# Mathematical Simulation of Lost Circulation in Fracture and Its Control

Authors: Xiao Cai; Boyun Guo; Qingfeng Guo; Hongwei Jiang
Journal: Geofluids
 (2021)

---

## Abstract

Lost circulation has been one of the major problems that impede efficient and cost-saving drilling operations. The nature of lost circulation and its control is not yet fully understood. A method to characterize the mud loss in fracture and the plugging process of lost circulation materials is highly desired to obtain a thorough understanding of mud losses in fracture and provide reference for lost circulation control. This paper presents an easy-to-use method to identify types of lost circulation in fracture and the corresponding control. Three analytical models are presented based on three loss mechanisms, namely, seepage/filtration in a fracture, pipe flow in a fracture, and gravity displacement in a fracture. A numerical model is developed to simulate the deposition of lost circulation materials in fractures and predict the time and the volume of drilling fluid needed for lost circulation control. Case studies with these analytical models provide a deeper insight of this subject. Sensitivity analyses with the numerical model identify the major factors responsible for lost circulation control. High viscosity of drilling fluid may prevent lost circulation, while low viscosity is desired for a fast control of lost circulation. Lowering the density of drilling fluid is another way to prevent the lost circulation and facilitate the deposition of lost circulation materials. Lost circulation materials with high density could deposit faster close to the wellbore and therefore accelerating the control process. High concentration of lost circulation materials is likely to shorten the plugging time, which should be determined referring to the severity of loss. This work provides drilling engineers a practical method for simulating the lost circulation and selecting lost circulation material.


