Propagation of Uncertainty from Geological Discrete Fracture Networks (DFNs) to Downstream Models

Transcription

1 Propagation of Uncertainty from Geological Discrete Fracture Networks (DFNs) to Downstream Models Aaron Fox Golder Associates AB, Stockholm, Sweden Paul La Pointe Golder Associates, Inc., Redmond, Washington Cristian Enachescu Golder Associates GmbH, Celle, Germany SPE ATW Upscaling of Fractured Carbonate Reservoirs, June / July 2009

2 Presentation Outline 1. DFN models and upscaling to reservoir simulators 2. The language of uncertainty 3. Quantifying uncertainty and propagation through the modeling process 4. Case studies: SDM Forsmark and SDM Laxemar 5. Group discussion and questions

3 What is a Discrete Fracture Network Model?

4 Upscaling DFN to EPM Simulators DFN Model Equivalent Continuum Model Directional permeabilities (k x, k y, k z ) or full permeability tensor Fracture pore volumes (equivalent porosity) Shape (σ) factor (matrix-fracture exchange) Mean fracture spacings (in grid-local space)

5 Effects of DFN Uncertainty when Upscaling Uncertainty in Orientation --- Permeability tensor, current stress field orientation (with corresponding effects on wellbore stability and frac design), compartmentalization Uncertainty in Size --- Reservoir compartmentalization, connectivity, fracture volumes (OOIP, EUR, and possible production rates), frac design and potential performance Uncertainty in Intensity --- Sigma factor, fracture volumes, wellbore stability, potential performance of frac, effectiveness of EOR efforts, avoidance / targeting of karst features, accessible volumes (OOIP / EUR) Uncertainty in Location --- Sigma factor, compartmentalization, location and spacing of sweet spots, avoidance / targeting of karst features, wellbore stabilty Uncertainty in the static fracture model has the potential to affect *all* aspects of the upscaled full-field model

6 Components of Uncertainty 1. Aleatoric: Inherent or irreducible uncertainty. The inclusion of additional data or processes cannot reduce this component of uncertainty An example of aleatoric uncertainty would be predicting whether a coin will turn up heads or tails on the next coin toss; if the coin is fair, then the probability of successfully predicting the outcome is 0.5, and this cannot be changed with additional knowledge. 2. Epistemic: External or reducible uncertainty. This can be reduced by the inclusion of additional data or knowledge about rates, processes, boundary conditions, etc. An example of epistemic uncertainty would be predicting the winner of a horse race; initially the odds might be equal for all horses, but with additional knowledge of past performance or other factors, the uncertainty regarding the prediction could be reduced. Result: Reservoir models will *never* be free of all uncertainty, no matter how much data we have!

7 Classes of Uncertainty Three Base Classes* 1. Conceptual Model Uncertainty: arises from incomplete understanding of the physical processes being modeled. 2. Representational (mathematical) Model Uncertainty: arises from the approximations and simplifications necessary to translate the conceptual model into a form suitable for numerical simulation 3. Parameter (data) Uncertainty: arises from imperfect knowledge of the system being modeled, measurement error, or because of limited data. *Source: Uncertainty Analyses and Strategy SA011481M4 REV 00, November 2001, Bechtel-SAIC.

8 Conceptual Uncertainties in DFN models Fracture size-intensity dependence on lithological / stratigraphic / structural position? 1000 Fractal Weibull Exponential Fracture intensity scaling (Euclidean, Fractal, multi-fractal, Gaussian?) Fracture location model (fractal clustering vs. Poissonian vs. correlated) Mass (Number of Fractures) Fracture orientations (random, Fisher, or correlated to reservoir property? E E E E E E E+03 Spacing (m) Conceptual uncertainties tend to dominate the uncertainty space of a static DFN model, making alternative model hypothesis testing almost a necessity!

