Flow of cement slurry through an eccentric annulus

A step by step procedure has been developed to design cement hydraulic during cement job to minimize non cemented channels or bad cement job by taking into consideration the non-flow area throughout the eccentric annulus. Since cement slurry exhibit nonNewtonian rheological behavior, a good description of the slurry rheology is required in order to estimate accurately the velocity profile across the annulus. To achieve this goal, six rheological models have been adopted: Power-law, Robertson-Stiff, Bingham, Casson, Modified Power-law, and Modified Robertson-Stiff Models. Using the last five rheological models, new frictional pressure gradient equations for laminar flow of cement slurry through an eccentric annulus have been derived by the author based on slit-approach approximation.


Introduction:
For all oil or gas wells, there can be three or more casing namely surface, intermediate, and production casing running into the well during the progress of drilling to reach the target reservoir. Each casing should be cemented base on specific cement program to get well bore support and perfect zonal isolation to avoid down hole blowout i.e. fluid flow from one permeable zone to another. After the casing reach to its setting depth, a series of fluid will be pumped inside casing and coming back to the annulus during cement operation.
Typically, a spacer fluid is pumped first followed by one or more cement slurries followed by drilling mud to displace final cement slurry, the circulation stopped leaving few meters of cement inside the bottom of the casing.
If the mud in the annulus did not completely displaced by cement slurry, due to the presence of casing hole eccentricity, this may lead in most cases to create a continuous unconsolidated mud channels which may not allow perfect zone isolation.
Base on the graphical relationship between shear stress (τ) on y-axes and shear rate (ɣ) on x-axes, fluid can be divided into two categories as Newtonian and non-Newtonian fluids.
This (τ-ɣ) relationship is defined as flow curve or rheogram of fluid.
The rheogram of Newtonian fluid will be straight line passing through the origin , the slop of this line is equal to constant viscosity, otherwise the fluid will be consider as Non-  For fluid with a yield stress, the flow will not start before shear stress applied exceeding yield stress, which is equal to the intercept with y-axes on Cartesian co-ordinates, the rheogram of this type of fluid will be straight line or curve intersecting the shear stress axes at yield stress, while for fluid without yield stress, the flow occur immediately after applying any shear stress regardless on its value, the flow curve of this type of fluid can be represented by curve passing through origin on Cartesian co-ordinate.
The Non-Newtonian fluid in which the shear stress is a function of the value of shear rate and its duration and the past history of the fluid called Time dependent fluid which can be subdivide into ,Thixotropic fluids and Rheopectic fluids, depending on the shape of the shear stress -shear rate curve.
The equation used to approximate the relationship between shear stress and shear rate called rheological models such as; Bingham, Power-law, Casson, Robertson Stiff, Modified Power-law, Modified Robertson-Stiff, Ellis, Meter, Eyring, Powell-Ering, Seely, Growley-Kitzes, Dhaven, Prandtl-Eyring, Reiner-Philippoff, Sisko, and Sutter by models [9].For each fluid we should determine which of these rheological model approximates its flow curve by calculating AAPE for the adopted models, the one that gives the lowest AAPE will be selected as the best represent the flow curve of that fluid.

Literature review:
The problem of Non-Newtonian fluid flow through an eccentric annulus has been investigated by different authors started at 1965 by Vaudgn [16], they used the following approaches to solve this problem: 1-Consider eccentric annulus as the slit of variable height [4,15,16].
2-Used bipolar co-ordinate system [2,6].  The difference among these approaches are the complexity of the calculations and accuracy of the results, slit of variable height approach is the most commonly used in the field industry because of its simplicity and acceptable accuracy of the results as compare with the other approaches.

Difinitions Of Eccentric Annulus parameters:
Eccentric annulus is defined as the annulus formed by two cylinders their centers are not located on same point as in case of concentric annulus. Offset (e); is the distance between inner and outer cylinder centers ,while concentric annular clearance can be defined as the inner radius of outer cylinder minus the outer radius of inner cylinder (c =r o -ri ), the ratio between these two radii called radius ratio ( r * =r i /r o ),finally, eccentricity can be defined as the ratio of the offset to the concentric annular clearance, it is dimensionless number as ϵ = e/c or in term of percent eccentricity as ϵ = 100 e / c. Its value ranging from 0 i.e. concentric annulus to 1 or 100% in which the wall of inner cylinder will touch the outer cylinder in specific point. Figure (1) illustrates three values of eccentricity [9].

Eccentric Annulus
Fully eccentric annulus

The problem of laminar flow through an eccentric annulus:
There are different approaches have been presented in the literatures [4,5,12,14] to solve the problem of axial laminar flow through an eccentric annulus. The difference among them is related to the complexity of the method and accuracy of the results.
The slit approach will be adopted essentially because it is more appropriate for field application due to the simplicity of this approach and its accuracy as compare with the other approaches.

Assumptions Used to Derive flow equations:
Base on slit approach [4, 10, and 15]. A slit of variable height as illustrated in Figures (2&3) can be used to simulate eccentric annulus geometry. The following assumptions will be used to derive flow equations; 1-Non-Newtonian, time-independent, incompressible fluid.
3-Fluid velocity at the eccentric annulus walls equal to zero.
4-Velocity profile is symmetrical in both halves of the annulus.

5-
The value of angle β is equal to 0 o correspond to wide portion of the annulus, and equal to correspond to the narrow cap of the annulus ,the slit height as derived by Iyoho [4] is given by; Where; The governing differential equation for laminar flow through slit of variable height shown in Figure (3) can be written as [4,9,10]; …………………………………………………………………. flow of non-Newtonian flow through an eccentric annulus for the adopted rheological models will be presented here, the derivation of these equations available with author.

2-Using rheological models with yield stress;
A-Using Bingham Model; This model takes the following form; The method of calculating integrals I 1 ,I 2 & T o is presented in the next section.

…………………..…(21)
The details for derivation haven't been presented here, only the final form of the flow equation will be presented here, the final form in the field units is;

C-Using Modified Power-Law Model;
This model has the following form; The procedure used for calculating some functions presented in the above equations are outlined now: a-The functions E 1 and E 2 are complete elliptic integrals for the first and second kinds, respectively. They are calculated by using the following series [13].  The integrals J, J1, J2, I1, I2, I3, Z1, Z2, Z3, and Z 4 are evaluated numerically using the 15 points Gauss-Legender quadrature method [9].

Model Description:
The minimum flow rate Q min. is the minimum rate required to start flow through the smallest gap in the eccentric annulus, in order to prevent channel formation as illustrated in To calculate the value of Q min. , the value of pressure gradient is now substituted into one of the equations that derived to relate the frictional pressure gradient with the flow rate depending on the rheology of cement slurry.

Hydraulic Design for cement job:
Using the equations that have been derived and presented in this paper to design step by step procedure to determine the minimum rate required to start flow through the smallest gap in the eccentric annulus to avoid bad cement job.
3-Casing outside diameter and hole size.
4-Real hole path (inclination angle as a function of depth).

5-Casing centralizers program.
Application procedure; 1-Calculate rheological parameters for all adopted models, the equations presented in Reference (13)  5-A step by step procedure has been presented to determine the minimum flow rate that we need to use during cementing operation to avoid bad cement behind the casing as well as, the determination of minimum pump capacity during cementing operation.