---

## Body

## 1. Introduction

Lost circulation is a condition that the drilling fluid uncontrollably flows into the formation instead of returning back to the surface while drilling a well. In addition to the loss of drilling fluid, lost circulation may decrease the bottom hole pressure, which will lead to well control [1] and wellbore instability issues (collapse). Besides, lost circulation may result in dry drilling (a condition that drilling fluid is totally lost from the wellbore), which will greatly damage the drill bit and the wellbore. Cost overruns will be generated by the loss of material and loss of productive time. Worldwide, the lost circulation leads to annual 2-4 billion dollars loss due to decreased productive time, lost drilling fluid, and materials used to prevent the loss [2].Many researchers have investigated the mathematical models of lost circulation. Majidi et al. [3, 4] presented a model of non-Newtonian drill mud loss in naturally fractured formation based on the principles of conservation of mass and linear momentum for drilling fluid and pressure diffusion for reservoir fluid. Majidi et al. (2010) proposed a quantitative analysis of mud loss in naturally fractured reservoirs considering the rheology properties of drill mud. They opined that the mud loss could be minimized by optimizing the rheology of drill mud such as the yield stress and shear-thinning/-thickening effect. Li et al. [5] established a lost circulation pressure model suitable for different lost circulations in carbonate formations based on leakage mechanisms. Wang et al. [6] developed a numerical simulation to investigate drilling fluid loss in fractured formations considering fracture wall roughness. Feng and Gray [7] proposed a numerical model considering the actual condition of dynamic mud circulation and the induced-fracture propagation in the drilling process to estimate the rate of loss and fracture profile.Due to the serious consequences of lost circulation, numerous methods and techniques have been developed to prevent and remediate the issue and thereby minimize the drilling cost. Sanders et al. [8] proposed a new high-fluid-loss and high-strength lost circulation system to address the lost circulation. Castro et al. [9] introduced an application of heat-activated and rigid rapid-fluid system to deal with the severe loss of circulation in deep-water environment and expand the gradient window. Ay et al. [10] developed an environmentally friendly silicate-polymer gel system with adjustable viscosity and density to seal shallow water flow and lost circulation in top hole drilling. Zhao et al. [11] presented a fully coupled hydraulic fracturing model based on the cohesive zone model, which describes the dynamic process of fracture growth and wellbore behavior during lost circulation. Mehrabian and Abousleiman [12] presented a wellbore-stability analysis of a fractured and LCM-treated wellbore. Abdila et al. [13] introduced a successful application of casing drilling technique to solve the issue of losses in the Louise fields, demonstrating its benefits of wellbore strengthening and efficiency improvement. Hou et al. [14] did a pioneering work to forecast lost circulations in south China sea by using an artificial neural network model. Pazziuagan et al. [15] highlighted a geomechanics modeling software to prevent lost circulation in offshore wells with narrow pressure margin and depleted reservoir.Lost circulation materials are commonly used in strengthening the wellbore and thereby remediate the lost circulation by bridging, plugging, or sealing the fractures, where the lost circulation occurs [7, 16]. A number of studies on LCM materials to mitigate the lost circulation condition have been proposed. Kefi et al. [17] proposed a novel composite blend system for controlling lost circulation based on a new four-step methodology. Javeri et al. [18] proposed a method of using silicon nanoparticles in drilling fluid to form a more integrated and thinner mud cake for mitigating lost circulation and differential sticking problems. Guo et al. [19] performed an investigation of mitigating lost circulation of oil-based drilling fluids by using gilsonite. Savari et al. [20] determined the plug-breaking pressure of different LCM combinations based on a permeability-plugging apparatus with tapered slots to evaluate their ability to experience displacement and failure pressures. Razavi et al. [21] proposed a method to determine the optimal LCM particle size distribution for LCM blend, which can maximize the wellbore strengthening effect obtained from fracture sealing. Nasiri et al. [22] used an advanced experimental method to determine the effectiveness of various LCMs on loss control in bentonite mud. Kulkarni et al. [23] performed an experimental study to evaluate the suspension characteristics of lost-circulation materials in different drilling fluids and analyzed the effects of suspending agents. Jaffery et al. [24] proposed an engineered fiber-based lost circulation control pills based on special fiber system and particle size distribution principle to solve the lost circulation challenges in a depleted formation with natural fractures. Xu et al. [25] experimentally analyzed the effects of multiple factors, such as low rate, particle size distribution, and particle geometry, on the transport and captured characteristics of lost circulation material. Savari and Whitfill [26] provided three types of lost circulation materials which were demonstrated by case studies to be efficient to manage severe losses of circulation. In 2020, a high fluid loss squeeze and reticulated foam lost circulation material is provided by Savari to deal with serious losses in naturally fractured/vugular formations with supporting data of experimental tests. Feng et al. [27] illustrated a novel method to inject the lost circulation materials, which is named as graded multiple injection method and proposed a visualization experimental means to support their conclusions.For the simulation of fluid loss, previous models assumed the pipe flow of single-phase liquid in fractures, while the seepage flow (filtration) and gravity displacement in the fractures have not been studied. This paper proposed three analytical models to individually simulate different types of lost circulation processes considering the seepage flow, pipe flow, and gravity displacement in fracture. Previous works mainly focus on the improvement of LCM to prevent and mitigate the lost circulation condition. No studies so far can provide a description of the LCM particle transport in the fracture and give out a time at which the fracture is completely sealed. This paper fills the gap by providing a numerical model to simulate the sealing process of LCMs in fracture and predict the time for controlling the lost circulation as well as the volume for curing the well.


## 2. Mathematical Model

In the case of lost circulation caused by fractures, the drill mud flowing in fracture can be generally characterized by fluid filtration, pipe flow, and gravity displacement in fracture. A simple method to identify the type of losses for LCM selection and predict the losses is highly desired. In order to elaborate the lost circulations caused by fractures, three analytical models are derived to simulate three loss conditions in fracture. A numerical model is presented to simulate the curing processes.


### 2.1. Analytical Models for Diagnosis of Types of Lost Circulation

#### 2.1.1. Loss of Circulation due to Fracture Seepage Flow

Darcy’s law applies in the liquid seepage/filtration in a fracture depicted in Figure1. The loss rate in Darcy’s units is expressed as (see Appendix A for derivation):
(1)q=2whfv=2ckwhft−1/2,where q is the total loss rate from two wings of a linear fracture, w is fracture width, hf is fracture height, t is time, and ck is defined as
(2)ck=kfpw−pfμ,where kf is fracture permeability, μ is drilling fluid viscosity, and pw and pf are pressures in the wellbore and fracture, respectively.Figure 1
Cross-sectional view of liquid seepage in a fracture.The cumulative volume of lost fluid is expressed by:(3)VL=2whfx=22ckwhft1/2.


