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Dynamic wellbore pressure prediction

Calculating dynamic pressures in a wellbore are significantly more difficult than calculating steady-state flowing conditions. In a dynamic calculation, there are two effects not considered in steady flow: fluid inertia and fluid accumulation.

Overview

In steady-state mass conservation, flow of fluid into a volume was matched by an equivalent flow out of the volume. In the dynamic calculation, there may not be equal inflow and outflow, but fluid may accumulate within the volume. For fluid accumulation to occur, either the fluid must compress, or the wellbore must expand. When considering the momentum equation, the fluid at rest must be accelerated to its final flow rate. The fluid inertia resists the change in velocity.

Typically, dynamic fluid flow is not a consideration. One exception is the operation of running pipe or casing into the wellbore, where dynamic pressure variation may be as important as pressures because of fluid friction. A second area of interest might be water-hammer effects during production startup.

Governing equations

The fluid pressures and velocities in open hole are determined by solving two coupled partial differential equations: the balance of mass and the balance of momentum.

Balance of mass ....................(1)

where

A = cross-sectional area, m2;

P = pressure, Pa;

Kb = fluid bulk modulus, Pa;

and

v = fluid velocity, m/s.

The term ....................(2)

is the compressibility, C, of the wellbore/fluid system (i.e., the change in wellbore volume per unit change in pressure). The balance of mass consists of three effects:

• The expansion of the hole because of internal fluid pressure.
• The compression of the fluid because of changes in fluid pressure.
• The influx or outflux of the fluid.

The expansion of the hole is governed by the elastic response of the formation and any casing cemented between the fluid and the formation. The fluid volume change is given by the bulk modulus K. For drilling muds, K varies as a function of composition, pressure, and temperature. The reciprocal of the bulk modulus is called the compressibility.

Balance of momentum ....................(3)

where

ρ = fluid density, kg/m3;

f = Fanning friction factor;

Dh = wellbore diameter, m;

g = gravitational constant, m/s2;

Φ = angle of inclination from the vertical;

and

The balance of momentum equation consists of four terms. The first term in Eq. 3 represents the inertia of the fluid [ i.e., the acceleration of the fluid (left side of Eq. 3 ) equals the sum of the forces on the fluid (right side of Eq. 3 )] . The last three terms are the forces on the fluid. The first of these terms is the pressure gradient. The second is the drag on the fluid because of frictional or viscous forces. The friction factor f is a function of the type of fluid and the velocity of the fluid. Frictional drag is discussed in the section on rheology (See Fluid Rheology). The last force is the gravitational force. The balance equations for flow with a pipe in the wellbore are similar to the equations for the openhole model with two important differences. First, the expansivity terms in the balance of mass equations depend on the pressures both inside and outside the pipe. For instance, increased annulus pressure can decrease the cross-sectional area inside the pipe, and increased pipe pressure can increase the cross-sectional area because of pipe elastic deformation. The second major difference is the effect of pipe speed on the frictional pressure drop in the annulus, as discussed in the steady-state surge article. Consult papers on dynamic surge pressures for more detail concerning the wellbore/pipe problem.

Borehole expansion

The balance of mass equation contains a term that relates the flow cross-sectional area to the fluid pressures. This section discusses the application of elasticity theory to the determination of the coefficients in the balance of mass equation. If we assume that the formation outside the wellbore is elastic, then the displacement of the borehole wall because of change in internal pressure is given by the elastic formula. ....................(4)

where

υf = Poisson ’ s ratio for the formation; and

Ef = Young ’ s modulus for the formation, Pa. The cross-sectional area of the annulus is given by ....................(5)

If we assume u is small compared to D, we can calculate the following formula from Eqs. 4 and 5. ....................(6)

Using typical values of formation elastic modulus, the borehole expansion term is the same order of magnitude as the fluid compressibility and cannot be neglected.

Solution method—fluid dynamics

The method of characteristics is the method most commonly used to solve the dynamic pressure-flow equations. This method has been extensively used in the analysis of dynamic fluid flow. However, applying the method of characteristics to realistic wellbore flow problems has the following difficulties:

• Iteration may be necessary to solve for characteristics and flow variables when properties and geometry vary in space.
• Multiple coordinate systems must be computed and related to a fixed coordinate system.
• Interpolation is necessary when characteristic curves do not intersect the spatial point of interest.
• Moving coordinate systems must be continually updated so that only points within the fixed-coordinate system are computed.

