WORKFLOW TO DIRECTLY RELATE RATE OF PENETRATION (ROP) TO HOLE CLEANING BASED ON REAL-TIME DENSITY AND RHEOLOGY OF DRILLING FLUIDS

Abstract

A method includes performing a drilling operation using a drilling fluid; determining pressure losses of the drilling fluid in the drilling operation; updating non-linear regression coefficients to model the drilling operation; and performing gel strength tests on the drilling fluid. The method also includes determining optimal drilling parameters and the non-linear regression coefficients and determining optimal hole cleaning energy by balancing a mechanical energy level of the drilling fluid based on a maximum pump pressure of the mud pump and the optimal drilling parameters. The maximum pump pressure is determined based on the pressure losses and a range. The method further includes determining pressure changes created by the drilling operation using the optimal hole cleaning energy and results of the gel strength tests; determining an estimated equivalent circulating density based on the pressure changes; and modifying or maintaining the drilling operation based on the estimated equivalent circulating density.

Description

BACKGROUND

Hydrocarbons are located in porous rock formations located far beneath the surface of the Earth. Wells are drilled into these formations to access and produce the hydrocarbons. Wells are drilled using a drill string having a drill bit. The drill bit breaks down the rock in the formation. Drilling fluid, also known in the art as mud, is used in a myriad of ways to aid the drilling operation. For example, drilling fluid is used to control bottom hole pressures, cool down and lubricate the drill bit, remove rock cuttings from the well, etc.

Hole cleaning is related to how well the rock cuttings are being circulated out of the well using the drilling fluid. When a well is inefficiently cleaned, cuttings build up in the mud and can cause problems that impede the drilling operation, such as stuck pipe and differential sticking. Rate of penetration (ROP) is the rate at which the drill bit breaks down the rock and extend the depth of the well. When the rate of penetration is high, the rate of hole cleaning may suffer. However, it is important to maintain the highest possible rate of penetration in order to efficiently perform the drilling operation. Thus, methods and systems that enable optimization of both the rate of hole cleaning and the rate of penetration in a drilling operation is beneficial.

SUMMARY

This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter.

This disclosure presents, in accordance with one or more embodiments methods and systems for drilling a well. The method, in accordance with one or more embodiments, includes obtaining real-time drilling data from drilling equipment, including a mud pump, performing a drilling operation using a drilling fluid; determining, using a computer processor, pressure losses of the drilling fluid in the drilling operation using the real-time drilling data; updating, using the computer processor, non-linear regression coefficients to model the drilling operation using results obtained during field experiments; and performing, using the computer processor, gel strength tests on the drilling fluid. The method also includes determining, using the computer processor, optimal drilling parameters based on the real-time drilling data and the non-linear regression coefficients and determining, using the computer processor, optimal hole cleaning energy by balancing a mechanical energy level of the drilling fluid based on a maximum pump pressure of the mud pump and the optimal drilling parameters. The maximum pump pressure is determined based on the pressure losses and a range. The method further includes determining, using the computer processor, pressure changes created by the drilling operation using the optimal hole cleaning energy and results of the gel strength tests; determining, using the computer processor, an estimated equivalent circulating density based on the pressure changes; and modifying or maintaining the drilling operation based on the estimated equivalent circulating density.

The system, in accordance with one or more embodiments, includes a drilling system comprising drilling equipment, including a mud pump, configured to perform a drilling operation using a drilling fluid and a computer processor. The computer processor is configured to, iteratively and repeatedly: obtain real-time drilling data from the drilling equipment; determine pressure losses of the drilling fluid in the drilling operation using the real-time drilling data; update non-linear regression coefficients to model the drilling operation using results obtained during field experiments; perform gel strength tests on the drilling fluid; determine optimal drilling parameters based on the real-time drilling data and the non-linear regression coefficients; determine optimal hole cleaning energy by balancing a mechanical energy level of the drilling fluid based on a maximum pump pressure of the mud pump and the optimal drilling parameters, wherein the maximum pump pressure is determined based on the pressure losses and a range; determine pressure changes created by the drilling operation using the optimal hole cleaning energy and results of the gel strength tests; determine an estimated equivalent circulating density based on the pressure changes; and modify or maintain the drilling operation based on the estimated equivalent circulating density.

Other aspects and advantages of the claimed subject matter will be apparent from the following description and the appended claims.

BRIEF DESCRIPTION OF DRAWINGS

Specific embodiments of the disclosed technology will now be described in detail with reference to the accompanying figures. Like elements in the various figures are denoted by like reference numerals for consistency. The sizes and relative positions of elements in the drawings are not necessarily drawn to scale. For example, the shapes of various elements and angles are not necessarily drawn to scale, and some of these elements may be arbitrarily enlarged and positioned to improve drawing legibility. Further, the particular shapes of the elements as drawn are not necessarily intended to convey any information regarding the actual shape of the particular elements and have been solely selected for ease of recognition in the drawing.

FIG. 1 shows an example well site in accordance with one or more embodiments.

FIG. 2 shows a flowchart in accordance with one or more embodiments.

FIG. 3 shows a graph outlining the three types of gel strength in accordance with one or more embodiments.

FIG. 4 shows torsion flow movement under a drill bit in accordance with one or more embodiments.

FIG. 5 shows a graph of hydraulic friction as a function of mud pump pressure and flow rate of the drilling fluid in accordance with one or more embodiments.

FIG. 6 shows a graph of theoretical pressure loss vs. real pressure loss of a mud pump in accordance with one or more embodiments.

FIG. 7 shows torque on a graph of WOB vs. ROP in accordance with one or more embodiments.

FIG. 8 shows a linear line plotted on a graph of WOB vs. torque in accordance with one or more embodiments.

FIG. 9 shows a curve plotted on a graph of WOB vs. PPR in accordance with one or more embodiments.

FIG. 10 shows a computer system in accordance with one or more embodiments.

DETAILED DESCRIPTION

In the following detailed description of embodiments of the disclosure, numerous specific details are set forth in order to provide a more thorough understanding of the disclosure. However, it will be apparent to one of ordinary skill in the art that the disclosure may be practiced without these specific details. In other instances, well-known features have not been described in detail to avoid unnecessarily complicating the description.

Throughout the application, ordinal numbers (e.g., first, second, third, etc.) may be used as an adjective for an element (i.e., any noun in the application). The use of ordinal numbers is not to imply or create any particular ordering of the elements nor to limit any element to being only a single element unless expressly disclosed, such as using the terms “before”, “after”, “single”, and other such terminology. Rather, the use of ordinal numbers is to distinguish between the elements. By way of an example, a first element is distinct from a second element, and the first element may encompass more than one element and succeed (or precede) the second element in an ordering of elements.

FIG. 1 shows an example well site (100) in accordance with one or more embodiments. In general, well sites may be configured in a myriad of ways. Therefore, well site (100) is not intended to be limiting with respect to the particular configuration of the drilling equipment. The well site (100) is depicted as being on land. In other examples, the well site (100) may be offshore, and drilling may be carried out with or without use of a marine riser. A drilling operation at well site (100) may include drilling a wellbore (102) into a subsurface including various formations (104, 106). For the purpose of drilling a new section of wellbore (102), a drill string (108) is suspended within the wellbore (102).

The drill string (108) may include one or more drill pipes (109) connected to form conduit and a bottom hole assembly (BHA) (110) disposed at the distal end of the conduit. The BHA (110) may include a drill bit (112) to cut into the subsurface rock. The BHA (110) may include measurement tools, such as a measurement-while-drilling (MWD) tool (114) and logging-while-drilling (LWD) tool 116. Measurement tools (114, 116) may include sensors and hardware to measure downhole drilling parameters, and these measurements may be transmitted to the surface using any suitable telemetry system known in the art. Herein, the term surface is defined as any location located outside of the wellbore (102), such as somewhere on the Earth's surface, on a man-made object located on the Earth's surface, etc. The BHA (110) and the drill string (108) may include other drilling tools known in the art but not specifically shown.

The drill string (108) may be suspended in wellbore (102) by a derrick (118). A crown block (120) may be mounted at the top of the derrick (118), and a traveling block (122) may hang down from the crown block (120) by means of a cable or drilling line (124). One end of the cable (124) may be connected to a draw works (126), which is a reeling device that may be used to adjust the length of the cable (124) so that the traveling block (122) may move up or down the derrick (118). The traveling block (122) may include a hook (128) on which a top drive (130) is supported.

The top drive (130) is coupled to the top of the drill string (108) and is operable to rotate the drill string (108). Alternatively, the drill string (108) may be rotated by means of a rotary table (not shown) on the drilling floor (131). Drilling fluid (commonly called mud) may be stored in a mud pit (132), and at least one pump (134) may pump the mud from the mud pit (132) into the drill string (108). The mud may flow into the drill string (108) through appropriate flow paths in the top drive (130) (or a rotary swivel if a rotary table is used instead of a top drive to rotate the drill string (108)).

In one implementation, a system (199) may be disposed at or communicate with the well site (100). System (199) may control at least a portion of a drilling operation at the well site (100) by providing controls to various components of the drilling operation. In one or more embodiments, system (199) may receive data from one or more sensors (160) arranged to measure controllable parameters of the drilling operation. As a non-limiting example, sensors (160) may be arranged to measure WOB (weight on bit), RPM (drill string rotational speed), GPM (flow rate of the mud pumps), and ROP (rate of penetration of the drilling operation). The system (199) may be a computer (1002) system, outlined below in FIG. 10, comprising a computer processor (1005) in accordance with one or more embodiments.

