (12) between the particle velocity and the pressure, after averaging the velocity over the height of a cross-section, obtains a factor
where for sufficiently small
(iv) The continuity equation is integrated over the cross-sectional height. As a result, the total flux through a cross-section and the total leak-off replace, respectively, the flux and leak-off per unit height. The average velocity
we have the modified continuity equation (15) in the unchanged form. The BC (18) at the inlet is also the same when denoting
Take into account that by the definition of the average opening, we have
where to simplify notation, we have omitted the subscript in the averaged velocity
Finally, by introducing the variables
where
is a function to be found in advance through the functions
The problem (43)-(49) differs from the problem (27)-(33) in the only detail: equation (44) for the velocity contains the function
6. Conclusions
The discussion above demonstrates the analytical and computational advantages of using the modified formulation. The analytical advantages are evident from the obtained simple analytical solutions for the PKN and KGD models, which otherwise require involved calculations. The computational advantages include: (i) the possibility to use the well-established theory of propagating surfaces, (ii) avoiding deterioration of numerical solution caused by ill-posedness of the problem when neglecting the lag and fixing the fracture contour at a time step, (iii) avoiding singularities on the fluid front, (iv) the possibility to use highly efficient numerical schemes for the PKN and P3D models. These beneficial features are of significance when developing simulators able to efficiently solve truly 3D and pseudo-three-dimensional problems in real time. The work on simulators of both types is in progress.
Acknowledgments
The authors appreciate support of the EU Marie Curie IAPP program (Grant # 251475). The first author (AL) also thankfully acknowledges support of the Russian Fund of Fundamental Investigations as concerns with the results on non-Newtonian fluids (Grant # 12-05-00140). Both authors are grateful to Piotr Kusmierczyk and Michal Wrobel for the help in performing many numerical experiments for the Nordgren’s problem with Carter’s leak-off.
References
- 1.
Mack MG., Warpinski NR. Mechanics of hydraulic fracturing. In: Economides MJ., Nolte KG. (eds) Reservoir simulation, 3-rd edn. John Willey & Sons; 2000. p6.1-6.49. - 2.
Linkov AM. Speed equation and its application for solving ill-posed problem of hydraulic fracturing. Doklady Physics 2011; 56 (8) 436-438. - 3.
Linkov AM. Use of speed equation for numerical simulation of hydraulic fractures. Available at: http://arxiv.org/abs/1108.6146. - 4.
Linkov AM. On efficient simulation of hydraulic fracturing in terms of particle velocity. Int. J. Engineering Sci. 2012; 52, 77-88. - 5.
Linkov AM. Numerical modeling of hydraulic fractures: State of art and new results. In: Advanced Problems in Mechanics, APM 2012: proceedings of International XL Summer School-Conference, 1-7 July 2012, Institute for Problems of Mechanical Engineering, RAS; 2012. CD-ROM. p225-236. - 6.
Mishuris G., Wrobel M., Linkov A. On modeling hydraulic fracture in proper variables: stiffness, accuracy, sensitivity. Int. J. Engineering Sci. 2012; 61, 10-23. - 7.
Adachi J., Siebrits E., Peirce A., Desroches J. Computer simulation of hydraulic fractures. Int. J. Rock Mech. Mining Sci. 2007; 44, 739-757. - 8.
Jamamoto K., Shimamoto T., Sukemura S. Multi fracture propagation model for a three-dimensional hydraulic fracture simulator. Int. J. Geomech. ASCE 2004; 1, 46-57. - 9.
Nordgren RP. Propagation of a vertical hydraulic fracture. SPEJ Aug., 1972; 10(4), 306-314. - 10.
Spence DA., Sharp PW. Self-similar solutions for elastohydrodynamic cavity flow. Proc. Roy Soc. London, Ser. A 1985; 400, 289-313. - 11.
Kemp LF. Study of Nordgren’s equation of hydraulic fracturing. SPE 18959, Society of Petroleum Engineers 1989; 287-298. - 12.
Adachi J., Detournay E. Self-similar solution of a plane-strain fracture driven by a power law fluid. Int. J. Numer. Anal. Methods Geomechanics 2002; 26, 579-604. - 13.
Perkins TK., Kern LF. Widths of hydraulic fractures. J. Pet. Tech. September 1961; 937-949. - 14.
Khristianovich SA., Zheltov VP. Formation of vertical fractures by means of highly viscous liquid. In: Proc. 4-th World Petroleum Congress, Rome, 1955. p. 579-586 - 15.
Geertsma J., de Klerk F. A rapid method of predicting width and extent of hydraulically induced fractures. J. Pet. Tech. December 1969; 1571-1581. - 16.
Kresse O., Cohen C., Weng X. et al. Numerical modeling of hydraulic fracturing in naturally fractured formations. In: Proc. 5-th US Rock Mechanics Symposium, San Francisco, CA, June 26-29, 2011, American Rock Mechanics Association. Paper ARMA 11-363. 11p. - 17.
Sethian JA. Level set methods and fast marching methods. Cambridge: Cambridge University Press; 1998. - 18.
Hadamard J. Sur les problemes aux derivees partielles et leur signification physique. Princeton University Bulletin 1902; 49-52. - 19.
Linkov AM. Analytical solution of PKN problem for non-Newtonian fluid. Russian Mining Science 2013; 1 (in press). - 20.
Peirce A., Detournay E. An implicit level set method for modeling hydraulically driven fractures. Comput. Methods Appl. Mech. Engng. 2008; 197, 2858-2885. - 21.
Kovalyshen Y., Detournay E. A reexamination of the classical PKN model of hydraulic fracture. Transp Porous Med. 2010; 81, 317–339.
Notes
- To the authors’knowledge, Kemp [11] was the first who clearly distinguished the speed equation when revisiting the Nordgren’s problem. When introducing the SE, numbered (5) in his paper, Kemp wrote (p. 289): „Nowhere is (5) mentioned. (5) is called the Stefan condition and is always present in moving boundary problems“. Kemp used the SE to solve the Nordgren’s problem as an initial value rather than boundary value problem. This excluded solving ill-posed boundary value problem.
- Mack and Warpinski [1] have noticed the beneficial property of the velocity. On p. 6-21 of their fundamental work they wrote: „Detailed numerical simulations have shown that the velocity varies much more slowly than the flow rate“. These authors made the best of this property by using the velocity as the unknown variable; they actually employed the speed equations, as well, although not writing it explicitly.
- Previously, Kovalyshen and Detournay [21] also have solved the same problem in the range of time wider than that covered by Nordgren. The authors performed calculations starting from the small-time asymptote as an initial condition at τ =10-8 and presented numerical results in the range 10-5 < τ < 500. Their results are shown by circles in Figure 2. They are indistinguishable from our numerical solution. The authors do not discuss the accuracy, stability and robustness of their calculations. From our results, to which the relative error does not exceed 10-3, we may conclude that the relative error of the results, given for the fracture length in Table 1 of the paper [21], does not exceed 1% in the whole range of the calculations.