elastic properties of microporous grainstones
DESCRIPTION
Fournier, Leonide, BorgomanoTRANSCRIPT
Elastic properties of microporous cemented grainstones
François Fournier1, Philippe Leonide2, Kévin Biscarrat1, Arnaud Gallois1,Jean Borgomano1, and Anneleen Foubert3
ABSTRACT
We investigated the effect of porosity, pore geometry, anddiagenetic history on the elastic properties of dry, tightly cemen-ted grainstones whose pore space consists dominantly of intra-granular microporosity within micritic grains. The integration oflaboratory petrophysical measurements (porosity, P- andS-wave velocity), petrographic analysis and scanning electronmicroscope (SEM) imaging of micropore space of 80 LowerCretaceous microporous carbonate samples from Provence(south-east France) allows (1) the changes in porosity and poregeometry during the diagenetic history to be related to changesin elastic properties, and (2) the impact of micritic grain diag-enesis on the elastic properties of microporous grainstones to bequantified by means of fitting parameters derived from equiva-lent elastic medium modeling. The Urgonian microporous
cemented grainstones are elastically equivalent to a homoge-neous calcitic host with spherical calcitic inclusions comprisingspheroidal pores. The best fit is obtained when porous spheresare modelled using the differential effective medium (DEM) ap-proach and the whole composite using the self-consistent (SC)method (DEM-SC model). At lower porosity values (<20%),when the micropore volume is controlled by intercrystalline ce-mentation processes without compaction, the equivalent poreaspect ratio (EPAR) derived from DEM-SC modelling is nearlyconstant and averages 0.15. At higher porosities, changes in mi-cropore space architecture related to leaching processes result inslightly increasing EPAR. The recognition of EPAR-preservingversus EPAR-non preserving elastic property evolution is pro-posed as a tool for diagenetic pattern detection in microporouscarbonate reservoirs.
INTRODUCTION
A major challenge in carbonate reservoir characterization is tocorrelate geological data with petrophysical properties that canbe used to populate sedimentary bodies in reservoir modeling(Grammer et al., 2004). Petrophysical, textural, diagenetic, and geo-chemical properties of carbonate reservoirs are heterogeneous dueto the complexity and diversity of the primary carbonate factory andcementation and dissolution processes (diagenesis) modifying themineralogy and pore structure of carbonate sediments (Moore,1989; Morse and Mackenzie, 1990; Tucker and Bathurst, 1990;Tucker and Wright, 1990).Numerous carbonate reservoirs in the Middle East are domi-
nantly micritic and/or are characterized by a mud-supported
microporous facies (Wilson, 1975; Wilson, 1980; Budd, 1989; Wittand Gokdag, 1994). In such reservoirs, porosity ranges from 0 to25%, whereas permeability values can reach up to several hundredmD (Volery et al., 2009). Their microporous nature makes oil ex-traction even more difficult because of the heterogeneous distribu-tion of reservoir properties, and strong capillary forces due tonarrow pore throats, which retain much of the oil in place (Kirkhamet al., 1996).Despite their economic importance, the genesis of microporous
carbonate reservoir rocks is poorly understood. Many hypothesesexplaining the origin of porosity of these carbonates have been dis-cussed (e.g., Budd, 1989; Ahr, 1989; Kaldi, 1989; Moshier, 1989;Saller and Moore, 1989; Cantrell and Hagerty, 1999; Lambert et al.,2006; Richard et al., 2007; Volery et al., 2009). Moreover, the
Manuscript received by the Editor 4 February 2011; revised manuscript received 18 April 2011; published online 13 January 2012.1Université de Provence, Geology of Carbonate Systems and Reservoirs Laboratory, Marseille, France. E-mail: [email protected];
[email protected]; [email protected]; [email protected] University Amsterdam, Faculty of Earth and Life Sciences, Department of Sedimentology and Marine Geology, Amsterdam, The Netherlands. E-mail:
[email protected] Katholieke Universiteit Leuven, Belgium. E-mail: [email protected].
© 2012 Society of Exploration Geophysicists. All rights reserved.
E211
GEOPHYSICS. VOL. 76, NO. 6 (NOVEMBER-DECEMBER 2011); P. E211–E226, 16 FIGS., 2 TABLES.10.1190/GEO2011-0047.1
mixture of different porosity types in most carbonate rocks make itdifficult to establish simple laws between elastic properties(velocity, density) and porosity. The compressional to shear wavevelocity ratio (VP∕VS) is an important parameter for interpretinggeophysical field data, and has been claimed to provide lithologicinformation (Wilkens et al., 1984; Duffaut and Landrø, 2007).Laboratory studies (Rafavich et al., 1984; Anselmetti and Eberli,1993; Wang, 1997; Eberli et al., 2003; Baechle et al., 2008) haveshown that porosity and pore type are the two main factors control-ling the seismic response in carbonate reservoirs. However, at pre-sent, no well-constrained correlation exists between the porestructure and VP∕VS ratio. To understand the influence of mineral-ogy, pore shape parameters, and pressure on the VP∕VS ratio in car-bonate rocks, relations between acoustic properties and porosity incarbonates should be established. This especially should be done incarbonate rocks that have a large range in porosity values, but withone dominant pore shape. Microporous micrites (with intercrystal-line microporosity) have been shown to exhibit a similar behaviorto siliciclastic sands, but with a smaller critical porosity around15–20% (Fournier and Borgomano, 2009).The Urgonian limestones from Provence are excellent outcrop
analogues of microporous carbonate reservoirs which are encoun-tered in the Middle East (Thamama, Kharaib and Shuaiba forma-tions), this in terms of both facies (age, depositional facies,and environment), and reservoir properties (Masse, 1976). TheProvence Urgonian Platform is located on the southern marginof the Vocontian Basin, and developed from Valanginian to EarlyAptian times (Masse, 1993). Shallow-water carbonate environmentsreached their largest extent during the Late Barremian andEarly Aptian. The Urgonian carbonates provide the unique oppor-tunity to (1) define the impact of diagenetic transformations on
petrophysical properties in a well-understood area, (2) performmeasurements on samples characterized by the same diagenetic his-tory and the same dominant pore types, and (3) compare, in thesame carbonate system, nonporous versus microporous carbonatesfrom diagenetic and petrophysical approaches.
DATA SET AND METHODS
Carbonate data set
The data set consists of 85 microporous limestone samples fromLower Cretaceous platform carbonates, collected in various out-crops in south-east France (Figure 1). Porosity values range from0.5 to 25.5%. Rock samples were selected using the following cri-teria: (1) a grainstone texture and (2) the absence of intergranular,intercrystalline, or moldic macroporosity.
