Mechanical effect of absorption Carbon sequestration and swelling of coal

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Mechanical effect of absorption Carbon sequestration and swelling of coal

Laurent Brochard

To cite this version:

Laurent Brochard. Mechanical effect of absorption Carbon sequestration and swelling of coal. TODO, Jan 2011, Lyon, France. �hal-00563374�

Mechanical effect of adsorption

Carbon sequestration and swelling of coal

Laurent B ^ROCHARD

Université Paris-Est. Laboratoire Navier (UMR CNRS 8205). Ecole des Ponts ParisTech

C ^ONTEXT - C ARBON SEQUESTRATION AND SWELLING OF COAL

In most scenarios for

stabilization of atmospheric greenhouse gas

concentrations [...] CCS

contributes 15 - 55% to the cumulative mitigation effort worldwide

(From: IPCC report on Carbon Capture and Sequestration (2005))

Pressure,psi

Time, year

Pressure 0

500 1000 1500 2000 2500

Rate of injection

Rateofinjection,Mcf/mo

0

10000 20000 30000 40000 50000 60000

Sequestration in coalbeds is promising (long term storage is safe and natural gas can be recovered, which improves the financial viability), but it is affected by a permeability issue. The injection pilots have encountered an important loss of permeability of the reservoir, after a few months of injection.

Left: case of the Allison Unit, San Juan Basin (NM), US DOE.

(adapted from Pekot & Reeves (2002))

Cause: coal swells more in a CO

₂

atmosphere than in a CH

₄

atmosphere

Experiment:

inject either CO

₂

or CH

₄

in a coal sample free of stress

P

_CO2

small P

_CO2

high

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0 2 4 6 8 10 12 14

Pressure, MPa

V olumetric strain, %

CH

₄

CO

₂

He

(adapted from Ottiger et al. (2008))

Conventional poromechanics fails to explain the swelling

Helmholtz free energy of the solid matrix of a saturated isotropic porous medium:

f

_solid

= 1

2 K + b

²

N

²

− bN ϕ + N

2 ϕ

²

+ X

i,j∈{1,2,3}

G 2 e

²_ij

where:

is the volumetric strain, ϕ the change of porosity, G the shear modulus, and N the Biot modulus.

e_ij the deviatoric strains, K the bulk modulus,

b the Biot coefficient,

The volumetric stress is obtained with the state equation:

σ = ∂ f

_solid

∂

ϕ,e_ij

= K − bP

(Coussy (2010))

Unjacketed experiment:

P

_CO2

small P

_CO2

high

Conventional poromechanics predicts a shrinkage, and the same whatever the gas!

σ = −P ⇒ = − 1 − b

K P < 0

Objective: Understand the physics of swelling and predict the permeability loss

C ONVENTIONAL POROMECHANICS EXTENDED TO SURFACE EFFECTS

Bulk density Γ = N^s/A

Density

Solid

Contrary to CO

₂

and CH

₄

, helium behaves as predicted by poromechanics.

The difference stems from the adsorption (low for Helium, high for CO

₂

and CH

₄

) which may have an impact on the mechanics of a solid, since it modifies the fluid-solid interface energy γ

_FS

(Gibbs adsorption equation, at fixed

temperature and interface area): d γ

_FS

= −Γd µ. The mechanical impact of adsorption can be sketched in the case of a thin plate:

Adsorbed

layer

Helmholtz free energy of the (solid matrix + interface):

f_solid = 1

2 K + b²N

² − bNϕ + N

2ϕ² + X

i,j∈{1,2,3}

G

2e_ij²+γs

s the specific surface., and γ_FS the fluid-solid interface energy.

σ = K − bP + σ e

_s

∂ s

∂

P

where e σ

_s

= γ + s ∂γ

∂s is the interface stress.

(Vandamme et al. (2010))

Predicts a possible swelling, specific to each fluid:

= − 1 − b

K P − ∆ σ e

_s

K

∂s

∂

P

(1)

with ∆γ = − R

µ µ₀

N^Ex

A

d µ

and

∆ σ e

_s

= ∆γ + s

^∂∆γ_∂s

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 10 20 30 40 50 60 70

Z position, Å NumberDensity,10−2 Å−1

CO₂ CH₄

CO

₂

and CH

₄

density profiles on the surface of coal matrix were obtained by molecular simulation.The swelling

predicted by the model (Equation 1) is compared to the experimental swelling.

