Gas-Driven Fracture Propagation
2021-08-30
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R. H. Nilson Gas-Driven Fracture Propagation1 Department of Fluid and Thermal Sciences, Sandla National Laboratories Division 5512, A one-dimensional gas-flow drives a wedge-shaped fracture into a linearly elastic, imper-Albuquerque, New Mex. 87185 meable half space which is in uniform compression,
•<=) which ex-ploits the disparity between the fracture length and penetration length of the flow. Since the seepage losses to a surrounding porous medium are shown to be negligable in the late-time long-fracture limit, the results have application to geologic problems such as: con-tainment evaluation of underground nuclear tests, stimulation of oil and gas wells, and permeability enhancement prior to in situ combustion processes. 1 Introduction 破裂端位移场的调整In a fluid-driven fracture process, it is the internal fluid pressure fracture velocity is controlled by the dynamic interaction of a com-which wedges open the fracture and, hence, controls the rate of pressible fluid with the elastic solid. Only the elastic field is quasi-propagation. The fluid velocity and the induced fracture-tip velocity steady. The driving pressure, po, may greatly exceed the resisting are generally small compared to the velocity of stress waves; so, the compressive stress, DECEMBER 1981, VOL. 48 / 757 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 04/17/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use(i.e., Kl^pU —• 0), because the fracture-closing effect of the in situ compressive stress is generally much greater than the local action of molecular cohesion. Transient compressible flow in the fracture is governed by con-servation of mass and momentum, here written in the one-dimen-sional form [9] 0 — (pw) H (pwu) dt dx (3) — (pwu) H (pwu2) = - pw f- X dt dx \\p dx in which p and u are density and velocity, each averaged across the channel. The viscous shear stress X is approximated as follows for the limiting cases of low and high Reynolds number (Reo = u0w0po/p.o), respectively d d (IdP d d 粘性剪切应力_ 2TW 1 12/xu Fig. 1 Schematic of wedge-shaped fracture driven by internal gas pres-sure p w' (laminar) (4) — (turbulent). w 2 Formulation 标准的泊肃叶表达In the planar geometry of Fig. 1, a constant pressure po drives a wedge-shaped fracture to a penetration depth l(t). The internal gas pressure p(x,t) spreads the walls of the fracture, producing a flow channel with aperture width w(x,t). The considered geometry de-scribes the vertical-plane fractures which occur in the geologic ap-plications where the least principal stress is horizontal, and it places an upper bound on fracture growth because it neglects the fracture-closing effect of a finite fracture height and it eliminates the flow di-vergence which occurs in axisymmetric penny-shaped configura-垂直面tions. 压裂闭合效应Since the velocity of stress waves greatly exceeds the fracture ve-locity in a subsonic gas-drive, the displacement field adjusts almost instantaneously to the changing pressure loading. A quasi-steady stress analysis is therefore appropriate. Suppose that the tectonic stress . But, there is an offsetting effect {Pa term) due to the temporal divergence (wT > 0) of the walls at any fixed 8 and this is dominant near 8 = 0, where W —*• 0. The flow, therefore, sees a diverging-converging channel. Evacuation of the tip region due to wall divergence is favored by the smooth closure, and hence small aperture, within the leading zone. The resulting viscous stresses (~ 1/Wn, n = 2, oi 1 + 6) prevent the flow from overtaking the tip. An analogous situation occurs in liq-uid-driven fractures where the fracture-fluid is unable to wet the tip [7,8]. In spite of these expectations of an evacuated tip, the only jus-tifiable a priori assertion is that no mass crosses the impermeable walls and hence p(U(i CAip) ~* 0, or equivalently P(U-a6)-*0 as (19) which implies either a finite fluid velocity l/-»aoia full vacuum P 2N [P(f) - N~)d{ ada 2 —2l•K In (2y/e N/r) Je Jo V (turbulent) argument given in [4] rules out any other possibility on the basis of the ODE and the condition that VK'(l) = 0, and hence W ~ C^(C = The time variable T is subscripted here as a reminder that the time constant > 0, J = 1 - 8, j3 > 1) near 8 = 1. The discontinuity in P' has scale (to = lo/uo) is different for the laminar and turbulent periods. the character of a degenerate shock. Pressure jump is precluded by Since / and g are increasing with T, the flow is accelerating from the absence of inertial terms at low Mach number, and the trailing laminar dominance at early times to turbulent dominance at inter-fluid velocity is the same as the velocity of the discontinuity. The same mediate times, followed by inviscid inertial dominance even later. conditions are found to occur at a contact interface in gas dynamics Thus the constant of integration in g is taken as unity in the laminar and at the leading edge of a porous-media flow which expands into case to have I = lo when t = 0, and in the turbulent case the constant a vacuum. C, which drops out at late times, is determined by matching consid-The numerical solution procedure consists of successive iterations erations at laminar/turbulent transition. 