Applied Hydro-Aeromechanics in Oil and Gas Drilling

CHAPTER 1

MAIN RESULTS AND DEVELOPMENT LINES OF HYDRO-AEROMECHANICS OF DRILLING PROCESSES

Intensive investigation of forms and laws of fluid flow in wells began in 1901 when in the United States application of the mechanical rotary drilling with washing, the so-called rotary drilling, was found on Spindletop field in Texas state. In 1911, for the first time in Russia's Suruchan region several wells were bored by rotary method with washing of well bottom by mud solution. After nationalization of the oil industry, the rotary boring began to develop quickly.

With steady increase in well depth and complexity of mine geological conditions, widespread use of jet drilling bit and downhole motors resulted in the washing and plugging back in hydro-aeromechanical well-bed system becoming more costly and power intensive. Since under real hydro-aeromechanical system it is understood that the whole set of well elements and uncovered beds connected with each other in a unified technological set have a complex structure, it is necessary to build a mathematical model of this system. The model was developed in two directions: the description of main hydro-aeromechanical properties of separate elements and the structure of the system as a whole.

Investigation of basic element properties is aimed at finding correlations between pressure, flow rate, and time through relations of theoretical hydro-aeromechanics and applied hydraulics. Let us point out the most significant results of hydro-aeromechanics in drilling.

Rheological equations formulated for viscous fluids by Newton in 1685 (Krilow, 1936), for viscous-plastic media by Shvedoff in 1889 (Reiner, 1960) and Bingham in 1916 (Bingham, 1922), and for pseudo-plastic media by Ostwald in 1924 (Reiner, 1960) are of profound importance in solving problems of drilling hydro-aeromechanics. With the help of these equations, formulas were obtained for pressure distribution in stationary laminar flow of viscous (Poiseuille, 1840, 1841; Stokes, 1845, 1850, 1901), viscous-plastic (Buckingham, 1921), and pseudoplastic (Rabinowitch, 1929; Mooney, 1931) fluids in circular pipes. Solutions have also been obtained for flows in concentric circular channels of viscous (Lamb, 1945), viscous-plastic (Volarovich and Gutkin, 1946), and pseudoplastic (Fredrickson and Bird, 1958) fluids.

On the basis of Bukingham and Volarovich and Gutkin formulas for the flow of viscous-plastic fluids in circular and concentric circular pipes, Grodde (1960) applied convenient graphic method to calculate pressure drop.

Schelkachev (1931) considered laminar stationary flow of viscous fluid in eccentric circular channel and obtained formula for pressure distribution. McLean et al. (1967) gave a general scheme for approximate calculation of pressure distribution in laminar flow of rheologic stationary fluid in concentric circular channel with cross section replaced by conventional sections of concentric channels with independent flows.

The stability of laminar flows of viscous fluid in circular pipes was experimentally investigated by Reynolds during 1876-1883 (Reynolds, 1883). He established transition criterion from laminar to turbulent flow. Hedström (1952) characterized the loss of viscous-plastic fluid laminar flow stability by Reynolds and Saint Venant numbers.

On the basis of boundary layer theory developed by Prandtl during 1904-1925 (Prandtl and Tietjens, 1929, 1931) for turbulent flow of viscous fluid in pipes with smooth and rough walls, Altshul (Altshul and Kiselev, 1975) obtained dependences for hydraulic resistance factors.

In developing the theory of multistage turbine, Shumilov (1943) gave formula for pressure drop in turbo-drill. To derive the pressure change in local resistances of circulation system, Herrick (1932) used the equivalent length method. Shumilov (1943) applied Borda-Karno formula for locks and Torricelli formula for drill bit orifice when determining pressure drop. Laminar flow of viscous fluid around a sphere was considered by Stokes (1845). Experimental investigations of flows around rigid spherical particles in a wide range of Reynolds numbers were generalized in the form of Rayleigh curve. Shischenko and Baklanov (1933) investigated conditions of stability and flow of mud solution around particles.

Targ (1951) found pressure distribution in laminar stationary flow of viscous fluid in an axially symmetric circular channel, one of the wall of which moves with constant velocity. Gukasov (1976) considered laminar flow of viscous-plastic fluid in concentric circular channel with movable internal wall.

