Simulation of Naturally Fractured Reservoirs with Dual Porosity Models

Transcription

1 Simulation of Naturally Fractured Reservoirs ith Dual Porosity Models Pallav Sarma Prof. Khalid Aziz Stanford University SUPRI-HW

2 Motivations! NFRs represent at least 0%of orld reserves, but difficult to produce! Unfeasible to model typical massively fractured NFRs through discrete fracture models! Many limitations of existing dual porosity models Circle Ridge Fractured Reservoir, Wyoming

3 The Dual Porosity Model Matrix Continuum Transfer Function Fracture Continuum 3

4 Outline Single Phase Transfer Functions:! Limitations of Existing Shape Factors! Shape Factors for Transient/Non-orthogonal Systems! Numerical Algorithm for Non-orthogonal Netorks! Validation and Comparison To Phase Transfer Functions:! The Complete Transfer Function! Limitations of the Existing Transfer Function! Ne Shape Factors for To Phase Compressible Flo! Validation, Comparison and Case Study 4

5 Single φ Transfer Function 5 The Single Phase Transfer Function TF = Rate of mass transfer beteen matrix and fracture q = Vρφc mf ρk q = σ p p µ ( ) m mf m f t p m σ = a L V, φ ρ

6 Single φ Transfer Function 6 Limitations of Existing Shape Factors! Assumes pseudo-steady state (PSS)! Only for cubic matrix blocks or orthogonal fracture systems

7 Single φ Transfer Function 7 Average Pressure (psi) Errors due to PSS Shape Factor Transient PSS Comparison of Discrete Fracture and Dual Porosity Model for D Fracture Time (days) ECLIPSE Discrete DP: Lim And Aziz DP: Warren and Root DP: Kazemi Single Block (10X10 ft Matrix) DP Model Fractures L

8 Single φ Transfer Function TheTransient Shape Factor p = BC and IC D p x p(0, t) = p p(, t) = p p( x,0) = p f m m 1 1 σ = 1 Dt P( η ) σ 1 t p f L x p m 8

9 Single φ Transfer Function 9 PSS σ for Non-Orthogonal Fractures = + τ X sin α Y P P 1 P BC and IC P(0, Y, τ ) = 0, P(1, Y, τ) = 0, P( X,0, τ) = 0, PX (,1, τ ) = 0, PXY (,,0) = 1 σ π σ = 1 sin L sin α + 1+ sin α α R = α σ sin α = = C.5 30 o p f y α L p f p f p f x

10 Single φ Transfer Function Generic Numerical Technique ρk q = σ ( p p ) q = ρφc µ m mf m f mf t σ = 1 m ( p ) D p m f p p m 10

11 Single φ Transfer Function Generic Numerical Technique σ = D p 1 m ( p m f ) p p m and p m 11

12 Single φ Transfer Function 1 Ave r age Pressur e (p s i ) Results using Numerical Technique Comparison of Discrete Fracture and Dual Porosity Model for D Fracture Time (days) ECLIPSE Discrete DP: Lim And Aziz DP: Warren and Root DP: Kazemi DP:Variable Sigma

13 Mechanisms of Phase Mass Transfer To φ Transfer Function! Pressure gradients due to sources and sinks! Pressure diffusion due to compressibility! Saturation diffusion due to capillary forces P x P y 13

14 Complete Transfer Function To φ Transfer Function TF = Rate of mass transfer beteen matrix and fracture dm = Vφ( S + ds )( ρ + dρ ) VφS ρ! VφS dρ + VφρdS p S q = VφS ρ c + Vφρ mf q = q + q mf mf 1 mf V, φ S ρ 14

15 Limitations of Existing Models To φ Transfer Function Existing simulation models: Single φ p S q = VφS ρ c + Vφρ mf Multi φ k q = Vρ σ p p µ k q = Vρ k σ p p mf µ ( ) r PD f ( ) mf PD f 15

16 Equations Governing Flo To φ Transfer Function ω λ ( p γ D) ω q ( φs ω ) " = p 0 cp, p p p cp, p p cp, Assumptions: Immiscible, no gravity, sources and sinks insignificant Assumptions: Density and mobility functions of average quantities φ S φs c p p = + p p p p p λ λ p p 16

17 To φ Transfer Function 17 Derivation of φ S φs c p λ λ p = + φ S φs c p λ λ o o o o p = + o o S = S T o S D() t = S S t 0 Dt * Transform: T = D( τ ) dτ 1 φc dp 1 1 = φ + λ ds λ λ av o () av c * From Crank,

18 To φ Transfer Function 18 Derivation of Typical Imbibition Process: 1D imbibition through a matrix face, matrix initially at a constant saturation, fracture instantly filled ith etting phase. S T = S x S ( x,0) = S ; S (0, T) = S = S ; S (, T) = S i max f i S ( S S ) = " σ SD i mf S " σ ( ) q = Vφρ " σ S S SD i SD Dt () = t D( τ ) dτ 0