积分常数of the sequence listed as follows, starting from the initial guess that Temporal wall divergence (i.e., dw/dt) has a first-order influence P{8) is linear and that 8* ^ ir/N (which is based on the asymptotic on the flow field, as illustrated by the following rearrangement of the large-iV solution to be given shortly) continuity equation (equation (11) with fig' = l/a from (16a) or 1 Calculate W(8) = W[P{8), N] from the elastic representation (166)) (13) using most-recent P(8) in the integrand. Averaging this result with previous W{8)PU'+ (U- a0)P' = -P (U-•ad) —W affords advantageous damping. W + a (18) 2 By application of a two-parameter shooting method, solve the problem (21), (22a), (236), and (24) forP(0) using most-recent W(8) in which U — a8 is the local velocity of the fluid relative to the moving in the ODE. It is elected to regard 8* and a as shooting parameters coordinate system (since (dx/dt)e/uQf = a8). Whenever U - ad > 0, (rather than P'(0) and a). Starting with a leftward expansion from 移动坐标系Journal of Applied Mechanics DECEMBER 1981, VOL. 48 / 759 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 04/17/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-useTable 1 Laminar flow results N = po/< /20 6 _ > V *s 5 °-CO N= 2^ \"^^ LU 6^ cc - 20-^ a. 0.4 OG^~ \\^u\"* _ ** -HI a! 0.4 < 0.2 0.2 0.0 0.0 ll i 0.2 0.4 0.6 5-*^ 0.8 1.0 n I I t v i r^^—. 0 0.4 0.8 1.2 1.6 2.0 BOUNDARY LAYER COORDINATE, 77 = a2'3 x/i Fig. 4 Convergence of turbulent pressure distribution to the large-W, boundary-layer solution (also shown is velocity 0 tor N —• co) one of the extensive measures is continuous. The matching is not, however, a critical issue since the turbulent solution eventually be-comes independent of the initial data. The dependence upon initial data is qualitatively different in the laminar and turbulent flow regimes. In the laminar solution (from (17a), (6), (7), and (15a)) I = IQ exp atM*M l0l 12/itoY/ (35) there is a lasting dependence on the initial data, lo- But in the late-time limit of the turbulent solution, which is when it becomes asymptoti-cally valid, (from (176), (6), (7), and (156) asr^») ; -at 7^oU+6 £0^ lt\\b 1/2)2/(2-6) (36) there is no longer any dependence on the initial length scale lo, since 0.0 0.2 0.4 0.6 0.8 1.0 the ratio WQIIO is given by (9) in terms of po,N, and the elastic con-stants. LONGITUDINAL POSITION, 6 = x/Kt) The laminar-dominated and turbulent-dominated flows belong to Fig. 3 Aperture and pressure distributions in turbulent flow for various values the class called self-similar asymptotics, each being strictly valid only of the pressure ratio, N = p0lum when the Reynolds number is either very low or very high everywhere within the channel. The transition from laminar to turbulent flow will therefore be a gradual one with turbulence taking over a longer and The requirement that 0 —* T](U —• ad) enforces agreement between longer section of the fracture, and there will always be a laminar zone the outer behavior of the inner (boundary-layer) solution and the near the tip. The character of such a transition and the validity of the inner behavior (leading terms) of the outer solution (tip expansion) asymptotic solutions have been investigated for the closely related as anticipated from the theory of matched asymptotic expansions. problem of the high-Re to low-Re transitional flow which occurs The numerical solution, which goes like 0 = r) + 2/T)2 as f) —• <», is during transient fluid flow in porous media, or equivalently in narrow capillary tubes or in constant-aperture fractures [11]. plotted in Fig. 4 where the large-W convergence of P(rj) is also pre-sented. The corresponding asymptotic entries in Table 2 are based Seepage interactions between a hydrofracture and a surrounding on the following observations (from (146) and from examination of permeable medium (e.g., geologic media) may also be viewed within Fig. 4): the context of asymptotic analysis, as described in [4] and as briefly outlined as follows. The continuity equation for the fracture (3a) is P(6)dQ 1 -2/3 modified to include a lateral seepage velocity, v, which depends on C'pdrf. 2N the lateral pressure gradient; u = -(in/KPo)y/Vt- It is found that the seepage interaction be-comes negligible when T has become sufficiently large that the fol-one matching condition at each transition. This single constant-of-lowing parameter becomes small (here c6 is porosity). integration can be interpreted as any extensive measure of the frac-ture/flow system such as length, volume, or elastic strain energy. The lo\\ 1 PO4>K-(37) early laminar solution depends linearly upon the nonzero integration — V7 «l ,Wo) Uo IXWo\\ 0constant lo which characterizes the extent of the fracture at the onset of gas-drive, which might be deduced from: a preexisting fracture Physically, the seepage interaction diminishes at late times because length, energy considerations at breakdown, or an analysis of the an increase in aperture, w, enhances the through-flow much more than early-time stress-controlled elongation. At the transition