Laboratory petrophysical measurements
We prepared in the laboratory, using a water-cooled diamond cor-ing drill, 3.81-cm (1.5-inch) diameter, vertically oriented, cylindri-cal plugs from rock samples collected in the field. Sample ends wereground flat and paralleled to within 0.01 mm. The samples werefirst dried in a 60°C oven for at least 72 hours and equilibrated48 hours to room temperature and humidity conditions (20–23°C, 50–60%) before dry measurements where performed becauseless than 1% of water can significantly reduce the bulk and shearmoduli (Clark et al., 1980; Mavko et al., 1995, 1998).Dry mass of the samples was measured on 80 samples and dry
bulk density calculated from the dry mass and measured cylindervolume. Grain densities (ρ) were measured using a MicromeriticsAccuPyc 1330 helium pycnometer. Total porosity Φ was calculatedfrom the computed dry bulk density and measured graindensity. More detailed procedures are described by Kenter andIvanov (1995).Acoustic velocity, density, and porosity were measured on 80
samples (Petrophysical laboratory, VU University). Ultrasonic com-pressional P- (VP) and S-wave (VS) velocities were measured as afunction of pressure using a transducer arrangement (VerdeGeoscience) that propagated one compressional and two indepen-dent and orthogonally polarized shear waves (VS1 and VS2) alongthe core axis. The transducer consists of a source crystal excited bya fast rise-time electrical voltage pulse, producing an ultrasonicpulse with a frequency of 1 MHz, which was recorded by a receivercrystal. Measuring the one-way traveltime of the acoustic wavealong the sample axis and dividing by the sample length producedthe acoustic velocities. The arrival time of the one-way traveltimewas picked when the signal exceeded a threshold voltage equal to3% of the overall peak-to-peak amplitude of the first three halfcycles of the signal. Uncertainty in velocity measurements forlow-porosity (<30% of total porosity) cemented carbonates is with-in approximately 1%. Uncertainties in density and velocities mea-surements result in an error in bulk and shear modulus ofapproximately 5% and 3%, respectively.The ultrasonic measurements were conducted at five differential
stresses (effective pressure) that ranged from 0 to 40 MPa (confin-ing pressures: 2.5, 5, 10, 20, and 40 MPa). Pore pressure was kept atatmospheric pressure (0.1 Mpa).Laboratory measurements from this study are summarized in
Table 1.Figure 1. Location map of Urgonian limestone outcrops and sam-pling localities.
E212 Fournier et al.
Tab
le1.
Petroph
ysical
labo
ratory
measurements.Sa
mplinglocalities(1:Cassis;
2:LaFareMassif;3:
Orgon
quarry;4:
Fon
t-Jo
uval)arerepo
rted
inFigure1.
SampleLocal-
ity
Poros-
ity (%)
Grain
density
(103
kg∕m
3)
Micrite
volume
%
VP
ðm∕sÞ
2.5Mpa
VP
ðm∕sÞ
5MPa
VP
ðm∕sÞ
10MPa
VP
ðm∕sÞ
20MPa
VP
ðm∕sÞ
40MPa
VS1
ðm∕sÞ
2.5MPa
VS1
ðm∕sÞ
5MPa
VS1
ðm∕sÞ
10MPa
VS1
ðm∕sÞ
20MPa
VS1
ðm∕sÞ
40MPa
VS2
ðm∕sÞ
2.5MPa
VS2
ðm∕sÞ
5MPa
VS2
ðm∕sÞ
10MPa
VS2
ðm∕sÞ
20MPa
VS2
ðm∕sÞ
40MPa
B01
216
.82.69
65.5
3610
3941
4104
4227
4288
2024
2142
2253
2341
2377
2032
2146
2246
2329
2353
B02
216
.02.69
58.9
3887
3997
4054
4091
4108
2219
2261
2310
2334
2326
2244
2287
2317
2336
2340
B03
215
.32.71
57.5
4014
4103
4168
4236
4233
2328
2357
2392
2409
2412
2333
2362
2397
2414
2413
B04
A2
15.3
2.72
62.0
4432
4452
4504
4546
4577
2469
2512
2545
2568
2574
2450
2498
2542
2561
2567
B04
B2
16.6
2.71
49.1
4242
4255
4299
4373
4401
2376
2401
2426
2447
2457
2399
2417
2440
2454
2462
B05
A2
18.2
2.71
58.6
4258
4267
4298
4328
4358
2431
2430
2446
2470
2471
2407
2420
2439
2463
2468
B05
B2
19.8
2.69
68.3
4373
4372
4371
4323
4326
2406
2437
2456
2462
2447
2465
2484
2490
2476
2448
B06
A2
14.5
2.70
44.1
3157
3588
3753
4007
4286
1969
2120
2208
2340
2461
1981
2138
2228
2346
2461
B06
B2
15.9
2.72
40.9
3535
3648
3855
4168
4440
2186
2224
2300
2416
2521
2060
2132
2233
2376
2502
C01
221
.12.70
59.3
3740
3819
3907
3950
3929
2154
2223
2251
2259
2237
2081
2159
2183
2207
2166
C03
221
.02.71
64.3
3451
3978
4171
4234
4216
2126
2345
2430
2442
2455
2001
2200
2320
2377
2380
C06
222
.72.72
48.7
3401
3673
3789
3929
4027
1992
2128
2202
2232
2176
2013
2124
2189
2236
2149
C07
220
.92.70
63.0
3481
3676
3848
3924
3868
2130
2205
2278
2315
2274
2066
2190
2242
2278
2211
C08
219
.62.72
62.7
3729
3892
4043
4159
4199
2197
2247
2342
2395
2385
2043
2149
2239
2290
2292
C11
220
.22.70
67.7
3804
3937
3944
3968
3956
2270
2305
2327
2327
2307
2281
2313
2315
2319
2303
C12
223
.12.69
69.3
3360
3739
3827
3888
3865
1992
2078
2143
2198
2240
2002
2103
2206
2235
2230
C13
b2
16.7
2.69
48.3
4193
4353
4408
4499
4523
2336
2395
2439
2472
2495
2384
2444
2497
2526
2538
C14
a2
15.8
2.69
55.0
4639
4666
4773
4757
4806
2502
2524
2562
2560
2564
2621
2593
2597
2615
2617
C14
b2
18.4
2.71
63.0
3982
4200
4280
4315
4320
2290
2381
2411
2435
2426
2365
2434
2466
2480
2481
C19
a2
23.8
2.72
70.0
—32
8535
8939
8640
18—
1997
2124
2305
2358
—20
5121
1722
4623
24
C20
214
.52.74
74.0
3690
3813
4030
4392
4675
2178
2234
2343
2489
2606
2206
2283
2381
2523
2638
C21
218
.92.70
62.0
3906
3978
4065
4164
4206
2299
2331
2368
2403
2416
2300
2337
2369
2405
2421
C22
217
.52.69
75.0
4036
4116
4175
4241
4250
2380
2403
2426
2444
2452
2360
2387
2409
2426
2433
C23
221
.32.69
76.1
3761
3803
3831
3846
3846
2212
2239
2261
2267
2258
2207
2240
2256
2262
2255
C24
217
.52.70
50.6
4086
4247
4532
4617
4670
2353
2460
2515
2547
2570
2364
2471
2533
2577
2588
C25
a2
23.6
2.73
68.0
2803
3129
3347
3597
3744
1734
1879
1987
2095
2153
1768
1917
2012
2106
2159
C25
b2
21.6
2.71
51.0
—30
9633
4936
8138
02—
1888
2054
2176
2233
—18
9220
2121
4722
29
C26
216
.92.73
64.0
3928
4036
4229
4444
4565
2320
2364
2443
2516
2565
2298
2355
2446
2524
2569
COU13
24.5
2.70
84.1
5757
5756
5822
5843
5886
3012
3042
3066
3078
3089
3012
3048
3072
3078
3089
COU14
211
.12.70
79.3
4878
4930
4997
5014
5027
2703
2744
2764
2779
2787
2691
2715
2739
2753
2757
D01
320
.32.71
72.7
3009
3406
3627
3837
—17
6819
1820
8121
81—
1748
1910
2059
2150
—D02
318
.12.70
65.7
—40
0442
0742
9843
91—
2300
2363
2392
2417
—22
8323
5824
0824
32
(con
tinued)
Microporous grainstone petrophysics E213
Tab
le1.