19 To φ Transfer Function 19 Derivation of p λ S φs c S c 1 = p p p T p = α + () t p f() t = p + Transform: T g" ( T ) t * = α( τ) dτ 0 * From Crank, 1975

20 To φ Transfer Function 0 Derivation of Typical PD Process: 1D fracture system, matrix initially at constant pressure, fractures suddenly reduced and maintained at a constant pressure (Lim and Aziz) p p T p = + gt "( ) x p 8 " σ = σ α() t p p + S S π S c p (0, T) = p ( L, T) = p f p ( x,0) = p SD ( ) ( ) PD f i m π σ = PD L 8 q = Vρλσ p p Vφρ σ S S mf 1 π " σ ( ) " ( ) PD f SD i SD D() t = t D( τ ) dτ 0

21 The Complete Transfer Function To φ Transfer Function mf ( ) φρσ ( ) q = Vρλσ p p V S S PD f SD i For the particular case of 1D parallel fractures ith PSS pressure diffusion and instantaneously filled fractures, e have: σ PD π 8 Dt ( ) = σ = 1 SD + t! at L π D( τ) dτ 0 1 1

22 The Complete Transfer Function To φ Transfer Function σ σ σ σ PD PD SD SD ( ) φρσ ( ) q = Vρλσ p p V S S PD f SD i mf a = Cubic Matrix and PSS (Lim and Aziz) L = f( t) Any shape and Tran + PSS (Numeric) 1 = at m = at Instantly filled fracture Gradually filling fracture * By comparison to results by Rangel-German *

23 Validation To φ Transfer Function Fractures Matrix Water Imbibing L Dimens: 00X00X00 cu.ft. Porosity: 5% Matrix Perm: 1 md Fracture Perm: 10 d Initial Pressure: 1000 psi Fracture Pressure: 500 psi Capillary Pressure: < 100 psi Compressibility: /psi Initial Water Saturation: 0. Relative Perm: Corey Type 3

24 Validation To φ Transfer Function 4 Oil production rate for single porosity fine grid, using the complete dual porosity function and only the first term

25 Validation To φ Transfer Function 5 Water imbibition rate for single porosity fine grid and using the complete dual porosity function

26 Case Study - Model To φ Transfer Function 6 Size: 8X8X Oil-Water DX = DY = 75ft DZ = 30ft Km = 1md Kf = 10d Porm = 19% Porf = 1% SigmaPD = 0.08 (10X10X30) Pc = 0-15 psi Kazemi et al., 76

27 Case Study - SigmaSD To φ Transfer Function 7 SigmaSD (1/day) SigmaSD vs S S SigmaSD related to S by using a single block model, that is, one 10x10x30 matrix block

28 Case Study - Results To φ Transfer Function 8 Rate (bbl/day) Oil and Water Production Rates Time (days) ECLIPSE Oil Rate GPRS Oil Rate ECLIPSE Water Rate GPRS Water Rate Additional Oil Recovery = 10% Reduced Water Production = 15%

29 Case Study - Results To φ Transfer Function 9 Water Cut Water Cut for Producer days Time (days) ECLIPSE GPRS Earlier Breakthrough by 150 days

30 Implementation into GPRS General Modifications: # # F ( X ) = { [ T ( X Φ ) ] + WI [ X ( p p )]} ns W W n, n+ 1 s λ ρ p p cp p s λ ρ p p cp p s= 1 p p n+ 1 n+ 1 n n φ ( S ρ X ) φ ( S ρ X ) p p cp p p cp p p V ± τ = cmf t 0 Transfer Function = τ cmf ( ) ( ) τ = Vk σ λ ρ X Φ Φ Vφ σ ρ X S S cmf m PD p p cp pm pf m SD p cp pm pmi p p 30

31 Implementation into GPRS mthfloeqnmodel mthbofloeqnmodel mthcompfloeqnmodel mthdpbofloeqnmodel mthdpcompfloeqnmodel! Object oriented approach through inheritence and polymorphism! Minimum modifications to existing code! Code structured, maintain compatibility and ensure bug-free code 31

32 Summary! Existing single phase shape factors inaccurate and limited in scope! Numerical technique for non-orthogonal systems and Transient+PSS flo! Existing to phase transfer function inaccurate! Ne transfer function for to phase compressible flo! Accurate modeling of fracture-matrix imbibition! Ne model implemented in GPRS 3

33 Single φ Transfer Function 33 Non-orthogonal Fracture Netorks

34 Single φ Transfer Function 34 Non-orthogonal Fracture Netorks Average Pressure (psi) Pressure Response of Rhombus and Square Difference ~ 8% Time (days) Rhombic Matrix Square Matrix

35 Single φ Transfer Function 35 Generic Numerical Technique Input Fracture Pattern Calculate Shape Factors Solve for Pressure Using any Commercial PDE Solver Calculate Average Pressure and Derivative

36 Validation To φ Transfer Function 36 k q = Vρ k σ p p mf µ ( ) r PD f