Solids Production
The source for sand production is the presence of disintegrated sand grains around the wellbore or perforation walls due to rock failure around the hole.
The Solids Production module calculates:
- Tangential and shear stresses at wellbore wall (or perforation)
- Sanding is predicted if maximum effective stress exceeds the effective rock strength
- Critical BHFP,
- Critical Drawdown
Outputs:
- Sanding evaluation log
- Mass of produced sand
- Sand-free operating envelope
- Optimum perforation and wellbore trajectory for both Open Hole and Cased perforated completions
Governing Equation in production
Sand Failure Criterion [Ref. 1]
Sand grains around an arbitrarily oriented wellbore or perforation walls are assume to disintegrate by shear failure when the maximum effective tangential compressive stress, 
, exceeds the effective strength of the formation, U.
Eq. 1
where LF is the called load factor. The maximum effective tangential compressive
stress, 
is calculated
as 
in
which 
is the wellbore
pressure during production, which is usually known as the bottomhole flowing
pressure (BHFP), and 
is
Biot's constant. During production from a permeable formation, the near well pore
pressure is equal to 
, and hence,
the effective stress is calculated by subtracting
. The compressive stress on an
arbitrarily oriented wellbore perforation that is likely to cause sand failure is the
maximum principal stress, 
(defined in equation 2). The location of this failure is on the hole circumference
where 
has its maximum
value, 
. This value is
found by solving the following equation for
. varying
values of 
with a small
interval from 0 to 180 degrees.
Eq. 2
For openhole completions, Equation 1 is directly applicable to calculate LF for a given drawdown or BHFP at a certain depth using the well's inclination and azimuth.
Thick Walled Cylinder [Ref. 4]
Thick Walled Cylinder (TWC) or Hollow Cylinder (HC) rock strength is a parameter measured in the laboratory and routinely used in analytical and numerical sanding evaluation required for sand control and well completion decisions through the identification of sand production risks, quantification of sanding rate and also as a scale model of wellbore or perforation stabilities. In these tests, a hollow cylinder is loaded under increasing hydrostatic stress until collapse occurs in the walls of the cylinder.
A literature list of formulas can be found in Appendix C, Theory and equations under TWC equations.
Effective Strength Factor
The Effective Strength factor is a parameter to empirically account for post-elastic residual strength and size effect of a TWC sample (SPE 116633, & 78618).
Scale Effect: To upscale the laboratory-measured TWC strength to reservoir conditions. In the laboratory, TWC strength is measured with a sample OD/ID ratio of ~3. In the reservoir, the OD/ID ratio is very high and tends towards infinity. The experimental results with typical perforation diameter as the sample ID showed that the TWC value of a sample with an OD/ID ratio of 3 needs to be up-scaled by ~1.4-1.7 (average 1.55) to consider the effect of a large OD/ID ratio.
Residual Strength Effect: The rock failure criterion is based on linear poro-elastic theory and shear failure assumption. It does not consider the stabilization effect by the post-elastic residual strength of the rock. This effect can be considered by multiplying the TWC strength approximately by 2.
The combined scale factor becomes ~2.8-3.1 (2× 1.4 or 1.55) for perforations. When the effect of sample ID on rock strength measurement was considered, the scale factor could become lower: ~2.0-2.5 for a typical open hole wellbore. These default ESF values can be used as a first order but practically ESF should be calibrated against documented field sanding or no sanding observations.
Default values in the software of the effective strength for open hole and cased hole are 2.0 and 3.1 respectively.
Accounting the effect in reservoir pressure declination [Ref. 1]
The effect of reservoir-pressure decline caused by production can be accounted for in the preceding computation by updating the in-situ stresses. For a laterally large reservoir compared to its thickness, the change in vertical stress, is considered negligible, and therefore it is usually kept constant. The maximum and minimum horizontal stresses are updated as follows, respectively:
where Pc is the current reservoir pressure and Pi is the initial reservoir pressure. While using updated SigmaH and sigma h stresses, it is also important to use current reservoir pressure Pc in place of Pr.
Depletion Stress Factor or poroelastic stress coefficient (reference 1): is defined as the change of horizontal stresses with reservoir-pressure variations in a passive and normal-faulting stress regime. Different equations have been suggested for the determination of stress-path factors in other stress regimes and tectonically active areas (Addis 1997). Nevertheless, published literature and the worldwide data set indicate a range of 0.5–0.9 for the stress-path factor.
Production zones
A production zone is the depth interval where the rock contains certain physical characteristics that make it commercially suitable to be produced.
Depletion
Depletion is the difference between the initial reservoir pressure and the current reservoir pressure. When depletion is zero, it means that the reservoir has not been started to produced yet.
