Downhole formation evaluation using nuclear magnetic resonance inversion matrix reduction
A formation evaluation system reduces inversion matrixes used to determine formation properties, thereby increasing the memory management and processing efficiency of the evaluation system. NMR data is acquired from a wellbore and expressed mathematically by the system as a least squares solution to a linear system. The least squares solution is approximated using a numerical decomposition method and the evaluation system determines a formation property using the approximated least squares solution. Thereafter, a downhole operation may be planned, analyzed or conducted using the determined formation property.
1. A formation evaluation method, comprising:
acquiring nuclear magnetic resonance (“NMR”) data of a formation surrounding a wellbore, the NMR data having a plurality of echo trains expressed as a matrix A i ;
expressing an inversion of the NMR data as a least squares solution to a linear system;
approximating the least squares solution using numerical decomposition by compressing a projection of the NMR data into a transpose of the matrix A i using an expression A′b, wherein the approximation is selectively applied to each echo train separately; and
determining a property of the formation using the approximated least squares solution.
2. The method as defined in claim 1 , wherein the approximation is performed using an orthogonal-triangular decomposition method.
3. The method as defined in claim 1 , wherein selectively applying the approximation to each echo train comprises determining which echo trains to approximate.
4. The method as defined in claim 1 , wherein the approximated least squares solution is expressed as:
x ls =(Σ i=1 P A′ i A i +αL′L ) −1 (Σ i=1 P Q i R i E i b i )=(Σ i=1 P A′ i A i +αL′L ) −1 (Σ i=1 P Σ k=1 N ( r i,k ( E i b i )) q i,k )≈(Σ i=1 P A′ i A i +αL′L ) −1 (Σ i=1 P Σ k=1 K i ( r i,k ( E i b i )) q i,k ).
5. The method as defined in claim 1 , wherein the property is at least one of a porosity, permeability, or pore size of the formation.
6. The method as defined in claim 1 , wherein the method is performed in real-time during a downhole logging operation.
7. A system for formation evaluation, comprising:
a logging assembly to be positioned along a wellbore; and
processing circuitry coupled to the logging assembly and configured to implement a method comprising:
acquiring nuclear magnetic resonance (“NMR”) data of a formation surrounding a wellbore, the NMR data having a plurality of echo trains expressed as a matrix A i ;
expressing an inversion of the NMR data as a least squares solution to a linear system;
approximating the least squares solution using numerical decomposition by compressing a projection of the NMR data into a transpose of the matrix A i using an expression A′b, wherein the approximation is selectively applied to each echo train separately; and
determining a property of the formation using the approximated least squares solution.
8. The system as defined in claim 7 , wherein the approximation is performed using an orthogonal-triangular decomposition method.
9. The system as defined in claim 7 , wherein selectively applying the approximation to each echo train comprises determining which echo trains to approximate.
10. The system as defined in claim 7 , wherein the property is at least one of a porosity, permeability, or pore size of the formation.
11. A non-transitory computer readable medium comprising instructions which, when executed by at least one processor, causes the processor to perform the method comprising:
acquiring nuclear magnetic resonance (“NMR”) data of a formation surrounding a wellbore, the NMR data having a plurality of echo trains expressed as a matrix A i ;
expressing an inversion of the NMR data as a least squares solution to a linear system;
approximating the least squares solution using numerical decomposition by compressing a projection of the NMR data into a transpose of the matrix A i using an expression A′b, wherein the approximation is selectively applied to each echo train separately; and
determining a property of the formation using the approximated least squares solution.
12. The computer readable medium of claim 11 , wherein the approximation is performed using an orthogonal-triangular decomposition method.
13. The computer readable medium of claim 11 , wherein selectively applying the approximation to each echo train comprises determining which echo trains to approximate.