Compressibility from Core

www.senergyworld.com

Compressibility from Core ☺ Phil McCurdy and Colin McPhee

☺and from logs too

Why is compressibility important??? Hydrocarbon recovery Reservoir depletion causes increase in effective stress Pore volume compacts and adds energy to reservoir Pore volume compressibility used in material balance calculations

Porosity and permeability reduction

1 0.95 In-Situ

0.9 0.85 0.8 0.75 0.7 0.65 0.6

5000

4500

4000

3500

3000

2500

2000

1500

1000

500

0

Inferred Reservoir Pressure, psi

Compaction and subsidence (weak sands & HPHT) Compaction can lead to casing and tubular failures Compaction can lead to surface subsidence Compaction linked to compressibility 2

Permeability Multiplier

Reduction in porosity and permeability with increasing effective stress on depletion Productivity reduction in depleting reservoirs

1.05

Compressibility terms and calculations Compressibility units 10-6psi-1 referred to as “microsips”

Grain compressibility, Cma or Cg Cg ~ 0.16 – 0.20 microsips

Cup for a “microsip”

Bulk Modulus, K related to rock stiffness inverse of compressibility

E K = 3(1 − 2ν )

Cb =

1 K

Bulk Compressibility, Cb Cbc –constant pore pressure and changing confining pressure Cbc =

1 ⎧ ∂Vb ⎫ ⎬ ⎨ Vb ⎩ ∂Pc ⎭ Pp

Vb = bulk volume

Pc= confining pressure

Pp = pore pressure

Cbp - under constant confining pressure and changing pore pressure (depletion) 1 ⎧ ∂Vb ⎫ Cbp =

⎨ ⎬ Vb ⎩ ∂Pp ⎭ Pc

3

Compressibility terms and calculations Bulk and Grain Compressibility

Cbp = Cbc − Cg As Cg is small in comparison, Cbc ≈ Cbp

Pore Volume Compressibility, Cf (Dake) or Cp Cpc - isostatic pore volume compressibility under constant pore pressure and changing confining pressure

Cpc =

1 ⎧ ∂Vp ⎫ ⎨ ⎬ Vp ⎩ ∂Pc ⎭ Pp

⎡ Cbc − Cg ⎤ Cpc = ⎢ ⎥ φ ⎣ ⎦

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

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

Cpp =

1 ⎧ ∂Vp ⎫ ⎨ ⎬ Vp ⎩ ∂Pp ⎭ Pc

Cpp = Cpc − Cg

As Cg is small in comparison, Cpp ≈ Cpc

4

Measurement Conditions σv = σz

Reservoir (Triaxial) three principal stresses uniaxial loading

SCAL Labs isostatic loading radial stress = axial stress

Rock Mechanics Labs

σhmin = σy σhmax = σx Axial Radial

biaxial loading radial stress < axial stress 5

Compressibility terms and calculations Isostatic and Uniaxial Compressibility, Cpu uniaxial loading assumes reservoir formations behave elastically and are boundary constrained in horizontal direction assumes strain is entirely vertical assumes no tectonic strain during burial loading Reservoir has stiff lateral restraints

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

⎡α (1 +ν ) ⎤ Cpu = Cpp ⎢ ⎥ ⎣ 3(1 −ν ) ⎦ For example, Biot factor (α) = 1 and ν = 0.3 then Cpu = 0.62*Cpp 6

Typical Lab Presentation ⎡α (1 +ν ) ⎤ Cpu = Cpp ⎢ ⎥ ⎣ 3(1 −ν ) ⎦

Note neither α nor υ are measured!

7

Core Test Methods Direct Measure change in pore volume as a function of increasing effective stress

σ 'iso = σ iso − αp p Effective stress method – SCAL labs Increase σ to increase σ’

Simulated depletion method – SCAL labs Reduce Pp to increase σ’

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

Indirect From E and υ from triaxial tests 8

Direct Measurements – SCAL Lab

Effective Stress Method SCAL lab method (porosity/FF at overburden) pore pressure constant, radial pressure increased effective stress increased by increasing confinement 1 ⎧ ∂Vp ⎫ 1 ⎧δVp ⎫ pore volume by squeeze-out = Cpc = ⎨ ⎬ ⎨ ⎬ Vp ⎩ ∂Pc ⎭ Pp