These difficulties can be reduced or eliminated by using the following approach:

• Adopt a fixed spatial grid.
• For a given time step, integrate the characteristic curves and flow equations from each gridpoint. Note that the flow equations are now evaluated at the new spatial point obtained from the characteristic curves.
• Interpolate the flow equations back to the fixed grid and solve for the flow variables.

This method eliminates the moving coordinate systems and replaces them with a set of interpolation factors. Because the grid is fixed, fluid properties and well geometry are known at each gridpoint, and no iteration is necessary. Most of the equations can be "presolved" so that they only need to be evaluated at each timestep. The disadvantages of this method are that the fluid variables must be evaluated at each gridpoint rather than only at points of interest, and that a maximum timestep size is required for stability.

The characteristic equations are developed using the methods given in Chap. 1 of Ref. 3. For the open hole below the moving pipe, the fluid motion is governed by the system of equations shown in Eq. 7. ....................(7)

where the first two equations are the balance of mass, with C equal to the wellbore-fluid compressibility, and the balance of momentum, with friction and gravitation terms lumped together as h. ....................(8)

The last two equations describe the variation of p and v along the characteristic curve ξ = z ± at, where a is the acoustic velocity. Subscripts here denote partial derivatives (e.g.,vz = ∂v/∂z). This system of equations is overdetermined; that is, there are more equations than unknowns. For this system to have a solution, the following condition must hold. ....................(9)

Evaluating the determinant (Eq. 9) defines the acoustic velocity. ....................(10)

The second condition that the equations have a solution requires ....................(11)

This determinant produces the following differential equations along the characteristic curve. ....................(12)

The characteristic equations are solved to give p(x,t) and v(x,t) in the following way. Eq. 12 is integrated along the characteristics for time step Δt. ....................(13)

and ....................(14)

Eqs. 13 and 14 can be solved simultaneously to give ....................(15)

and ....................(16)

Generally, c + and c must be interpolated to give values at the points of interest.

Nomenclature

 a = acoustic velocity, m/s αvs , bvs = constants that include the viscometer dimensions, the spring constant, and all conversion factors A = flow area (see subscripts), m2 c = average concentration of cuttings overall C = compressibility dv/dr = velocity gradient, s –1 dv/dt = total derivative of velocity with respect to time, Pa/s D = characteristic length in Reynolds number, m Dh = wellbore diameter, m Ef = Young’ s modulus for the formation, Pa E(k) = complete elliptic integral of the second kind, parameter k ƒ = Fanning friction factor, dimensionless g = acceleration of gravity, m/s2 h = specific enthalpy, J/kg h = total friction pressure drop, Pa/m n = power law exponent for pseudoplastic fluids pn = pressure in bit nozzle, Pa pr = pressure in bit annular area, Pa P = pressure, Pa t = time, s T = absolute temperature, °K u = radial displacement, m v* = characteristic velocity for turbulent flow calculations, m/s v = average velocity, m/s x = parameter in settling velocity equation y = parameter in settling velocity equation z = measure depth, ft Z = true vertical depth, ft ΔP = pressure drop, Pa Δt = time increment, s Δv = change in velocity, m/s Δz = length of flow increment, m ε = internal energy, J/kg ζ = measured depth integration variable, m θ = viscometer reading, degrees ϑ = integration variable μ = Newtonian viscosity of the fluid, Pa-s ξ = integration variable corresponding to depth z, m ρ = fluid density, kg/m3 υƒ = Poisson’ s ratio for the formation Φ = angle of inclination from the vertical Φ = viscous dissipation, W

Subscripts

1 = properties inside pipe, surge calculations

2 = properties inside annulus, surge calculations

3 = properties of moving pipe, surge calculation

c = concentric

e = eccentric

n = properties in bit nozzle, surge calculations

o = upstream, initial, or inlet

r = properties in annulus outside bit, surge calculations

Superscripts

- = upstream properties