The model does not capture the experimental swelling

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 1 2 3 4 5 6

Bulk pressure, MPa

Volumetricstrain,% Experimental swelling CH₄

Experimental swelling CO₂ Predicted swelling CH₄

Predicted swelling CO₂

(Experimental data from Levine (1996))

P OROMECHANICS OF NANOPORE ADSORPTION

Explanation: Adsorption occurs also in the nanopores of the solid matrix.

Guideline for the poromechanics of nanopore adsorption: Use known and measurable quantities only. Questionable quantities (porosity, specific surface) should not intervene.

Behavior law of a porous solid subjected to a fluid under any form (bulk, surface

adsorption, nanopore adsorption...) σ = df

_S

d − ∂

∂

Z

µ

−∞

N

V

₀

d µ

µ

(2)

V₀ is the volume of the porous solid under unstressed conditions,

N is the number of fluid molecules whatever their state (adsorbed in nanopores, on surfaces, bulk...).

f_S = F_S/V₀ and F_S is the free energy of the sole solid, that is when there is no fluid molecule in the pores.

f_S = ¹₂K² for an elastic solid.

Requires to know the adsorbed amount as a function of both µ and .

Case of a 1D chain subjected to fluid adsorption

Fugacity

−3

−2

−1 0 1 2

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

Strain Confining stress σ , 10

−10

N

2 × 10

⁻¹¹

1 × 10

⁻¹¹

5 × 10

⁻¹²

1 × 10

⁻¹²

0

The uneven adsorption behavior can be explained by the pore commensurability

Fugacity

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

-0.1 -0.05 0 0.05 0.1

Strain

NumberofgasmoleculesN/V0,Å−1

2×10⁻¹¹ 1×10⁻¹¹ 7.5 ×10⁻¹² 5×10⁻¹² 3×10⁻¹² 2×10⁻¹² 1×10⁻¹² 7.5 ×10⁻¹³ 5×10⁻¹³ 2.5 ×10⁻¹³ 1×10⁻¹³ 1×10⁻¹⁴ 0

Fugacity 2 × 10⁻¹¹N

-2 -1 0 1 2 3 4 5 6 7

5.4 5.6 5.8 6 6.2 6.4 6.6

Pore size, Å

0.25 0.27 0.29 0.31 0.33 0.35 0.37 0.39 0.41

DensityN/V,Å−1

Excess pressure Density

−[σ−σ(f=0)],10−11 NExcessconfiningpressure

The excess stress directly computed (virial estimate) and the excess stress predicted

by the proposed thermodynamics (Equation 2) are consistent.

σ − σ (f = 0) = − ∂

∂

Z

µ

−∞

N

V

₀

d µ

µ

−6

−5

−4

−3

−2

−1 0 1 2

10⁻¹⁴ 10⁻¹³ 10⁻¹² 10⁻¹¹ 10⁻¹⁰

Fugacity - f, N

Excessconfiningstress

Measure Thermodynamics Measure Thermodynamics Measure Thermodynamics Measure Thermodynamics Measure Thermodynamics = 0.09

= 0.05 = 0

= −0.05 = −0.09 [σ−σ(f=0)],10−11 N

Coal is a disordered nanoporous matrix.

What about a disordered chain?

0 0.05 0.1 0.15 0.2 0.25 0.3

0 2 4 6 8 10 12

Poresizedistribution,Å−1

Pore size, Å

Fugacity

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-0.1 -0.05 0 0.05 0.1

Strain

NumberofmoleculesN/V0,Å−1

2 ×10⁻¹¹ 1 ×10⁻¹¹ 7.5×10⁻¹² 5 ×10⁻¹² 3 ×10⁻¹² 2 ×10⁻¹² 1 ×10⁻¹² 7.5×10⁻¹³ 5 ×10⁻¹³ 2.5×10⁻¹³ 1 ×10⁻¹⁴ 0

1×10⁻¹³

The adsorption behavior in the disordered chain is ordered!

1

The amount adsorbed is a linear function of strain,

2

and the slope is proportional to the number of fluid atoms.

Simplified model σ = df

_S

d − C

Z

µ

−∞

N ( = 0, µ) V

₀

d µ

Swellings predicted with the simplified model are satisfying.

Perspective: predict the reservoir permeability

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.2 0.4 0.6 0.8 1

Bulk CO₂ mole fraction

Permeabilitychangek/k0

300 m 600 m 900 m 1200 m 1500 m

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0 2 4 6 8 10 12 14

Pressure, MPa

Volumetricstrain,%

CH₄ Computed CO₂ Computed CH₄ Experimental CO₂ Experimental

(Experimental results from Ottiger et al. (2008))

L. BROCHARD (Laboratoire Navier) Adsorption induced swelling of coal January 27^th, 2011 1 / 1