Basic hydrodynamic equations for multiphase fluids using empirical relations for concentrations and hydraulic resistance factor were derived by Teletov (1958). On this basis were obtained pressure distributions in pipes and circular channels in well washing by aerated fluid or gas blowdown.

A fundamental contribution to solving the problem of nonstationary flows in hydraulic systems with regard to compressibility of fluids and elasticity of walls was made by Zhukowski (1899-1921), who developed the theory of one-dimensional nonstationary flow of viscous fluid to solve many problems (Zhukowski, 1948).

In connection with problems of oil- and gas-field development in works of Pavlowski (1922), Leibenson (1934), Schelkachev (1990), Charniy (1963), Muskat (1963), and many others, the flow of reservoir fluid in porous medium has been extensively studied to solve problems with opening up of productive buildup and problems with drilling.

Along with the investigation of hydro-aeromechanic properties of system elements, methods to investigate well-bed system as a whole have also been developed. In doing so, there have been established correlations between elements of the system needed to simultaneously solve all equations characterizing separate elements. For exampfe, Herrick (1932) had considered a problem on feed and pressure of drilling pump for circulation of washing fluid and Shazov (1938) devised a scheme of procedure in choosing number and parameters of cementing aggregates for one-step well plugging. Mirzadjanzadeh and his collaborators (Mirzadjanzadeh, 1959) developed a method for analyzing hydro-aerodynamic processes with the help of stochastic and adaptive training models.

Shischenko and Baklanov (1933) were first to systematically outline a number of washing fluid hydraulic problems. Many aspects of hydro-aeromechanics of drilling processes were considered in monographs (Gukasov, 1976; Gukasov and Kochnev, 1991; Goins and Sheffield, 1983; Esman, 1982; Mezshlumov, 1976; Mezshlumov and Makurin, 1967; Mirzadjanzadeh, 1959; Mirzadjanzadeh and Entov, 1985; Shischenko et al., 1976; Macovei, 1982; and others), handbooks (Mittelman, 1963; Filatov, 1973; Gabolde and Nguyen, 1991; and others) and the periodic literature.

At present, there has been a tendency to develop systems approach to drilling hydro-aeromechanics chiefly in building well-bed system models both simplified and more complex ones demanding application of various mathematical methods with regard to designing, building, and operation of wells.

CHAPTER 2

BASIC PROBLEMS OF HYDRO-AEROMECHANICS IN DRILLING PROCESSES

Hydro-aeromechanic processes in drilling occur in the well-bed system consisting in the simplest case of two parts: circulation system of the well along which fluid, gas, or their mixture including rigid particles flow and one or several opened up bed formations.

In general, the hydro-aeromechanic program of well-bed operation would be engineered when distributions of the following parameters are determined and reconciled: (1) flow rates; (2) pressures; (3) densities; (4) stresses; (5) concentrations; (6) temperatures; (7) geometric sizes of system elements (length, diameter, and spatial arrangement of each circulation system including level depth, radius, and thickness of beds); (8) characteristics of compressors and pumps, cementing units, and mixers (deliveries and pumps); (9) strength characteristics of system elements; (10) characteristics of the lifting mechanism of the drilling rig (velocities and accelerations in round trips); (11) characteristics of downhole motors (pressure drops at different flow rates of the flushing fluid); and (12) granulometric compositions of the cutting drilled and carried out from the well.

Distributions 1-6 are connected with each other by common hydro-aeromechanic equations in the region of distributions 7-12 taking place in drilling. The description of hydro-aeromechanic processes of drilling reduces to finding relations connecting distributions listed in 1-12.

Depending on the goal of technological operation, any distribution from 1 to 12 can be sought or given in the form of technical, technological, economic, or ecological restrictions. In designing and handling hydro-aeromechanic program or its parts, it is required to get distributions or separate values of some of them as functions of flow rate and pressure at given values of the rest.

Figure 2.1 presents a list of main processes 1.1-1.3 and 2.1-2.5 and problems 1.1.1-1.3.3 and 2.1.1-2.5.1 associated with them, which usually happen to be considered in drilling. In order to solve them, it is necessary to investigate distributions 1-12 for stationary and nonstationary flows in well-bed system elements. When solving a concrete problem, one finds one or more distributions among 1-12 so that they would not contradict the rest of them.

