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: Francois.Fournier@univ-provence.fr;

Jean.Borgomano@univ-provence.fr; k.biscarrat@aliceadsl.fr; peycors13@hotmail.com.2VU University Amsterdam, Faculty of Earth and Life Sciences, Department of Sedimentology and Marine Geology, Amsterdam, The Netherlands. E-mail:

philippe.leonide@univ-provence.fr.3Geo-Instituut Katholieke Universiteit Leuven, Belgium. E-mail: Anneleen.Foubert@ees.kleuven.be.

© 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)

REFERENCES

Ahr, W. M., 1989, Early diagenetic microporosity in the cotton valley lime-stone of east Texas: Sedimentary Geology, 63, 275–292, doi: 10.1016/0037-0738(89)90136-X.

Anselmetti, F. S., and G. P. Eberli, 1993, Controls on sonic velocity in car-bonates: Pure and Applied Geophysics, 141, 287–323, doi: 10.1007/BF00998333.

Assefa, S., C. McCann, and J. Sothcott, 2003, Velocities of compressionaland shear waves in limestones: Geophysical prospecting, 51, 1–13, doi:10.1046/j.1365-2478.2003.00349.x.

Baechle, G. T., A. Colpaert, G. P. Eberli, and R. J. Weger, 2008, Effects ofmicroporosity on sonic velocity in carbonate rocks: The Leading Edge,27, 1012–1018, doi: 10.1190/1.2967554.

Baronnet, A., 1982, Ostwald ripening in solution— The case of calcite andmica: Estudios Geologicos, 38, 185–198.

Berryman, J. G., 1980, Long-wavelength propagation in composite elasticmedia II. Ellipsoidal inclusions: Journal of the Acoustical Society ofAmerica, 68, 1809–1831, doi: 10.1121/1.385172.

Berryman, J. G., 1995, Mixture theories for rock properties, in T. J. Ahrens,ed., Handbook of physical constants: AGU, 205–228.

Budd, D. A., 1989, Micro-rhombic calcite and microporosity in limestones:A geochemical study of the lower cretaceous thamama group, U.A.E.:Sedimentary Geology, 63, no. 3–4, 293–311, doi: 10.1016/0037-0738(89)90137-1.

Budiansky, B., 1965, On the elastic moduli of some heterogeneous materi-als: Journal of the Mechanics and Physics of Solids, 13, 223–227, doi:10.1016/0022-5096(65)90011-6.

Cantrell, D. L., and R. M. Hagerty, 1999, Microporosity in Arab formationcarbonates, Saudi Arabia: GeoArabia, 4, no. 2, 129–154.

Clark, V. A., B. R. Tittman, and T. W. Spencer, 1980, Effects of volatiles onattenuation and velocity in sedimentary rocks: Journal of GeophysicalResearch, 85, 5190–5198, doi: 10.1029/JB085iB10p05190.

Cleary, M. P., I. W. Chen, and S. M. Lee, 1980, Self-consistent techniquesfor heterogeneous media: Journal of the Engineering Mechanics Division,American Society of Civil Engineers, 106, 861–887.

Duffaut, A., and M. Landrø, 2007, VP∕VS ratio versus differential stress androck consolidation— A comparison between rock models and time-lapseAVO data: Geophysics, 72, no. 5, C81–C94, doi: 10.1190/1.2752175.

Eberli, G. P., B. Baechle, F. Anselmetti, and M. Incze, 2003, Factors con-trolling elastic properties in carbonate sediments and rocks: The LeadingEdge, 22, 654–660, doi: 10.1190/1.1599691.

Fournier, F., and J. Borgomano, 2009, Critical porosity and elastic propertiesof microporous mixed carbonate-siliciclastic rocks: Geophysics, 74, no. 2,93–109, doi: 10.1190/1.3043727.

Gardner, G. H. F., L. W. Gardner, and R. W. Gregory, 1974, Formation ve-locity and density: The diagnostic basis for stratigraphic traps: Geophy-sics, 39, 770–780, doi: 10.1190/1.1440465.

Grammer, G. M., P. M. Harris, and G. P. Eberli, 2004, Integration of outcropand modern analogs in reservoir modeling: Overview with examples fromthe Bahamas, in G. M. Grammer, P. M. Harris, and G. P. Eberli, eds.,Integration of Outcrop and Modern Analogs in Reservoir Modeling:AAPG Memoir, 80, 1–22.

Hashin, Z., and S. Shtrikman, 1963, A variational approach to the elasticbehaviour of multiphase materials: Journal of the Mechanics and Physicsof Solids, 11, 127–140, doi: 10.1016/0022-5096(63)90060-7.

