Modeling Lost-Circulation into Fractured Formation in Rock Drilling Operations

Loss of circulation while drilling is a challenging problem that may interrupt drilling operations, reduce efficiency, and increases cost. When a drilled borehole intercepts conductive faults or fractures, lost circulation manifests as a partial or total escape of drilling, workover, or cementing fluids into the surrounding rock formations. Studying drilling fluid loss into a fractured system has been investigated using laboratory experiments, analytical modeling, and numerical simulations. Analytical modeling of fluid flow is a tool that can be quickly deployed to assess lost circulation and perform diagnostics, including leakage rate decline and fracture conductivity. In this chapter, various analytical methods developed to model the flow of non-Newtonian drilling fluid in a fractured medium are discussed. The solution methods are applicable for yield-power-law, including shear-thinning, shear-thickening, and Bingham plastic fluids. Numerical solutions of the Cauchy equation are used to verify the analytical solutions. Type-curves are also described using dimensionless groups. The solution methods are used to estimate the range of fracture conductivity and time-dependent fluid loss rate, and the ultimate total volume of lost fluid. The applicability of the proposed models is demonstrated for several field cases encountering lost circulations.


Introduction
Drilling technology has been widely deployed in many industries, such as oil and gas, geothermal, environmental remediation, mining, carbon dioxide sequestration, gas storage, water well, infrastructure development, among others [1]. In many situations, drilling entails various technical challenges and difficulties, often causing economic, safety, and environmental disturbances. Due to the complex nature of subsurface geological formations, technical problems often emerge unexpectedly. One of the most pressing problems is lost circulation into fractured formation. Preserving the drilling fluid within the borehole is crucial for removing cuttings, lubrication, hydraulic rotations, pressure control, among others. Fluid total or partial loss into the wellbore surrounding formation may result in wellbore instability.
Drilling fluid-loss is a costly problem. This phenomenon may obstruct operations, increase the nonproductive time (NPT), contaminate water tables, and cause formation damage and safety hazard [2][3][4][5][6][7][8][9]. Drilling fluid typically accounts for 25-40% of the total drilling costs [8]. Furthermore, lost circulation may cause other issues, such as wellbore instability, sloughing shale, and washout [10]. Such problems cost the industry about one billion US dollars per annum worldwide [11,12]. For instance, a study including 1500 gas wells in the Gulf of Mexico showed that lost-circulation accounts for 13% of the total drilling problems [5], as appears in Figure 1. Another study for 103 wells in the Duvernay area in Canada reported a loss of $2.6 million and 27.5 days of NPT because of lost circulation (Figure 2). In the Middle East, a group of 144 wells in Rumaila field, Iraq, encountered major lostcirculation problems [14], causing 48% of all drilling issues and loss of 295 days NPT, as illustrated in Figure 3. This high occurrence of lost-circulation was attributed to the presence of conductive natural fractures in the carbonate formations. For instance, more than 35% of drilled wells in a fractured carbonate formation in Iran experienced lost circulation [15]. Similarly, in Saudi Arabia, one-third of the Contribution of various drilling problems reported for 1500 wells in the Gulf of Mexico [5], where lost circulation accounts for 13% of the total.