FIGURE 2.1 List of main processes and problems associated with them.

For example, let us consider in more detail a distribution of pressure in underground part of the circulation system, which happens to be often determined in carrying out hydro-mechanical process of drilling with fluid washing. Figure 2.2 shows the sought pressure distribution (diagram) in circulation system of a vertical well in boring with washing of incompressible fluid at a given arrangement of the drill pipe string (the arrows show directions of the circulation). The pressure in the diagram is determined under the following conditions:

FIGURE 2.2 Diagram of underground part of vertical well circulation system and pressure distribution in bed-well system. (a) Diagram of the underground part of circulation system: 1—annular system; 2—drill pipe; 3—drill collar; 4—downhole motor; 5—drilling bit; 6—joint; 7—the last lowered casing string; 8—opened borehole; 9—covered weak bed; 10—rock under shoe of the last lowered casing string; 11—opening bed. (b) Pressure distribution in system elements (I—hydrostatic; II—at circulation in annular system; III—at circulation in drill stem): 1-2, 3-4, 4-7—after drill pipe; 2-3—after joints; 7-8—after drill collar; 8-9—after motor; 9-12—in drilling bit; 12-13—in downhole motor; 13-14—in drill collar; 14-15, 16-17—in drill pipe; 15-16—in joints. Values of pressure: 1—pressure in annular system at well head; 5, 10—formation pressures Psb1 and Psb2; 6, 11—hydro-fracturing (absorption) pressures Pp1 and Pp2 in rock and bottom opening bed; 18—bottom-hole hydrostatic pressure; 9—bottom-hole pressure in circulation (washing); 19—hydrostatic pressure in annular system under shoe of casing string; 20—pressure in annular system in washing under shoe of casing string; 21—pressure in ascending pipe.

(a) Pressure in the ascending pipe (pap) does not exceed the allowed pressure of the drill pump (pa1); that is, it satisfies the consistency of distributions 2 and 9.

(b) Pressure in uncased parts of the well is higher than pressure in showing beds (psb1 and psb2) but does not exceed absorption or hydro-fracturing pressures (pp1 and pp2):

(2.1)

that is, it is a valid consistency of distributions 2 and 9.

(c) Flow rates of fluid in the annular space (Qas) and at the well bore (Qwb) ensure the cutting recovery; these flow rates provide distribution 1.

(d) Difference of pressures in pipes (ppd) and in the annular space (pasd) satisfies condition of the pipe strength (pst):

(2.2) equation

that is, it is a valid consistency of distributions 2 and 9.

In a variety of problems, the expected pressure depends on the characteristics of items 1-12. In performing calculations, it is not necessary to find the whole pressure distribution (diagram). For example, in the absence of weak or showing beds, it is enough to determine the pressure only in the ascending pipe, which should not exceed permissible pressure in the pump. In the process considered, other distributions 1-12 are not mentioned, but it is meant that they satisfy the diagram in Fig. 2.2. And yet the existence of such distributions should be kept in mind, and they must be taken into account when solving concrete problems.

From what has been said, it follows that the basis of all hydrodynamic calculations consists of the facility to find pressure distributions in circulation system elements of the well. In order to calculate pressure distribution and to build pressure diagram, one should be able to determine pressure drop both in concrete circulation system elements and in given cross section of the well element.

CHAPTER 3

MULTIPHASE MEDIA IN DRILLING PROCESSES

Drilling fluid, grouting mortars, special solutions, for example, spacers, reservoir fluids, and skeleton represent complex media consisting of more simple elements. Therefore, such fluids are mixtures of several media with definite properties inherent to each of them.

By setting up hydro-aeromechanic problem, the medium taking part in the process considered happened to be homogeneous or heterogeneous, single phase or multiphase, one component or multicomponent depending on the type of technological operation (Basarov, 1991).

There are macroscopic systems in which ingredients are vastly superior in sizes to molecular sizes. From macroscopic systems are set apart two systems: homogeneous and heterogeneous. Homogeneous (uniform) systems possess identical properties in any arbitrarily chosen part equal in volume to another part. For example, water and in many cases mud and cement solution may be considered uniform or homogeneous. Rocks can be approximately taken as homogeneous in salt and mud beds. Heterogeneous (nonuniform, multiphase) systems consist of several different physically homogeneous media. In such systems, one or several physical properties may be safely assumed to undergo a sudden change when going from one point of the volume to another. For example, in gas-liquid (aerated) flushing fluid, it is often assumed that the density instantly changes while going through bubble or air plug boundary.