#### 2.1.2. Loss of Circulation due to Fracture Pipe Flow

Figure2 shows the liquid pipe flow in a fracture. Darcy’s law does not apply to the liquid pipe flow. Friction factor dominates loss rate and the expression for the loss rate in SI units is (see Appendix A for derivation)
(4)q=4332cf2/3whft−1/3,where
(5)cf=hfpw−pfπfρL,where ρL is liquid density and f is friction factor.Figure 2
Cross-sectional view of liquid pipe flow in a fracture.The cumulative volume of lost liquid is expressed as(6)VL=232cf2/3whft2/3.


#### 2.1.3. Loss of Circulation due to Gravity Displacement

For steady liquid intrusion into a fracture due to gravity segregation depicted in Figure3, an analytical model was derived. Model derivation is shown in the Appendix A. The loss rate in SI units is expressed as
(7)q=wρL2πg3fLfpw−pf2,where g is gravitational acceleration and Lf is travel distance of the liquid front propagating in the fracture (it is also referred as the fracture length invaded by the liquid). The cumulative volume of lost liquid can be calculated based on numerical integration of Equation (7) over time.Figure 3
Cross-sectional view of liquid flow in a fracture due to gravity displacement.


### 2.2. Numerical Model for Curing Lost Circulation

For the remedial treatment, it is highly desirable to figure out the time for lost circulation control and thus the total volume of drilling fluid needed. In this study, a new mathematical model is developed to predict the location of LCMs and the plugging time for lost circulation control. Figure4 shows a sketch of single LCM particle flowing into the fracture with an inclination angle of δ, where vo is the initial velocity of particle, vx and vy represent the velocity components in the vertical and horizontal directions, respectively.Figure 4
Trajectory of LCM particle flowing inside a fracture.The settling distance of LCM particle is (see AppendixB for derivation)
(8)y=1βW−Bt+2αlnCy+e−αtCy+1,where
(9)α=2gcβmmW−B,(10)β=fAρf2gc,(11)Cy=W−B+vyoβW−B−vyoβ.W is the LCM particle weight, lbf; B is the buoyant force, lbf. They are defined as:
(12)W=ggcρmVm,(13)B=ggcρfVm,where mm is the LCM mass, f is the friction factor, ρf is drilling fluid density, ρm is the LCM density, A is the characteristic area of particle, vyo initial velocity of LCM particle in y-direction, Vm is the volume of LCM particle, gc is the unit conversion factor, which is 32.17ft‐lbm/lbf‐s2.The traveling distance of LCM particle is expressed as(14)x=vLt+2mmfAρflnCxt+Cx,where
(15)Cx=2mmfAρmvL−vxo.vL is the fracturing fluid velocity, vxo is the initial LCM particle velocity in x-direction, mm is the LCM particle mass, lbm, f is the friction factor, ρf is drilling fluid density, and A is the characteristic area of LCM particle.Set the settling distance in the vertical direction in Equation (8) to be the fracture height, the settling time is expressed as
(16)t=−2αlnCy+1eα/2Hfβ/W−B+Cy+12eαHfβ/W−B−4Cy2.When the LCM particles flow into the fracture, pile of LCM will form in the fracture. In order to mathematically simulate the forming process of the LCM pile, it is considered as an integration of rectangular pile elements. Figure5 is a sketch of rectangular-slice model and single slice of LCM pile. The volume of slice i, which is composed of LCM particles of size dmi, can be expressed as
(17)Vi=fViVt,where fVi is the volume fraction of LCM with size dmi, Vi is the volume of the slice composed of the LCM of size dmi, and Vt is the total volume of LCM pile.Figure 5
A sketch of the rectangular-slice model and a single LCM slice.The volume of slicei containing the LCM of size dmi is formulated as
(18)Vi=ϖihiTi,where ϖi is the width of slice composed of LCM with size dmi, which is assumed to be the fracture width; hi is the height of slice containing LCM of size dmi; Ti is the thickness of slice composed of the LCM of sizedmi.The thickness of the slice composed of LCM of sizedmi can be expressed as
(19)Ti=xi−xi+1,where xi is the horizontal drifting distance of LCM with size dmi and xi+1 is the horizontal drifting distance of LCM with sizedmi+1.The height of pile slice containing the LCM of sizedmi can be expressed as
(20)hi=ViϖiTi.The configuration of LCM pile in fracture is the integration of pile element.