Vp ⎩ δσ ' ⎭

Simulated Depletion Method raise stresses and pore pressure to reservoir values total stress (Pc) constant – Pp reduced 1 ⎧ ∂Vp ⎫ 1 ⎧δVp ⎫ depletion Cpp = ⎬ ⎨ ⎨ ⎬ = Vp ⎩ ∂Pp ⎭ Pc Vp ⎩ δσ ' ⎭ isostatic pore volume compressibility (SCAL) 9

Uniaxial Ko Test Sample instrumented with axial and radial strain gauges Core Compaction Sample loaded to same total vertical (axial) and total horizontal (radial) stresses as in reservoir Pore pressure increased to reservoir value Pore pressure reduction ∆σh vertical stress stays the same horizontal stress adjusted to maintain εh = 0 zero radial strain rock mechanics labs only uniaxial pore volume compressibility (K0) ∆pp 1 ⎧ ∂Vp ⎫

Cpu =

⎨ ⎬ Vp ⎩ ∂Pp ⎭ε radial =0

10

Example PV calculation – SCAL data

14.20

14.00

cf ( hyd ) =

Pore Volume (ml)

13.80

1 (Vpi − Vpd ) Vp i (σ 'i −σ 'd )

Data Model

13.60

13.40 Initial Reservoir Pressure Depleted Reservoir Pressure 13.20

13.00 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Effective Hydrostatic Pressure (psi)

11

Stress Hysteresis Effective Stress Method initial loading cycle microcracks in plug close higher pore volume reduction OK for φ stress correction Simulated Depletion Method extended loading cycle load to initial conditions (cracks close) depletion stage (Cp from matrix pore volume compaction) GAUGE

ROSETTE

more reliable pore volume compressibility data

Uniaxial KO Method potentially most reliable data closest representation of stresses/pressures during depletion 12

Stress Hysteresis Example

50.0

45.0

Pore Volume Compressibility (x10-6psi-1)

40.0

1A 2A 3A 4A 5A 6A 7A 8A 1D 2D 3D 4D 5D 6D 7D 8D

35.0

30.0

25.0

20.0

Suffix A: Effective Stress Method Suffix D: Simulated Stress Mrthod

15.0

10.0

5.0

0.0 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Effective Overburden Stress (psi)

13

Indirect Method Triaxial data Determine E and υ over equivalent deviatoric stress range associated with depletion

⎡ Cbc − Cg ⎤ Cpc = ⎢ ⎥ φ ⎣ ⎦

1 Cbc = K

K =

E 3(1 − 2ν ) 14

Compressibility from Logs DSI Logs DTS (∆ts), DTCO (∆tc)

Obtain dynamic (elastic) moduli Poisson’s Ratio, ν

1 (∆t s / ∆tc )2 − 1 2 (∆t s / ∆tc )2 − 1

1.34 x1010

Shear Modulus, G (psi)

ρb ∆t s2

2G (1 + ν )

Young’s Modulus, E (psi)

K=

Bulk Modulus, Kb (psi)

1 Kb

-1

Bulk Compressibility, Cbc (psi )

-1

Pore Volume Compressibility, Cpc (psi )

ρb in g/cc ∆t in µsecs/ft

E 3(1 − 2ν )

⎡ Cbc − Cg ⎤ Cpc = ⎢ ⎥ φ ⎣ ⎦ 15

Scaling Dynamic and Static Moduli Dynamic elastic and perfectly reversible

Static (core) large strains irreversible

Scaling static ε < dynamic ε Esta = 0.15 - 0.5 Edyn − νsta = 0.8 - 1.2 νdyn

16

Compaction and Subsidence Compaction change in reservoir thickness (Hres) as a result of depletion (Geertsma) ∆H = C m H res ( Pi − Pfinal )

Depth, D

Compaction coefficient 1 ⎡1 +ν ⎤ (1 − β )Cb Cm = ⎢ 3 ⎣1 −ν ⎥⎦

Subsidence

Mud Line

Compaction

Cg β= Cb

Casing compressive strain

Thickness, H H

ε c = 0.5(1 + cos 2θ )Cm ∆p Reservoir Radius, R

Subsidence (Bruno)

[

]

S = 2C m (1 − ν ) H res − (R 2 + ( D + H res ) 2 ) + ( R 2 + D 2 ) 0.5 ∆p 0 .5

17

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

18