JOINT INVERSION OF BOREHOLE ACOUSTIC RADIAL PROFILES FOR IN SITU STRESSES AS WELL AS THIRD-ORDER NONLINEAR DYNAMIC MODULI, LINEAR DYNAMIC ELASTIC MODULI, AND STATIC ELASTIC MODULI IN AN ISOTROPICALLY STRESSED REFERENCE STATE

• US 20090070042A1
• Filed: 08/27/2008
• Published: 03/12/2009
• Est. Priority Date: 09/11/2007
• Status: Abandoned Application

• Current Assignee: Schlumberger Technology Corporation (SLB)
• Original Assignee: Schlumberger Technology Corporation (SLB)
• Inventors: Kisra, Saad, Birchwood, Richard
• Application Number:
  US12/199,243
• Publication Number:
  US 20090070042A1
• Time in Patent Office: Days
• US Class Current: 702/11
• CPC Class Codes: G01V 1/50 Analysing data

Abstract

A computer-implemented method of estimating mechanical properties and stresses of rocks around a borehole. In situ stresses and dynamic and static moduli of a rock are jointly inferred from acoustic radial profiles measured with a borehole sonic tool. Inversion is performed on equations that govern the near-borehole distributions of the compressional wave slowness, the fast shear wave slowness, the slow shear wave slowness, and the shear wave modulus in the plane perpendicular to the borehole axis for in situ stresses, the dynamic shear modulus, the dynamic Lamé parameter, λ, the static drained Young'"'"'s modulus, and the static drained Poisson'"'"'s ratio. Third-order nonlinear dynamic moduli are also inferred by the procedure. Material properties are retrieved in an isotropically stressed reference state. The input data used by the inversion is appropriately prescribed. Methods for constraining the solution in the event of noisy or limited data are demonstrated.

Claims

1. A computer-implemented method of estimating mechanical properties and stresses of material surrounding a borehole, the computer-implemented method comprising:

inferring in situ, a static Young'"'"'s modulus and a static Poisson'"'"'s ratio of the material surrounding the borehole;

generating a geomechanical model of the material using the static elastic property; and

storing the geomechanical model in a memory of a data processing system.

2. The computer-implemented method of claim 1 further comprising using one of the static elastic property or the geomechanical model to determine at least one production improvement for the borehole.

3. A computer-implemented method of modeling stresses and mechanical properties of rocks surrounding a borehole, the computer-implemented method comprising:

inferring at least one dynamic elastic property in an isotropically stressed reference state for a rock formation, wherein the inference is made on the basis of at least one of sonic tool data, seismic data, and combinations thereof;

using the at least one dynamic elastic property in an isotropically stressed reference state to estimate pore pressure;

inverting for at least one static elastic property in an isotropically stressed reference state of the material based on a measurement taken by a sonic tool;

generating a borehole model using the at least one dynamic elastic property and the at least one static elastic property; and

storing the borehole model in a memory of a data processing system.

4. The computer-implemented method of claim 3 further comprising:

inverting at least one radial profile to accomplish determining the at least one static elastic property of the material.

5. The computer-implemented method of claim 3 further comprising using the borehole model to determine at least one production improvement for the well.

6. A computer-implemented method of determining stresses and material properties around a borehole of a well, the computer-implemented method comprising:

specifying a first set of equations for compressional radial profiles, a second set of equations for shear radial profiles, and third set of equations for c₆₆ radial profiles, wherein the first set of equations, second set of equations, and third set of equations all adhere to a first form comprising

$\mathrm{\alpha <br><br>}+\frac{\mathrm{\beta <br><br>}}{x^2}+\frac{\mathrm{\gamma <br><br>}}{x^4},$

wherein “

x”

is a dimensionless radius comprising a fourth equation of a second form comprising (r/R), wherein “

r”

is a radial distance from a center of the borehole and “

R”

is a radius of the borehole, and wherein α

, β

, and γ

are coefficients of sequential powers of a fifth equation of a third form comprising (1/x²);

constructing independent equations of the first form for unknown variables by finding values of α

, β

, and γ

that best match the compressional radial profile, the shear radial profile and the c₆₆ radial profile;

constraining at least one of the unknown variables;

inverting for at least one of the unknown variables using at least one of the independent equations;

employing at least one of a deterministic inversion scheme and a probabilistic inversion schemes to expedite inversion;

responsive to a full complement of radial profiles being unavailable, employing one or more constraints to solve for at least one of the unknown variables; and

storing corresponding solved variables in a memory of a data processing system.