### 2.3. Model Analysis

The procedure of applying the proposed method includes 3 steps:(i)
Step 1: identify the mechanism of loss of circulation. During application, first, we need to identify the type of lost circulation. Identification of the loss mechanisms can be made by plotting the lost volume data from pit loss measurement. For the seepage type of lost circulation, the cumulative lost volume of drilling fluid is proportional to the square root of time. This means that the relationship between the pit loss data divided by the square root of time versus the time should shows a zero-slope trend. Besides, the lost circulation due to gravity displacement also shows a zero-slope trend between the lost volume of drilling fluid divided by time versus time. However, the gas or oil cut in the drilling fluid can explicitly demonstrate a lost circulation of gravity displacement type. Therefore, the lost circulation is characterized as the fracture pipe flow, when the plot of lost volume of drilling fluid divided by the square root of time versus time does not show a data trend of zero-slope(ii)
Step 2: predict the loss rate and volume of drilling fluid at a given time using the corresponding lost circulation model(iii)
Step 3: predict the time of lost circulation control and the lost volume of drilling fluid used to control the lost circulation based on the proposed modelThe lost circulation models are validated using the field data and experimental data obtained from a real size testing setup. The field pit loss data divided by the square root of time versus time is plotted in Figure6, demonstrating a trend of zero slope. The zero slope of the plot shown in Figure 6 indicates a fact that the lost circulation is due to the fracture seepage flow. After identifying the type of lost circulation, the loss flow rate and lost volume of drilling fluid can be predicted based on the corresponding model. The model for lost circulation due to fracture seepage flow is validated using the data shown in Table 1. Figure 7 shows the trends of loss rate calculated by Equation (1) and cumulative volume calculated by Equation (3). The rapid decline in loss rate indicates that there is no need to use high concentrations of LCM to control the lost circulation.Figure 6
Plot of lost volume data divided by the square root of time versus time showing a data trend of zero slope.Table 1
Fluid and fracture properties for model analysis.
Well depth (D)3000mLiquid density (ρL)1400kg/m3Liquid viscosity (μ)5cpFracture height (hf)500cmFracture width (w)1cmFracture permeability (kf)10DarcyPore pressure gradient (Gp)0.12Atm/mFigure 7
The trends of loss rate and cumulative volume for seepage type of lost circulation.The other two lost circulation models were validated with the experimental data shown in Table2 obtained from a real size testing setup, which is described elsewhere.Table 2
Geometry and operating conditions in an experimental test.
Liquid density (ρL)1000kg/m3Liquid viscosity (μ)5cpFracture height (hf)0.7mFracture width (w)0.1cmInvaded fracture length (Lf)1.2mBorehole pressure (pw)15kPaFracture pressure (pf)10kPaThe profiles of loss rate and cumulative volume calculated by the fracture pipe flow model using the data in Table2 are presented by Figure 8. A rapid decline in loss rate is observed by the model. However, the high sustainable loss requires the use of high concentrations of LCM.Figure 8
The trends of loss rate and cumulative volume for pipe flow type of lost circulation.Figure9 represents a graph of the trends of loss rate and cumulative volume using the gravity displacement model. High concentration of LCM is suggested because a slow decline in loss rate is illustrated by the model.Figure 9
The trends of loss rate and cumulative volume for gravity displacement type of lost circulation.The plugging model is validated using the fluid and fracture properties in Table1 and the properties of LCM in Table 3. Size distribution of LCM is shown in Table 4. After 6.42 hours, the lost circulation is controlled; the volume of lost circulation is 9.44 m3. Figure 10 demonstrates the final configuration of LCM pile in fracture, which totally seals the fracture.Table 3
Properties of LCM used in case study.
ParameterValueUnitDrilling fluid lost rate1.4697m3/hLCM density1650kg/m3LCM sphericity0.82Number of fracture1LCM discharge angle45DegreeRatio of LCM to drilling fluid0.05Number of injection point1Table 4
Side distribution of lost circulation material.
MeshμmFraction %1001498.366025010.334042032.862084029.4816119017.141216801.26823800.57Figure 10
Configuration of LCM pile in fracture when the lost circulation is controlled.