from laminar a corresponding increase in length, I, enhances the seepage losses (i.e., to turbulent flow, roughly Re = Reo/(r)g(r) ^ 2-103, the integration enhances the surface area for seepage loss). Thus, with respect to this constant of the turbulent flow, C in (176), should be chosen so that consideration, the present results must be viewed as late-time as-s1 r = Journal of Applied Mechanics DECEMBER 1981, VOL. 48 / 761 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 04/17/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-useymptotics in permeable media. Although the seepage interaction will always result in a finite tip pressure, the zero-pressure analysis should give a good approximation so long as the pressure ratio, Pdriving/Ptip, is large (roughly 5 or greater) as observed in the closely related problem of transient flow-through porous media or slender channels [11]. Quantitative results provide convenient estimates of fracture propagation rates and other engineering data. 1 Temporal variation of length, aperture and velocity is described by the analytic expressions l = log(r), w(0,t) = w0W(0)g{T), u(0,t) = uoU(0)f(T) in which g(r) and /(T) are given by (16) and (17); T = t/tn and t0 = 7 Summary 'o/\"o; lo is the initial fracture length; wv and tin are defined in (9) and A problem of gas-driven fracture propagation has been solved by (15); and Tables 1 and 2 provide a, W(0), and (7(0) as functions of JV; separation of the time and position variables, followed by numerical see in particular equations (35) and (36) for l(t). integration of the resulting ordinary differential equations. The 2 Spatial variations in pressure, aperture, and velocity are pre-considered case is a prototypic one, which reveals a number of inter-sented graphically. esting qualitative features. 3 Asymptotic results for large N are particularly simple and, by demonstrating convergence, they lend verification to the numerics. 1 The flow accelerates through a sequence a self-similar asymp-totic regimes: laminar, turbulent, and inviscid; only the first two are References reported here. 1 Howard, G. C, and Fast, C. R., Hydraulic Fracturing, Society of Pe-2 A strong vacuum exists at the tip of the fracture, either in an troleum Engineers of AIME, Dallas, 1970. impermeable medium or in a permeable medium at late times. 2 Pitts, J. H., and Brandt, H., \"Gas Flow in a Permeable Earth Formation 3 The flow effectively experiences a diverging/converging channel, Containing a Crack,\" ASME JOURNAL OF APPLIED MECHANICS, Vol. 44,1977, because of the competing effects of timewise wall divergence, dw/dt, pp. S53-558. 3 Keller, C. E., Davis, A. H., and Stewart, J. N., \"The Calculation of Steam and spacewise wall convergence, dw/dx. Flow and Hydraulic Fracturing in a Porous Medium With the KRAK Code,\" 4 If the pressure ratio {N = po/c») is large, the flow is confined Los Alamos Scientific Laboratory Report LA-5602-MS, 1974. to a narrow entry-layer region at the entrance to the channel. 4 Nilson, R. H., \"Gas-Driven Fracture Propagation,\" SAND79-2379, 5 In laminar-dominated flow at early times: the fracture grows Sandia National Laboratories, Albuquerque, New Mex., 1981. 5 Daneshy, A. A., \"On the Design of Vertical Hydraulic Fractures,\" exponentially, the pressure goes to zero at midspan, and the pressure Journal of Petroleum Technology, Vol. 25, Jan. 1973, pp. 83-97. distribution is linear for large values of N. 6 England, A. H., and Green, A. E., \"Some Two-Dimensional Punch and 6 In turbulent-dominated flow at late times: the fracture grows Crack Problems in Classical Elasticity,\" Proceedings, Cambridge Philosophical like a near-unity power of time, the pressure goes to zero at the tip, Society, Vol. 59,1963, pp. 489-500. 7 Geertsma, J., and De Klerk, F., \" A Rapid Method of Predicting Width and the pressure distribution is exponential for large values of and Extent of Hydraulically Induced Fractures,\" Journal of Petroleum time. Technology, Vol. 21, Dec. 1969, pp. 1571-1581. 7 The initial data (i.e., length l0 at the onset of gas-drive) has a 8 Barenblatt, G. I., \"The Mathematical Theory of Equilibrium Cracks lasting influence in the early laminar period but not in the late-time in Brittle Fracture,\" Advances in Applied Mechanics, Vol. 7, 1962, pp. 55-129. limit of the turbulent period. 9 Shapiro, A. H., \"The Dynamics and Thermodynamics of Compressible Assumptions and approximations in the analysis are self-consistent Fluid Flow,\" II, Ronald Press, New York, 1954. 10 Huitt, J. L., \"Fluid Flow in Simulated Fractures,\" AIChE Journal, Vol. with respect to two different criteria: late-time, long-fracture limit 2, (2), 1956, pp. 259-264. (quasi-steady stress field, impermeable media, no tensile strength); 11 Nilson, R. H., \"Transient Fluid Flow in Porous Media: Inertia-Domi-and upper-bound on fracture extension (all above, isothermal, nated to Viscous-Dominated Transition,\" Journal of Fluids Engineering, in press, 1981. wedge-shaped configuration). 762 / VOL. 48, DECEMBER 1981 Transactions of the ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 04/17/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use