Sensors (160) may be positioned to measure parameter(s) related to the rotation of the drill string (108), parameter(s) related to travel of the traveling block (122), which may be used to determine ROP of the drilling operation, and parameter(s) related to flow rate of the pump (134). For illustration purposes, sensors (160) are shown on drill string (108) and proximate mud pump (134). The illustrated locations of sensors (160) are not intended to be limiting, and sensors (160) could be disposed wherever drilling parameters need to be measured. Moreover, there may be many more sensors (160) than shown in FIG. 1 to measure various other parameters of the drilling operation. Each sensor (160) may be configured to measure a desired physical stimulus.

During a drilling operation at the well site (100), the drill string (108) is rotated relative to the wellbore (102), and weight is applied to the drill bit (112) to enable the drill bit (112) to break rock as the drill string (108) is rotated. In some cases, the drill bit (112) may be rotated independently with a drilling motor. In further embodiments, the drill bit (112) may be rotated using a combination of the drilling motor and the top drive (130) (or a rotary swivel if a rotary table is used instead of a top drive to rotate the drill string (108)). While cutting rock with the drill bit (112), mud is pumped into the drill string (108).

The mud flows down the drill string (108) and exits into the bottom of the wellbore (102) through nozzles in the drill bit (112). The mud in the wellbore (102) then flows back up to the surface in an annular space between the drill string (108) and the wellbore (102) with entrained cuttings. The mud with the cuttings is returned to the pit (132) to be circulated back again into the drill string (108). Typically, the cuttings are removed from the mud, and the mud is reconditioned as necessary, before pumping the mud again into the drill string (108). In one or more embodiments, the drilling operation may be controlled by the system (199).

Mud, or drilling fluid, has a myriad of properties that effect parameters in a drilling operation, such as hole cleaning and rate of penetration (ROP). In order for drilling fluid to achieve optimal levels of hole cleaning and ROP, the drilling fluid properties should be at certain levels such that the drilling fluid maintains the effects of frictional pressure drop and maintains the solids bearing capacity at an optimal level throughout the drilling operation.

Specifically, fluid viscosity and yield stress are some drilling fluid properties that control the aforementioned effect of drilling fluid. Efficient drilling fluid lift capacity requires low viscosity during circulation and requires adequate yield stress to maintain cuttings suspension during circulation breaks. As such, the rheology of the drilling fluid may be controlled using different types of viscosifiers, such as clay or polymeric materials, depending on the type of drilling fluid and the requirements of the drilling operation.

Clay viscosifiers create suspended clay particles within the drilling fluid. The suspended clay particles form a loose structure or semirigid colloidal dispersion of a solid in a liquid. As such, the suspended clay particles are responsible for the gel strength of the drilling fluid when the drilling fluid is at rest and are responsible for the thinning behavior after the fluid is sheared. This loose structure and the resulting bulk rheological properties are dependent on temperature, pressure, composition, and shear history of the bulk fluid properties.

In order to optimize the drilling performance, especially during surge and swab operations, gel strengths are taken into account to indicate the drilling fluid quality. The results of the gel strength tests give indications as to whether or not the drilling fluid is good or bad for tripping a drill string (108) or running in casing.

When relating drilling fluid flow rate to ROP, as outlined further in this disclosure, optimal flow rate, optimal pressure drop across the drill bit (112), and optimal equivalent diameter are able to be determined. Using these determinations, the cuttings rate may be determined enabling ROP to be related to hole cleaning and real-time mud weight. This relationship between ROP and hole cleaning may be then used to optimize both ROP and hole cleaning at the same time. As such, the present disclosure presents systems and methods for improving the quality of downhole predictions using the relationship between ROP and hole cleaning.

FIG. 2 shows a flowchart in accordance with one or more embodiments. The flowchart outlines a method for improving the quality of downhole predictions by relating ROP to hole cleaning. The method further utilizes the downhole predictions to make decisions regarding the drilling operation. While the various blocks in FIG. 2 are presented and described sequentially, one of ordinary skill in the art will appreciate that some or all of the blocks may be executed in different orders, may be combined or omitted, and some or all of the blocks may be executed in parallel. Furthermore, the blocks may be performed actively or passively.

In step 300, real-time drilling data is obtained from drilling equipment. The drilling equipment includes a mud pump (134) and is performing a drilling operation using a drilling fluid. Herein, drilling equipment may include any drilling equipment known in the art such as the system (199), the drill bit (112), the drill string (108), the mud pump (134), a real-time Density and Rheology Unit, surface mud equipment, etc. The drilling fluid may be any type of drilling fluid known in the art such as water-based mud, oil-based mud, etc.

The drilling operation may be any type of drilling operation known in the art such as conventional drilling, extended reach (horizontal) drilling, onshore/offshore drilling, etc. The real-time drilling data includes any drilling related data, such as real-time drilling parameters, specifications of the drilling equipment, etc. For purposes of example only, the real-time drilling data may include drill bit (112) specifications, drill string (108) specifications, bottom hole assembly (BHA) specifications, mud pump (134) specifications, ROP, weight on bit (WOB), rate of rotation of the top drive and/or drill bit (RPM), drilling fluid rheology such as gel strength and density, depth of the well, etc.

In step 302, pressure losses of the drilling fluid are determined using a computer processor (1005) and the real-time drilling data. Specifically, the pressure losses may be determined using Equation (1) and Equation (2), below. In the equations below, K_L=loss coefficient; ΔP=change in pressure; ρ=ρ_mud=density of the drilling fluid; v=velocity of the drilling fluid at the surface; ΔP_bit=change in pressure at the drill bit (112); v_n=velocity of the drilling fluid at the nozzle of the drill bit (112); d_n=equivalent diameter of the nozzle of the drill bit (112).

$\begin{array}{cc}{K}_L=\Delta P/\left(\frac{1}{2}\rho {\overline{v}}^2\right)& \mathrm{Equation}\left(1\right)\end{array}$ $\begin{array}{cc}\Delta P_{\mathrm{bit}}=1\text{.11}·\left(\frac{1}{2}{\rho }_{mud}{\overline{v}}_n^2\right)\mathrm{where}{\overline{v}}_n=33.36\sqrt{\frac{\Delta P_{\mathrm{bit}}}{{\rho }_{mud}}}& \mathrm{Equation}\left(2\right)\end{array}$

Assuming tri-cone bit, total area of nozzles of equal sizes A_n (in.²):

$A_n=3A=\frac{3\pi }{4}{d}_n^2$ $d_n=32\sqrt{\frac{4A_n}{3\pi }}$

In step 304, non-linear regression coefficients are updated to model the drilling operation using results obtained during field experiments. Specifically, the field experiments are performed on nearby wells and change depending on the depth of the well. As such, the non-linear regression coefficients must be updated depending on the current depth of the drilling operation. The non-linear regression field real-time data collected from the nearby wells represents the ROP-WOB behaviors.

In step 306, gel strength tests are performed on the drilling fluid in real time suing the computer processor (1005). Gel strength of a fluid is the fluid shear stress measured at low shear rate after it has been static for a considerable period of time. Gelling occurs in a well once the circulation of the drilling fluid stops, and it is a measurement of the inter-particle forces that would prevent the cuttings settling at the bottom of the well. Thus, this fluid property reveals the drilling fluid's ability to suspend the cuttings and the weighting materials, such as clay, during static periods and during circulation. More pressure from the mud pumps (134) is required to start up circulation when the gel strength increases, and more pressure from the mud pumps (134) tends to increase the ROP.

Gel strengths are conventionally measured by running the drilling fluid in a viscometer at 3 rotations per minute (rpm). The gel strength of the drilling fluid may be measured after 10-second and 10-minute periods. A third reading may also be measured after 30 minutes to determine if the gelling is significant over long static periods, such as during tripping out of hole operations.

In accordance with one or more embodiments, a field rotational viscometer may be replaced with a real-time Density and Rheology Unit (DRU). The DRU measures periodically laboratory-grade pressurized mud density and six-speed rheology. The DRU minimizes the fluctuation in the equivalent circulating density (ECD) of the drilling fluid and prevent losses by providing on-the-fly monitoring of the drilling fluids density and rheology deviations. When optimizing the drilling fluid, the yield stress needed to re-initiate the flow and the strain to break the gel structure should be analyzed using the DRU.

In accordance with one or more embodiments, the drilling fluid is inserted into the DRU for the gel-strength tests. The rpm of the DRU is recorded and 10 second, 10 minute, and 30 minute gel periods are started. The gel strength measurements are recorded using a 3 rpm reading after the 10 second, 10 minute, and 30 minute periods. The gel strengths are analyzed to determine if the rheology of the drilling fluid is desirable or not.

FIG. 3 shows a graph outlining the three types of gel strength: high flat (300), progressive (302), and low flat (304) in accordance with one or more embodiments. FIG. 3 shows gel strength on the y-axis and time on the x-axis. High flat (300) is when the gel strength remains relatively high throughout the gel strength experiments. Progressive (302) is when the gel strength changes from a low gel strength to a high gel strength through the gel strength experiments. Low flat (304) is when the gel strength remains relatively low throughout the gel strength experiments. The low flat gel strength mud is the most efficient fluid from the hole cleaning perspective, mud pumps efficiency, and fuel consumption economy.