### 2.4. Sensitivity Analysis

To minimize drilling cost, it is highly desirable to figure out factors that affect the plugging time and the lost volume of drilling fluid. In this study, sensitivity analysis was performed with the proposed numerical model to identify the major factors responsible for the lost circulation using the data in the case study with one parameter value changed at a time.Figure11 shows the time of controlling the lost circulation using different viscosities of drilling fluid. Increasing the viscosity of drilling fluid is one of the major ways to prevent lost circulation by enhancing the filter cake (Liu, 2008; Deng, 2008). The proposed analytical models for simulating the lost circulation also indicate that the loss rate can be reduced by increasing the viscosity of drilling fluid. However, increasing the viscosity of drilling fluid allows more LCM to stay in the annulus and increases the settling time of LCM in fracture. As a result, the time for completely controlling the lost circulation increases. Therefore, it is better to decrease the viscosity of drilling fluid when LCM is added in the fracturing fluid to seal the fractures.Figure 11
Plugging time versus viscosity of drilling fluid.High density of drilling fluid creates a high mud pressure which can also cause the loss of circulation; therefore, reducing the density is another way to deal with the loss of circulation (Liu, 2008), which is consistent with Figure12. Reducing the density of drilling fluid can not only avoid generating a high mud pressure but also facilitate the deposition of LCM and thereby accelerate fracture sealing.Figure 12
Plugging time versus density of drilling fluid.Properties of LCM are also needed to be considered during bridging the fractures. Figure13 presents the time for completing bridging the fracture using different LCM densities. LCM with high density can deposit in the mouth of fracture faster, forming a pile of LCM to plug the fracture and remediate the loss of circulation. Hence, LCM with higher density is preferred during curing the lost circulation.Figure 13
Plugging time versus density of LCM.The concentration of LCM is another important factor because it determines the amount of LCM within the drilling fluid entering the fracture. Figure14 shows that increasing the concentration of LCM is more likely to shorten the time to plug the fracture face. The concentration of LCM should be determined based on the severity of lost circulation. A low concentration can be used to prevent the loss of circulation, and a high concentration is desired to mitigate the issue.Figure 14
Plugging time versus ratio of LCM to fluid.


## 2.1. Analytical Models for Diagnosis of Types of Lost Circulation

### 2.1.1. Loss of Circulation due to Fracture Seepage Flow

Darcy’s law applies in the liquid seepage/filtration in a fracture depicted in Figure1. The loss rate in Darcy’s units is expressed as (see Appendix A for derivation):
(1)q=2whfv=2ckwhft−1/2,where q is the total loss rate from two wings of a linear fracture, w is fracture width, hf is fracture height, t is time, and ck is defined as
(2)ck=kfpw−pfμ,where kf is fracture permeability, μ is drilling fluid viscosity, and pw and pf are pressures in the wellbore and fracture, respectively.Figure 1
Cross-sectional view of liquid seepage in a fracture.The cumulative volume of lost fluid is expressed by:(3)VL=2whfx=22ckwhft1/2.


### 2.1.2. Loss of Circulation due to Fracture Pipe Flow

Figure2 shows the liquid pipe flow in a fracture. Darcy’s law does not apply to the liquid pipe flow. Friction factor dominates loss rate and the expression for the loss rate in SI units is (see Appendix A for derivation)
(4)q=4332cf2/3whft−1/3,where
(5)cf=hfpw−pfπfρL,where ρL is liquid density and f is friction factor.Figure 2
Cross-sectional view of liquid pipe flow in a fracture.The cumulative volume of lost liquid is expressed as(6)VL=232cf2/3whft2/3.


### 2.1.3. Loss of Circulation due to Gravity Displacement

For steady liquid intrusion into a fracture due to gravity segregation depicted in Figure3, an analytical model was derived. Model derivation is shown in the Appendix A. The loss rate in SI units is expressed as
(7)q=wρL2πg3fLfpw−pf2,where g is gravitational acceleration and Lf is travel distance of the liquid front propagating in the fracture (it is also referred as the fracture length invaded by the liquid). The cumulative volume of lost liquid can be calculated based on numerical integration of Equation (7) over time.Figure 3
Cross-sectional view of liquid flow in a fracture due to gravity displacement.


