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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
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