Figure 2.
Cost overhead from various drilling problems for 103 wells in the Duvernay area in Canada [13], where lost circulation account for $2.6 m.
drilled well underwent lost circulation [16]. Lost circulation has also been reported in other places [17,18].
Lost circulation can be classified based on the severity of the flow loss rate [19], as shown in Table 1. These categories provide general guidance to set the mitigation plan depending on the severity of the fluid loss. Seepage loss can sometimes be tolerable to continue drilling without interruption. However, major loss demands a careful response to regain a full circulation fluid.
Fluid loss is often encountered in fractured formations, which can be subdivided into four types: induced-drilling fracture, caves or vugs, natural fracture, and high permeable formation [20]. An illustration of the four types is shown in Figure 4. Each formation type exhibits different fluid loss behavior and, therefore, may require a specific mitigation plan.
Naturally fractured formation, which is the focus of this chapter, is often prone to severe loss during either drilling, cementing, or completion/workover job [21]. Consequently, multiple problems may emanate because of the loss severity, such as kicks, wellbore instability, environmental contamination, and formation damage. To mitigate this problem, one procedure is to add lost circulation material (LCM) to the circulation fluid [22][23][24][25]. The LCM fluid properties such as viscosity and density are selected according to the formation type and the subsurface conditions, such as the depth, pressure, and fracture conductivity, among other factors [26]. Fluid loss rate into natural fractures mostly commences with a sudden spurt followed by a gradual declining loss [27]. The ultimately lost volume is dependent on several factors, such as fluid mobility, fracture conductivity, pore-volume, and fracture extension [28].
Because of the nature of this problem that requires immediate intervention, there is a need in the industry to establish an accurate and efficient modeling tool Contribution of various drilling problems to NPT in the Rumaila field in Iraq, total lost circulation is the top drilling issue encountered in that field, data source from [14].  [19].

Lost
that is feasible at real-time drilling operations to perform diagnostics and predictions. During the last two decades, many analytical solutions have been introduced in the literature to model mud loss into fractured formation. Early modeling attempts for simplified fractured cases were based on Darcy's Law at steady-state conditions [29,30]. Afterward, a semi-analytical solution was introduced for the Newtonian fluid model into a horizontal fracture by combining the diffusivity equation and mass conservation in a one-dimensional (1D) radial system [31]. Since this derived ordinary differential equation (ODE) was solved numerically, an analytical solution of the diffusivity equation for a fluid with a constant viscosity at steady-state conditions was introduced [32]. Another approach based on type-curves, which were generated by numerical solutions to describe mud loss volume as a function of time into a horizontal fracture, was established [33]. The type-curves are applicable for non-Newtonian fluid and follow a model that exhibits Bingham plastic rheological behavior. The numerically generated type-curves are based on dimensionless parameters that depend on the effective fracture hydraulic aperture, fluid properties, and differential pressure. However, these models inherit the limitations of numerical methods in introducing numerical artifacts such as numerical dispersion and grid dependency. Later, an analytical solution was proposed [34,35]. Estimation of hydraulic fracture aperture by simplifying insignificant terms in the final equation form was analyzed [36]. Following the same workflow, a solution was developed based on Yield-Power-Law fluid by reducing a Taylor expansion of the governing nonlinear flow equation into its linear terms [37].
This chapter is intended to give an overview of laboratory and modeling tools applicable for lost circulation in fractured media. Two main subjects are discussed.  nonlinearity of the problem, the system of equations is reformulated and transformed into ODE's, which is then computed numerically with an efficient ODE solver [38]. Based on the semi-analytical solution, type-curves are generated, capturing dimensionless fluid loss volume as a function of time. High-resolution finite element methods are used to verify the analytical approach. The applicability of the method is then demonstrated for field cases exhibiting loss of circulation, where formation and fluid uncertainties are addressed with Monte Carlo simulations. In the second subject, an experimental study designed to mimic fluid leakage in a horizontal fracture is discussed. These experiments are used to study the steadystate flow conditions of non-Newtonian fluids into the fracture and demonstrate the flow stoppage process. Simulations are used to replicate the physics, including the effect of fracture deformation. Type-curves are also derived from Cauchy equation of motion to capture the effect of fracture ballooning.