## 2.1.1. Loss of Circulation due to Fracture Seepage Flow

Darcy’s law applies in the liquid seepage/filtration in a fracture depicted in Figure1. The loss rate in Darcy’s units is expressed as (see Appendix A for derivation):
(1)q=2whfv=2ckwhft−1/2,where q is the total loss rate from two wings of a linear fracture, w is fracture width, hf is fracture height, t is time, and ck is defined as
(2)ck=kfpw−pfμ,where kf is fracture permeability, μ is drilling fluid viscosity, and pw and pf are pressures in the wellbore and fracture, respectively.Figure 1
Cross-sectional view of liquid seepage in a fracture.The cumulative volume of lost fluid is expressed by:(3)VL=2whfx=22ckwhft1/2.


## 2.1.2. Loss of Circulation due to Fracture Pipe Flow

Figure2 shows the liquid pipe flow in a fracture. Darcy’s law does not apply to the liquid pipe flow. Friction factor dominates loss rate and the expression for the loss rate in SI units is (see Appendix A for derivation)
(4)q=4332cf2/3whft−1/3,where
(5)cf=hfpw−pfπfρL,where ρL is liquid density and f is friction factor.Figure 2
Cross-sectional view of liquid pipe flow in a fracture.The cumulative volume of lost liquid is expressed as(6)VL=232cf2/3whft2/3.


## 2.1.3. Loss of Circulation due to Gravity Displacement

For steady liquid intrusion into a fracture due to gravity segregation depicted in Figure3, an analytical model was derived. Model derivation is shown in the Appendix A. The loss rate in SI units is expressed as
(7)q=wρL2πg3fLfpw−pf2,where g is gravitational acceleration and Lf is travel distance of the liquid front propagating in the fracture (it is also referred as the fracture length invaded by the liquid). The cumulative volume of lost liquid can be calculated based on numerical integration of Equation (7) over time.Figure 3
Cross-sectional view of liquid flow in a fracture due to gravity displacement.


## 2.2. Numerical Model for Curing Lost Circulation

For the remedial treatment, it is highly desirable to figure out the time for lost circulation control and thus the total volume of drilling fluid needed. In this study, a new mathematical model is developed to predict the location of LCMs and the plugging time for lost circulation control. Figure4 shows a sketch of single LCM particle flowing into the fracture with an inclination angle of δ, where vo is the initial velocity of particle, vx and vy represent the velocity components in the vertical and horizontal directions, respectively.Figure 4
Trajectory of LCM particle flowing inside a fracture.The settling distance of LCM particle is (see AppendixB for derivation)
(8)y=1βW−Bt+2αlnCy+e−αtCy+1,where
(9)α=2gcβmmW−B,(10)β=fAρf2gc,(11)Cy=W−B+vyoβW−B−vyoβ.W is the LCM particle weight, lbf; B is the buoyant force, lbf. They are defined as:
(12)W=ggcρmVm,(13)B=ggcρfVm,where mm is the LCM mass, f is the friction factor, ρf is drilling fluid density, ρm is the LCM density, A is the characteristic area of particle, vyo initial velocity of LCM particle in y-direction, Vm is the volume of LCM particle, gc is the unit conversion factor, which is 32.17ft‐lbm/lbf‐s2.The traveling distance of LCM particle is expressed as(14)x=vLt+2mmfAρflnCxt+Cx,where
(15)Cx=2mmfAρmvL−vxo.vL is the fracturing fluid velocity, vxo is the initial LCM particle velocity in x-direction, mm is the LCM particle mass, lbm, f is the friction factor, ρf is drilling fluid density, and A is the characteristic area of LCM particle.Set the settling distance in the vertical direction in Equation (8) to be the fracture height, the settling time is expressed as
(16)t=−2αlnCy+1eα/2Hfβ/W−B+Cy+12eαHfβ/W−B−4Cy2.When the LCM particles flow into the fracture, pile of LCM will form in the fracture. In order to mathematically simulate the forming process of the LCM pile, it is considered as an integration of rectangular pile elements. Figure5 is a sketch of rectangular-slice model and single slice of LCM pile. The volume of slice i, which is composed of LCM particles of size dmi, can be expressed as
(17)Vi=fViVt,where fVi is the volume fraction of LCM with size dmi, Vi is the volume of the slice composed of the LCM of size dmi, and Vt is the total volume of LCM pile.Figure 5
A sketch of the rectangular-slice model and a single LCM slice.The volume of slicei containing the LCM of size dmi is formulated as
(18)Vi=ϖihiTi,where ϖi is the width of slice composed of LCM with size dmi, which is assumed to be the fracture width; hi is the height of slice containing LCM of size dmi; Ti is the thickness of slice composed of the LCM of sizedmi.The thickness of the slice composed of LCM of sizedmi can be expressed as
(19)Ti=xi−xi+1,where xi is the horizontal drifting distance of LCM with size dmi and xi+1 is the horizontal drifting distance of LCM with sizedmi+1.The height of pile slice containing the LCM of sizedmi can be expressed as
(20)hi=ViϖiTi.The configuration of LCM pile in fracture is the integration of pile element.