Drawdown
Drawdown is the Difference between the pore pressure and the bottom hole flowing pressure
Perforation Orientation
Angle of the perforation orientation with respect to a specific reference. For vertical wells is reference to the wellbore azimuth. For deviated wells, is reference to the top of the hole
Sanding Evaluation
This Workflow helps determines the Loading factor, which means the sanding likelihood, based on a specific user defined drawdown. If LF is > than 1 means that sanding will occur. Additionally, critical bottom hole flowing pressure and critical drawdown based on well logs inputs.
Operating Envelope
The operating envelope lets a user determine the bottom hole flowing pressure or drawdown necessary to produce a sand free operating wellbore for a proposed inclination, azimuth, depth, completion type and rock strength.
Polar Plot
A sanding polar plot, allows to establish the optimum well trajectory for open hole completion in the field with the initial reservoir pressure based on the variation of critical bottomhole flowing pressure (CBHFP). A lower CBHFP value allows a higher critical drawdown (CDD) value, and thus is considered a less sand-prone, or more stable well design
These graphical outputs show the weak intervals in the reservoir section and the maximum drawdown that can be achieved in these intervals without producing sand with the reservoir pressure at a particular production stage (depletion)
TWC Correlations TWC-UCS Relationships Jaeger et al 2007 presented mathematical and rock mechanical solutions for failure of hollow cylinder tests as a function of hole size and stress concentration at inner and outer walls relating to the applied stress. They showed that TWC strength is generally higher than the UCS because of strengthening effect and elasto-plastic nature of porous rocks. Ewy et al. 2001 used a Modified Lade failure criteria to relate TWC and UCS strengths and to calculateTWC strength from the failure envelope parameters defined from triaxial compressive strength tests. Experimental data also show that TWC and UCS strength are related and several empirical equations are developed for different rock types for TWC as a function of UCS in a general form of TWC = A*UCSB where A and B are rock type related empirical constants. In the absence of measured TWC but the availability UCS data these equations could give a reasonable estimate of TWC from UCS providing that the UCS input itself is robust and reliable, that is the UCS tests are adequately quality controlled and the UCS profile is calibrated and accurate. However, the applicability of these empirical equations which are derived from a specific dataset with specific rock type and geological characters and their use to other areas have to be justified. Also, if a specific equation is used, extra care should be taken to not to extrapolate the equation beyond the range of its original dataset. Nevertheless, all of these equations show a common trend for a given UCS, the TWC strength can be 1.0 to 6 times greater than the UCS strength. The lower the UCS (or the weaker the rock) the greater the difference between UCS and TWC. This may be related to the strengthening effect of hydrostatic loading in TWC tests and high plasticity of weak and poorly consolidated rocks. In contrast, strong and competent rocks will behave more elastically with a lesser plastic deformation before catastrophic failure and hence the difference between UCS and TWC will be lower for stronger rocks.
TWC-Porosity Relationships Similar to UCS and Young’s modulus, TWC strength also show a strong reverse relationship with rock porosity. However, no single relationship exists that could be applicable to all rock types and porosity ranges.
TWC-Elastic Moduli and TWC-Acoustic Velocity Relationships TWC strength correlates well with elastic moduli and acoustic velocities as they are also related to overall rock mechanics properties. Khaksar et al. 2009 have listed similar strong correlations between UCS with elastic moduli and acoustic velocities.
References
1. RAHMAN, K., KHAKSAR, A. AND KAYES, T., 2008—Minimizing sanding risk by optimizing well and perforation trajectory using an integrated geomechanical and passive sand-control approach. Annual Technical Conference and Exhibition, Denver, Colorado, USA, 21–4 September, SPE 116633.
2. KHAKSAR, A., RAHMAN, K., GHANI, J. AND MANGOR, H., 2008—Integrated geomechanical study for hole stability, sanding potential and completion selection: a case study from south east Asia. SPE Annual Technical Conference and Exhibition, Denver, Colorado, USA, 21–4 September, SPE 115915.
3. A geomechanical approach for sanding risk assessment Applied to three field cases for completion optimisation K. Rahman, A. Khaksar and T. Kayes. APPEA Journal 2010 50th ANNIVERSARY ISSUE—1
4. THICK WALL CYLINDER STRENGTH AND CRITICAL STRAIN LIMIT FROM CORE TESTS AND WELL LOGS, IMPLICATIONS FOR SAND CONTROLDECISIONS Abbas Khaksar1, Sadegh Asadi1, Ahmadreza Younessi1, Feng Gui1&Yan Zheng1. Baker Hughes. The 2nd SPWLA Asia Pacific Technical Symposium – Indonesia, November 7-8, 2018