7. The computer-implemented method of claim 6 further comprising:

inverting for the unknown variables by matching modeled and predicted radial profiles without recourse to the intermediate step of finding values α

, β

, and γ

.

8. The computer-implemented method of claim 6 wherein the first set of equations, the second set of equations, and the third set of equations are all based on an elastic solution for stresses around the borehole.

9. The computer-implemented method of claim 6 further comprising constructing additional dependent equations for the unknown variables.

10. The computer-implemented method of claim 6 wherein:

the first set of equations is derived from;

$\mathrm{Vp}<br><br>\left(r,\mathrm{\theta <br><br>}\right)=\sqrt{\frac{\mathrm{\lambda <br><br>}+2<br><br><br><br>\mathrm{\mu <br><br>}}{\mathrm{\rho <br><br>}}}<br><br>\left[\begin{array}{c}1+\frac{\begin{array}{c}\left(1+v\right)<br><br>\left({\mathrm{\sigma <br><br>}}_h+{\mathrm{\sigma <br><br>}}_H-2<br><br><br><br>{\mathrm{\sigma <br><br>}}_v\right)\\ \left(2<br><br><br><br>c_{155}+5<br><br><br><br>\mathrm{\mu <br><br>}+2<br><br><br><br>\mathrm{\lambda <br><br>}\right)\end{array}}{3<br><br><br><br>E<br><br>\left(\mathrm{\lambda <br><br>}+2<br><br><br><br>\mathrm{\mu <br><br>}\right)}+\\ \frac{\begin{array}{c}\left(c_{111}-4<br><br><br><br>c_{155}-<br><br>2<br><br><br><br>\mathrm{\mu <br><br>}\right)<br><br>\left(1+v\right)\\ \left(2<br><br><br><br>v-1\right)<br><br>\left({\mathrm{\sigma <br><br>}}_h-{\mathrm{\sigma <br><br>}}_H\right)<br><br>\mathrm{Cos}<br><br>\left(2<br><br><br><br>\mathrm{\theta <br><br>}\right)\end{array}}{E<br><br>\left(\mathrm{\lambda <br><br>}+2<br><br><br><br>\mathrm{\mu <br><br>}\right)}<br><br>\frac{R^2}{r^2}\end{array}\right];<br><br>$

the second set of equations is derived from;

$\mathrm{Vs}<br><br>\left(r,\mathrm{\theta <br><br>}\right)=\sqrt{\frac{\mathrm{\mu <br><br>}}{\mathrm{\rho <br><br>}}}<br><br>\left[\begin{array}{c}1+\frac{\left(1+v\right)<br><br>\left[\begin{array}{c}\left(c_{155}-c_{144}+6<br><br><br><br>\mathrm{\mu <br><br>}\right)<br><br>\left({\mathrm{\sigma <br><br>}}_h+{\mathrm{\sigma <br><br>}}_H-2<br><br><br><br>{\mathrm{\sigma <br><br>}}_v\right)+\\ \left(3<br><br><br><br>c_{155}-3<br><br><br><br>c_{144}+6<br><br><br><br>\mathrm{\mu <br><br>}\right)<br><br>\left({\mathrm{\sigma <br><br>}}_h-{\mathrm{\sigma <br><br>}}_H\right)<br><br>\mathrm{Cos}<br><br>\left(2<br><br><br><br>\mathrm{\theta <br><br>}\right)\end{array}\right]}{12<br><br><br><br>\mathrm{\mu <br><br>}<br><br><br><br>E}+\\ \frac{\left(1+v\right)<br><br>\left[\begin{array}{c}\left(c_{155}-c_{144}+2<br><br><br><br>\mathrm{\mu <br><br>}\right)<br><br>\left({\mathrm{\sigma <br><br>}}_h+{\mathrm{\sigma <br><br>}}_H-2<br><br><br><br>p_{\mathrm{mud}}\right)+\\ 4<br><br>\left(c_{144}<br><br>v-c_{155}<br><br>\left(1-v\right)-\mathrm{\mu <br><br>}-\mathrm{\lambda <br><br>}<br><br>\left(1-2<br><br><br><br>v\right)\right)\\ \left({\mathrm{\sigma <br><br>}}_h-{\mathrm{\sigma <br><br>}}_H\right)<br><br>\mathrm{Cos}<br><br>\left(2<br><br><br><br>\mathrm{\theta <br><br>}\right)\end{array}\right]}{4<br><br><br><br>\mathrm{\mu <br><br>}<br><br><br><br>E}<br><br>\frac{R^2}{r^2}+\\ \frac{3<br><br>\left(-c_{144}+c_{155}+2<br><br><br><br>\mathrm{\mu <br><br>}\right)<br><br>\left(1+v\right)<br><br>\left({\mathrm{\sigma <br><br>}}_h-{\mathrm{\sigma <br><br>}}_H\right)<br><br>\mathrm{Cos}<br><br>\left(2<br><br><br><br>\mathrm{\theta <br><br>}\right)}{4<br><br><br><br>\mathrm{\mu <br><br>}<br><br><br><br>E}<br><br>\frac{R^4}{r^4}\end{array}\right];<br><br>$