## 2.3. Model Analysis

The procedure of applying the proposed method includes 3 steps:(i)
Step 1: identify the mechanism of loss of circulation. During application, first, we need to identify the type of lost circulation. Identification of the loss mechanisms can be made by plotting the lost volume data from pit loss measurement. For the seepage type of lost circulation, the cumulative lost volume of drilling fluid is proportional to the square root of time. This means that the relationship between the pit loss data divided by the square root of time versus the time should shows a zero-slope trend. Besides, the lost circulation due to gravity displacement also shows a zero-slope trend between the lost volume of drilling fluid divided by time versus time. However, the gas or oil cut in the drilling fluid can explicitly demonstrate a lost circulation of gravity displacement type. Therefore, the lost circulation is characterized as the fracture pipe flow, when the plot of lost volume of drilling fluid divided by the square root of time versus time does not show a data trend of zero-slope(ii)
Step 2: predict the loss rate and volume of drilling fluid at a given time using the corresponding lost circulation model(iii)
Step 3: predict the time of lost circulation control and the lost volume of drilling fluid used to control the lost circulation based on the proposed modelThe lost circulation models are validated using the field data and experimental data obtained from a real size testing setup. The field pit loss data divided by the square root of time versus time is plotted in Figure6, demonstrating a trend of zero slope. The zero slope of the plot shown in Figure 6 indicates a fact that the lost circulation is due to the fracture seepage flow. After identifying the type of lost circulation, the loss flow rate and lost volume of drilling fluid can be predicted based on the corresponding model. The model for lost circulation due to fracture seepage flow is validated using the data shown in Table 1. Figure 7 shows the trends of loss rate calculated by Equation (1) and cumulative volume calculated by Equation (3). The rapid decline in loss rate indicates that there is no need to use high concentrations of LCM to control the lost circulation.Figure 6
Plot of lost volume data divided by the square root of time versus time showing a data trend of zero slope.Table 1
Fluid and fracture properties for model analysis.
Well depth (D)3000mLiquid density (ρL)1400kg/m3Liquid viscosity (μ)5cpFracture height (hf)500cmFracture width (w)1cmFracture permeability (kf)10DarcyPore pressure gradient (Gp)0.12Atm/mFigure 7
The trends of loss rate and cumulative volume for seepage type of lost circulation.The other two lost circulation models were validated with the experimental data shown in Table2 obtained from a real size testing setup, which is described elsewhere.Table 2
Geometry and operating conditions in an experimental test.
Liquid density (ρL)1000kg/m3Liquid viscosity (μ)5cpFracture height (hf)0.7mFracture width (w)0.1cmInvaded fracture length (Lf)1.2mBorehole pressure (pw)15kPaFracture pressure (pf)10kPaThe profiles of loss rate and cumulative volume calculated by the fracture pipe flow model using the data in Table2 are presented by Figure 8. A rapid decline in loss rate is observed by the model. However, the high sustainable loss requires the use of high concentrations of LCM.Figure 8
The trends of loss rate and cumulative volume for pipe flow type of lost circulation.Figure9 represents a graph of the trends of loss rate and cumulative volume using the gravity displacement model. High concentration of LCM is suggested because a slow decline in loss rate is illustrated by the model.Figure 9
The trends of loss rate and cumulative volume for gravity displacement type of lost circulation.The plugging model is validated using the fluid and fracture properties in Table1 and the properties of LCM in Table 3. Size distribution of LCM is shown in Table 4. After 6.42 hours, the lost circulation is controlled; the volume of lost circulation is 9.44 m3. Figure 10 demonstrates the final configuration of LCM pile in fracture, which totally seals the fracture.Table 3
Properties of LCM used in case study.
ParameterValueUnitDrilling fluid lost rate1.4697m3/hLCM density1650kg/m3LCM sphericity0.82Number of fracture1LCM discharge angle45DegreeRatio of LCM to drilling fluid0.05Number of injection point1Table 4
Side distribution of lost circulation material.
MeshμmFraction %1001498.366025010.334042032.862084029.4816119017.141216801.26823800.57Figure 10
Configuration of LCM pile in fracture when the lost circulation is controlled.


