What can Single-Well Constant-Rate Pump Tests really tell about Fractured Rocks?

Transcription

1 What can Single-Well Constant-Rate Pump Tests really tell about Fractured Rocks? Stephan K. Matthäi, Sebastian Geiger, Centre for Petroleum Studies, Department of Earth Science and Engineering at Imperial Workshop on Model Concepts for Fluid-Fluid and Fluid-Solid Interactions Freudenstadt-Lauterbad, 20-22/3/2006 Imperial College London

2 Background Dynamic data from well tests often provide the only clues about the large-scale permeability and-or storativity of fractured reservoirs Rock matrix properties tend to be relatively well constrained by special core analysis (SCAL) The assumption that the permeability of the rock matrix is relatively uniform as compared with the fractures is often reasonable Single-well, constant-rate tests are the only ones which are carried out on fractured reservoirs on a routine basis Although common, shut-in tests are useless for fractured reservoirs because they only sample the fracture system Matthäi/Geiger-Lauterbad 21/3/06, slide 2

3 Outline 1. Rationale 1. ABC of well testing 2. Flow velocity spectra and the fracture-matrix flux ratio 2. Utility of fracture-matrix flux ratio for upscaling 2-phase flow 3. Estimation of fracture-matrix flux ratio in well tests Matthäi/Geiger-Lauterbad 21/3/06, slide 3

4 Well testing = transient fluid pressure diffusion p k = 2 t φµ c t p + fluid Standard plots employ dimensionless variables: Dimensionless time, t D : q where c t is the total-system compressibility and r w is the well bore radius. k h 2 Dimensionless pressure, p D : p D = q B where h is the height of the reservoir layer and B is the formation factor. t D = c t = V k t φµ c v p t r π µ w f Matthäi/Geiger-Lauterbad 21/3/06, slide 4

5 Theory (A. Gringarten, 1973) well (planview) Dimensionless time vs. pressure, log-log plot p isobars pd pd radial drawdown 0.5 infinite acting The fluid pressure-time derivative, dp/dt, at the well head is indicative of the flow regime dp/dt decreases when drawdown spreads into high permeability regions which discharge fluid more rapidly to the well dp/dt increases when internal of external impermeable boundaries get within the radius of drawdown dp/dt is constant while drawdown spreads across regions in which the bulk properties are uniform; in this infinite-acting case k can be inferred from dp/dt and the pumping rate Matthäi/Geiger-Lauterbad 21/3/06, slide 5

6 Previous work: transient response of faults Fault Properties from Outcrop Synthetic Well Test (2D) Matthäi et al., 1998 (SPE J ) Matthäi/Geiger-Lauterbad 21/3/06, slide 6

7 Simulated deviations from radial drawdown (Arches model) Topview of reservoir t = 8 hrs Def. Band Fault Joint 3 wells Matthäi et al., 1998 (Royal Soc. London, Special Publication 147, ) Matthäi/Geiger-Lauterbad 21/3/06, slide 7

8 Outline 1. Rationale 1. ABC of well testing 2. Flow velocity spectra and the fracture-matrix flux ratio 2. Utility of fracture-matrix flux ratio for upscaling 2-phase flow 3. Estimation of fracture-matrix flux ratio from well tests Matthäi/Geiger-Lauterbad 21/3/06, slide 8

9 Stochastic model with 2000 disc-shaped fractures generated using field data from San Andreas formation 1000 m 3.5 mm Fracture Aperture (m) 1 mm 1000 m Model built with FRED (Golder Associates) Matthäi/Geiger-Lauterbad 21/3/06, slide 9

10 Geometric and material properties Parameter Dimensions Porosity, Φ Φ f V f A, fracture-matrix interface Specific A f (P32) Units km - - m 3 km 2 m 2 m -3 FRACS x 1 x log k m md 10 1,468 k eff in flow direction q f /q m in x-direction log c t D - m 3 m -3 Pa Matthäi/Geiger-Lauterbad 21/3/06, slide 10

11 Single-phase flow velocity spectrum k eff = 1,468 D q f /q m (x) = 150 analysis method described in Matthäi & Belayneh, 2004, GRL 31:7, Matthäi/Geiger-Lauterbad 21/3/06, slide 11

12 Ratio q f / q m is determined together with effective permeability Single-phase flow analysis: MODEL k eff = qµ L w ( ( ) ( )) A p u p d f f q in A q out p f (u) p f (d) NB: k eff value will be different to that determined by ODA (1984) method Results depend on boundary setup (Flodin et al. 2004) L Matthäi/Geiger-Lauterbad 21/3/06, slide 12