(Con
tinued)
SampleLocal-
ity
Poros-
ity (%)
Grain
density
(103
kg∕m
3)
Micrite
volume
%
VP
ðm∕sÞ
2.5Mpa
VP
ðm∕sÞ
5MPa
VP
ðm∕sÞ
10MPa
VP
ðm∕sÞ
20MPa
VP
ðm∕sÞ
40MPa
VS1
ðm∕sÞ
2.5MPa
VS1
ðm∕sÞ
5MPa
VS1
ðm∕sÞ
10MPa
VS1
ðm∕sÞ
20MPa
VS1
ðm∕sÞ
40MPa
VS2
ðm∕sÞ
2.5MPa
VS2
ðm∕sÞ
5MPa
VS2
ðm∕sÞ
10MPa
VS2
ðm∕sÞ
20MPa
VS2
ðm∕sÞ
40MPa
D03
322
.22.73
78.7
3577
3724
3931
4026
—20
1521
5922
3922
94—
2033
2164
2271
2329
—D04
316
.72.69
77.7
—43
9944
5145
6246
0322
5123
7024
9425
5725
8022
7123
9724
8225
2425
75
D05
320
.72.70
76.7
3395
3677
—39
25—
2017
2056
2147
2241
—20
2220
4721
3922
38—
D07
39.7
2.69
69.0
—50
0450
5651
7552
17—
2747
2791
2843
2849
—27
1927
8128
5028
66
D09
39.6
2.71
64.7
5089
5223
5183
5191
5248
2803
2839
2848
2862
2870
2775
2848
2830
2853
2864
D10
39.6
2.70
81.3
4618
4680
4817
4910
5062
2529
2595
2673
2752
2801
2557
2600
2665
2749
2781
D11
317
.62.73
81.7
4411
4449
4477
4540
4653
2487
2571
2605
2568
2586
2461
2504
2540
2551
2546
D12
311
.32.69
85.3
5022
5044
5111
5121
5130
2724
2774
2797
2817
2830
2744
2767
2800
2800
2809
D13
312
.82.70
70.0
4690
4708
4766
4815
4844
2593
2602
2640
2665
2679
2606
2632
2649
2667
2685
D14
33.2
2.71
59.1
5795
5840
5848
5801
——
3095
3071
3072
——
3078
3021
3024
—D15
311
.82.70
65.8
4898
5019
5062
5129
5184
2599
2658
2712
2747
2797
2645
2732
2755
2798
2824
D16
34.3
2.70
74.3
5550
5677
5711
5805
5865
2912
2999
3062
3075
3088
2915
3006
3049
3078
3102
D17
314
.72.73
71.3
—42
0942
4543
5144
6222
8923
7024
5524
8025
1022
3923
4924
0724
8125
25
D18
34.9
2.70
71.3
5522
5590
5622
5655
5690
2962
3015
3039
3058
3085
2976
3013
3029
3047
3059
D20
38.3
2.72
62.3
5141
5220
5245
5315
5335
2837
2864
2888
2916
2923
2835
2858
2874
2901
2913
D21
313
.52.73
81.3
4648
4682
4716
4777
4802
2620
2642
2675
2697
2716
2591
2617
2647
2674
2687
D22
319
.32.75
82.0
—37
9138
9340
2041
3420
7221
4022
1123
0023
5520
0921
5222
0723
2723
77
D23
34.1
2.68
52.0
5534
5668
5706
5768
5813
2967
3002
3046
3077
3097
2956
2997
3046
3083
3102
F01
216
.32.70
60.0
3396
3509
3760
4019
4250
2104
2157
2261
2354
2437
2136
2180
2261
2364
2455
F02
214
.52.70
62.0
4327
4400
4450
4527
4648
2386
2437
2474
2507
2552
2424
2483
2506
2551
2584
F03
216
.42.71
60.7
3774
3801
3874
4136
4396
2106
2188
2247
2358
2457
1984
2118
2188
2307
2408
F04
213
.12.70
70.0
3855
3976
4219
4470
4684
2175
2249
2389
2508
2590
2238
2313
2435
2551
2619
F05
29.4
2.71
60.7
4695
4762
4934
5118
5247
2634
2701
2736
2801
2849
2628
2697
2755
2816
2864
F06
213
.32.70
42.1
3907
4035
4185
4369
4552
2352
2418
2479
2547
2604
2345
2392
2466
2541
2607
F07
210
.62.69
82.3
4681
4749
4858
4955
5013
2582
2638
2691
2732
2760
2594
2638
2693
2732
2754
F08
210
.52.70
67.7
4591
4662
4766
4874
4912
2578
2616
2661
2703
2726
2570
2606
2663
2704
2723
F09
210
.42.72
58.3
4704
4762
4838
4916
4979
2591
2625
2676
2706
2737
2578
2625
2673
2714
2740
F10
213
.02.68
48.3
—42
5243
1244
7246
31—
2306
2408
2502
2580
—23
5524
3825
2025
99
F11
23.9
2.66
50.1
5490
5504
5599
5714
5798
2933
2969
3019
3062
3100
2933
2974
3024
3072
3090
F12
213
.42.72
52.0
3796
3924
4024
4282
4487
2196
2249
2327
2416
2513
2218
2260
2321
2417
2499
FJ02
40.4
2.69
73.0
6096
6162
6210
6223
—32
4032
9832
3732
29—
3295
3338
3352
3291
—FJ06
413
.62.73
79.7
3995
4170
4316
4521
4650
2298
2385
2472
2537
2584
2310
2381
2472
2557
2590
FJ07
414
.52.69
61.0
4207
4234
4262
4275
4339
2389
2411
2424
2440
2457
2386
2409
2422
2437
2453
(con
tinued)
E214 Fournier et al.
Tab
le1.