## 2.4. Sensitivity Analysis

To minimize drilling cost, it is highly desirable to figure out factors that affect the plugging time and the lost volume of drilling fluid. In this study, sensitivity analysis was performed with the proposed numerical model to identify the major factors responsible for the lost circulation using the data in the case study with one parameter value changed at a time.Figure11 shows the time of controlling the lost circulation using different viscosities of drilling fluid. Increasing the viscosity of drilling fluid is one of the major ways to prevent lost circulation by enhancing the filter cake (Liu, 2008; Deng, 2008). The proposed analytical models for simulating the lost circulation also indicate that the loss rate can be reduced by increasing the viscosity of drilling fluid. However, increasing the viscosity of drilling fluid allows more LCM to stay in the annulus and increases the settling time of LCM in fracture. As a result, the time for completely controlling the lost circulation increases. Therefore, it is better to decrease the viscosity of drilling fluid when LCM is added in the fracturing fluid to seal the fractures.Figure 11
Plugging time versus viscosity of drilling fluid.High density of drilling fluid creates a high mud pressure which can also cause the loss of circulation; therefore, reducing the density is another way to deal with the loss of circulation (Liu, 2008), which is consistent with Figure12. Reducing the density of drilling fluid can not only avoid generating a high mud pressure but also facilitate the deposition of LCM and thereby accelerate fracture sealing.Figure 12
Plugging time versus density of drilling fluid.Properties of LCM are also needed to be considered during bridging the fractures. Figure13 presents the time for completing bridging the fracture using different LCM densities. LCM with high density can deposit in the mouth of fracture faster, forming a pile of LCM to plug the fracture and remediate the loss of circulation. Hence, LCM with higher density is preferred during curing the lost circulation.Figure 13
Plugging time versus density of LCM.The concentration of LCM is another important factor because it determines the amount of LCM within the drilling fluid entering the fracture. Figure14 shows that increasing the concentration of LCM is more likely to shorten the time to plug the fracture face. The concentration of LCM should be determined based on the severity of lost circulation. A low concentration can be used to prevent the loss of circulation, and a high concentration is desired to mitigate the issue.Figure 14
Plugging time versus ratio of LCM to fluid.


## 3. Conclusion

Three analytical models were developed to describe the lost circulation with three mechanisms in an inherently fractured reservoir or a reservoir with induced fracture. A simple method was used to identify the loss mechanisms. A numerical model was proposed to simulate the process of LCM deposition in fractures and predict the time for lost circulation control and the lost volume of drilling fluid. Case studies with these four models were conducted. Sensitivity analysis with the model for curing the lost circulation was performed. The following conclusions are drawn:(1)
For the three types of loss of circulations, the loss rate can be reduced using drilling fluids of low density and high viscosity(2)
Use of low fluid viscosity can control the lost circulation faster(3)
Use of LCM with high density can facilitate the sealing process in fracture and reduce the time for controlling the lost circulation(4)
Reducing mud density can mitigate the loss of circulation, but not as effective as the other factors investigated in this study(5)
The concentration of LCM can determine the time needed to control the lost circulation; it should be selected based on the severity of lost circulation



---

*Source: 6691385-2021-03-12.xml*