13 Geologically conditioned, simulation-assisted upscaling From core plug to grid block scale Larger features must have a discrete model representation We have a new comprehensive strategy but it requires the fracture-matrix flux ratio to upscale relative permeability / fractional flow, water cut at breakthrough, rate of countercurrent imbibition Subseismic (mm-hm)-scale Seismic (hm-hm)-scale Oman reservoir model, p f contours 2 km Matthäi/Geiger-Lauterbad 21/3/06, slide 13

14 Outline 1. Rationale 2. Utility of fracture-matrix flux ratio for upscaling 2-phase flow 3. Estimation of fracture-matrix flux ratio from well tests Matthäi/Geiger-Lauterbad 21/3/06, slide 14

15 Numerically predicted kr i erratic? IRREGULAR-x200y20z50 mobility ratio, m =2 krw krn kri Saturation water Matthäi/Geiger-Lauterbad 21/3/06, slide 15

16 Prediction of relative permeability at ( s w(avg) ) Numerically determined total mobility, λ t is related to relative permeability λ t kr o = = kr µ k w w r t + λ µ µ o kr µ w + o o µ o µ ; w ; k kr r w = = kr kr k k r r w o λ µ µ t o w + µ µ o w Our experiments show that during water flooding, the kr w / kr o ratio is approximated closely by the q f / q m ratio Matthäi/Geiger-Lauterbad 21/3/06, slide 16

17 krw q f kr o q m Makes sense because the water immediately imbibes the fractures in a model it flows at the rate reflecting their contribution to the total flux Oil flows at the matrix rate Is determined by cheap single phase steady-state experiments Tensor property, range: 0 to ~10 4 in experiments conducted so far, anisotropy 100. Matthäi/Geiger-Lauterbad 21/3/06, slide 17

18 Matthäi/Geiger-Lauterbad 21/3/06, slide 18 Predicted grid-block scale relative permeability Example, using error functions: ( ) ( ) = + = m f w f w w w m f w f w w o q q s s s kr q q s s s kr 1 erf, 1 erfc φ φ

19 Implications of up-scaled relative permeability Derivative of fractional flow function no longer switches sign with increasing saturation but becomes monotonously increasing There is no shock in the volume integrated saturation but a long leading edge: f w f w, upscaled s w integrated over grid block s w water flood Matthäi/Geiger-Lauterbad 21/3/06, slide 19

20 Matthäi/Geiger-Lauterbad 21/3/06, slide 20

21 Summary, Utility of q f /q m Ratio for Upscaling The fracture-matrix flux ratio found by steadystate single phase analysis helps to predict gridblock scale relative permeability, fractional flow, and water cut at breakthrough Matthäi/Geiger-Lauterbad 21/3/06, slide 21

22 Outline 1. Rationale 2. Utility of fracture-matrix flux ratio for upscaling 2-phase flow 3. Estimation of fracture-matrix flux ratio from well tests Matthäi/Geiger-Lauterbad 21/3/06, slide 22

23 Well testing of fractured reservoirs Common wisdom Radial drawdown leads to a constant derivative of ½ Presence of finite length fractures is indicated by ¼ slope Presence of infinite length fractures is indicated by ½ slope Complications Real fractures have a diameter spectrum Heterogeneous rock matrix Anisotropy Storage capacity contrast between fractures and matrix Matthäi/Geiger-Lauterbad 21/3/06, slide 23

24 Closer to reality (horizontal well in Kilve a.r.) p f contours p f drop blue=5 x 10-6 ms -1 red= 0.1 ms -1 L = 2.32 κt well Belayneh, Geiger, Matthai, AAPG Bulletin, in press Matthäi/Geiger-Lauterbad 21/3/06, slide 24

25 Matrix and fracture storativity Hydraulic diffusivity Matrix storativity S κ = k φµ c t = αφ + β 1 k S µ ( φ ) Fracture storativity derived from toughness of mode I joint: S f = V f a 3 π µ ( ν 1) V f = 2 x x = = d 0 a ( x ) dx Matthäi/Geiger-Lauterbad 21/3/06, slide 25

26 Relative importance of fracture vs. matrix storativity Storativity light oil (API=45) saturated Troll sandstone with single fracture of variable length α = 3.1 x β = 2.6 x µ = 15 GPa ν = 0.25 Matthäi/Geiger-Lauterbad 21/3/06, slide 26

27 A fresh look at well tests We perform numerical well-test experiments on fracture models with known effective permeability and fracture-matrix flux ratio Key question: can we determine k eff and q f / q m in constant-rate, single-well pump tests? We will assume that the permeability of the rock matrix is known from core plug testing (SCAL), is isotropic, and lacks a correlation structure on the grid-block scale Matthäi/Geiger-Lauterbad 21/3/06, slide 27