In accordance with one or more embodiments, a low flat (304) gel strength is desirable because circulation is easy to break, and drilling is able to continue with minimal mud pump (134) pressure output. Depending on the results of the gel strength tests, the rheology of the drilling fluid may be modified or maintained. For example, if the gel strength is high flat (300) or progressive (302), then the rheology may need to be modified. If the gel strength is low flat (304), then the rheology of the drilling mud needs only be maintained during the drilling operation.

In step 308, optimal drilling parameters based on the real-time drilling data and the non-linear regression coefficients are determined using the computer processor (1005). In step 310, optimal hole cleaning is determined using the computer processor (1005) by balancing a mechanical energy level of the drilling fluid based on a maximum pump pressure of the mud pump (134) and the optimal drilling parameters. The maximum pump pressure is determined based on the pressure losses and a range.

In accordance with one or more embodiments, the range comprises a first range and a second range. The first range is based on a maximum operating capacity of the mud pump (134) and the second range is based on a maximum energy of the mud pump (134). The first range assumes the rate of penetration of the drill bit (112) is linearly related to the maximum operating capacity of the mud pump (134). The second range assumes rate of penetration is a function of bottom hole cleaning.

FIG. 4 shows torsion flow movement under a drill bit (112) in accordance with one or more embodiments. Specifically, the drill bit (112) is shown in relation to the floor (400) of the wellbore. The distance (402) drilled along the wellbore, the direction of torque (404) being applied to the drill bit (112), the angular rotation (406) of the drill bit (112), and the radius (408) of the drill bit (112) are shown. The distance (402) drilled along the wellbore can be calculated using Equation (3) below. In the equations below, Δh=the distance (402) drilled along the wellbore; RPM=rotations per minute; ROP=rate of penetration; PPR=penetration per revolution; MSE=mechanical specific energy; and WOB=weight on bit.

$\begin{array}{cc}\Delta h=\frac{\mathrm{Penetration}\mathrm{Per}\mathrm{Minute}\left(\mathrm{in}/\mathrm{hr}\right)}{R\mathrm{PM}\left(\mathrm{rev}/\min \right)}=\frac{ROP}{RPM}=\mathrm{PPR}& \mathrm{Equation}\left(3\right)\end{array}$ $\begin{array}{cc}\mathrm{MS}{E}_{axial-\mathrm{rotary}}=\frac{WOB*\Delta h}{\mathrm{Area}*\Delta h}+\frac{\mathrm{Tourque}*2\pi *\mathrm{num}\mathrm{of}\mathrm{Rotations}}{\mathrm{Area}*\Delta h}& \mathrm{Equation}\left(4\right)\end{array}$ $\begin{array}{cc}\mathrm{PPR}=\frac{ROP}{RPM}& \mathrm{Equation}\left(5\right)\end{array}$ $\begin{array}{cc}{\mathrm{MSE}}_{axial-\mathrm{rotary}}=\frac{WOB}{{\mathrm{Area}}_{bit}}+\frac{\mathrm{Tourque}*2\pi }{{\mathrm{Area}}_{bit}*PPR}& \mathrm{Equation}\left(6\right)\end{array}$

Ambient field stress is related to the rock break down pressure as shown in Equation (7) below. The rock break down pressure is the pressure at which a tensile fracture occurs as well as the instantaneous shut-in pressure (ISIP). The ISIP is the pressure required to hold the fracture open around the wellbore. The radial stress is negligible since the wellbore thickness is infinite. Therefore, the radial strain will approach zero. The stress field at any point within the drilled trajectory can be represented as three principal stresses denoted as S₁, S₂, and S₃ for the maximum, intermediate, and minimum stresses respectively.

These stresses are considered positively compressive rather than tensile stresses and are represented as a vertical principal overburden stress and two horizontal principal stresses. In the equations below, P_break-down=rock break down pressure; T=tensile stress; S_Hmin=minimum horizontal stress; S_Hmax=maximum horizontal stress; P_pores=pore pressure; S_v=vertical principal overburden stress; g=gravity; h=wellbore vertical depth; v=poison ratio τ=shear stress; τ₀=cohesive strength; μ=coefficient of friction; and S_n=compressive stress acting in a direction normal to the plane of failure.

$$\begin{gathered} \begin{array}{cc}{P}_{\mathrm{break}-\mathrm{down}}=T+3·S_{H_{\min }}-S_{H_{\max }}-P_{\mathrm{pores}}\mathrm{where}\text{ \\ Sv=ρmud·g·h⁢ \\ SHmax=(ν/1-ν)·Sv⁢ \\ τ=τ0+μ·Sn}& \mathrm{Equation}\left(7\right)\end{array} \end{gathered}$$

In normal drilling operations, the drill bit (112) is breaking down the rock and the drilling fluid is transporting the drilled cuttings to the surface utilizing the lift forces of the drilling fluid. The cleaning action under the drill bit (112) is directly proportional to the shear rate around the drill bit (112), drilling fluid flow rate, and the pump pressure of the mud pump (134). The cleaning action under the drill bit (112) is inversely proportional to the drill bit (112) nozzle diameter. ROP is exponentially proportional to the Uniaxial Bit Compression Pressure/Differential Bottom Hole Pressure (DBHP). DBHP slightly influences ROP in very permeable and tight zones. DBHP highly influences ROP in medium permeable zones, such as shales and sedimentary rocks.

During drilling, the drill string (108) acts as a viscometer spindle and the wellbore acts as the measurement chamber, as depicted in FIG. 4. The fluid viscosity is being sheared at two different stages: through the jet impact of fluid flowing out of the drill bit (112) nozzles and as a result of the rotating drill bit (112) pushing solids in a helical flow around the drill bit (112).

From the rheology theory, the drilling fluid effective viscosity (shown below in Equation (8)) is a result of the torsion flow between two parallel plates. This is depicted in FIG. 4 with the drill bit (112) and the floor (400) of the wellbore acting as parallel plates. In the equations below: μ_mud=drilling fluid effective viscosity; σ_bit=shear stress applied by the drill bit (112); {dot over (γ)}_bit=shear strain under the drill bit (112); θ=angular rotation of the drill bit (112); K_σ=stress constant; K_γ=strain constant; and K_σ/K_γ=geometric shape constants of the drill bit (112).

$\begin{array}{cc}{\mu }_{mud}=\frac{{\sigma }_{\mathrm{bit}}}{{\stackrel{.}{\gamma }}_{\mathrm{bit}}}=\frac{\mathrm{Tour}que}{\mathrm{RPM}\left(\theta \right)}·\frac{K_{\sigma }}{K_{\gamma }}& \mathrm{Equation}\left(8\right)\end{array}$

Assuming the bottom hole cleaning is adequate, then ROP is proportional to the drilling fluid flow rate occupying the removed volume of rocks divided by the equivalent nozzle diameter. Thus, the hole cleaning is optimal if:

$\mathrm{ROP}\propto \frac{\mathrm{Removed}\mathrm{Volume}\mathrm{of}\mathrm{Rocks}}{\mathrm{time}\mathrm{unit}}\propto \frac{\begin{array}{c}\mathrm{Amount}\mathrm{of}\mathrm{lifting}\mathrm{fluid}\mathrm{occupying}\mathrm{Removed}\\ \mathrm{Volume}\mathrm{of}\mathrm{Rocks}\mathrm{to}\mathrm{transport}\mathrm{cuttings}\end{array}}{\mathrm{time}\mathrm{unit}}$ $\mathrm{ROP}\propto \frac{\begin{array}{c}\mathrm{Amount}\mathrm{of}\mathrm{lifting}\mathrm{fluid}\mathrm{occupying}\mathrm{Removed}\\ \mathrm{Volume}\mathrm{of}\mathrm{Rocks}\mathrm{to}\mathrm{transport}\mathrm{cuttings}\end{array}}{\mathrm{time}\mathrm{unit}}$ $\mathrm{ROP}=\left[\frac{\mathrm{Removed}\mathrm{Volume}\mathrm{of}\mathrm{Rock}{V}_{\mathrm{rock}-\mathrm{removed}}}{\pi ·{\left(\mathrm{bit}\mathrm{diameter}\right)}^2·\mathrm{time}\mathrm{unit}}\right]=\left[\frac{\begin{array}{c}\mathrm{fluid}\mathrm{shear}\mathrm{force}·\mathrm{Amount}\mathrm{of}\mathrm{lifting}\mathrm{fluid}\mathrm{occupying}\\ V_{\mathrm{rock}-\mathrm{removed}}\mathrm{to}\mathrm{transport}\mathrm{cuttings}\mathrm{under}\mathrm{bit}\end{array}}{\mathrm{time}\mathrm{unit}}\right]$

Equation (9) below shows an ordinary differential equation and the subsequent solution of the ordinary differential equation. The solution of the ordinary differential equation is mathematical relationship between hole cleaning, ROP, and flow rate, as shown in Equation (10).

