The modeling of hydraulic fractures applies three fundamental equations:

Transcription

1 Te modelin of ydraulic fractures applies tree fundamental equations: 1. Continuity. Momentum (Fracture Fluid Flow) 3. LEFM (Linear Elastic Fracture Mecanics)

2 Solution Tecnique Te tree sets of equations need to be coupled to simulate te propaation of te fracture. Te material balance and fluid flow are coupled usin te relation between te fracture widt and fluid pressure. Te resultin deformation is modeled trou LEFM. Complex matematical problem requires sopisticated numerical scemes. D models provide tractable solutions but are limited by assumptions 3D and pseudo-3d are less restrictive but require computer analysis

3 Perkins-Kern-Nordren Model (PKN) witout leakoff Te followin assumptions simplify te complex problem: 1. Te fracture eit, f, is fixed and independent of fracture lent.. Te fracture fluid pressure is constant in te vertical cross sections perpendicular to te direction of propaation. 3. Reservoir rock stiffness, its resistance to deformation prevails in te vertical plane; i.e, D plane-strain deformation in te vertical plane 4. Eac plane obtains an elliptic sape wit maximum widt in te center, w(x, t) 1 p f G Scematic representation of linearly propaatin fracture wit laminar fluid flow accordin to PKN model

4 Perkins-Kern-Nordren Model (PKN) witout leakoff 5. Te fluid pressure radient in te x-direction can be written in terms of a narrow, elliptical flow cannel, p 64 q x 3 w f 6. Te fluid pressure in te fracture falls off at te tip, suc tat at x = L and tus p =. 7. Flow rate is a function of te rowt rate of te fracture widt, q x f 4 w t 8. Combinin provides a non-linear PDE in terms of w(x,t): G w w 0 subject to te followin conditions, 64(1 ) w(x,0) = 0 for t = 0 f t x w(x,t) = 0 for x > L(t) q(0,t) = q i / for two fracture wins

5 Geertsma-de Klerk (GDK) Model witout leakoff Assumptions: 1. Fixed fracture eit, f.. Rock stiffness is taken into account in te orizontal plane only. D plane strain deformation in te orizontal plane. 3. Tus fracture widt does not depend on fracture eit and is constant in te vertical direction. 4. Te fluid pressure radient is wit respect to a narrow, rectanular slit of variable widt, p(0, t) p(x, t) 1q i f x dx 3 0 w (x,t) Scematic representation of linearly propaatin fracture wit laminar fluid flow accordin to GDK model

6 Geertsma-de Klerk (GDK) Model witout leakoff Assumptions: 5. Te sape of te fracture in te orizontal plane is elliptic wit maximum widt at te wellbore w(0, t) (1 )L(p f G ) Scematic representation of linearly propaatin fracture wit laminar fluid flow accordin to GDK model

7 Net pressure at wellbore, psi Stimulation Comparison PKN KGD Fluid volume, als

8 fracture lent, ft Stimulation Comparison PKN KGD Fluid volume, als

9 maximum widt at wellbore, in Stimulation Comparison PKN KGD Fluid volume, als

10 3D Applications Primarily for complex reservoir conditions Multiple zones wit varyin elastic or leakoff properties Closure stress profiles indicate complex eometries Vertical fracture profile illustratin te canes in widt across te fracture

11 3D Components Assumptions 1. 3D stress distribution linear elastic beavior propaation criterion iven by fracture touness. D fluid flow in fracture laminar flow of newtonian or non-newtonian fluid 3. D proppant transport 4. Heat transfer 5. Leakoff Leakoff is 1D, to fracture face

12 3D Formulation Elliptic D.E. for elasticity Convective-diffusive eq. for eat transfer Parabolic D.E. for leakoff Solution Finite element metod discretization of formation to solve for stresses and displacements Boundary interal metod discretization of boundary

13 3D Pseudo 3D models (P3D) Crack eit variations are approximate dependent on position and time 1D fracture fluid flow Similar to PKN, i.e., vertical planes deform independently D P3D 3D

14 3D Comparison to validate D models Example A: Stron stress barriers, neliible leakoff More examples in Capter 5 of SPE monorap Vol 1 3D simulator

15 Dynamic Fracture Propaation Desin PKN Model Includes effects of non-newtonian fluids and net-to-ross eit 1. Initial uess of maximum wellbore widt, w wb = 0.10 in.. Calculate te averae widt, w w 4 wb 3. Calculate te effective viscosity, e n q 47880K i w 4. Calculate dimensionless time, 41 B x10 e 5 3C q i G /3 5 n t D t B

16 Dynamic Fracture Propaation Desin Includes effects of non-newtonian fluids and net-to-ross eit 5. Calculate dimensionless widt, w D 0.78t D 6. Calculate te maximum wellbore widt, 7. Test for converence, 161 q e 5.078x10 e i C G w wb ew D n 1/3 PKN Model n w wb n1 w wb TOL YES Continue NO Go to step ) wit updated w wb.

17 Dynamic Fracture Propaation Desin Includes effects of non-newtonian fluids and net-to-ross eit 8. Calculate te fracture lent, 1. Calculate te fracture volume, 10. Calculate te fracture pressure 5 1 q a x10 e i C G.695 L D t D L al D w L V 1 n 1/3 8 PKN Model P f (0,t) 1/ G q i e L 1 3,min 11. Update pumpin time and repeat te procedure, startin at step 1).

18 Dynamic Fracture Propaation Desin GDK Model 1. Initialize te procedure by uessin w wb = 0.1 in.. Calculate te dimensionless fluid loss parameter and fracture lent, n sp 8V 1 we w n t 8C L ) ( L erfc L e L n sp V we w n i q C L 3. Averae widt, wb w 4 w 4. Calculate te effective viscosity, 1 n w i 80.84q 47880K e

19 Dynamic Fracture Propaation Desin GDK Model 5. Simplified expression for fracture widt, w wb 84(1 ) e G q i L 1/ 4 6. Test for converence, n w wb n1 w wb TOL YES Continue NO Go to step ) wit updated w wb. 7. Volume of one win of te fracture, V Lw wb Bottomole fracture pressure,

20 wb wb Stimulation YES Continue NO Go to step ) wit updated w wb. 7. Volume of one win of te fracture, Dynamic Fracture Propaation Desin V Lw wb 48 GDK Model 8. Bottomole fracture pressure, P f (0, t) 1/ G q i e 1 3 L,min 9.Update pumpin time and repeat te procedure, startin at step 1).

21 Nomenclature a = lent constant, ft. B = time constant, min. C = fluid loss coefficient, ft/(min) 1/ E = widt constant, in. G = sear modulus, psi n = ross fracture eit, ft. = net permeable sand tickness, ft. K = consistency index, (lbf-sec n )/ft L = fracture lent, ft. L D P f q i t t D = dimensionless fracture lent = bottomole fracture pressure, psi = flow rate into sinle win of fracture, bpm = pumpin time, min. = dimensionless time V = volume of sinle win, ft 3 V sp = spurt loss, ft 3 /ft w = volumetric averae fracture widt, in. L D P f q i = dimensionless fracture lent = bottomole fracture pressure, psi = flow rate into sinle win of fracture, bpm t = pumpin time, min. t D = dimensionless time V = volume of sinle win, ft 3 V sp = spurt loss, ft 3 /ft w = volumetric averae fracture widt, in. w D w wb w we L e = dimensionless fracture widt = fracture widt at wellbore, in. = fracture widt at wellbore at end of pumpin, in. = dimensionless fluid-loss parameter includin spurt loss = effective fracture fluid viscosity, cp = orizontal, minimum stress, psi = poisson s ratio