(Con
tinued)
SampleLocal-
ity
Poros-
ity (%)
Grain
density
(103
kg∕m
3)
Micrite
volume
%
VP
ðm∕sÞ
2.5Mpa
VP
ðm∕sÞ
5MPa
VP
ðm∕sÞ
10MPa
VP
ðm∕sÞ
20MPa
VP
ðm∕sÞ
40MPa
VS1
ðm∕sÞ
2.5MPa
VS1
ðm∕sÞ
5MPa
VS1
ðm∕sÞ
10MPa
VS1
ðm∕sÞ
20MPa
VS1
ðm∕sÞ
40MPa
VS2
ðm∕sÞ
2.5MPa
VS2
ðm∕sÞ
5MPa
VS2
ðm∕sÞ
10MPa
VS2
ðm∕sÞ
20MPa
VS2
ðm∕sÞ
40MPa
FJ09
418
.92.70
68.0
3624
3656
3714
3813
3917
2117
2140
2169
2208
2257
2114
2135
2158
2206
2249
FJ12
415
.92.71
80.0
3910
3958
3974
3997
4063
2213
2233
2254
2258
2281
2203
2231
2246
2258
2267
FJ13
48.4
2.71
57.0
4974
5070
5204
5248
5291
2713
2762
2815
2845
2862
2705
2758
2810
2854
2880
FJ16
420
.32.73
60.5
—40
1740
5041
1842
4322
0122
4922
6722
9823
5022
1322
4422
7823
1823
56
FJ17
413
.02.71
55.7
4704
4733
4793
4900
5011
2549
2607
2633
2669
2701
2571
2616
2638
2689
2721
FJ23
425
.52.70
67.0
3247
3406
3571
3732
—18
2619
2120
0320
89—
1868
1906
2012
2114
—O12
317
.32.71
73.0
3847
4012
4096
4177
4222
2190
2292
2339
2376
2398
2250
2308
2352
2387
2406
O74
320
.92.71
81.0
3344
3480
3659
3781
3843
1934
2021
2107
2166
2208
1931
1993
2094
2157
2194
RR01
212
.82.71
72.7
4398
4550
4688
4792
4831
2474
2540
2606
2642
2659
2458
2531
2592
2632
2651
RR05
210
.72.70
50.8
4994
5078
5152
5175
5224
2638
2749
2792
2819
2843
2645
2749
2799
2821
2843
RR06
212
.02.69
75.7
4105
4168
4324
4479
4658
2275
2325
2401
2466
2557
2298
2348
2406
2472
2537
RR07
A2
8.4
2.70
63.7
4833
4887
4951
4999
5031
2664
2698
2737
2767
2779
2647
2685
2724
2753
2767
RR07
B2
10.8
2.71
53.8
4677
4756
4874
4973
5048
2597
2649
2696
2743
2773
2588
2644
2694
2743
2773
RR07
C2
10.0
2.70
72.0
5117
5181
5201
5245
5276
2781
2803
2828
2847
2864
2775
2787
2812
2827
2849
S02
10.6
2.69
85.0
6178
6212
6222
6282
——
—32
3132
44—
——
——
—
Microporous grainstone petrophysics E215
Petrographic analysis on thin sections and SEM
Thin section study under polarized-light microscopy provides thesedimentologic and petrographic framework for this study. Blueepoxy-stained thin sections were prepared from all of the 80 carbo-nate samples used for the petrophysical laboratory measurements.All thin sections were point-counted on the basis of 400 points to
estimate the proportion of micrite, and microsparite/sparite. Aftervan der Plas and Tobi (1965), for a 400 points counting, the half-width of uncertainty on percentage estimation is less than 5%, withintwo-sided 95% confidence bounds. Considering the relatively highhomogeneity of the studied rocks compared to the sample size, therelative abundance ofmicrite evaluated by point-counting is assumedto reflect the actual composition of thewhole plug sample, within theanalytical uncertainty bounds. Point-counting allowed to estimatethe micritic volume fraction fm for all samples.Scanning electron microscopy (SEM) was performed on 14 gold-
coated samples using a PHILIPS XL30 ESEM with a current set at20kV. These observations made it possible to characterize the mi-crite morphology and the micropore network structure. Microporesare defined by pores with diameter lower than 10 microns (Cantrelland Hagerty, 1999).
MICROFABRICS OF THE URGONIANMICROPOROUS GRAINSTONES
Micrite characterization
The carbonate grainstone samples used in this study are well-sorted and medium to very-coarse grained (Figure 2). More than50% of the grain population consists of rounded micritic peloids(Figure 2a, 2b, and 2c). Most of these peloids probably representbroken and micritized bioclasts, such as foraminifers, red algae, and
molluscs (Samankassou et al., 2005). Nearly the whole intergranu-lar and intraskeletal space is filled with calcitic cements (Figure 2d).The most common cement fabrics encountered in the studied sam-ples include (1) isopacheous rim of fibrous to prismatic cementsaround grains, (2) blocky calcite cements, and (3) syntaxial cementsaround echinoderm fragments. SEM observations revealed that porespace consisted almost exclusively of intercrystalline microporositybetween calcitic micrite crystals (Figure 3a). Minor intercrystallinemicroporosity is observed between sparry calcite crystals in blockycements (Figure 3a and 3b).Four micrite microfabrics were defined in the selected samples
from the Urgonian limestone, on the basis of the crystal shape, sort-ing, and contacts by using Loreau’s terminology (Loreau, 1972): (1)Microfabric 1 (MF1): subhedral mosaic micrite, (2)Microfabric 2(MF2): serrate subhedral/euhedral micrite (3) Microfabric 3(MF3): punctic to serrate subhedral/euhedral micrite, showingsubrounded crystals with subhedral/euhedral overgrowths, and(4) Microfabric 4 (MF4): punctic, loosely packed, and locallycoalescent subrounded micrite.The MF1 consists of a dense mosaic of small subhedral crystals
(1–2 μm) of low magnesium calcite (LMC) with dominantly serratecontacts (Figure 4a). The enfacial junctions between crystals, thedominance of subhedral crystal morphologies and serrate contactscompared to the relative scarcity of anhedral morphologies andcoalescent contacts, suggest that this microfabric results more fromthe cementation than from the compaction of a micritic precursor.In sample S2, the estimated average micrite microporosity islow (0.7%).In MF2, micritic LMC crystals are subhedral to euhedral, poorly
sorted, with mainly serrate contacts and locally punctic contacts(Figure 4b). This microfabric is interpreted to result from moderatecementation of a micrite precursor. Serrate subhedral micrites dis-
play a moderate residual porosity, ranging from5.3 to 16% in samples observed under SEM.The MF3 is characterized by poorly sorted,
dominantly subhedral to euhedral LMC crystalswith punctic to serrate contact (Figure 4c, 4e).This microfabric commonly exhibits smallrounded crystals (< 2 μm) surrounded by largereuhedral overgrowth (up to 5 μm), thus suggest-ing dissolution-reprecipitation by Ostwald ripen-ing process (Baronnet, 1982; Morse and Casey,1988). In carbonates of homogeneous mineralo-gical composition, the smallest crystals are themost unstable and are dissolved, thus leading torounded-shaped micrites, in favor of euhedral/subhedral overgrowths of larger crystal. In ourdata set, the diagenetic environment and thetiming of occurrence of calcite overgrowths arepoorly constrained. Such diagenetic feature hasbeen reported in lacustrine micrites from theMadrid basin (Volery et al., 2010a; 2010b) andwas interpreted as resulting from fluctuating me-teoric phreatic lenses. In samples observed underSEM, MF3 micrites display porosity values ran-ging from 14 to 26%.The MF4 consists of fine-grained (<1 μm),
well-sorted LMC subrounded crystals with punc-tic contacts (Figure 4d). MF4 micrites are highly
Figure 2. Thin-section photomicrographs, under polarized-light of typical microporousgrainstones from the Urgonian limestone: (a) sample C20 from La Fare: well-sortedcoarse-grained grainstone dominantly composed of dark micritic grains (p: peloids,m: miliolids) and sparitic grains (s) with thick micritic envelope (arrow); (b) sampleD01 from Le Défens quarry: medium-coarse-grained grainstone with dark micriticgrains (p: peloids, t: textularids); (c) sample FJ23 from Font-Jouval: highly porous fineto medium-grained peloidal; (d) sample C2 from La Fare: coarse-grained grainstonewith peloids (p), echinoid fragment (ech.) and sparitic/microsparitic grains with micriteenvelope (me); grains are rimmed by an isopacheous prismatic cement (ir) whereas rem-nant intergranular space is filled by blocky sparry calcite (bs).