Mud invasion into a fractured system
Various studies have been conducted in the literature to investigate the flow behavior of drilling fluids in fractured systems [39][40][41], where a horizontal, radial fracture is considered, as shown in Figure 5. The choice of the fracture geometry is motivated by its convenience to be replicated with experimental apparatus and analytical modeling. Even though the fracture geometry may seem simplistic, it can provide useful insights at lab and field scales [43,44].
Consider two parallel radial plates to mimic a horizontal fracture, intercepting a wellbore, as illustrated in Figure 5. Note that horizontal fractures could occur at shallow depths and over-pressurized formations [45,46].
The general governing equation describing the dynamics of non-Newtonian fluid flow in an open fracture is given by the Cauchy momentum Equation [47,48], such that, Where p denotes the fluid pressure, t is time, τ is the shear stress, g is the gravity term, ρ is the density, I is identity matrix, and v is the flow velocity. Divergence delta operator is ∇Á.

Figure 5.
Physical domain illustrating a horizontal fracture intercepted by a wellbore. The shaded brown area reflects the invaded mud zone, r w is the wellbore radius, w is the fracture aperture, V mud is the mud loss volume, and t is the time [42].
Assuming steady-state conditions and neglecting the gravity and inertial effects, Eq. (1) simplifies to, The above equation reduces into two fundamental forces; pressure forces and shear stress force. In a 1D radial system with polar coordinates, Eq. (2) simplifies to (see [49]), Where τ z, r ð Þ is the radial shear stress component perpendicular to the z-directionand r is radial distance which is the variable argument.
On the other hand, the Herschel-Bulkley model is expressed by [50], The fluid yield stress, which determines the fluidity state, is denoted by τ 0 , and the consistency multiplier and behavioral flow index are mand n, respectively. The flow index is a positive number, reflecting the fluid rheological behavior where shearthinning (n < 1) and shear-thickening (n > 1) can occur. Typical values of this dimensionless parameter for drilling fluids range from 0.3 to 1.0 [51]. Shear rate, which is the derivative of the radial velocity v r in the z-direction, is nonlinear due to the flow index.
The mechanisms corresponding to the solution method of the mud invasion phenomenon in a fractured system are shown on a cross-section in Figure 6. The radial velocity decreases as the fluid propagates away from the wellbore within the fracture, and therefore shear stress lessens. Therefore, shear-thinning is expected to be maximum near the wellbore and reduces gradually with radial distance, which induces shear-thickening from yield stress. Furthermore, flow velocity and shear stress variations in the z-direction create layers of fluid rheological properties and fluid self-friction that becomes maximum at the walls of the fracture. Fluid selffriction is minimum at the centerline of the fracture, as shown in Figure 6, resulting in a region at the fracture center with zero shear rate, that is, dv r =dz ¼ 0. In this zone, the yield shear stress corresponds to τ 0 (see Eq. (4)). This fluid flow region in the fracture is subdivided into plug-flow and free-flow regions. The plug-region extends toward the walls of the fracture as the shear-stress reduces, and the fluid flows further from the wellbore. The plug region can eventually reach the fracture wall leading to a complete stoppage of the fluid leakage (see Figure 7a). In other words, mud leakage stalls in the fracture when the pressure gradient between the fracture inlet and the mud front becomes smaller than the yield stress τ 0 (see Figure 7b).
The following boundary conditions are considered for the two flow regions, In the above equation, z plug represents the extension of the plug region in the zdirection. v r,plug , and v r,free are, respectively, the flow velocities within the plug-and free-region. The no-slip boundary condition is described by the last equation in (5).
Eq. (4) and Eq. (3) are combined to express the solution of the velocity, such that, The plug-region is modeled by imposing the condition, dv r =dz ¼ 0. Therefore, Eq. (6) can be expressed for each region individually, as follows, From the definition of the total volumetric flow rate Q total , one obtains, Applying surface integral, Substituting Eq. (7) into Eq. (9) and arranging to obtain, The above equation is rearranged with the following quadratic equation to express the pressure term explicitly, that is, dp dr Where, Solving the differential pressure and integrating along the radial domain by implementing a moving boundary condition, a final ODE system is reached, as follows, Eq. (12) is nonlinear for a general value of n, and it cannot be solved analytically. However, a general semi-analytical solution can be derived [42]. This solution is a generalization to other particular solutions in the literature. For instance, when n ¼ 1, reflecting a Bingham plastic fluid, a closed-form solution can be obtained, as demonstrated by Lietard et al. [33]. Figure 8 shows that the proposed general semianalytical solution is in perfect agreement with the analytical solution by Lietard et al. [33]. For general cases of n, numerical simulations could be used to verify the semi-analytical model, as shown in Figure 9.