28 Governing equations p = κ 2 t p Time discretization: + q Backward Euler, finite difference, progressive t: ([ φ S t ] + [ K ]){ p } = {[ S t ]} p q Spatial discretization: t + t φ t + Linear finite-element basis, adaptively refined hybrid element mesh, variable representations of wells Solution method: Algebraic Multigrid (SAMG, SCAI Fraunhofer), grid-reuse + p t as initial guess Matthäi/Geiger-Lauterbad 21/3/06, slide 28

29 Benchmarks uniform 3D model For different well shapes and hybrid element meshes our numerical method tracks the expected properties of homogeneos models closely 0.5 plot normalized by well rate and permeability assigned to grid Matthäi/Geiger-Lauterbad 21/3/06, slide 29

30 Benchmarks (continued) 3D experiments with triangular well Well intersects the fracture finite-length fractures infinite fractures Matthäi/Geiger-Lauterbad 21/3/06, slide 30

31 benchmark case well next to fracture Spatial drawdown patterns in the presence of fractures finite length fracture Matthäi/Geiger-Lauterbad 21/3/06, slide 31

32 Model FRACS2000 revisited well does not intersect fracture drawdown is not radial k m =1 x m 2 k f = 1 x to 4 x m 2 q f /q m =~150, anisotropy=1.3 Matthäi/Geiger-Lauterbad 21/3/06, slide 32

33 Derivative plot, normalized using matrix permeability FRACS2000? drawdown begins to affect model boundary Matthäi/Geiger-Lauterbad 21/3/06, slide 33

34 Single set of fractures with powerlaw diameter distribution Model POWERLAW200 a( x,..) = 2 ( ) ( 1 ν σ ) d yy p f µ Matthäi/Geiger-Lauterbad 21/3/06, slide 34 2 x 2

35 How should one interpret this test? Matthäi/Geiger-Lauterbad 21/3/06, slide 35

36 Summary 3, Well Test Experiments So far we have not found a way to deduce the q f /q m ratio from the results of a single-well constant rate test Radial drawdown needed to measure k eff may occur in the presence of fractures but only if the nature and abundance of fracture heterogeneities near the well does not change in a sufficiently large area around the well the effective permeability of this region is close to isotropic therefore - a separation of scales is possible Provided that a sufficiently long period of radial flow occurs once the effects of well-bore storage have ceased, q f / q m may determined in this largely hypothetical case but only if the permeability of the host rock is known Matthäi/Geiger-Lauterbad 21/3/06, slide 36

37 Mandefro Belayneh, Paul LaPointe, Miguel Gomez, Tanja Clees, Klaus Stüben, Alain Gringarten, Thomas von Schröter, Mike Homeyer Thank You FRED Matthäi/Geiger-Lauterbad 21/3/06, slide 37

38 CSP Complex Systems Platform API by Stephan K Matthäi (IC), Stephen G. Roberts (ANU), Sebastian Geiger (ETH), + contributions from: Thomas Driesner, Chris Pain, Andrey Mezentsev, Dim Coumou, Adriana Paluszny 1994: Idea of CSP at Cornell University, NY Spring 1995: SKM s first implementation of CSP Autumn 1995: Sabbatical of S.G. Roberts at Stanford University, CA, CSP with Meschach & AMG solver 1996: Review of CSP and suggestions by Bruce Eckel : SKM s main development and implementation of CSP : S. Geiger develops FV capabilities 2001: CSP3D 4.0 in Std C++ using meta-template programming techniques 2001: ICEM CSP interface 2002: Design of IMP-IMPS capabilities in collaboration with C. Pain, integration and interfacing with CFD tools 2003: SAMG-based large-scale mechanical calculations 2003: Interface to anisotropic adaptivity module 2003: SAMG-based large-scale mechanical calculations 2003: Interface to anisotropic adaptivity module 2003: Generalized 3D IMP-IMPS multiphase flow capabilities including gravity drainage and capillary driven flow 2004: Introduction of DFEM methods to deal with discontinua, EOS for H20-NaCl mixtures 2005: Parallelisation including SAMGp, work on advanced graphical user-interface, Navier-Stokes compressible flows Matthäi/Geiger-Lauterbad 21/3/06, slide 38

39 At beginning of water flood fractures are not water saturated Is this a problem? Case 1: well interconnected fractures q f >> q m, but fracture contribution to total porosity is only % (FRACS2000) At a high q f / q m, water breaks through the fractures instantaneously (compare slide 15). In this case the approximation applies Case 2: poorly interconnected fractures As q f approaches a fraction of q m, flow is controlled by the relative permeability-saturation curve of the rock matrix Matthäi/Geiger-Lauterbad 21/3/06, slide 39