the third set of equations is derived from;

$c_{66}<br><br>\left(r,\mathrm{\theta <br><br>}\right)=\mathrm{\mu <br><br>}-\frac{\left(1+v\right)<br><br>\left[\begin{array}{c}\left(c_{155}-c_{144}+3<br><br><br><br>\mathrm{\mu <br><br>}\right)<br><br>\left({\mathrm{\sigma <br><br>}}_h+{\mathrm{\sigma <br><br>}}_H-2<br><br><br><br>{\mathrm{\sigma <br><br>}}_v\right)-\\ 3<br><br><br><br>\mathrm{\mu <br><br>}<br><br>\left({\mathrm{\sigma <br><br>}}_h-{\mathrm{\sigma <br><br>}}_H\right)<br><br>\mathrm{Cos}<br><br>\left(2<br><br><br><br>\mathrm{\theta <br><br>}\right)\end{array}\right]}{3<br><br><br><br>E}-\frac{\left(1+v\right)<br><br>\left[\begin{array}{c}-\mathrm{\mu <br><br>}<br><br>\left({\mathrm{\sigma <br><br>}}_h+{\mathrm{\sigma <br><br>}}_H-2<br><br><br><br>p_{\mathrm{mud}}\right)+\\ \left(2<br><br><br><br>c_{155}+2<br><br><br><br>\mathrm{\lambda <br><br>}<br><br>\left(1-2<br><br><br><br>v\right)+8<br><br><br><br>\mathrm{\mu <br><br>}-4<br><br><br><br>c_{155}<br><br>v-12<br><br><br><br>\mathrm{\mu <br><br>}<br><br><br><br>v\right)\\ \left({\mathrm{\sigma <br><br>}}_h-{\mathrm{\sigma <br><br>}}_H\right)<br><br>\mathrm{Cos}<br><br>\left(2<br><br><br><br>\mathrm{\theta <br><br>}\right)\end{array}\right]}{E}<br><br>\frac{R^2}{r^2}+\frac{3<br><br>\left(1+v\right)<br><br>\mathrm{\mu <br><br>}<br><br>\left({\mathrm{\sigma <br><br>}}_h-{\mathrm{\sigma <br><br>}}_H\right)<br><br>\mathrm{Cos}<br><br>\left(2<br><br><br><br>\mathrm{\theta <br><br>}\right)}{E}<br><br>\frac{R^4}{r^4};<br><br>$

wherein Vp(r,θ

) comprises a compressional wave velocity, Vs(r,θ

) comprises a shear wave velocity, and c₆₆(r,θ

) comprises a modulus of a shear wave propagating in a plane perpendicular to the borehole axis;

wherein Vp(r,θ

), Vs(r,θ

), and c₆₆(r,θ

) are expressed as functions of a radial distance from the borehole centerline, r, and an azimuthal angle, θ

;

wherein the azimuthal angle is an angle measured relative to an azimuth of the maximum horizontal stress;

wherein λ

is a dynamic Lamé

'"'"'s constant, μ

is a dynamic shear modulus, ρ

is a bulk density, ν

is a static drained Poisson'"'"'s ratio, E is a static drained Young'"'"'s modulus, σ

_v is a vertical stress, σ

_H is a maximum horizontal stress, σ

_h is a minimum horizontal stress, and wherein λ

, μ

, ν

, and E, are referenced to an isotropic reference state;

wherein c₁₁₁, c₁₄₄, and c₁₅₅ are third-order nonlinear elastic constants, and wherein c₁₁₁, c₁₄₄, and c₁₅₅ are referenced to the isotropic reference state; and

wherein P_mud is a mud pressure and R is the radius of the borehole.