In the below equations, α₁ . . . α₃ . . . β=the non-linear regression coefficients described above; Y_nozzle=drilling fluid shear rate across the nozzles; q=flow rate; d_bit=diameter of the drill bit (112); d_e=equivalent nozzle diameter of the drill bit (112)

$\mathrm{nozzles}\left(\mathrm{WOB}/\frac{\pi }{4}{d}_{\mathrm{bit}}^2\right)/\left(P_{\mathrm{break}-\mathrm{down}}\right)=\mathrm{rock}\mathrm{bulk}\mathrm{compressibility}$

; W_dcdrill collar nominal weight. If sonic data is available, then

$\beta =\frac{c_B}{c_t},c_B=\mathrm{Rock}\mathrm{Bulk}\mathrm{Compressibility},\mathrm{and}$ $c_t=\mathrm{Total}\mathrm{Compressibility}.$

$\begin{array}{cc}\frac{\partial \mathrm{ROP}}{\partial \mathrm{WOB}}=\beta ·{\left(\left[{\gamma }_{nozzle}·\left(q/d_e\right)·\left({\mu }_{mud}·\frac{\mathrm{RPM}}{\mathrm{Torque}}\right)\right]\right)}^{\alpha }·\left[\frac{\frac{1}{\left(\pi /4\right)d_{\mathrm{bit}}^2}}{P_{\mathrm{break}-\mathrm{down}}}\right]·\mathrm{ROP}& \mathrm{Equation}9\end{array}$ $\frac{\partial \mathrm{ROP}}{\mathrm{ROP}}=\beta ·{\left(\left[{\gamma }_{nozzle}·\left(q/d_e\right)·\left({\mu }_{mud}·\frac{\mathrm{RPM}}{\mathrm{Torque}}\right)\right]\right)}^{\alpha }·\left[\frac{\frac{1}{\left(\pi /4\right)d_{\mathrm{bit}}^2}}{P_{\mathrm{break}-\mathrm{down}}}\right]·\partial \mathrm{WOB}·\left(\sin \theta +\cos \theta \right)$ $\underset{RO{P}_1}{\overset{RO{P}_2}{\int }}\frac{\partial \mathrm{ROP}}{\mathrm{ROP}}=\beta ·{\left(\left[{\gamma }_{nozzle}·\left(q/d_e\right)·\left({\mu }_{mud}·\frac{\mathrm{RPM}}{\mathrm{Torque}}\right)\right]\right)}^{\alpha }·\left[\frac{\frac{1}{\left(\pi /4\right)d_{\mathrm{bit}}^2}}{P_{\mathrm{break}-\mathrm{down}}}\right]·\left(\sin \theta +\cos \theta \right)·\underset{{\mathrm{WOB}}_1}{\overset{{\mathrm{WOB}}_2}{\int }}\partial \mathrm{WOB}$ $\ln \mathrm{ROP}=\beta ·{\left(\left[{\gamma }_{nozzle}·\left(q/d_e\right)·\left({\mu }_{mud}·\frac{\mathrm{RPM}}{\mathrm{Torque}}\right)\right]\right)}^{\alpha }·\left[\frac{\frac{1}{\left(\pi /4\right)d_{\mathrm{bit}}^2}}{P_{\mathrm{break}-\mathrm{down}}}\right]·\left(\sin \theta +\cos \theta \right)·\left(\mathrm{WOB}+C\right)$ $\mathrm{ROP}=e^{\beta ·{\left[{\gamma }_{nozzle}·\left(q/d_e\right)·\left({\mu }_{mud}·\frac{\mathrm{RPM}}{\mathrm{Torque}}\right)\right]}^{\alpha }·\left[\left(\mathrm{WOB}/\frac{\pi }{4}{d}_{\mathrm{bit}}^2\right)/\left(P_{\mathrm{break}-\mathrm{down}}\right)\right]·\left(\sin \theta +\cos \theta \right)+\beta ·C}$ $\mathrm{ROP}=e^{\beta ·\left[{\stackrel{.}{\gamma }}_{nozzle}^{{\alpha }_1}·{\left(q/d_{e_{\mathrm{nozzle}}}\right)}^{{\alpha }_2}·{\left({\mu }_{mud}·\frac{\mathrm{RPM}}{\mathrm{Torque}}\right)}^{{\alpha }_3}\right]·\left[\left(\mathrm{WOB}/\frac{\pi }{4}{d}_{\mathrm{bit}}^2\right)/\left(P_{\mathrm{break}-\mathrm{down}}\right)\right]·\left(\sin \theta +\cos \theta \right)+\beta ·C}$ $\mathrm{ROP}=e^{\left[\left(\mathrm{Uniaxial}\mathrm{pressure}\mathrm{exerted}\mathrm{by}\mathrm{bit}\right)/\mathrm{Breakdown}\mathrm{Bottom}\mathrm{Hole}\mathrm{Pressure}\right]·\left[\mathrm{fluid}\mathrm{shear}\mathrm{force}.\mathrm{fluid}\mathrm{rate}/\mathrm{equivalent}\mathrm{nozzle}\mathrm{diameter}\right]}$ $\begin{array}{cc}\mathrm{ROP}=e^{\beta ·\left[{\stackrel{.}{\gamma }}_{\mathrm{nozzle}}^{{\alpha }_1}·{\left(q/d_{e_{\mathrm{nozzle}}}\right)}^{{\alpha }_2}·{\left({\mu }_{mud}·\frac{\mathrm{RPM}}{\mathrm{Torque}}\right)}^{{\alpha }_3}\right]·\left[\left(\mathrm{WOB}/\frac{\pi }{4}{d}_{\mathrm{bit}}^2\right)/\left(P_{\mathrm{break}-\mathrm{down}}\right)\right]·\left(\sin \theta +\cos \theta \right)+\beta ·C}& \mathrm{Equation}10\end{array}$ $\begin{array}{cc}\mathrm{WOP}=\frac{W_{\mathrm{dc}}\times \mathrm{Buoyancy}\mathrm{Factor}\times \mathrm{Length}\mathrm{of}\mathrm{BHA}}{\mathrm{Safety}\mathrm{Factor}}& \mathrm{Equation}11\end{array}$

FIG. 5 shows a graph of hydraulic friction as a function of mud pump (134) pressure and flow rate of the drilling fluid in accordance with one or more embodiments. The y axis shows mud pump (134) pressure. The x-axis shows flow rate of the drilling fluid. The maximum working pressure (500) of the mud pump (134) is labeled on the y-axis.

The lowest recommended flow rate (502)of the drilling fluid and the maximum flow rate (504) of the drilling fluid are depicted on the x-axis. A first line (506) depicting hydraulic friction as a function of pump pressure and a second line (508) depicting hydraulic friction as a function of pressure loss are shown on the graph. The second line also represents overall pressure loss due to hydraulic friction at each flow rate.

The change in bit pressure (510) is shown on the graph as the difference in mud pump (134) pressures at the same flow rate between the first line (506) and the second line (508). Where the first line (506) and the second line (508) intersect is the theoretical operating point of the mud pump (134). The first range (512) and the second range (514), introduced above, are shown on FIG. 5.

The mud pump (134) pressure is shown below in Equation (12). The hydraulic friction loss is shown below in Equation (13). In the equations below, P_pump=mud pump (134) pressure; ΔP_bit=change in pressure at the drill bit (112); ΔP_loss=overall pressure loss due to hydraulic friction; and D=measured depth of the wellbore.

$\begin{array}{cc}{P}_{\mathrm{pump}}=\Delta P_{\mathrm{bit}}+\Delta P_{\mathrm{loss}}& \mathrm{Equation}\left(12\right)\end{array}$ $\begin{array}{cc}\Delta P_{loss}=K_1·D·q^m& \mathrm{Equation}\left(13\right)\end{array}$ $\mathrm{where}$ $m=\frac{\log \left(\Delta P_{los{s}_2}\right)-\log \left(\Delta P_{los{s}_1}\right)}{\log \left(q_2\right)-\log \left(q_1\right)},K_1=\frac{\Delta P_{los{s}_2}}{D·q_2^m}$ $\Delta P_{los{s}_1}=P_{pum{p}_1}-\Delta P_{{\mathrm{bit}}_1}$ $\Delta P_{los{s}_2}=P_{pum{p}_2}-\Delta P_{{\mathrm{bit}}_2}$

FIG. 6 shows a graph of theoretical pressure loss vs. real pressure loss of a mud pump (134) in accordance with one or more embodiments. The pressure of the mud pump (134) is shown on the y-axis and the flow rate of the drilling fluid is shown on the x-axis. A theoretical pressure loss line (600) and a real pressure loss line (602) are shown on the graph.

The theoretical pressure loss line (600) is shown as a straight line that slowly increases as mud pump (134) pressure and flow rate increase. However, the real pressure loss line (602) deviates from the theoretical pressure loss line (600). The real pressure loss line (602) shows higher pressure losses at low pump pressures/flow rates and at higher pump pressures/flow rates when compared to the theoretical pressure loss line (600). As such, it is more efficient to operate the mud pumps (134) at 80-90% of the maximum flow rate (504).

As stated above, the optimal hole cleaning energy is optimized, in part, using two different ranges. The first range (512) and the second range (514). The first range (512) is based on a maximum operating capacity of the mud pump (134) and assumes the rate of penetration of the drill bit (112) is linearly related to the maximum operating capacity of the mud pump (134). However, as shown in FIG. 6, it is not recommended to operate the mud pumps (134) faster than 80-90% of their maximum operating capacity.