In mud solution, discontinuity of density is also considered, in particular in passage through boundary between rigid particle of mud weighting material and fluid of water-mud solution. Since it is considered a boundary or a surface of definite thickness between two physically inhomogeneous media, and some properties undergo a great change at this surface, it is called interface and the media are called phases. Thus, for example, the aerated fluid is a two-phase heterogeneous in which one phase is fluid (water, oil) and the other one is gas (air, natural gas). In drilling with flushing, such solution can also contain particles of a slurry when flowing in annular channel.

Hence, in annular channel, the heterogeneous flows three-phase mixture: fluid—phase 1, gas—phase 2, and particles of slurry—phase 3. If it is an aerated mud solution, then in some problems one more phase should be taken into account since mud in water is usually dispersed not up to molecular level, and in some investigations, the system should be considered as a four-phase system.

One should not identify aggregate state with phase state. There are three aggregate states, solid, liquid, and gaseous, but phases may be unbounded in number. For example, many-colored immiscible fluids are in one and the same aggregate state (liquid) but represent separate phases distinguished by a determined property, namely, by color. In water-oil or spacer solution displacement, matters are in one and the same aggregate state, that is, liquid, but it is clear that oil and water and spacer differ essentially in properties; that is, they are different phases.

One should also not identify notions of phases and components. A system has as many components as there are chemical elements or their compounds. Mixture of gases is a single-phase but multicomponent system. In a mixture of chemically nonreacting gases, there are as many components as it has different gases. For example, when the drilling is performed at great depth with the help of aerated water, it may happen that all gas would be dissolved in fluid at molecular level and the resulting solution becomes homogeneous and uniform in properties but multicomponent, containing water and gas. As such solution moves to the well mouth, the pressure reduces, the gas liberates, and the solution transits into heterogeneous two-phase multi-component state. Other situations are also possible when a homogeneous system is multicomponent and one-component system is multiphase.

A variety of media being used and encountered in well drilling require their properties to be studied. Properties of multiphase systems, in particular of two-phase media, may be different depending on to what degree each phase is dispersed. If one or several phases are dispersed and surrounded by another phase, then such heterogeneous system is sometimes called dispersed system, the crushed phase is called dispersed (discontinuous) phase, and the surrounding phase is called dispersion (continuous) phase. For example, air bubbles in aerated fluid represent dispersed phase and the fluid is continuous phase.

In addition, in some two-phase systems, it is impossible to determine which of the phases is a dispersed phase and which is a continuous phase since it is impossible to find which phase is surrounded by another one. For example, in porous media (rocks) with communicating pores, in gas-liquid and water-oil mixtures with near-equal volume concentration of both phases, they can have continuous distribution.

Classification of heterogeneous systems in dispersivity is presented in physical chemistry. If particles of dispersed phase have sizes 10-7, the system is called microheterogeneous. The word micro denotes dispersivity up to indicated size. If particles of the dispersed phase have sizes from 10-9 to 10-7 m, the system is called ultraheterogeneous or fine grained. In these systems, particles of dispersed phase are called colloidal particles. One should distinguish colloid systems from true solutions. Recall that true solutions are solutions in which substances are distributed at molecular level and form homogeneous systems, while colloid system is a variant of heterogeneous systems.

True systems can be one component or multicomponent. Heterogeneous systems are suspensions (rigid particles suspended in fluid), emulsions (droplets of one fluid suspended in another one), aerosols (droplets suspended in gas), and so on.

It is required to determine quantitative physical characteristics inherent to homogeneous and heterogeneous systems. On the basis of continuum mechanics (Sedov, 1983; Nigmatullin, 1987), all considered media are taken as macroscopic systems; that is, any volume of medium under consideration is taken as homogeneous or heterogeneous.

Arbitrary macroscopic system or a part of it possesses a mass; that is, it contains a definite amount of substance. In a system let us consider a volume V with mass m. If this system is homogeneous, its density is a continuous function of point location M and can be defined as

(3.1) equation

Thereby, the density of the system is determined at each point.