Dimensionless type-curves
For general applications, type-curves are used as a diagnostic tool to assess the solution by matching the trends of observed data to the type-curves. This approach is commonly used in well testing [52]. To enable scalability of the solution for a wide range of problem conditions, type-curves are expressed in terms of dimensionless groups. In this problem, the following dimensionless variables are considered,  Where, r D denotes the dimensionless radial mud-front, V D is the dimensionless mud-loss volume, and t D is the dimensionless time. The generated type-curves from Eq. (13) can represent a range of fluid properties, captured by two parameters. The first parameter, α, represents the mud rheological properties and flow behavior, while the second parameter, β, reflects the criteria for mud loss stoppage. The final forms of the type-curves are illustrated in Figure 10. These type-curves for mudloss are more accurate than the ones proposed by Majidi et al. [37], which showed lower accuracy for small α. Comparisons of the proposed solution with Majidi et al. is plotted in Figure 11, which show that the simplified model by Majidi et al. could overestimate the radial distance of mud invasion.

Demonstration for real field cases
The applicability of the discussed modeling approach is demonstrated for field cases that encountered lost circulation. The field data for four wells include the leakage rates and the fluid types corresponding to Bingham plastic and Herschel-Bulkley fluids. Two modeling approaches are discussed, where the first is a deterministic approach that is matched with the proposed analytical solution, and the second is a probabilistic approach based on Monte Carlo simulations. The utilization of the probabilistic approach is motivated by the subsurface uncertainty of the downhole parameters.

Field case 1
The lost circulation data correspond to two wells, Machar 18 and Machar 20 in Machar field in North Sea [33]. The fluid loss occurred due to the presence of natural fractures. Using the proposed type-curves, the fracture apertures were    Real field data matched with the semi-analytical solutions for two wells; well data are from Lietard et al. [33].   [42].

Field case 2
In this case, two sets of data were reported for two wells in the Gulf of Mexico [53]. The drilling fluid is Herschel-Bulkley fluid. The leakage data is provided as a volumetric flow rate in gallon per minute versus time. The provided data is converted into volumetric flow rates in m 3 per second as a function of time. Due to the uncertainty of subsurface parameters such as pressure drop, and rock properties, the semi-analytical solution is combined Monte Carlo simulations. The solutions and the data match are shown in Figure 13. Various useful information can be generated from these simulations. For instance, the uncertainty range of the hydraulic fracture aperture for the natural fractures can be predicted. Furthermore, the total mud-loss volumes can be estimated in the form of a histogram covering the range of uncertainty for P10, P50, and P90, percentiles, as shown in Figure 14. The relevant data are provided in Table 3.

Experimental study
The flow of drilling mud into an artificial fracture was studied by Majidi et al. [54]. The fracture consisted of two horizontal plates placed in parallel within a 1 mm opening, mimicking a fracture aperture. An illustration and photograph of the experimental apparatus are shown in Figure 15. Different fluid types and flow conditions were investigated. The inner hole, used for fluid injection, is 3 inches (in) in diameter located at the center of the upper plate. The radius of the fracture plates is 36 in. Pressure transducers were installed at different locations along the fracture radial distance. The pressure at the fracture outlet was maintained at the  Table 3. Wells condition and fluid properties for field case 2 at surface conditions. atmospheric conditions. The non-Newtonian fluid was injected by gravity from a supply tank placed at a certain height.