E216 Fournier et al.
porous (up to 41% in the samples observed underSEM), loosely packed, and locally coalescent. Asevidenced in microporous carbonate reservoirsfrom the Middle East (Lambert et al., 2006) andin mixed carbonate-siliciclastics from Provence(Fournier and Borgomano, 2009), the roundedshape of these micrites probably results fromdissolution processes. The various stages ofedge roundness in MF4 microfabric, fromsharp-edged to well-rounded crystals, suggestthat rounded micrites have subhedral to euhedralprecursors. The coalescent aggregates of roundedcrystals are interpreted to result from reprecipita-tion of the dissolved carbonates around the crys-tal junction (Figure 4f). The nature and origin ofthe fluids responsible of the leaching in MF4micrites have not been investigated.As suggested in Figure 5a, a general trend of
increasing porosity is observed from tight MF1microfabric to highly porous MF4 microfabric.The petrographic observations under SEM indi-cate that the intragranular micrite did not undergosignificant mechanical and chemical compaction.Changes in intragranular micrite porosity may berelated to cementation and leaching processes(Figure 5b). The low compaction of intragranularmicrite may be due to the formation of early mar-ine or shallow burial cements, which may haveprevented the intragranular micrite from latercompaction. Indeed, micritized allochems donot display any mechanical compaction featuressuch as grain deformation, sutured and conca-vo-convex grain contacts. In addition, as alreadyreported in lacustrine and marine micrites (Voleryet al., 2010a; 2010b), the formation of calciteovergrowth in porousmicrite during early diagen-esis could have created a rigid framework thatcould have reduced the effects of compaction.
ELASTIC PROPERTIES OFMICROPOROUS CARBONATES
Effect of effective pressure and porosityon VP, VS and VP∕VS
In the low-effective-pressure range (2.5–10 MPa), most samples show a rapid, nonlinearincrease in P-wave velocity (Figure 6a), whichcould be attributed to closing of microcracks(Gardner et al., 1974; Vernik, 1994). At highereffective pressures, most of the samples exhibitalmost no pressure dependence on the P-wavevelocity (lower than 6 m:s−1:MPa−1), indicatingthat at effective pressures greater than 10–20 MPa, most of the microcracks are closed.As illustrated by Figure 6c and 6d, the rate of
velocity increase with increasing effectivepressure is not correlated with porosity or withmicrite or sparite/microsparite content. This im-plies that the contacts between the micrite crys-tals (Figure 3) do not behave elastically like
Figure 3. (a) SEMphotomicrograph of a polish thin-sectionned sample showing roundedpeloidal grains with high intercrystalline microporosity (black) between micrite crystals(white); intergranular space is filled with blocky sparry calcite exhibiting rare and sparseintercrystalline pores; (b) SEM photomicrograph of a micritic grain boundary: micropor-ousmicrite (M) is rimmedbya thin layerofmicrospar crystals (MS) and intergranular spaceis filled with coarse sparry calcite cements (S); pore space in sparite and microsparite isrestricted to very flat, crack-like pores, at the contact between crystals.
Figure 4. SEM photomicrographs of the main micrite microfabrics of the micritic grainsfrom theUrgonian limestone : (a)Microfabric 1 (MF1): fine grained (1–2 μm), tight anhe-dral compact micrite (sample S2: measured sample porosity ¼ 0.6%, estimatedmicrite porosity ¼ 0.7%); (b) Microfabric 2 (MF2): serrate subhedral micrite (sampleCOU13: measured sample porosity ¼ 4.4%, estimated micrite porosity ¼ 5.3%);(c) Microfabric 3 (MF3): poorly sorted, punctic to serrate micrite, showing subroundedcrystals (white arrow) with euhedral overgrowths (black arrow) (sample D07:measured sample porosity ¼ 9.7%, estimated micrite porosity ¼ 14%); (d)Microfabric4 (MF4): fine grained (<1 μm) punctic, loosely packed and locally coalescent subroundedmicrite (sample FJ23: measured sample porosity ¼ 25.5%, estimated micriteporosity ¼ 38%); (e) close-up on the MF3 microfabric (Figure 4c) showing a euhedralcalcite overgrowth (black arrow) around a rounded crystal (white arrow); (f) exampleof coalescent aggregates of subrounded micrite commonly found in MF4 microfabric.
Microporous grainstone petrophysics E217
microcracks. Cracklike behavior in these samples may be due tostress relief and cooling related to natural uplift and erosion, or ar-tifacts from core and plug recovery (Vernik, 1997). Importantly, athigher effective stresses, these samples do not behave like a crackedmedium (Smith et al., 2009).As opposed to the experiments by Anselmetti and Eberli (1993)
on carbonate samples from the Bahamas and Maiella (south Italy),we did not observe a decrease in velocity with increasing pressure inUrgonian limestone samples, thus suggesting that stress-inducedcracking and fracturing did not occur during our experiments.As documented in Upper Cretaceous carbonates from Provence(Fournier and Borgomano, 2009), values of VP∕VS show almostno change with increases in effective stress (Figure 6b).P- and S-wave velocities display a steep decrease with increasing
porosity from 0% to 15% and a more gentle decrease above 15%(Figure 7). Such a nonlinear velocity-porosity transform is commonin carbonate rocks (Anselmetti and Eberli, 1993). Figure 8 displaysthe linear trend of the P-wave velocity versus S-wave velocity trans-form and the changes in VP∕VS ratios as a function of porosity. Inspite of the very few papers documenting the porosity dependencyof VP∕VS ratios for dry carbonates, a similar decreasing linear trendwith increasing porosity has been evidenced by Assefa et al. (2003)in oolitic and skeletal grainstones and packstones, and by Røgenet al. (2005) in North Sea chalks.
Estimates of the dry micriteelastic moduli
The petrographic observations discussed above support modelingthe carbonate rocks used in this study as a mixture of microporouscalcitic micrite, and nonporous calcitic microsparite and sparite.The microporous micrite is the dominant constituent of the grains(peloids, micritized bioclasts) and can as well be found forming mi-
crite rims on sparitized bioclasts. The microspar-ry and sparry calcite is found in the form ofequigranular cements filling intergranular porespace and in the form of recrystallized bioclastsresulting from aragonite to calcite transforma-tion. As indicated by SEM observations, the por-osity in intergranular cements and recrystallizedbioclasts is assumed to be negligible.The micrite porosity Φm can be therefore
estimated as
Φm ¼ Φf m
; (1)
whereΦ is the total porosity of the sample and fmthe micrite volume fraction. Equation 1 is validonly for samples devoid of macropores such asmoldic, intergranular, or intraskeletal pores.Effective properties of a mixture of distinct
elastic media cannot be predicted exactly, butupper and lower bounds can be calculated for agiven composition. The bounds that define thenarrowest range of possible values, indepen-dently of the geometry of the constituents, arethe HS bounds (Hashin and Shtrikman, 1963).We consider the Urgonian grainstones as atwo-constituent medium, made of microporous
Figure 6. (a) Compressional-wave velocity (VP) versus effective pressure; (b) velocityratio (VP∕VS) versus effective pressure; (c) compressional-wave velocity increment ratio(between 2.5 and 40Mpa effective pressure) versus total sample porosity; (d) compres-sional-wave velocity increment ratio (between 2.5 and 40 Mpa effective pressure)versusmicrospariteþ sparite volume content.
Figure 5. (a) Relationship between grain micrite microfabric andmicritic grain microporosity for the samples analyzed underSEM; (b) scenario of diagenetic evolution of the micritic grainsshowing the genetic relationship between the micrite microfabrics.