Density functions of the type (3.1) will be as many as the number of phases since when an arbitrary volume V of such system tends to be zero one gets the density of one or another phase. In doing so in multiphase system with N phases, N densities are obtained. When investigating heterogeneous system motion, it is required to use a notion of density of a volume containing all or several phases. In this connection, introduce a notion of true phase content in the following way. Let V be volume of the system part. Then,

(3.2) equation

where Vi is the volume of ith phase. If any kth phase does not enter in this volume, then Vk = 0.

Relation

(3.3) equation

is called true volume content of ith phase or concentration of ith phase in volume V. The sum of all phase concentrations φi is equal to

(3.4)

It is evident that

(3.5) equation

The true density of the system in the volume V may be determined as follows:

(3.6) equation

where ρi is density of each phase.

Now, find the phase velocity at point M. Let given phase at the instant of time t be at point M and at t + Δt shifts to the point M'. The way moved by the phase is |Δl| = MM', where Δl is vector of phase displacement in time Δt. Then, the velocity is equal to

(3.7) equation

The magnitude of the velocity is independent of the frame of reference in which the velocity is considered, but velocity projections on coordinate axes in one coordinate system differ from velocity projections in another coordinate system.

In order to solve problems on motion of multiphase medium, one should know velocities of the system. If at point M, the velocity of phase is wi, the true velocity of the mixture can be represented as

(3.8) equation

The velocity is a vector quantity distinct from density, which is a scalar quantity.

The motion of media will be studied in cylindrical coordinate system because in well drilling the flow takes place chiefly in pipes, annular channels, and beds.

In cylindrical coordinate system, variables are r,φ,z (Fig. 3.1). In accordance with (3.7), velocity projections are

FIGURE 3.1 Cylindrical coordinate system.

(3.9) equation

Then, phase velocity is

(3.10) equation

where i,j,k are unit vectors.

Thus, at each point the velocity w is defined as vector quantity, the projections of which depend on point location

(3.11)

Take in the medium a surface element S with normal n. The flow rate of the medium through this element is

(3.12) equation

where wn is the velocity projection on the normal n.

If the integrand in the expression (3.12) is a projection of an arbitrary vector (it need not be a velocity) on the normal to the surface element, expression (3.12) is called vector flux through the surface S. The flow rate of a phase through normal cross section Sn(Q = vSn), where v is mean velocity of the medium and Sn is the area of the channel cross section, represents a particular case of vector flux.

If the surface is closed, that is, it restricts a volume ΔV, then the relation

(3.13) equation

is called vector divergence, which is vector flux through the surface of infinitely small volume surrounding the considered point.

From (3.13), it ensues that if the flow rate through any closed surface S vanishes, then ∇ w = 0.

It is possible to show that

(3.14) equation

This means that the divergence characterizes the relative increase or decrease in medium volume, that is, medium compressibility.

From mathematics, it is known that expression (3.13) in cylindrical coordinates has the form

(3.15) equation

The motion and internal stress in media are caused by forces that can be classified into internal and external forces. External forces relative to the system are those that are induced by other systems, whereas internal forces are conditioned by another parts of the same system.

If at arbitrarily chosen point M of the medium takes an elementary surface ΔS with normal n, on this surface an external force ΔF will act produced by a part of the medium located, as viewed from the surface, in the normal direction. The surface exhibits a stress equal to the ratio between force and surface area

(3.16) equation

Upon contraction of the area into a point, one obtains stress at point M

(3.17) equation

Designate ideal medium (ideal fluid or ideal gas) as such a medium in which the stress vector pn at any surface element with normal n is orthogonal to the surface, that is, vector pn is parallel to n.

Figure 3.2 demonstrates decomposition of the vector pn on normal pnn and tangential pnτcomponents. In ideal fluid, there is by definition

FIGURE 3.2 Decomposition of the stress vector pnτ = pnτ(M, n) on components.

(3.18) equation

that is, tangential stresses in ideal fluid are absent.