Kachanov, M., 1999, Solids with cracks and non-spherical pores: Properparameters of defect density and effective elastic properties: InternationalJournal of Fracture, 97, 1–32, doi: 10.1023/A:1018345702490.

Kachanov, M., and I. Sevostianov, 2005, On quantitative characterization ofmicrostuctures and effective properties: International Journal of Solidsand Structures, 42, 309–336, doi: 10.1016/j.ijsolstr.2004.06.016.

Kaldi, J., 1989, Diagenetic microporosity (chalky porosity), Middle Devo-nian Kee Scarp Reef Complex, Norman Wells, Northwest Territories,Canada: Sedimentary Geology, 63, 241–252, doi: 10.1016/0037-0738(89)90134-6.

Kenter, J. A. M., and M. Ivanov, 1995, Parameters controlling acoustic prop-erties of carbonate and volcaniclastic sediments at sites 866 and 869, in E.L. Winterer, W. W. Sager, J. V. Firth, and J. M. Sinton, eds., Proceedingsof the ocean drilling program, scientific results, ocean drilling program,143, 287–303.

Kirkham, A., M. Bin Juma, T. A. M. McKean, A. F. Palmer, M. J. Smith, A.H. Thomas, and B. N. Twombley, 1996, Fluid saturation prediction in the“transition zone” carbonate reservoir, Abu Dhabi: GeoArabia, 1,551–566.

Microporous grainstone petrophysics E225

Lambert, L., C. Durlet, J. P. Loreau, and G. Marnier, 2006, Burial dissolutionof micrite in Middle East carbonate reservoirs (Jurassic-Cretaceous): Keysfor recognition and timing: Marine and Petroleum Geology, 23, 79–92,doi: 10.1016/j.marpetgeo.2005.04.003.

Loreau, J. P., 1972, Pétrographie des calcaires fins au microscope électro-nique à balayage introduction à une classification des “micrites”: ComptesRendus de l’Académie des Sciences, Paris, 274, 810–813.

Masse, J. P., 1976, Les calcaires urgoniens de Provence: Ph.D. thesis,Aix-Marseille University.

Masse, J. P., 1993, Valanginian to early Aptian carbonate platforms fromProvence (S.E. France), in A. Simo, R. W. Scott, and J. P. Masse,eds., Cretaceous carbonate platforms: AAPG Memoir, 56, 363–374.

Mavko, G., T. Mukerji, and J. Dvorkin, 1998, The rock physicshandbook: Tool for seismic analysis in porous media: CambridgeUniversity Press.

Mavko, G., T. Mukerji, and N. Godfrey, 1995, Predicting stress-induced ve-locity anisotropy in rocks: Geophysics, 60, 1081–1087, doi: 10.1190/1.1443836.

Moore, C. H., 1989, Carbonate diagenesis and porosity: Elsevier, Develop-ments in Sedimentology, 46.

Morse, J. W., and W. H. Casey, 1988, Ostwald processes and mineral para-genesis in sediments: American Journal of Science, 288, 537–560, doi:10.2475/ajs.288.6.537.

Morse, J. W., and F. T. Mackenzie, 1990, Geochemistry of sedimentary car-bonates: Elsevier, Developments in Sedimentology, 48.

Moshier, S. O., 1989, Development of microporosity in a micritic limestonereservoir, Lower Cretaceous, Middle East: Sedimentary Geology, 63,217–240, doi: 10.1016/0037-0738(89)90133-4.

Norris, A. N., 1985, A differential scheme for the effective moduli of com-posites: Mechanics of Materials, 4, 1–16, doi: 10.1016/0167-6636(85)90002-X.

Rafavich, F., C. H. S. C. Kendall, and T. P. Todd, 1984, The relationshipbetween acoustic properties and the petrographic character of carbonaterocks: Geophysics, 49, 1622–1636, doi: 10.1190/1.1441570.

Richard, J., J. P. Sizun, and L. Machhour, 2007, Development and compart-mentalization of chalky carbonate reservoirs: The Urgonian Jura-BasDauphiné platform model (Génissiat, southeastern France): SedimentaryGeology, 198, 195–207, doi: 10.1016/j.sedgeo.2006.12.003.

Røgen, B., I. L. Fabricius, P. Japsen, C. Høier, G. Mavko, andJ. M. Pedersen, 2005, Ultrasonic velocities of North Sea chalk samples— Influence of porosity, fluid content and texture: Geophysical Prospect-ing, 53, 481–496, doi: 10.1111/gpr.2005.53.issue-4.