Thus, the goal, when using the first range (512), is to determine the optimal flow rate of the drilling fluid and the optimal drill bit (112) nozzle size. This is determined by applying drill bit (112) hydraulic horsepower as the optimization criteria. Optimal hydraulic parameters estimated for vertical wells can be used as a first approach for inclined wells. The hydraulic program depends on selecting the correct mud pump (134). Specifically, the size of the mud pump (134) liner may be minimized to ensure the highest pressure available, as shown in Equations 14-21 below. In the equations below,

$A_n=\frac{\pi }{4}·{\left(d_e\right)}^2;$

K_bit=pressure loss coefficient across the drill bit (112); and d_eI=equivalent nozzle diameter.

$\begin{array}{cc}{P}_{{\mathrm{pump}}_{\max }}=1\text{.11}·\left(\frac{1}{2}{\rho }_{mud}{\overline{v}}_n^2\right)+K_1·D·q^m& \mathrm{Equation}\left(14\right)\end{array}$ $\mathrm{Multiply}\mathrm{Equation}\left(14\right)\mathrm{by}{q}^2\mathrm{and}\overline{v}=\frac{q}{A}\mathrm{to}\mathrm{get}:$ $P_{{\mathrm{pump}}_{\max }}·q^2=1\text{.11}·\left(\frac{1}{2}{\rho }_{mud}·{\left(\frac{q}{A_n}\right)}^2\right)·q^2+K_1·D·q^{m+2}$ $\begin{array}{cc}{P}_{{\mathrm{pump}}_{\max }}·q^2=1\text{.11}·\left(\frac{1}{2}{\rho }_{mud}·\frac{q^2·q^2}{{\left(\frac{\pi }{4}·d_e^2\right)}^2}\right)+K_1·D·q^{m+2}& \mathrm{Equation}\left(15\right)\end{array}$ $\begin{array}{cc}{K}_{\mathrm{bit}}=\Delta P/\left(\frac{1}{2}{\rho }_{mud}{\overline{v}}_n^2\right)& \mathrm{Equation}\left(16\right)\end{array}$ $P_{{\mathrm{pump}}_{\max }}·q^2=1\text{.11}·\left(\frac{1}{2}{\rho }_{mud}·\frac{q^2·q^2}{{\left(\frac{\pi }{4}·d_e^2\right)}^2}\right)+K_1·D·q^{m+2}$ $K_{\mathrm{bit}}=1\text{.11}·\frac{1}{2}·{\rho }_{mud}/{\left(\frac{\pi }{4}\right)}^2$ $P_{{\mathrm{pump}}_{\max }}·q^2=1\text{.11}·\left(\frac{1}{2}{\rho }_{mud}·\frac{q^2·q^2}{{\left(\frac{\pi }{4}·d_e^2\right)}^2}\right)+K_1·D·q^{m+2}$ $K_{\mathrm{bit}}=1\text{.11}·\left(\frac{\frac{1}{2}{\rho }_{mud}}{{\left(\frac{\pi }{4}\right)}^2}\right)$ $P_{{\mathrm{pump}}_{\max }}·q^2=K_{\mathrm{bit}}·\left(\frac{q^2·q^2}{d_e^4}\right)+K_1·D·q^{m+2}$ $P_{{\mathrm{pump}}_{\max }}·q^2=K_{\mathrm{bit}}·{\left(\frac{q}{d_e}\right)}^4+K_1·D·q^{m+2}$ $K_{\mathrm{bit}}·{\left(\frac{q}{d_e}\right)}^4=P_{{\mathrm{pump}}_{\max }}·q^2-K_1·D·q^{m+2}$ $\begin{array}{cc}\frac{q}{d_e}={\left[\frac{1}{K_{bit}}.\left(P_{{\mathrm{pump}}_{\max }}·q^2-K_1·D·q^{m+2}\right)\right]}^{1/4}& \mathrm{Equation}\left(17\right)\end{array}$ $\frac{\partial \mathrm{ROP}}{\partial q}=\frac{\partial }{\partial q}{d_e\left[\frac{1}{K_{\mathrm{bit}}}.\left(P_{{\mathrm{pump}}_{\max }}·q^2-K_1·D·q^{m+2}\right)\right]}^{1/4}=0$ $\begin{array}{cc}2·P_{{\mathrm{pump}}_{\max }}·q-\left(m+2\right)·K_1·D·q^{m+1}=0& \mathrm{Equation}\left(18\right)\end{array}$ $\begin{array}{cc}{q}_{{\mathrm{optimum}}_I}={\left(\frac{2·P_{{\mathrm{pump}}_{\max }}}{\left(m+2\right)·K_1·D}\right)}^{1/m}& \mathrm{Equation}\left(19\right)\end{array}$

Substitute Equation (18) into Equation (12) to get Equation (20) below:

$\begin{array}{cc}\Delta P_{\mathrm{bit},{\mathrm{optimum}}_I}={\left(\frac{m·P_{{\mathrm{pump}}_{\max }}}{\left(m+2\right)·K_1·D}\right)}^{1/m}& \mathrm{Equation}\left(20\right)\end{array}$ $\begin{array}{cc}{d}_{e_I}={\left[K_{\mathrm{bit}}·\frac{q_{\mathrm{optimu}{m}_I}^2}{\left(\Delta P_{bit,{\mathrm{optimum}}_I}\right)}\right]}^{1/4}& \mathrm{Equation}\left(21\right)\end{array}$

As stated above, the second range is based on a maximum energy of the mud pump (134) and assumes rate of penetration is a function of bottom hole cleaning. Thus, the mud pump (134) energy capacity must be fully utilized to optimize the ROP. This is done by increasing the energy of the mud pump (134) to the maximum limit (E_pumpmax=P_pumpmax·q_max=constant). The second range may be optimized by minimizing the mechanical specific energy using Equation (4).

The drilling parameters are optimized with respect to the change in the axial force (AF) on the drill bit (112) and the torsional force (TF). The relationship between WOB and Torque can be linear line within a specific range. If the drill bit (112) is 100% efficient, MSE (as in Equation (6)) approached the compressive strength of the rock.

The efficiency of the WOB increases when the relationship with ROP is linear as shown in FIG. 7. FIG. 7 shows torque (404) on a graph of WOB vs. ROP in accordance with one or more embodiments. Specifically, FIG. 7 shows ROP on the y-axis and WOB on the x-axis. A line representing torque (404) is displayed on the graph. The specific range where torque (404) is a linear line is the range of ROP and WOB that creates the most efficient drilling. This relationship is affected by the rock type, drill bit (112) type, drill bit (112) wear, hydrostatic pressure, and drilling fluids. Equation (22), below, shows torque as a linear function of WOB. FIG. 8 shows Equation (22) as a linear line (800) plotted on a graph of WOB vs. torque (404) in accordance with one or more embodiments.

Torque=f(WOB)=c+a·WOB   Equation (22)

Equation (23), below, shows PPR as a function of WOB. FIG. 9 shows Equation (23) as a curve (900) plotted on a graph of WOB vs. PPR in accordance with one or more embodiments. Specifically, WOB is shown along the x-axis and PPR is shown along the y-axis. FIG. 9 shows how PPR governs MSE in accordance with one or more embodiments. The curve (900) is a quadratic function based on the availability of datasets and the evaluation of the determination coefficient (R²) for the available dataset test fits.

PPR(WOB)=a·WOB²+b·WOB+c   Equation (23)

Assuming ROP is a function of the bottom hole cleaning, optimal ROP can be found by plugging Equation (17) into Equation (10) (see below) and assuming the non-linear regression coefficients=0.5.

$\mathrm{ROP}=e^{\beta ·\left[{\stackrel{.}{\gamma }}_{\mathrm{nozzle}}^{{\alpha }_1}·{\left(q/d_{e_{\mathrm{nozzle}}}\right)}^{{\alpha }_2}·{\left({\mu }_{mud}·\frac{\mathrm{RPM}}{\mathrm{Torque}}\right)}^{{\alpha }_3}\right]·\left[\left(\mathrm{WOB}/\frac{\pi }{4}{d}_{\mathrm{bit}}^2\right)/\left(P_{\mathrm{break}-\mathrm{down}}\right)\right]·\left(\sin \theta +\cos \theta \right)+\beta ·C}$ $\mathrm{ROP}=e^{\beta ·\left[{\stackrel{.}{\gamma }}_{\mathrm{nozzle}}^{{\alpha }_1}·{\left(\frac{1}{K_{\mathrm{bit}}}\right)}^{\frac{1}{4}{\alpha }_2}·{\left(q^2·P_{{\mathrm{pump}}_{\max }}-K_1·D·q^{m+2}\right)}^{\frac{1}{4}{\alpha }_2}·{\left({\mu }_{mud}·\frac{\mathrm{RPM}}{\mathrm{Torque}}\right)}^{{\alpha }_3}\right]·\left[\left(\mathrm{WOB}/\frac{\pi }{4}{d}_{\mathrm{bit}}^2\right)/\left(P_{\mathrm{break}-\mathrm{down}}\right)\right]·\left(\sin \theta +\cos \theta \right)+\beta ·C}$ $\mathrm{Where}A=\beta ·\left[{\stackrel{.}{\gamma }}_{\mathrm{nozzle}}^{{\alpha }_1}·{\left({\mu }_{mud}·\frac{\mathrm{RPM}}{\mathrm{Torque}}\right)}^{{\alpha }_3}\right]·\left[\left(\mathrm{WOB}/\frac{\pi }{4}{d}_{\mathrm{bit}}^2\right)/\left(P_{\mathrm{break}-\mathrm{down}}\right)\right]·\left(\sin \theta +\cos \theta \right)+\beta ·C$ $\mathrm{ROP}=e^{A·{\left(\frac{1}{K_{\mathrm{bit}}}\right)}^{\frac{1}{4}{\alpha }_2}·{\left(q^2·P_{{\mathrm{pump}}_{\max }}-K_1·D·q^{m+2}\right)}^{\frac{1}{4}{\alpha }_2}}$ ${\left(q/d_{e_{\mathrm{nozzle}}}\right)}^{{\alpha }_2}={\left(\frac{1}{K_{\mathrm{bit}}}\right)}^{\frac{1}{4}{\alpha }_2}·q^{\frac{1}{4}{\alpha }_2}·{\left(q·P_{{\mathrm{pump}}_{\max }}-K_1·D·q^{m+2}\right)}^{\frac{1}{4}{\alpha }_2}$