Steady-state flow test
Fluids were injected into the fracture system at a constant rate in a circulation mode until the steady-state condition was reached for the pressure within the fracture. The experiment was conducted with two different fluids, as provided in Table 4. A semi-analytical solution was developed to model steady-state radial flow for non-Newtonian fluids following a Yield-Power Law model described in the following, Where differential pressure with respect to radial distance dp dr equals to the operating conditions for inlet flow rate Q in , fluid properties, and fracture aperture w. The fluid is characterized by fluid yield stress τ 0 , consistency factor k, and flow behavioral index n. The limitation of this semi-analytical model is related to the assumption of constant fracture aperture, which is inconsistent with the varying fracture aperture caused by a slight deformation of the upper plate due to fluid pressure during injection.
A commercial simulator (COMSOL®) was used to investigate the effect of fracture wall deformation on the pressure behavior [55]. The results of viscosity and corresponding shear at three locations with respect to radial distance, reflecting shear thinning and thickening effects are shown in Figure 16.

Fracture ballooning effect
Fracture deformation and ballooning due to increased fluid pressure inside the fracture could occur [55]. Majidi et al. reported that there is a deformation happening somewhere between inlet and outlet caused by force distribution from fluid pressure. The physics behind fracture ballooning is related to mechanical deformation of the fracture resulting from fluid pressure and the surrounding stress field, causing the fracture to reshape its aperture by either opening or closing [56][57][58].
Simulations are used to investigate the fracture wall deformation and the corresponding pressure response. Simulations were conducted for deformed fracture plates, as illustrated in Figure 17. The results of pressure profiles for the deformed plate are in good agreement with the experimental measurements, as observed in Figure 18.

Plug flow test
The experiment was conducted to investigate the stoppage of flow invasion as a result of the fluid yield stress. The final invasion distances for two fluids were measured in the radial fracture system. Different injection pressures at the inlet were tested, for which the steady-state mud invasion front was measured. This    Figure 19. Understanding this effect is required to predict the maximum mud loss volume [59]. The two fluids are described in Table 5.
Because of the fracture ballooning, the aperture is not constant, as shown in Figure 18. The equation used to describe the varying aperture w r ð Þas a function of radial distancer is given in Eq. (15). The maximum opening of fracture aperture is captured by the parameter w m , distance from inlet where no occurred deformation is defined by L 1 and initial aperture where no deformation occurred is w i . Inlet and outlet radii are r w and r o .
When the mud stops, the flow rate is zero, that is, After integration of Eq. (17), the invasion radius, r f , is expressed in terms of the fluid rheological properties and the fracture geometry, as follows, The solution of this equation is plotted along with experimental measurements in Figure 20. The results are in excellent agreement with the experimental data and show significant improvement to the model by Majidi et al. [37], which assumes uniform fracture and, therefore, ignores fracture deformation.

Conclusions
Lost circulation during drilling operations is a common problem that requires immediate intervention to circumvent fluid loss. Diagnostic tools, based on simplified input data such as fluid properties, pressure, and rate trends, can be quickly deployed to quantify uncertainties related to the fluid leakage into the subsurface formation and to perform predictions. Semi-analytical solutions are used to model the leakage behavior of general Herschel-Bulkley fluids into a horizontal infiniteacting fracture, mimicking the effect of a fractured formation. The approach is applicable to different types of non-Newtonian fluids, including yield stress shearthinning and shear-thickening fluids. The model could predict the trend of mud leakage in a system with horizontal fractures as a function of time. It can estimate the effective hydraulic aperture of the fracture, the ultimate total mud-loss volume, and the expected duration before the leakage stalls, if conditions allow. Dimensionless groups are used to generate type-curves, which can provide quick diagnostics about the leakage behavior from matching the type-curve trends without a need for simulations. The applicability of the model was demonstrated for four wells from two different fields. A numerical procedure was described to couple the model with Monte-Carlo simulations to perform predictions under uncertainties. This approach is a practical diagnostic tool to perform quick predictions and to optimize LCM selection.