Stochastic Analysis of Aquifer Interconnectedness: Wilcox Group, Trawick Area, East Texas

RI0189. Stochastic Analysis of Aquifer Interconnectedness: Wilcox Group, Trawick Area, East Texas, by G. E. Fogg. 68 p., 41 figs., 4 tables, 5 appendices, 1989. ISSN: 0082335X: Print.

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ABSTRACT
Detailed characterization of the spatial distribution of hydraulic conductivity (K) by direct measurement is usually impractical. A viable alternative is to use geologic information and geostatistics to characterize interconnectedness of critical K facies that have a dominant influence on fluid flow. Conditional simulation is introduced herein as a method for estimating aquifer interconnectedness. Facies continuity is accounted for by the variogram, which can be determined by conventional means or synthesized from information on average dimensions and fractional content of the K facies of interest.

A test case in a 6 by 11 mi (10 by 18 km) region in the East Texas Wilcox Group, a fluvial multiple aquifer system, demonstrates the feasibility of this approach. Two-dimensional conditional and unconditional simulations of channel-sand distribution along a strike cross section indicate sensitivity of interconnection probability (P) and equivalent K to sand-bod) continuity, variogram range and model, flow-region length, and sand fraction (SF). Conditional simulations are generated efficiently with an improved matrix technique that takes advantage of the vertical continuity of geophysical log data.

Results show that if the variogram and SF can be estimated, continuity and interconnectedness of facies in stationary systems can be characterized with acceptable accuracy. However, P and equivalent K are much more sensitive to SF than to the variogram range or model. Removal of data from the simulations causes loss of local detail on interconnection frequency but does not noticeably change the overall average continuity of sands or equivalent K. P exhibits threshold behavior, increasing abruptly when SF exceeds 0.3 to 0.5. The relationship between SF and equivalent K differs from that predicted by percolation theory owing to the presence of spatial correlation and nonzero K values assigned to the interchannel facies.

Keywords: aquifer, geostatistics, ground water, ground-water modeling, heterogeneity, interconnectedness, reservoir characterization, stochastic, Wilcox Group

CONTENTS

Nomenclature

Abstract

Introduction

Conceptual Framework

Use of Geologic Data in Ground-Water Models

The Problem

A Stochastic Approach: Conditional Simulation

Geostatistical Theory

The Variogram

Stationarity, Intrinsic Hypothesis, and Ergodicity

Kriging

The variance and covariance of e

Conditional Simulation

Unconditional Simulation

Test Case: Wilcox Group, Trawick Area, East Texas

Hydrogeologic Setting

Depositional Framework

Recognition of Channel-Fill Sands

Methods: Estimating Aquifer Interconnectedness

Geologic Characterization

Choice and Regularization of Simulation Variable

Statistical Analysis

Simulation of Sand Thickness

Back-transforming the simulated variable (Y)

Scanning the simulation results

Flow Simulation

Experimental Strategy

Performance of Simulation Algorithms

Estimating Interconnectedness and Its Influence on Flow

Comparison with geologic description

Effects of data conditioning and variogram model

Interconnectedness and equivalent hydraulic conductivity (K')

Results and Interpretation

Preliminary Statistical Analysis

Transforming S, to a normal distribution

Experimental variograms

Fitting variogram models

Comparison of Sand Distributions Obtained by Simulation and Geologic Interpretation

Geologic interpretation of sand distribution

Comparison with simulated sands

Effects of Data Conditioning

Influence of Variogram Model

Comparison of spherical and Bessel-exponential models

Effects of variogram sill and range

Synthesizing the Variogram from Geologic Information

Estimating the variogram

Interconnection Probability

Equivalent Hydraulic Conductivity

K' as a function of sand-body continuity

K' as a function of sand fraction

Prediction uncertainty of K'

Discussion

Limitations of Method

Discretization errors and an alternate approach

Two versus three dimensions in the fluvial system

Stationarity and ergodicity

Delineating permeability facies and defining a cutoff

Incorporating Geologic Information in Models

Synthesizing the variogram in fluvial systems

Applying Monte Carlo simulation results to ground-water modeling

Perspective on Oakwood modeling study

Comparison with Percolation Theory

Analytic Methods of Estimating K'

