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Compressibility from Core ☺

Phil McCurdy and Colin McPhee

☺and from logs too

2

Why is compressibility important???

Hydrocarbon recoveryReservoir depletion causes increase in effective stressPore volume compacts and adds energy to reservoirPore volume compressibility used in material balance calculations

Porosity and permeability reductionReduction in porosity and permeability with increasing effective stress on depletionProductivity reduction in depleting reservoirs

Compaction and subsidence (weak sands & HPHT)Compaction can lead to casing and tubular failuresCompaction can lead to surface subsidenceCompaction linked to compressibility

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

0500100015002000250030003500400045005000

Inferred Reservoir Pressure, psi

Per

mea

bilit

y M

ultip

lierIn-Situ

3

Compressibility terms and calculations

Compressibility units10-6psi-1 referred to as “microsips”

Grain compressibility, Cma or CgCg ~ 0.16 – 0.20 microsips

Bulk Modulus, Krelated to rock stiffnessinverse of compressibility

Bulk Compressibility, CbCbc –constant pore pressure and changing confining pressure

Cbp - under constant confining pressure and changing pore pressure (depletion)

)21(3 ν−=

EKK

Cb 1=

PpPcVb

VbCbc

⎭⎬⎫

⎩⎨⎧∂∂

=1

PcPpVb

VbCbp

⎭⎬⎫

⎩⎨⎧∂∂

=1

Vb = bulk volume Pc= confining pressure Pp = pore pressure

Cup for a “microsip”

4

Compressibility terms and calculations

Bulk and Grain Compressibility

As Cg is small in comparison, Cbc ≈ CbpPore Volume Compressibility, Cf (Dake) or Cp

Cpc - isostatic pore volume compressibility under constant pore pressure and changing confining pressure

Cpp – isostatic pore volume compressibility under constant confining pressure and changing pore pressure (depletion)

As Cg is small in comparison, Cpp ≈ Cpc

⎥⎦

⎤⎢⎣

⎡ −=

φCgCbcCpc

PpPcVp

VpCpc

⎭⎬⎫

⎩⎨⎧∂∂

=1

PcPpVp

VpCpp

⎭⎬⎫

⎩⎨⎧∂∂

=1

CgCbcCbp −=

CgCpcCpp −=

i.e. pore volume compressibility is 3 to 5 times higher than bulk compressibility

5

Measurement Conditions

Reservoir (Triaxial)three principal stressesuniaxial loading

SCAL Labsisostatic loadingradial stress = axial stress

Rock Mechanics Labsbiaxial loadingradial stress < axial stress

Axial

Radial

σv = σz

σhmax = σx

σhmin = σy

6

Compressibility terms and calculations

Isostatic and Uniaxial Compressibility, Cpuuniaxial loading assumes reservoir formations behave elasticallyand are boundary constrained in horizontal direction

assumes strain is entirely verticalassumes no tectonic strain during burial loading

Cpu defined as uniaxial pore volume compressibility under producing conditions (from Teeuw)

For example, Biot factor (α) = 1 and ν = 0.3 then Cpu = 0.62*Cpp

( )( ) ⎥⎦

⎤⎢⎣

⎡−+

=ννα

131CppCpu

Reservoir has stiff lateral restraints

7

Typical Lab Presentation

( )( ) ⎥⎦

⎤⎢⎣

⎡−+

=ννα

131CppCpu Note neither α nor υ are measured!

8

Core Test Methods

DirectMeasure change in pore volume as a function of increasing effective stress

Effective stress method – SCAL labsIncrease σ to increase σ’

Simulated depletion method – SCAL labsReduce Pp to increase σ’

Uniaxial (K0) Test – Rock Mechanics labsReduce pp to increase σ’Instrument core to determine strains

IndirectFrom E and υ from triaxial tests

pisoiso pασσ −='

9

Direct Measurements – SCAL Lab

Effective Stress MethodSCAL lab method (porosity/FF at overburden)pore pressure constant, radial pressure increasedeffective stress increased by increasing confinementpore volume by squeeze-out

Simulated Depletion Methodraise stresses and pore pressure to reservoir valuestotal stress (Pc) constant – Pp reduceddepletionisostatic pore volume compressibility (SCAL)