Ruiz, F., and J. Dvorkin, 2009, Sediment with porous grains: Rock-physicsmodel and application to marine carbonate and opal: Geophysics,74, no. 1, E1–E15, doi: 10.1190/1.3033212.

Ruiz, F., and J. Dvorkin, 2010, Predicting elasticity in nonclastic rocks withdifferential effective medium model: Geophysics, 75, no. 1, E41–E53,doi: 10.1190/1.3267854.

Saller, A. H., and C. H. Moore, 1989, Meteoric diagenesis, marine diagen-esis, and microporosity in Pleistocene and Oligocene limestones, Enewe-tak Atoll, Marshall Islands: Sedimentary Geology, 63, 253–272, doi:10.1016/0037-0738(89)90135-8.

Samankassou, E., J. Tresch, and A. Strasser, 2005, Origin of peloidsin Early Cretaceous deposits, Dorset, South England: Facies, 51,264–273, doi: 10.1007/s10347-005-0002-8.

Smith, T. M., C. M. Sayers, and C. H. Sondergeld, 2009, Rock properties inlow-porosity/low permeability sandstones: The Leading Edge, 28, 48–59,doi: 10.1190/1.3064146.

Tsukrov, I., and M. Kachanov, 1993, Solids with holes of irregular shapes:Effective moduli and anisotropy: International Journal of Fracture, 64,R9–R12, doi: 10.1007/BF00019630.

Tucker, M. E., and R. G. C. Bathurst, 1990, Carbonate diagenesis, ReprintSeries 1 of the IAS: Blackwell Scientific Publications.

Tucker, M. E., and V. P. Wright, 1990, Carbonate Sedimentology: BlackwellScientific Publications.

van der Plas, J., and A. C. Tobi, 1965, A chart for judging the reliability ofpoint counting results: American Journal of Science, 263, 87–90, doi:10.2475/ajs.263.1.87.

Vernik, L., 1994, Predicting lithology and transport properties from acousticvelocities based on petrophysical classification of siliciclastics: Geophy-sics, 59, 420–427, doi: 10.1190/1.1443604.

Vernik, L., 1997, Prediction porosity from acoustic velocities in siliciclas-tics: A new look: Geophysics, 62, 118–128, doi: 10.1190/1.1444111.

Volery, C., E. Davaud, C. Durlet, B. Clavel, J. Charollais, and B. Caline,2010b, Microporous and tight limestones in the Urgonian Formation(late Hauterivian to early Aptian) of the French Jura Mountains:Focus on the factors controlling the formation of microporous facies:Sedimentary Geology, 230, no. 1–2, 21–34, doi: 10.1016/j.sedgeo.2010.06.017.

Volery, C., E. Davaud, A. Foubert, and B. Caline, 2009, Shallow-marinemicroporous carbonate reservoir rocks in the Middle East: relationshipwith seawater Mg/Ca ratio and eustatic sea level: Journal of PetroleumGeology, 32, 313–325, doi: 10.1111/jpg.2009.32.issue-4.

Volery, C., E. Davaud, A. Foubert, and B. Caline, 2010a, Lacustrinemicroporous micrites of the Madrid Basin (Late Miocene, Spain) asanalogues for shallow-marine carbonates of the Mishrif reservoirFormation (Cenomanian to Early Turoniam, Middle East): Facies, 56,385–397.

Wang, Z., 1997, Seismic properties of carbonate rocks, in I. Palaz, and K. J.Marfurt eds., Carbonate seismology: Geophysical Developments, 6,29–52.

Wilkens, R., G. Simmons, and L. Caruso, 1984, The ratio VP∕VS as a dis-criminant of composition for siliceous limestones: Geophysics, 49,1850–1860, doi: 10.1190/1.1441598.

Wilson, J. L., 1975, Carbonate facies in geologic history: Springer-Verlag.Wilson, J. L., 1980, Limestone and dolomite reservoirs, in G. D. Hobson,

ed.: Petroleum Geology: Applied Science Publisher Ltd, 2, 1–51.Witt, W., and H. Gokdag, 1994, Orbitolinid biostratigraphy of the Shuaiba

Formation (Aptian), Oman. Implications for reservoir development, inM. D. Simmons, ed., Micropalaeontology and hydrocarbon explorationin the Middle East: Chapman & Hall, 221–234.

Wu, T. T., 1966, The effect of inclusion shape on elastic moduli of a two-phase material: International Journal of Solids and Structures, 2, 1–18.

Zimmerman, R. W., 1991, Compressibility of sandstones: Elsevier.

E226 Fournier et al.