Summary and Conclusions

Acknowledgments

References

Appendices

A. Derivation of Covariance V_nm

B. Theory of Matrix Conditional Simulation

C. Conditional Simulation along Panels

D. Performance of Algorithms

E. Conditional Probability Theory of Facies Continuity

Text Figures

I. Log-normal hydraulic conductivity (K) distribution on arithmetic and log scales

2. Effect of a high-K channel on ground-water flow

3. Location of Trawick area and well control used in this study

4. Sketch illustrating the sand-body correlation problem between wells

5. Concept of conditional simulation illustrated in one dimension

6. Sketch illustrating computation of the experimental variogram

7. Schematic diagram of kriging

8. Maximum-sand map, Wilcox Group, Sabine Uplift area

9. Histograms showing K values of channel-fill and interchannel sediments of the Wilcox Group, Oakwood Dome area

10. Representative electric log, Wilcox Group, Trawick area

11. Cross section A-A'

12. Sand-isolith map, SU3 unit, Trawick area

13. Sand-isolith map, SU2 unit, Trawick area

14. Charts showing how sand thicknesses were regularized to 50-ft (15-m) intervals

15. Schematic diagram illustrating delineation of three-dimensional search zones for variogram analysis of S_rdata

16. Schematic diagram illustrating interconnected sand domains in the conditional simulation grid

17. Schematic diagram illustrating setup for flow simulations in which equivalent hydraulic conductivity (K') was computed

18. Histograms and normal probability plots of regularized sand thickness S_r for SU3

19. Histogram and normal probability plot of regularized sand thickness S_r for SU228

20. Directional experimental variograms, SU3 unit

21. Sketch illustrating potential causes of poorer than expected correlation in the dip direction in SU3

22. Directional experimental variograms, SU2 unit

23. Spherical and nested Bessel-exponential models fit to the experimental variograms from·SU3

24. Geologic interpretation of sand-body occurrence and continuity in cross section A-A'

25. Comparison of simulated and interpreted sands

26. Interconnectedness frequency (ICF) plots for simulations IIIA, IIIB, IIIC, and IV

27. ICF plot, simulation I

28. ICF plot, simulation V

29. Plot of average horizontal continuity of sand and horizontal interconnection probability versus variogram range, simulations IV through X

30. Plot of K’_h, versus variogram range, simulations IV through X

31. Sand fraction versus horizontal interconnection probability, runs IV, IV1-5; VII, VII 1,2,5; and VIII, Vl11 1,2,5

32. Sand fraction versus vertical interconnection probability, runs IV, IV1-5; VI, VI1,2,5; and VI II , VIII1 ,2,5

33. Equivalent length of flow region versus Ph, runs IV, VI-X; IV, , VII, , VIili; and IV2, V!i-X2

34. L/ C hmax•versus Ph, runs IV, VI-X; IV1, VII1, VIII; and IV2, Vh-X2

35. Average horizontal sand-body continuity versus horizontal equivalent K, runs IVA-XA; IVB-XB; IV 1B-X1B; and IV2B-X2B

36. Ch versus Kt,, runs IVC-XC

37. SF versus Kt,, runs IVA, IV1-sA: IVB, IV1-sB; VIIB, VII1,2,sB; VIIIB, VIII1,2,sB; and VIIC, VII1 ,2,sC . ..

38. SF versus K:., runs IVA, IV 1-sA

39. SF versus K ~, runs IVB, IV 1-sB

40. Ch versus coefficient of variation (Vh and Yv) of K', runs IVA-XA, IVB-XB, IV2B-X2B, and IVC-XC 41.

41. SF versus coefficient of variation (Vh and Vv) of K', runs IVA, IV1-5A;VIIB, VII1,2,5B; and VIIC, VII1-5C

Text Tables

1. Conditions imposed in each simulation

2. Calculated versus simulated horizontal interconnection probability

3. Interconnection probability and equivalent hydraulic conductivity

4. Fluvial-channel width/ depth ratios for selected simulations

Appendix Figures

C-1. Schematic diagram of conditional simulation along panels

D-1. Comparison of spherical variogram model and experimental variograms computed from results of conditional simulation I

D-2. Comparison of nested Bessel-exponential variogram model and mean experimental variogram from conditional simulation BIA

D-3. Interconnectedness frequency plots for conditional simulations IIA through IID

D-4. Plot of mean, average maximum, and ensemble maximum horizontal

continuity versus horizontal block size

D-5. Plot of norm of Sr versus number of realizations in the conditional simulation

Appendix Table

D-1. Ensemble statistics from conditional and unconditional simulations


Citation
Fogg, G. E., 1989, Stochastic Analysis of Aquifer Interconnectedness: Wilcox Group, Trawick Area, East Texas: The University of Texas at Austin, Bureau of Economic Geology, Report of Investigations No. 189, 68 p.