E218 Fournier et al.
calcitic micrite and nonporous mosaïcs of calcite sparite and micro-sparite. The upper bounds KHSþ and μHSþ and the lower boundsKHS− and μHS− for the dry bulk and shear moduli, respectivelyare given in Appendix A.Ranges of dry bulk and shear moduli of the microporous micrite
(Kmicrite, μmicrite) are estimated by resolving the inequalities
�KHS−ðKmicriteÞ < Kmeasured < KHSþðKmicriteÞμHS−ðKmicrite; μmicriteÞ < μmeasured < μHSþðKmicrite; μmicriteÞ ;
(2)
where KHS−, KHSþ and μHS−, μHSþ are the lower and upper HSbounds for the bulk and shear moduli of the mixture, respectively;and Kmeasured and μmeasured are the bulk and shear moduluscalculated from measured densities and P- and S-wave velocities,respectively.This approach applies if each constituent is isotropic, linear elas-
tic; and the rock is isotropic, linear, and elastic. This implies that (1)all the pore volume is located within the micrite fraction, and (2) thespatial distribution of the pore space is homogeneous within themicrite fraction. The volumetric fraction of each constituent isreported in Table 1. The elastic moduli of the calcite spar andmicrospar aggregates are assumed to be close to the moduli ofthe mineral, independently of the aggregate geometry.The computed ranges of microporous grain elastic moduli,
derived from equation 2 are plotted in Figure 9 as a function ofmicritic grain microporosity. In contrast to the results of Fournierand Borgomano (2009) in Upper Cretaceous mixed carbonate-siliciclastic rocks from Provence, the decrease in bulk and shearmoduli with increasing porosity does not display a significant breakand a critical porosity behavior is not evidenced.
EQUIVALENT ELASTIC MEDIUMMODELING OF URGONIAN MICROPOROUS
CEMENTED GRAINSTONES
We propose to model microporous grainstones as a compositematerial with two end-member constituents: (1) A pure calcite host(spary calcite cements and grains) assumed to be nonporous, and (2)porous spherical inclusions (micritic grains). Both constituents areassumed to be isotropic, linear, and elastic. In this model, all the
Figure 7. Compressional-wave velocity (VP) and shear-wavevelocity (VS) measured at 20 MPa effective pressure as a functionof total sample porosity.
Figure 8. (a) Compressional-wave velocity (VP) versus shear-wavevelocity (VS); (b) velocity ratio (VP∕VS) versus total sample por-osity. Measurements are performed at 20 MPa effective pressure.
Figure 9. Crossplot of estimated ranges of micrite bulk (a) andshear (b) modulus versus average micrite microporosity. Bars repre-sent the possible range of micrite moduli honoring the inequalities(equation 2): they integrate the uncertainty in mineralogic composi-tion and bulk moduli of analyzed samples at 20 MPa.
Microporous grainstone petrophysics E219
pore space is assumed to be located within the spherical micriteinclusion.To calculate the elastic moduli Km and μm of the inclusions
(microporous micritic grains), the spherical inclusion model(SIM) was tested (Figure 10).
Effective properties of dry micritic grains:Spheroidal inclusion model
Microporous micritic grains are modeled as a pure calcite host(bulk modulus Kc ¼ 71 GPa and shear modulus μc ¼ 30 GPa) con-taining oblate spheroidal pores (bulk and shear moduli are set tozero). Effective property computations of the microporous grainswere performed using two approaches: (1) the differential effectivemedium (DEM) theory (Norris, 1985) that assumes isolatedpores set within a continuous host material (Appendix B) and(2) the self-consistent (SC) approximation (Berryman, 1980) that
treats pores and host symmetrically (Appendix C). In both ap-proaches, models were computed for various pore aspect ratios.Elastic moduli — porosity curves derived from DEM and SC mod-eling are plotted in Figure 9 for aspect ratios ranging from 0.05 to0.3. The comparison with elastic moduli estimates derived fromequation 2 (Figure 9) indicates that, in most of the measured sam-ples, the micritic grains are elastically equivalent to idealized cal-citic media with spheroidal pore inclusions of aspect ratios rangingfrom 0.1 to 0.2, at least from 0 to 30% porosity.
Effective properties of the dry composite rock
The effective bulk and shear moduli K� and μ�, of the drycomposite rock, consisting of microporous grains embedded withina homogeneous nonporous calcitic host (bulk modulusKc ¼ 71 GPa and shear modulus μc ¼ 31 GPa), are estimatedusing three different methods: (1) the HS lower (HS−) and upper(HSþ) bounds (Appendix A; Hashin and Shtrikman, 1963), (2) thedifferential effective medium (DEM) theory (Appendix B; Norris,1985), and (3) the SC approximation (Appendix C; Berryman,1980). For HS bounds, no grain geometry is assumed, as it is justa mixture of two end-member components. In DEM and SCapproaches, grains are treated as spherical inclusions. The DEMmodels are computed by resolving the coupled system of equation(Appendix B), in which geometric factors P and Q are those ofspherical inclusions. In the SC approach, elastic moduli are foundby solving iteratively the equations presented in Appendix C and byusing the geometric factors P and Q given in Appendix B.Eight distinct types of effective property models are therefore
computed according to the grain property and the whole-rock prop-erty modeling method (Table 2). Figures 11 and 12 display the HS,DEM, and SC models of the composite rock for the two distinctgrain property modeling approaches (DEM and SC). Results areplotted for various micritic grain concentrations and for various mi-cropore aspect ratios.The models exhibit the following features (Figures 11 and 12):
• At low-porosity values (<10%), elastic moduli-porositytransforms are controlled by the pore aspect ratio, and areindependent of the micritic grain concentration and of themodeling approach.
• At higher porosity values (>20%), elastic moduli computedusing DEM-DEM, DEM-HSþ, SC-DEM and SC-HSþmethods reach a plateau whose value depends on the micritic
grain concentration only and elastic mod-uli computed using DEM-SC, DEM-HS−, SC-SC, and SC-HS− tend to zero.
Equivalent elastic models formicroporous cemented grainstones
The effective property models computed afterHS, DEM and SC methods are compared withlaboratory measurements performed on 85 mi-croporous cemented grainstone samples. Ruizand Dvorkin (2010) demonstrated that idealizedelastic models, such as DEM, could be predictivefor a certain type of rock, in spite of the unrea-listic spheroidal shape assumed by these models.
Figure 10. Summary of the methodology used for modeling theelastic properties of equivalent elastic media for microporous ce-mented grainstones.
Table 2. Classification of effective property models according to the grainproperty and the whole-rock property modeling method.
Whole-rock modelling method
DEM usingsphericalinclusion
SC usingsphericalinclusion
UpperHS bound
LowerHS bound
Grain-propertymethod
DEM usingspheroidalinclusion
DEM-DEM DEM-SC DEM-HS+ DEM-HS−
SC usingspheroidalinclusion
SC-DEM SC-SC SC-HS+ SC-HS−
E220 Fournier et al.
Figure 12. Hashin-Shtrikman (HS) bounds, DEM and SC models of a composite rock made of spherical inclusions of porous inclusions. Theelastic moduli of the porous inclusions are computed using the (SC) approximation applied for a SIM. Results are plotted for various micriticgrain concentrations and for various micropore aspect ratio.
Figure 11. Hashin-Shtrikman (HS) bounds, DEM and SC models of a composite rock made of spherical inclusions of porous inclusions. Theelastic moduli of the porous inclusions are computed using the DEM theory applied for a SIM. Results are plotted for various micritic grainconcentrations and for various micropore aspect ratio.