Using the second range, the pump pressure is assumed to be constant, thus:

${\left(q/d_{e_{nozzle}}\right)}^{{\alpha }_2}={\left(\frac{1}{K_{bit}}\right)}^{\frac{1}{4}{\alpha }_2}·q^{\frac{1}{4}{\alpha }_2}·{\left(E_{{\mathrm{pump}}_{\max }}-K_1·D·q^{m+2}\right)}^{\frac{1}{4}{\alpha }_2}$ $\frac{\partial \mathrm{ROP}}{\partial q}=A.{\left(\frac{1}{K_{\mathrm{bit}}}\right)}^{\frac{1}{4}{\alpha }_2}·\frac{1}{4}·\frac{1}{4}·{\alpha }_3·q^{\frac{{\alpha }_2-4}{4}}·{\left(E_{{\mathrm{pump}}_{\max }}-\left(m+2\right)·K_1·D·q^{m+1}\right)}^{\frac{{\alpha }_2-4}{4}}·\left(-\left(m+2\right)·K_1·D·q^{m+1}\right)·e^{A·{\left(\frac{1}{K_{\mathrm{bit}}}\right)}^{\frac{1}{4}{\alpha }_2}·{\left(q^2·P_{\mathrm{pump}}-K_1·D·q^{m+2}\right)}^{\frac{1}{4}{\alpha }_2}}=0$ $\begin{array}{cc}\frac{\partial ROP}{\partial q}=E_{{\mathrm{pump}}_{\max }}-\left(m+2\right)·K_1·D·q^{m+1}=0& \mathrm{Equation}\left(24\right)\end{array}$ $\begin{array}{cc}{q}_{{\mathrm{optimum}}_{\mathrm{II}}}={\left(\frac{E_{{\mathrm{pump}}_{\max }}}{\left(m+2\right)K_1·D}\right)}^{\frac{1}{m+1}}& \mathrm{Equation}\left(25\right)\end{array}$

ΔP_{bit,optimumII} can be found by plugging Equation (25) into Equation (12), and solving for the optimal ΔP_bit, see below:

P _pump =ΔP _bit +ΔP _loss

P _{pump max} =ΔP _bit +K ₁ ·D·q ^m

ΔP_bit=P_pumpmax−K₁·D·q^m

$\begin{array}{cc}\Delta P_{\mathrm{bit},{\mathrm{optimum}}_{\mathrm{II}}}=\frac{m+1}{m+2}{P}_{{\mathrm{pump}}_{\max }}& \mathrm{Equation}\left(26\right)\end{array}$ $\begin{array}{cc}{d}_{e_{\mathrm{II}}}={\left[K_{\mathrm{bit}}·\frac{q_{\mathrm{optimu}{m}_{\mathrm{II}}}^2}{\left(\Delta P_{\mathrm{bit},\mathrm{optimum}}\right)}\right]}^{1/4}& \mathrm{Equation}\left(27\right)\end{array}$

In step 312, pressure changes created by the drilling operation are determined using the optimal hole cleaning energy and results of the gel strength tests and the computer processor (1005). The pressure changes are determined, in part, by estimating an annular velocity of cuttings in the drilling fluid. In order to estimate the annular velocity of the cuttings, an average size of the cuttings should be estimated using the drilling equipment.

For purposes of example only, the drilling equipment that may be used to estimate the size of the cuttings may include hydrocyclones, desanders, desilters, shakers, and centrifuges. Classification of cuttings charts may be used to classify the cuttings by their size. The pressure change calculations are outlined below in Equations (28)-(56). In Equation (28) the ROP being used is the ROP calculated from Equation (10).

In the below equations, q_cuttings=flow rate of the cuttings; c_cuttings,0=original concentration of the cuttings; q_pump=flow rate of the pump; q_filling=flow rate required to fill the wellbore; ρ_particle=particle density of the cuttings which can estimated based on the porosity ϕ relationship; cuttings are assumed to be spheres; V_sphere=volume of a sphere; A_sphere; d_particle=average diameter of the cuttings; ρ_matrix=density of the rock matrix; ρ_effective=effective density of the cuttings and drilling fluid; and the slot flow is considered annular flow. This conversion is achieved through the conversion of the width to radius. The average width is considered the periphery (p).

$b=\pi ·\frac{\left(d_{i\mathrm{nner}}+d_{\mathrm{outer}}\right)}{2}$ $\mathrm{where}{d}_{\mathrm{hydraulic}}=\frac{4A}{\mathrm{perimeter}}=\frac{4·2\mathrm{hb}}{2b+4h},h=0.25$

v_slip=slip velocity; μ_eff=effective viscosity; N_Re,P=Reynold number of particle; F_Drag=drag force; C_Drag=drag coefficient Ψ=sphericity of a particle; R_transport=transport ratio; F_Lift=lift force; C_Lift=lift constant; F_cohesive=cohesive force; φ=angle of repose; F_g=force of gravity; W=weight of the particles in the cuttings; F_{net, lift}=force required to lift the cuttings; F_L=lift forces; F_cohesive=cohesive forces; α=wellbore inclination angle; Φ=angle of repose; and v_x=relative fluid−particle velocity.

$\begin{array}{cc}{q}_{\mathrm{cuttings}}=\frac{\pi }{4}·d_{\mathrm{bit}}^2·\mathrm{ROP}& \mathrm{Equation}\left(28\right)\end{array}$ $\begin{array}{cc}{c}_{\mathrm{cuttings},0}=\frac{q_{cut\mathrm{tings}}}{\left(q_{\mathrm{pump}}+q_{\mathrm{cuttings}}-q_{\mathrm{filling}}\right)}\cong \frac{q_{\mathrm{cuttings}}}{q_{\mathrm{pump}}}& \mathrm{Equation}\left(29\right)\end{array}$

Two forces are acting at stationary flow: Gravity Force=Shear Force

(ρ_particle−ρ_mud)·V_sphereg=τ·A_sphere   Equation (30)

$\begin{array}{cc}{V}_{\mathrm{sphere}}=\frac{4}{3}\pi r^3=\frac{\pi }{6}{d}_{\mathrm{particle}}^3=\frac{A_{\mathrm{sphere}}^{\frac{3}{2}}}{6\sqrt{\pi }}& \mathrm{Equation}\left(31\right)\end{array}$ A _sphere=4πr²=πd_particle²   Equation (32)

The shear stress (τ) of the drilling fluid is calculated from μ_eff and {dot over (γ)} based on the selected rheology model:

$\begin{array}{cc}\tau ={\mu }_{\mathrm{eff}}·\dot{\gamma }=\frac{\left({\rho }_{particle}-{\rho }_{mud}\right)}{6}·\sqrt{\frac{A_{\mathrm{sphere}}}{\pi }}& \mathrm{Equation}\left(33\right)\end{array}$ $\begin{array}{cc}\varphi =100-\frac{{\rho }_{\mathrm{bulk}}}{{\rho }_{\mathrm{particle}}}\times 100& \mathrm{Equation}\left(34\right)\end{array}$

The formation bulk density can be estimated based on the matrix density, porosity, and formation fluid density:

$\begin{array}{cc}\varphi =\frac{{\rho }_{matrix}-{\rho }_{bulk}}{{\rho }_{matrix}-{\rho }_{mud}}& \mathrm{Equation}\left(35\right)\end{array}$ $\begin{array}{cc}{\rho }_{\mathrm{effective}}={\rho }_{mud}+{\rho }_{\mathrm{cuttings}}& \mathrm{Equation}\left(36\right)\end{array}$ $\begin{array}{cc}{\rho }_{\mathrm{effective}}=\frac{\left({\rho }_{mud}\times q\right)+\left(141.4296\times 10^{-4}\times ROP\times d_{bit}^2\right)}{q+\left(6.7995\times 10^{-4}\times ROP\times d_{bit}^2\right)}& \mathrm{Equation}\left(37\right)\end{array}$

Equations (30)-(37) are combined to create the below equations:

$\begin{array}{cc}\left({\rho }_{particle}-{\rho }_{mud}\right)·g·\frac{\pi }{6}·d_{particle}^3={\mu }_{\mathrm{eff}}·\frac{\pi v_{slip}}{d_{\mathrm{particle}}}4{\pi \left(\frac{d_{\mathrm{particle}}}{2}\right)}^2& \mathrm{Equation}\left(38\right)\end{array}$ $\begin{array}{cc}\mathrm{Stokes}\mathrm{Law}:v_{\mathrm{slip}}=\frac{\left({\rho }_{particle}-{\rho }_{mud}\right)·g·d_P^2}{6\pi ·{\mu }_{\mathrm{eff}}}& \mathrm{Equation}\left(39\right)\end{array}$ $\begin{array}{cc}{N}_{\mathrm{Re},P}=\frac{v_{\mathrm{slip}}·d_{particle}·{\rho }_{mud}}{{\mu }_{\mathrm{eff}}}=\frac{\mathrm{Inertial}\mathrm{Forces}}{Vi\mathrm{scous}\mathrm{Forces}}& \mathrm{Equation}\left(40\right)\end{array}$ $\begin{array}{cc}{v}_{\mathrm{slip}}=\sqrt{\frac{4g·\left({\rho }_{\mathrm{particle}}-{\rho }_{\mathrm{mud}}\right)·d_{\mathrm{particle}}}{\left(3{\rho }_{mud}{C}_{Drag}\right)}}& \mathrm{Equation}\left(41\right)\end{array}$

In an intermediate flow regime: C_Drag is a function the particle shape:

$\begin{array}{cc}\Psi =\frac{\mathrm{area}\mathrm{of}a\mathrm{sphere}\mathrm{of}\mathrm{same}\mathrm{volume}\mathrm{as}\mathrm{particle}}{\mathrm{area}\mathrm{of}\mathrm{particle}}=\frac{{{\pi }^{\frac{1}{3}}\left(6V_{\mathrm{particle}}\right)}^{2/3}}{A_{\mathrm{particle}}}& \mathrm{Equation}\left(42\right)\end{array}$

If the flow average velocity is more than the resulting slip velocity, the particles will be carried out of the hole. This is represented by Equation (43) below.

v _transport =v _annular −v _slip   Equation (43)

$\begin{array}{cc}{R}_{\mathrm{transport}}=\frac{v_{tr\mathrm{ansport}}}{v_{annular}}=1-\frac{v_{\mathrm{slip}}}{v_{annular}}& \mathrm{Equation}\left(44\right)\end{array}$

The goal is to have the minimum required v_annular. Below this velocity, the cuttings will start to accumulate, and the problems related to accumulations will increase dramatically. Drilling practices show that this velocity corresponds to 4-6% of cuttings concentrations

$\begin{array}{cc}{C}_{\mathrm{cuttings}}=\frac{C_{c\mathrm{uttings},0}}{R_{tr\mathrm{ansport}}}& \mathrm{Equation}\left(45\right)\end{array}$ $\begin{array}{cc}{F}_{\mathrm{Drag}}=C_{\mathrm{Drag}}\frac{\pi d_{particle}^2}{8}·{\rho }_{mud}·v_{slip}^2& \mathrm{Equation}\left(46\right)\end{array}$ $\begin{array}{cc}{C}_{Drag}=\frac{24}{R_{e,\mathrm{particle}}}+\frac{6}{\left(1+R_{e,\mathrm{particle}}\right)}+0.4,& \mathrm{Equation}\left(47\right)\end{array}$ $R_{e,\mathrm{particle}}=\frac{v_x^2·{\rho }_{\mathrm{particle}}}{\tau }=\frac{{\left(\frac{\pi ·v_{slip}}{2}\right)}^2*{\rho }_{\mathrm{particle}}}{\tau }<2\times 10^5$

the Lift Force F_Lift caused by the local fluid flow velocity gradient v_x=relative fluid−particle velocity

$\begin{array}{cc}{F}_{\mathrm{Lift}}=C_{\mathrm{Lift}}\frac{\pi d_{\mathrm{particle}}^2}{8}·\rho ·v_x^2,C_{\mathrm{Lift}}=5.8{2\left[\frac{d_{\mathrm{particle}}}{2v_x{N}_{R_{e,\mathrm{particle}}}}·\frac{{\mathrm{dv}}_x}{\mathrm{dr}}\right]}^2& \mathrm{Equation}\left(48\right)\end{array}$ $\begin{array}{cc}{F}_{\mathrm{Cohesive}}=\frac{\pi d_{\mathrm{particle}}^2{\tau }_y}{4}·f_y& \mathrm{Equation}\left(49\right)\end{array}$ $\begin{array}{cc}{f}_y=2\left(\phi +\left(\frac{\pi }{2}-\phi \right){\sin }^2\phi -\cos \mathrm{\phi sin\phi }\right)F_g=g\frac{\pi d_{\mathrm{particle}}^3}{6}\left({\rho }_{particle}-{\rho }_{\mathrm{mud}}\right)=W& \mathrm{Equation}\left(50\right)\end{array}$

In order to sufficiently initiate the lifting/rolling of cuttings, the drilling fluid flow rate should eventually reach the minimum critical velocity. The force balance acting on a single bed particle in the direction normal to the bed is represented in Equation (51):

F _net,lift =F _L −F _cohesive −W·sin α>0   Equation (51)

$\begin{array}{cc}{F}_{net,rol\mathrm{ling}}=\frac{d_{\mathrm{particle}}}{2}\left[F_D·\sin \Phi +\left(F_L-F_{cohessive}\right)\cos \Phi +W·\sin \left(-\alpha -\Phi \right)\right]>0& \mathrm{Equation}\left(52\right)\end{array}$

In step 314, an estimated equivalent circulating density is determined based n the pressure changes. The estimated equivalent circulating density is shown below in Equation (53). In the below equations, TVD=true vertical depth of the drill bit (112); A_pipe=area of the drill pipe; A_annular=area of the annulus

$\begin{array}{cc}ECD={\rho }_{mud}+\frac{\begin{array}{c}\Delta P_{an\mathrm{nular}\mathrm{friction}}+\Delta P_{\mathrm{cuttings}}+\\ \Delta P_{surge\&swab}+\Delta P_{\mathrm{rotation}}+\Delta P_{\mathrm{acceleration}}\end{array}}{0.52·\mathrm{TVD}}& \mathrm{Equation}\left(53\right)\end{array}$

$\begin{array}{cc}\Delta P_{\mathrm{cuttings}}={\rho }_{\mathrm{mud},\mathrm{average}}·g·\mathrm{TVD}& \mathrm{Equation}\left(54\right)\end{array}$ ${\rho }_{\mathrm{mud},\mathrm{average}}={\rho }_{mud}\left(1-c_{\mathrm{cuttings},\mathrm{average}}\right)+{\rho }_{\mathrm{cuttings}}·c_{\mathrm{cuttings},\mathrm{average}}$ $\Delta P_{a\mathrm{cceleration}}=m·{\mathrm{acceleration}}_{\mathrm{pipe}}=\rho ·V·\frac{A_{\mathrm{pipe}}}{A_{ann\mathrm{ular}}}$

Due to the effect of gelling, pressure changes due to gel strength may be considered when calculating the equivalent circulating density because there is more pressure required to break the gel around the pipe's surface on the drill string (108). The equations below show pressure change calculations due to gel strength. In the below equations, Δh=the height of the drill string before gravity overcomes the gelled shear stress and D_pipe=diameter of the drill pipe.

$$\begin{gathered} \begin{array}{cc}\Delta P_{\mathrm{gel}}=\frac{4{\tau }_w·L}{D_{\mathrm{pipe}}}={\rho }_{mud}·g·\Delta h\text{ \\ ρm⁢u⁢d·g·Δ⁢h=4⁢τw·LDpipe}& \mathrm{Equation}\left(55\right)\end{array} \end{gathered}$$

In step 316, the drilling operation is modified or maintained based on the equivalent circulating density calculated above. Modifying the drilling operation includes pulling the drill string (108) out of hole and changing parameters and composition of the drilling fluid. Maintaining the drilling operation comprises continuing to drill and repeating the above methodology as the drilling operation progressives.

As the drilling operation progressives, the non-linear regression coefficients may need to be updated to properly model the drilling operation for that specific depth range. The method listed above may be performed as many times as required. For example, the methodology may be performed iteratively until the drilling operation is completed.

FIG. 10 shows a computer (1002) system in accordance with one or more embodiments. Specifically, FIG. 10 shows a block diagram of a computer (1002) system used to provide computational functionalities associated with described algorithms, methods, functions, processes, flows, and procedures as described in the instant disclosure, according to an implementation. The illustrated computer (1002) is intended to encompass any computing device such as a server, desktop computer, laptop/notebook computer, wireless data port, smart phone, personal data assistant (PDA), tablet computing device, one or more processors within these devices, or any other suitable processing device, including both physical or virtual instances (or both) of the computing device.

Additionally, the computer (1002) may include a computer that includes an input device, such as a keypad, keyboard, touch screen, or other device that can accept user information, and an output device that conveys information associated with the operation of the computer (1002), including digital data, visual, or audio information (or a combination of information), or a GUI.

The computer (1002) can serve in a role as a client, network component, a server, a database or other persistency, or any other component (or a combination of roles) of a computer system for performing the subject matter described in the instant disclosure. The illustrated computer (1002) is communicably coupled with a network (1030). In some implementations, one or more components of the computer (1002) may be configured to operate within environments, including cloud-computing-based, local, global, or other environment (or a combination of environments).

At a high level, the computer (1002) is an electronic computing device operable to receive, transmit, process, store, or manage data and information associated with the described subject matter. According to some implementations, the computer (1002) may also include or be communicably coupled with an application server, e-mail server, web server, caching server, streaming data server, business intelligence (BI) server, or other server (or a combination of servers).