9 Representational Uncertainty Most positive curvature from time surface Fracture intensity conditioned to coherence from volumetric seismics

10 Parameter Uncertainty N FRACTURE SET NAME ENE [6] Fisher concentration parameter (k) Fracture Domain FFM01 W E EW [8] NE [19] NNE [3] NS [16] NW [18] SH [20] SH2 [1] SH3 [2] 5 0 ENE EW NE NNE NS NW SH Fracture Set ID S Equal Area Lower Hemisphere 93 Poles 93 Entries 10 Unlinked Traces - WNW Set Best Fit Scaling Exponent All Lineaments: kt = % of Lineaments: kt = % of Lineaments: kt = 1.85 Cumulative Normalized Number ASM ASM ASM ASM Envelope Lineaments Best Fit - All Lineaments 50% of Lineaments 75% of Lineaments Best Fit - 50% Lineaments Best Fit - 75% Lineaments Trace Length (m)

11 Expressing Uncertainty Qualitative Deliberately vague statements (largely based on expert judgment ) Ranking without quantification Stating outcomes without expressing likelihoods Quantitative Numerical ranking of alternatives without expressing likelihoods Probabilities of events or outcomes Ranges of model parameters or variables (parametric sensitivity analysis) Confidence intervals on model parameters, variables, or results Probability distributions for model parameters, variables, or predicted outcomes Monte Carlo simulation

12 Propagating Uncertainty What are the target downstream models for a static DFN? What does it mean to propagate uncertainty? Is there a consensus? How much uncertainty can be dealt with before upscaling in the static model? What DFN uncertainties need to be propagated? Conductivity tensor, sigma factor, fracture pore volume, connectivity? Monte Carlo simulation / multiple alternative models: how far can we really go once the model has been upscaled to the full field / reservoir? Sensitivity: Do we have a gut feeling for the sensitivity of the various upscaled DFN properties to variability?

13 Case Study: SDM Laxemar and SDM Forsmark Two candidate sites for high-level radwaste and spent nuclear fuel from the Swedish nuclear industry. Site investigation consists of cored and percussion boreholes, surface outcrop mapping, airborne and ground EM (VLF, gravity, magnetics) geophysical surveys, with only limited seismic control (1-2 reflection surveys, no volumetrics) DFN models used for: Regional and local hydrogeological models (both FE and upscaled to EPM) Radionuclide transport Design and repository layout Seismic safety case

14 Case Study: Uncertainty Evaluation Main Conceptual Uncertainties Fracture Size Intensity as a function of lithology / fracture domain Intensity Scaling: Fractal, Euclidean, or other? Are lineaments (faults) part of the same population as outcrop and borehole fractures? Are fractures at the surface (size, locations) representative of the rock mass at depth? Location model for fractures: Poissonian, fractal, geostatistical, correlated or other? r 0 -fixed versus k r -fixed : two competing ways to compute scaling exponent Main Parameter Uncertainties Use of all surface lineaments versus some in size model (are all lineaments faults?) Spatial variability in fracture orientation and location (presence / absence of sets) Intensity / orientation of open versus sealed versus all fractures Value of scaling exponent (k r -fixed approach) of power law fracture size distribution Mathematical implementation of intensity-depth correlation (linear, exponential, polynomial) Hydraulic parameters (variability with depth, anisotropy inside faults, connectivity)

15 Case Study: Uncertainty Quantification Preferred model (out of 20+ alternative models) selected using: Ranking based on percentage error in prediction Kruskal-Wallis nonparametric ANOVA Uncertainty in size / intensity model quantified as ratios of P 32 of Base Model versus P 32 of Alternative Model(s) Intensity calculated over size range of interest : 2.7m 564.2m and 75m 564.2m Ratio is only an estimate of magnitude of uncertainty effect on model Uncertainty in small fault (MDZ) intensity further explored through Monte Carlo simulation with previously unused cored borehole Carrying uncertainty into upscaled model Moreland s (1974) block strain tensor approximation Oda s (1985) block hydraulic conductivity tensor approximation