The majority of flows in the drilling practice would be considered one dimensional in the sense that in appropriate coordinate system, Cartesian or cylindrical, only one velocity component plays a significant role. Such suggestion is true in many cases and gives needed accuracy in calculations. For example, flows in pipes and annular channels have only one velocity component wz directed along the pipe axis z, being dependent on pipe radius r, that is, wz = wz(r). The flow (inflow or outflow) of fluid in circular bed may be taken as one dimensional; that is, in cylindrical coordinate system, the flow has one velocity component wr directed along the radius r of the bed and being the function of only z in the limit of formation thickness H and radius r, that is, wr = wr(z,r). One dimensionality is of course a matter of convention since radial flow considered in Cartesian coordinates has two components, wx and wy.

Depending on the properties of fluids taking part in flows, one can consider them as incompressible or compressible. Incompressibility of fluid is defined as invariability of arbitrarily chosen fluid volume in the sense that the volume shape can be deformed but the volume by itself remains constant. One should distinguish between incompressibility and homogeneity notions. If incompressible fluid is homogeneous, then everywhere in the flow the density is constant (p = const). If the heterogeneous fluid is incompressible, then in passage through interface the density changes (p ≠ const). And yet the heterogeneous gas is compressible and in rare cases it can be taken as incompressible.

The flows, in what follows, will be mainly considered in circular pipes, in annular and concentric channels, and between parallel circular plates. It should be noted that all flows taking place in circulation system of well or in the whole system of well-bed are bounded.

CHAPTER 4

HYDRO-AEROMECHANIC EQUATIONS OF DRILLING PROCESSES

4.1 MASS CONSERVATION EQUATION

Mass conservation law states: the net mass ΔM of a mixture part occupying at time t the space volume ΔV remains constant and at following instants of time if mass change due to internal and external sources is absent (Loitsyansky, 1987)

(4.1.1) equation

By definition of density ΔM ã ρΔV, where , there is

(4.1.2) equation

Differentiation of the left part (4.1.1) gives

(4.1.3) equation

Substituting (3.14) in (4.1.3) and carrying all terms in the left part, one obtains

Since ΔV≠0, it is

or

(4.1.4)

In a similar manner, it is possible to derive equations for each phase. In doing so, one gets as many equations of the type (4.1.4) as there are phases. For example, for two-phase mixture, there is

(4.1.5) equation

In accordance with (3.15) for and definition of total derivative, the equation (4.1.4) may be rewritten as

As far as , where ω is angular velocity, it yields

(4.1.6) equation

From (4.1.6), it follows

(4.1.7)

In the case of stationary flow, that is, when ∂ρ/∂t = 0, (4.1.7) gives

(4.1.8)

Later on, it will be obtained for one-dimensional symmetric flows: in tubes where only wz ≠ 0, ρwz is a function only of the radial coordinate r; in circular slots where only wr ≠ 0, ρr is a function only of coordinates r and z; in flows induced by rotation of pipes, where only wφ≠0, ρwφ is a function only of r.

For incompressible homogeneous fluid, the density ρ in equation (4.1.8) can be removed from the derivative and thus be canceled.

4.2 MOMENTUM (MOTION) EQUATION

By definition of mixture density using phase volume contents φ1φ2, and so on, the expression

(4.2.1)

may be considered as vector of mixture momentum (Teletov, 1958). Then, the momentum of a mixture filling the volume ΔV will be

(4.2.2) equation

The vector of mass force (gravity force) distributed over the volume ΔV has the form

(4.2.3) equation

where g is the gravity acceleration.

The surface Δσ of volume ΔV under the action of external forces is exposed to surface tension. Denote through Πs total vector of surface forces. It will be determined further as applied to drilling problems. Supposing that the theorem of momentum change could be applied to elementary volume of the mixture as a whole, one obtains

(4.2.4)

and differentiation of the left side yields

In the case of the absence of additional mass sources, the second term owing to (4.1.1) vanishes and

(4.2.5)

For simplicity sake of the following mathematical treatment, let us introduce designations

(4.2.6) equation

Earlier, it has been mentioned that it makes sense to consider main drilling problems related to flows of washing solutions and grouting mortars in pipes, annular channels, circular slots, and beds with the use of cylindrical coordinates. In doing so, let us obtain concrete form of equation (4.2.5).

Take a point with coordinates r, φ, z in moving medium and mark out in the vicinity of