Microporous grainstone petrophysics E221
Figures 11 and 12 show that for microporous grain propertiescomputed using DEM and SC approaches, HS bounds, DEMand SC models of the composite rock match the whole data set,when the micropore aspect ratio is set at 0.15. In Figure 13a and13b, the measured P- and S-wave velocities are compared withvalues derived from DEM models of a composite rock comprising60% of microporous inclusions. Results show that for a given por-osity, the equivalent micropore aspect ratios derived from VP andVS measurements are slightly lower for DEM-DEM models com-pared to those derived from DEM-SC models. In addition, in DEM-DEM models, aspect ratios derived from VP values are slightlyhigher than those derived from VS values. In contrast, aspect ratiosderived from VP and VS values are consistent.The choice of the best equivalent elastic medium depends on the
consistency between VP and VS predictions for a given microporousgrain concentration and a given equivalent micropore aspect ratio.In Figure 14, the comparison between the measured VS–VS trans-forms and the various numerical models of equivalent elastic med-
ium leads to the following results: (1) in DEM-DEM and SC-DEMmodels, predicted VP values are underestimated for high VS valuesand for micritic grain concentrations higher than 40%, (2) SC-SCmodels provide good VP predictions for high micritic grain concen-trations and overestimated values at lower concentrations, and (3)DEM-SC models match the whole data set at any micritic grain con-centrations and for pore aspect ratios averaging 0.15. As a conse-quence, the best equivalent elastic model for the Urgonianmicroporous cemented grainstones can be defined as follows: (1)a composite rock made of spherical inclusions (microporous grains)embedded within a homogeneous calcitic host and whose elasticproperties are modeled using the DEM method; and (2) micropor-ous grains made of spheroidal pores of constant aspect ratio (aver-aging 0.15) set within a homogeneous calcitic host, and whoseelastic properties are computed using the SC method. In addition,this model predicts a decrease in VP∕VS ratio with increasing
Figure 13. (a) and (b) Models of DEM-DEM and DEM-SC of com-pressionnal-wave and shear-wave velocity (VP and VS), respec-tively, as a function of porosity, (b) DEM-DEM and DEM-SCmodels of velocity ratio (VP∕VS) as a function of porosity. Labora-tory velocity measurements at 20 MPa effective pressure arereported.
Figure 14. Models of DEM-DEM, SC-DEM, DEM-SC, and SC-SC of compressional-wave velocity (VP) versus shear-wave velocity(VS) for various pore aspect ratios and for 40% (a) and 70% (b)micritic grain concentrations.
E222 Fournier et al.
porosity, as observed in measured samples (Figure 13c). Theequivalent pore-aspect ratios derived from matching laboratorymeasurements with theoretical models are plotted in Figure 15.When using DEM-SC models (Figure 15a), equivalent pore aspectratios derived from bulk modulus matching are consistent with thevalues derived from shear moduli. Values range mainly between0.12 and 0.20 and display no significant change with porosity, ex-cept few values at grain porosities higher than 30%. In contrast,significant gaps are observed between aspect ratio values derivedfrom bulk and shear modulus models using SC-DEM method(Figure 15b).
DISCUSSION: EQUIVALENT ASPECT RATIO(EPAR) AS A DIAGENETIC INDEX IN
MICROPOROUS CARBONATE RESERVOIRS?
In Urgonian micritic grains, intercrystalline micropores exhibit acomplex geometry and the question arises on the applicability andsignificance of modeling results for “perfect” spheroidal or ellipsoi-dal shapes. Tsukrov and Kachanov (1993) demonstrated thatelongated rectangular-type pores could be replaced by ellipsoidsfor modeling effective moduli. The accuracy of the method in-creases with the elongation of pores and with the randomness ofpore orientation. However, as mentioned by Kachanov (1999),pores with concave shapes cannot be replaced by ellipsoids foraccurately modeling the effective properties of the rock. In turn,rugosity of pore boundaries, sharpness of corner points and noncir-cularity of planar cracks have minor effect on overall elastic proper-ties (Kachanov and Sevostianov, 2005). In addition, Ruiz andDvorkin (2009; 2010) demonstrated that DEMmodels of compositemedia with spheroidal inclusions may match experimental velocitydata, although actual pores are not inclusions and do not exhibitspheroidal shapes. They found that the required aspect ratio tomatch data in competent sand, shale, and calcite-quartz mixture isalmost constant with values averaging 0.13 (Ruiz and Dvorkin,2010). In the present database, we show that the elastic behaviorof Urgonian microporous grainstones is equivalent to that of acemented pack of microporous spherical grains: the microporeshave an aspect ratio averaging 0.15. It is important to recognize thatthe aspect ratio used in these computations should be consideredonly as a fitting parameter but should not be regarded as an estimateof the actual pore shape.As in most of the carbonate rocks, diagenetic processes largely
control the pore network structure in microporous Urgonian grain-stones. The earlier identified stage of micrite transformation withingrains is the dissolution of an initial micrite of undetermined miner-alogy that nourished overgrowths around the most stable low-Mgcalcite crystals (Volery et al., 2010a). This phase led to the forma-tion of subhedral/euhedral micrites (MF3) microfabric. Latercementation processes dominantly caused porosity destruction inmicrite, with a very minor influence of mechanical compaction, thusleading to the low-to-moderate porosity microfabric MF2 and to thetight microfabric MF1.After DEM-SC modeling, the diagenetic and pore network trans-
formations from MF3 to MF1 microfabrics occurred at almost con-stant equivalent pore aspect ratios, averaging 0.15 (Figure 15a). Athigher grain porosities (>30%), equivalent aspect ratios display ascattered pattern with values ranging from 0.15 to 0.27. This changein equivalent aspect ratio pattern results from the occurrence of the
MF4 microfabric related to porosity enhancement by micrite crystalleaching.The relationship between elastic properties and micrite-scale
diagenetic transformations was investigated by Fournier andBorgomano (2009) in Upper Cretaceous microporous carbonatesfrom Provence. In this particular case, intergranular and intragranu-lar tight micrites exhibit anhedral compact microfabric interpretedas a result of cementation and compaction processes. The porosityreduction and the steep increase in elastic moduli with decreasingporosity was interpreted in this case as resulting from compactionand cementation of an initial well-sorted euhedral micrite. Extrac-tion of micrite elastic moduli using HS bounds provides estimatesof equivalent pore aspect ratio, according to DEM or SC schemes(Figure 16a). The steep linear increase in bulk modulus with de-creasing porosity during compaction processes resulted in a signif-icant increase (from 0.05 to 0.2–0.3) in equivalent microporeaspect ratio.As a consequence, the use of the equivalent pore aspect
ratio (EPAR) approach allows us to define two main categoriesof diagenetic transformations with regards to changes in elasticproperties in microporous micritic media: (1) EPAR-preservingtransformations, such as euhedral (MF3) to mosaic micrite(MF1), related to micropore occlusion by cementation processes;and (2) non-EPAR-preserving transformations, such as euhedralto anhedral compact micrite by compaction processes, or euhe-dral (MF3) to subrounded micrite (MF4) by leaching processes.
Figure 15. Equivalent pore aspect ratios for Urgonian microporouscemented grainstones as a function of micritic grain microporosityusing (a) DEM-SC model and (b) SC-DEM model. Error bars in-tegrate the uncertainties in laboratory measurements.