⎭⎬⎫

⎩⎨⎧=

⎭⎬⎫

⎩⎨⎧∂∂

='

11δσδVp

VpPcVp

VpCpc

Pp

⎭⎬⎫

⎩⎨⎧=

⎭⎬⎫

⎩⎨⎧∂∂

='

11δσδVp

VpPpVp

VpCpp

Pc

10

Uniaxial Ko Test

Sample instrumented with axial and radial strain gaugesSample loaded to same total vertical (axial) and total horizontal (radial) stresses as in reservoirPore pressure increased to reservoir valuePore pressure reduction

vertical stress stays the samehorizontal stress adjusted to maintain zero radial strainrock mechanics labs onlyuniaxial pore volume compressibility (K0) ∆pp

∆σh

Core Compaction

εh = 0

0

1

=⎭⎬⎫

⎩⎨⎧∂∂

=radial

PpVp

VpCpu

ε

11

Example PV calculation – SCAL data

13.00

13.20

13.40

13.60

13.80

14.00

14.20

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Effective Hydrostatic Pressure (psi)

Pore

Vol

ume

(ml)

DataModel

Initial Reservoir Pressure

Depleted Reservoir Pressure

( )di

diihyd

VpVpVp

cf''

)(1)( σσ −

−=

12

Stress Hysteresis

Effective Stress Methodinitial loading cyclemicrocracks in plug closehigher pore volume reductionOK for φ stress correction

Simulated Depletion Methodextended loading cycle

load to initial conditions (cracks close)depletion stage (Cp from matrix pore volume compaction)more reliable pore volume compressibility data

Uniaxial KO Methodpotentially most reliable dataclosest representation of stresses/pressures during depletion

GAUGEROSETTE

13

Stress Hysteresis Example

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

50.0

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Effective Overburden Stress (psi)

Pore

Vol

ume

Com

pres

sibi

lity

(x10

-6ps

i-1)

1A2A3A4A5A6A7A8A1D2D3D4D5D6D7D8D

Suffix A: Effective Stress MethodSuffix D: Simulated Stress Mrthod

14

Indirect Method

Triaxial dataDetermine E and υ over equivalent deviatoric stress range associated with depletion

KCbc 1

=)21(3 ν−

=EK⎥

⎦

⎤⎢⎣

⎡ −=

φCgCbcCpc

15

Compressibility from Logs

DSI LogsDTS (∆ts), DTCO (∆tc)

Obtain dynamic (elastic) moduli

)21(3 ν−=

EK

Poisson’s Ratio, ν

Shear Modulus, G (psi)

Young’s Modulus, E (psi)

Bulk Modulus, Kb (psi)

Bulk Compressibility, Cbc (psi-1)

Pore Volume Compressibility, Cpc (psi-1)

( )( ) 1/

1/21

2

2

−∆∆

−∆∆

cs

cs

tt

tt

2101034.1

s

b

tx

∆ρ

( )ν+12G

bK1

ρb in g/cc

∆t in µsecs/ft

⎥⎦

⎤⎢⎣

⎡ −=

φCgCbcCpc

16

Scaling Dynamic and Static Moduli

Dynamicelastic and perfectly reversible

Static (core)large strainsirreversible

Scalingstatic ε < dynamic εEsta = 0.15 - 0.5 Edyn

− νsta = 0.8 - 1.2 νdyn

17

Compaction and Subsidence

Compactionchange in reservoir thickness (Hres) as a result of depletion (Geertsma)

Compaction coefficient

Casing compressive strain

Subsidence (Bruno)

( )CbCm βνν

−⎥⎦⎤

⎢⎣⎡−+

= 111

31

CbCg

=β

)( finaliresm PPHCH −=∆Depth, D

Thickness, HH

Subsidence

Compaction

Reservoir Radius, R

Mud Line

Depth, D

Thickness, HH

Subsidence

Compaction

Reservoir Radius, R

Mud Line

( )[ ] pDRHDRHCS resresm ∆++++−−= 5.0225.022 )()()1(2 ν

pCmc ∆+= )2cos1(5.0 θε

18

Conclusions

Common techniques for measuring compressibility and situations that they are most suited to