16 Case Study: Uncertainty Results Uncertainty Scaling Exponent Dependence on Fracture Domain Conceptual Uncertainty Euclidean vs. Fractal Scaling Conceptual Uncertainty Linked vs. Unlinked Traces Conceptual Uncertainty Number of lineaments used in sizeintensity model parameterisation Scaling Exponent Parameter Uncertainty Open vs. Sealed Parameter Uncertainty Range of the Ratio of Alternative Model Intensity to Base Model Intensity 0.14 to 2.79; generally ~ for subhorizontal sets 0.77 to 1.67; 0.96 for subhorizontal sets 0.63 to 1.03; 1.03 for subhorizontal sets 0.44 to 1.11; unable to test subhorizontal sets ; 0.35 for subhorizontal sets 0.05 to 0.10 for outcrops; 0.20 to 0.30 for boreholes; generally around 30% for subhorizontal set (borehole data) Comments Greatest variability is by set, not fracture domain. Varies by set Fairly minor impact, especially on subhorizontal fractures Smaller than uncertainty in scaling exponent parameter fit Not highly variable among different sets Varies by set and domain Major uncertainties: r 0 -fixed versus k r -fixed size-intensity model (most important) k r as a function of fracture domain Uncertainty in size-intensity quantified and provided to downstream modelers

17 Questions for Discussion What methods have you used for propagating uncertainty that have worked well or poorly? Please stress failures as well as successes! What has been the most important uncertainty in your own experience and why? Do you regularly run uncertainty analyses on the fractured reservoir models? If not, why not? Lack of methodology? Too expensive in terms of time and money? Lack of inhouse expertise? In cases where static fracture models have been upscaled to a full-field simulator, what uncertainties have had the largest impact in your history matching and flow simulations? Are scenario and mathematical uncertainties captured in your reserves calculations? Are they formally reviewed by the reserves auditors in your company? Who should do the uncertainty calculations? The geologists? The reservoir simulation engineers? An independent team of company experts who fly in and train/set up the framework?

18 Thank you very much!

19 extra slides

20 Scale in DFN modeling Matrix mm scale Micro fractures cm scale Background fractures m s-100 s m scale Scale of traditional reservoir cellular modeling Mega Fractures 100 s m -km scale Discrete Seismic Faults km scale

21 Data Sources Seismic interpretation Seismic attributes VSP Core Well logs Image Logs Surface lineament & faults Production history Well tests Microseismic data Structural modeling Analogues Data Analysis Leads to Conceptual Fracture Model Derivation of DFN parameters Fracture Orientation Fracture Intensity Spatial / Location Model Fracture Size Fracture Hydraulic Properties DFN Model Construction Typical discrete fracture network model workflow DFN Output Statistical recipe for modeling Static and dynamic models Fracture Porosity Directional Permeability Sigma Factor Stress Tensor Static / Dynamic Model Testing and Validation Simulated exploration > Can we match existing data from wells and trace planes > Can we predict conditions in unsampledvolumes? Hydraulic simulations (well test matching, connectivity analysis) Validation Loop

22 Fracture size Size is specified in terms of equivalent radius ; i.e. the radius of a circular diskshaped fracture with an area equivalent to the desired fracture polygon r

23 Static Upscaling: Shape (σ) Factor Slab Matchstick Block Matrix Block Size and Shape X-Spacing Y-Spacing Z-Spacing after Barenblatt, 1960; Warren and Root, 1963 = σ S 2 x S y Sz

24 Dynamic Upscaling: Oda Tensor k k ij 1 = kkδij 12 ( F F ) ij A F e F ij = 1 V N k= 1 A k k k e k n ik n jk Oda s method (1984) does not require flow simulations Does not take fracture network connectivity into account Therefore, this method is limited to well-connected fracture networks.

25 Dynamic Upscaling: Block-K k = q ν l Ag ρ p For all [i, j] indices There are several types of boundary conditions that might be applied: (No flow, Linear, Periodic) Computationally expensive but accurate and less affected by sparse networks 25