The computer (1002) can receive requests over network (1030) from a client application (for example, executing on another computer (1002)) and responding to the received requests by processing the said requests in an appropriate software application. In addition, requests may also be sent to the computer (1002) from internal users (for example, from a command console or by other appropriate access method), external or third-parties, other automated applications, as well as any other appropriate entities, individuals, systems, or computers.

Each of the components of the computer (1002) can communicate using a system bus (1003). In some implementations, any or all of the components of the computer (1002), both hardware or software (or a combination of hardware and software), may interface with each other or the interface (1004) (or a combination of both) over the system bus (1003) using an application programming interface (API) (1012) or a service layer (1013) (or a combination of the API (1012) and service layer (1013). The API (1012) may include specifications for routines, data structures, and object classes. The API (1012) may be either computer-language independent or dependent and refer to a complete interface, a single function, or even a set of APIs. The service layer (1013) provides software services to the computer (1002) or other components (whether or not illustrated) that are communicably coupled to the computer (1002).

The functionality of the computer (1002) may be accessible for all service consumers using this service layer. Software services, such as those provided by the service layer (1013), provide reusable, defined business functionalities through a defined interface. For example, the interface may be software written in JAVA, C++, or other suitable language providing data in extensible markup language (XML) format or other suitable format. While illustrated as an integrated component of the computer (1002), alternative implementations may illustrate the API (1012) or the service layer (1013) as stand-alone components in relation to other components of the computer (1002) or other components (whether or not illustrated) that are communicably coupled to the computer (1002). Moreover, any or all parts of the API (1012) or the service layer (1013) may be implemented as child or sub-modules of another software module, enterprise application. or hardware module without departing from the scope of this disclosure.

The computer (1002) includes an interface (1004). Although illustrated as a single interface (1004) in FIG. 10, two or more interfaces (1004) may be used according to particular needs, desires, or particular implementations of the computer (1002). The interface (1004) is used by the computer (1002) for communicating with other systems in a distributed environment that are connected to the network (1030). Generally, the interface (1004) includes logic encoded in software or hardware (or a combination of software and hardware) and operable to communicate with the network (1030). More specifically, the interface (1004) may include software supporting one or more communication protocols associated with communications such that the network (1030) or interface's hardware is operable to communicate physical signals within and outside of the illustrated computer (1002).

The computer (1002) includes at least one computer processor (1005). Although illustrated as a single computer processor (1005) in FIG. 10, two or more processors may be used according to particular needs, desires, or particular implementations of the computer (1002). Generally, the computer processor (1005) executes instructions and manipulates data to perform the operations of the computer (1002) and any algorithms, methods, functions, processes, flows, and procedures as described in the instant disclosure.

The computer (1002) also includes a non-transitory computer (1002) readable medium, or a memory (1006), that holds data for the computer (1002) or other components (or a combination of both) that can be connected to the network (1030). For example, memory (1006) can be a database storing data consistent with this disclosure. Although illustrated as a single memory (1006) in FIG. 10, two or more memories may be used according to particular needs, desires, or particular implementations of the computer (1002) and the described functionality. While memory (1006) is illustrated as an integral component of the computer (1002), in alternative implementations, memory (1006) can be external to the computer (1002).

The application (1007) is an algorithmic software engine providing functionality according to particular needs, desires, or particular implementations of the computer (1002), particularly with respect to functionality described in this disclosure. For example, application (1007) can serve as one or more components, modules, applications, etc. Further, although illustrated as a single application (1007), the application (1007) may be implemented as multiple applications (1007) on the computer (1002). In addition, although illustrated as integral to the computer (1002), in alternative implementations, the application (1007) can be external to the computer (1002).

There may be any number of computers (1002) associated with, or external to, a computer system containing computer (1002), each computer (1002) communicating over network (1030). Further, the term “client,” “user,” and other appropriate terminology may be used interchangeably as appropriate without departing from the scope of this disclosure. Moreover, this disclosure contemplates that many users may use one computer (1002), or that one user may use multiple computers (1002).

Although only a few example embodiments have been described in detail above, those skilled in the art will readily appreciate that many modifications are possible in the example embodiments without materially departing from this invention. Accordingly, all such modifications are intended to be included within the scope of this disclosure as defined in the following claims. In the claims, means-plus-function clauses are intended to cover the structures described herein as performing the recited function and not only structural equivalents, but also equivalent structures. Thus, although a nail and a screw may not be structural equivalents in that a nail employs a cylindrical surface to secure wooden parts together, whereas a screw employs a helical surface, in the environment of fastening wooden parts, a nail and a screw may be equivalent structures. It is the express intention of the applicant not to invoke 35 U.S.C. § 112(f) for any limitations of any of the claims herein, except for those in which the claim expressly uses the words ‘means for’ together with an associated function.

Claims

1. A method, performed iteratively and repeatedly, comprising: obtaining real-time drilling data from drilling equipment, including a mud pump, performing a drilling operation using a drilling fluid;determining, using a computer processor, pressure losses of the drilling fluid in the drilling operation using the real-time drilling data;updating, using the computer processor, non-linear regression coefficients to model the drilling operation using results obtained during field experiments;performing, using the computer processor, gel strength tests on the drilling fluid;determining, using the computer processor, optimal drilling parameters based on the real-time drilling data and the non-linear regression coefficients;determining, using the computer processor, optimal hole cleaning energy by balancing a mechanical energy level of the drilling fluid based on a maximum pump pressure of the mud pump and the optimal drilling parameters, wherein the maximum pump pressure is determined based on the pressure losses and a range;determining, using the computer processor, pressure changes created by the drilling operation using the optimal hole cleaning energy and results of the gel strength tests;determining, using the computer processor, an estimated equivalent circulating density based on the pressure changes; andmodifying or maintaining the drilling operation based on the estimated equivalent circulating density.

2. The method of claim 1, wherein performing the gel strength tests on the drilling fluid comprises modifying or maintaining a rheology of the drilling fluid.

3. The method of claim 1, wherein the range comprises a first range and a second range, the first range comprising a maximum operating capacity of the mud pump and the second range comprising a maximum energy of the mud pump.

4. The method of claim 3, wherein balancing the mechanical energy level of the drilling fluid using the first range comprises assuming rate of penetration is linearly related to the maximum operating capacity of the mud pump.

5. The method of claim 3, wherein balancing the mechanical energy level of the drilling fluid using the second range comprises assuming rate of penetration is a function of bottom hole cleaning.

6. The method of claim 1, wherein determining the pressure changes comprises estimating an annular velocity of cuttings in the drilling fluid.

7. The method of claim 6, wherein estimating the annular velocity of the cuttings in the drilling fluid comprises estimating an average size of the cuttings using the drilling equipment.

8. The method of claim 1, wherein the drilling equipment further comprises a drill string and modifying the drilling operation comprises pulling the drill string out of hole and changing parameters and composition of the drilling fluid.

9. The method of claim 1, wherein maintaining the drilling operation comprises updating the non-linear regression coefficients based on progression of the drilling operation.

10. The method of claim 1, wherein determining the optimal drilling parameters comprises optimizing the drilling parameters with respect to change in axial force on the drilling equipment and torsional force.

11. A system comprising: a drilling system comprising drilling equipment, including a mud pump, configured to perform a drilling operation using a drilling fluid anda computer processor configured to, iteratively and repeatedly: obtain real-time drilling data from the drilling equipment;determine pressure losses of the drilling fluid in the drilling operation using the real-time drilling data;update non-linear regression coefficients to model the drilling operation using results obtained during field experiments;perform gel strength tests on the drilling fluid;determine optimal drilling parameters based on the real-time drilling data and the non-linear regression coefficients;determine optimal hole cleaning energy by balancing a mechanical energy level of the drilling fluid based on a maximum pump pressure of the mud pump and the optimal drilling parameters, wherein the maximum pump pressure is determined based on the pressure losses and a range;determine pressure changes created by the drilling operation using the optimal hole cleaning energy and results of the gel strength tests;determine an estimated equivalent circulating density based on the pressure changes; andmodify or maintain the drilling operation based on the estimated equivalent circulating density.

12. The system of claim 11, wherein performing the gel strength tests on the drilling fluid comprises modifying or maintaining a rheology of the drilling fluid.

13. The system of claim 11, wherein the range comprises a first range and a second range, the first range comprising a maximum operating capacity of the mud pump and the second range comprising a maximum energy of the mud pump.

14. The system of claim 13, wherein balancing the mechanical energy level of the drilling fluid using the second range comprises assuming rate of penetration is linearly related to the maximum operating capacity of the mud pump.

15. The system of claim 13, wherein balancing the mechanical energy level of the drilling fluid using the second range comprises assuming rate of penetration is a function of bottom hole cleaning.

16. The system of claim 11, wherein determining the pressure changes comprises estimating an annular velocity of cuttings in the drilling fluid.

17. The system of claim 16, wherein estimating the annular velocity of the cuttings in the drilling fluid comprises estimating an average size of the cuttings using the drilling equipment.

18. The system of claim 11, wherein the drilling equipment further comprises a drill string and modifying the drilling operation comprises pulling the drill string out of hole and changing parameters and composition of the drilling fluid.

19. The system of claim 11, wherein maintaining the drilling operation comprises updating the non-linear regression coefficients based on progression of the drilling operation.

20. The system of claim 11, wherein determining the optimal drilling parameters comprises optimizing the drilling parameters with respect to change in axial force on the drilling equipment and torsional force.