Alternative equation for sorption data interpretation Patrick van Hemert & K-H.A.A. Wolf

transcript

Alternative equation for sorption data interpretation

Patrick van Hemert & K-H.A.A. Wolf

May 25, 2006

International Symposium CBM, Tuscaloosa

Technical GeosciencesDelft University of Technology, The Netherlands

Objectives

• For experts:• Introduce alternative equation for sorption

interpretation• Simpler calculation• Easier error analysis

• For non-experts:• Basic understanding of volumetric sorption

isotherms measurements

Outline

• Relevance • Introduction to volumetric sorption• Alternative formula for interpretation• Comparison to original formula• Conclusion

Reasons for measuring sorption isotherms

• Determine Gas Storage Capacity• Used for Gas In Place analysis

• Determine fluid-solid interaction• Reservoir models• Science

Typical laboratory output: The Langmuir

•X-axis is usually pressure in either bar, psia or Mpa.

•Y-axis is usually sorbed gas in mol/kg, or SCF/Ton

( )1ads

n Pn P

What are volumetric isotherms?

• Isothermal experiments.• Material balance calculations from P,T,V data• Procedure consist of two steps, isolated(1) and connected(2) •Sorbed amount is the difference between the total amount and the equilibrium amount of gas

Gas supply

Valve 1

Valve 2

sample cell

vacuum

Gas supply

Sorption experimental procedureRepresentation of the two experimental steps in a sorption experiment

From N. SiemonsFrom N. Siemons

Pressure development during an experiment

Sorption is a material balance difference calculation

M M MGibbs total void gasn mole n V

1 1 2 2 2 21

[ ] ( , ) ( , ) ( , )M

M i i i i M MGibbs ref void

n mole V P T P T V P T

1 1 2 21 1

[ ] 0 ( , ) ( , )M M

M i i i i itotal initial added ref

n mole n n V P T P T

M M Madsorbed total gasn mole n n

1 1 2 2 2 21

1[ ] ( , ) ( , ) ( , )

MM i i i i M MGibbs ref ref

molen V P T P T V P T

1 1 2 2 2 21

[ ] ( , ) ( , ) ( , )M

refM i i i i M MGibbs

Vmolen P T P T P T

M:Measured point

Determination of volume ratio from material balance conservation

2 2 1 11 1

2 2 2 2

( , ) ( , )

i i i ivoid

i i i iref

V P T P T

•For micro-porous solids needs to be experimentally determined

•Helium sorption experiments with the assumption that helium sorption is negligible

Alternative formulation for sorption interpretation

simplifies error analysis

1 1 2 2 2 21

( , ) ( , ) ( , )M

Vn P T P T P T

Void volume ratio sample and reference cell(defines the dividing surface)

Each parameter can separately be calculated

High pressure volumetric isotherms are very sensitive

• Changes in the volume ratio have a large effect • The use of an wrong densities has large effects

Previous formulation is unnecessarily difficult

1 1 1 1 2 2 2

1 1 11 2 2 2 2 2 2

i iMrefM

Gibbs i i i i i ii

i iMvoid

M M i M M ii

V P Pn

m Z P T T R Z P T T R

•Redundant terms in equilibrium amount•Real gas law formulation unnecessarily complicated•Void volume is not independently measured• This unnecessarily complicates error analysis!

The alternative equation for volumetric sorption interpretation has the following

advantages:

• Physical interpretation is straightforward• Easy to calculate• Simple error analysis

• Contains only independent parameters

Questions?

Alternative formulation for sorption simplifies error analysis

1 1 2 2 2 21

( , ) ( , ) ( , )M

Vn P T P T P T

1 1 1 1 2 2 2

1 1 11 2 2 2 2 2 2

i iMrefM

Gibbs i i i i i ii

i iMvoid

M M i M M ii

V P Pn

m Z P T T R Z P T T R

Experimental uncertainties from alternative equation

M MMsys M iGibbs Gibbs

Gibbs ref kii kref k

M M MMiGibbs Gibbs Gibbski

dn dnn V P

dn dn dnT m

dT d dm

2 22 2

1 1 1 1

M M MGibbs Gibbs Gibbs

refrefran M

GibbsM MM M

i iGibbs Gibbsk ki i

i k i kk k

n n nV m

n nP T

Alternative formulation for sorption interpretation

Volume ratio sample and reference cell

The sorbed amount is calculated from the difference in material balance between the total amount of gas and the equilibrium amount of gas.

Alternative formulations require unknown parameters

1 1 2 2 2 21

( , ) ( , ) ( , )M

refM i i i i M MAbs

Vn P T P T P T

( , )1

M Msorb

1 1 2 2 2 21

( , ) ( , ) ( ) ( , )M

refM i i i i M Mswell

Vn P T P T P T

sample

Gibbs sorption is the only rigorous thermodynamic framework for adsorption

equilibrium

• The extent and density of the sorbed phase can not be measured independently or directly.

• Gibbs suggested the mathematical transformation that all fluid-solid interaction is attributed to a dividing surface.

• In micro-porous solids measuring the dividing surface is an experimental challenge.

• This constitutes an ill-defined problem.

What causes the considerable differences between inter-laboratory sorption

experiment?

• Investigation into the nature of the measurements

• Quantification of the accuracy and precision of the measurements

Introduce alternative formula for interpretation of volumetric sorption

experiments

•Calculation is simpler•Error analysis is easier•More straightforward interpretation

1 1 2 2 2 21

( , ) ( , ) ( , )M

Vn P T P T P T

Enhanced Coalbed Methane production may be a commercial energy source

• Underground coal contains ,mainly sorbed, methane

• Coalbed Methane projects (CBM) produce methane by pumping gas and water from the reservoir

• Enhanced Coalbed Methane (ECBM) projects may enhance production by injecting gas

• Gas-solid interactions determine production behavior

Optimizing ECBM requires knowledge of gas-coal interactions

• Equilibrium adsorption data quantify fluid-solid interaction

• Pure component isotherms are the simplest form of adsorption equilibrium data• Methane isotherms on coal are already used

to quantify the amount of methane producible

Sorption isotherms are measured volumetrically

• Volumetric technique is very widespread• Procedures and apparatus are easily obtainable• Actively used in the field of (E)CBM

• Gravimetric sorption measurements are less common

• More complex equipment and procedures

Reliability of sorption isotherms is often omitted

• Every measurement has an associated uncertainty

• This uncertainty quantifies the measurements’ significance

• Reliability estimates from error analysis and repeated experiments should concur.

Inter-laboratory isotherms differ considerably

• Repeated (inter-laboratory) experiments are the best reliability estimators

• The associated experimental errors must be specified.

• The cause of the difference must be investigated

Sorption is a material balance difference calculation

Gibbs total void gasn mole n V

1 1 2 2 2 21

[ ] ( , ) ( , ) ( , )M

M i i i i M MGibbs ref void

n mole V P T P T V P T

1 1 2 21 1

[ ] 0 ( , ) ( , )M M

M i i i i itotal initial added ref

n mole n n V P T P T

adsorbed total gasn mole n n

1 1 2 2 2 21

[ ] ( , ) ( , ) ( , )M

Vmolen P T P T P T

Alternative equation for sorption data interpretation Patrick van Hemert & K-H.A.A. Wolf

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