Microporous grainstone petrophysics E223
In addition, the study of the Urgonian (intragranular) micriteelastic properties shows that the critical porosity model, proposedby Fournier and Borgomano (2009) for Upper Cretaceous (inter-and intragranular) micrites from the Upper Cretaceous of Prov-ence, should not be generalized to all types of microporousmicritic media. Indeed, the application of the critical porosityconcept to micrites required the combination of two diageneticconditions: (1) a very soft initial micrite (i.e., with very low elas-tic moduli) is required and (2) the diagenetic processes of por-osity destruction should be combined with an increase inequivalent micropore aspect ratio (non-EPAR-preserving transfor-mation), as illustrated in Figure 16b.
CONCLUSIONS
The diagenetic characterization of micritic media is a major issuefor the determination of elastic moduli-porosity transforms in
microporous carbonate reservoirs. The computation of elastic prop-erty models using DEM or SC theory allows the equivalent elasticmedium for the microporous elastic medium and for the whole car-bonate rock to be defined. A major achievement of the petrographi-cal and petrophysical analysis of the Urgonian limestone is theestablishment of the link between equivalent elastic media and mi-crite diagenetic pattern in microporous carbonates. In other words,in spite of the unrealistic structure of the equivalent elastic mediacompared to the petrographic observations of micrite, the fittingparameters derived from them, such as the equivalent pore aspectratio (EPAR) can be used as index for diagenetic evolution patternin microporous micrites. Such an approach could be used practi-cally in subsurface studies after diagenetic and petrophysicalcalibration to (1) detect the diagenetic evolution patterns of micro-porous carbonate reservoirs, and (2) predict VS from VP, by usingsonic and neutron-porosity logs in uncored intervals.Finally, critical porosity concepts are not applicable to all mi-
crites. Such concepts could provide correct estimations of micriteelastic moduli only in specific diagenetic settings.
APPENDIX A
HASHIN-SHTRIKMAN BOUNDS
The exact prediction of the effective elastic moduli of a mixtureof various constituents requires the input of (1) the volume fractionof each phase, (2) the bulk and shear moduli of each phase, and (3)the spatial architecture of the mixture. If the spatial architecture ofthe mixture is unknown, the effective elastic moduli can be approxi-mated by lower and upper bounds. The HS lower and upper bounds(Hashin and Shtrikman, 1963) provide the narrowest possible rangeof elastic moduli when geometrical parameters of the mixture areunknown:(
KHSþ ¼ Kc þ f mðKm−KcÞ−1þð1−f mÞðKcþ4
3μcÞ−1
KHS− ¼ Km þ ð1−f mÞðKc−KmÞ−1þf mðKmþ4
3μmÞ−1
(A-1)
and 8>><>>:
μHSþ ¼ μc þ f mðμm−μcÞ−1þ2ð1−f mÞðKcþ2μcÞ
5μcðKcþ43μcÞ
KHS− ¼ μm þ ð1−f mÞðμc−μmÞ−1þ2f mðKmþ2μmÞ
5μmðKmþ43μmÞ
; (A-2)
where KHS− and KHSþ, respectively, are lower and upper HSbounds for bulk modulus; μHS− and μHSþ, respectively, are lowerand upper HS bounds for shear modulus; Kc and μc, respectively,are bulk and shear moduli for pure calcite; Km, μm, respectively, arebulk and shear moduli for microporous micritic grains; and f m is themicritic grain volume fraction.
APPENDIX B
DEM THEORY
The DEM theory models the effective elastic moduli of two-phase composites by adding infinitesimal quantities of inclusionsto the host phase (Cleary et al., 1980; Norris, 1985; Zimmerman,1991). In this theory, the effective bulk and shear moduli of thecomposite, K�ðyÞ and μ�ðyÞ, respectively, are governed by a coupledsystem of ordinary differential equations (Mavko et al., 1998)
Figure 16. (a) Trends of micrite bulk modulus changes with micriteporosity as a result of diagenetic processes and microtexture. Urgo-nian intragranular micrite is compared with Upper Cretaceous mi-crite from La Ciotat wells (after Fournier and Borgomano, 2009).SIM models are reported for various aspect ratios (dotted lines). InUrgonian grainstones, the changes in micrite bulk modulus withporosity for micrite porosity values lower than 25% are consistentwith diagenetic transformations at constant aspect ratio (around0.1–0.2. In contrast, in La Ciotat micrites, the steep linear increasein bulk modulus with decreasing porosity is consistent with a sig-nificant increase (from 0.05 to 0.2–0.3) in micropore aspect ratio.(b) Theoretical micrite bulk modulus versus micrite porosity in thehypothesis of a critical porosity behavior shows that comparisonwith SIM models indicates that a linear decrease in bulk moduluswith increasing porosity should be related to a decrease in micro-pore aspect ratio.
E224 Fournier et al.
ð1 − yÞ ddy
½K�ðyÞ� ¼ PðK2 − K�ÞðyÞ
ð1 − yÞ ddy
½μ�ðyÞ� ¼ Qðμ2 − μ�ÞðyÞ; (B-1)
with initial conditions K�ð0Þ ¼ K1 and μ�ð0Þ ¼ μ1; where K1, μ1 ¼bulk and shear moduli of the initial host material; respectively;K2, μ2 ¼ bulk and shear moduli of the inclusion; respectively;y ¼ concentration of the inclusions. The coefficients P and Q de-pend upon the shape of the inclusion and upon the elastic moduli ofthe host and inclusion phases. For ellipsoidal inclusions of a givenaspect ratio α, P and Q are given by (Wu, 1966)
P ¼ 1∕3 Tiijj and Q ¼ 1∕5ðT ijij − 1∕3 TiijjÞ; (B-2)
where the tensor Tijkl relates the uniform far-field strain to the strainwithin the ellipsoidal inclusion. Tijkl are functions of the inclusionaspect ratio α and of the bulk and shear moduli of the initial host, K1
and μ1, respectively, and of the inclusions, K2 and μ2, respectively(Mavko et al., 1998). For spherical inclusions, P and Q are given by(Berryman, 1995)
P ¼ Kc þ 43μc
Km þ 43μc
(B-3)
and
Q ¼ Kc þ ξcKm þ ξc
; with ξc ¼μc6
ð9Kc þ 8μcÞðKc þ 2μcÞ
(B-4)
APPENDIX C
SELF-CONSISTENT APPROXIMATION
The SC approximation (Budiansky, 1965; Wu, 1966) allows us topredict the elastic moduli of a composite materials with inclusions.In this approach, the interaction of the inclusions is approximatedby replacing the background medium with an as-yet-unknown ef-fective medium and each constituent is treated symmetrically.The SC formulas for bulk K and shear μ moduli of a 2C
rock (one host phase, one inclusion phase) are
f iðKi − KÞPi þ ð1 − f iÞðKh − KÞPh ¼ 0 (C-1)
f iðμi − μÞQi þ ð1 − f iÞðμh − μÞQh ¼ 0; (C-2)
where Kh and μh, respectively, are bulk and shear moduli of the hostmaterial; Ki and μi, respectively, are bulk and shear moduli of theinclusion; f i: volume fraction of the inclusion; and P, Q are geo-metrical factors where the superscript i (respectively, h) indicatesthat the factor is for the material of elastic moduli Ki and μi (respec-tively, Kh and μh) in a background medium of elastic moduli K andm.The equations C-1 and C-2 are solved iteratively as follows:
(Knþ1 ¼ ΦKiPi
nþð1−ΦÞKhPhn
ΦPinþð1−ΦÞPh
n
μnþ1 ¼ ΦμiQinþð1−ΦÞμhQh
n
ΦQinþð1−ΦÞQh
n
(C-3)
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