# New Leakage Model for Naturally Fractured Formations Considering the Effects of Dual-System Hydro-Mechanical Coupling

Authors: Hao Peng; Qian Li; JiaJia Gao; Hu Yin; Jun Wang
Journal: Geofluids
 (2021)

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## Abstract

Within the existing leakage model accounting for drilling mud loss in naturally-fractured formations, the leak-off velocity is assigned to a fixed value or described by the Cater model, which does not consider the influence of dual-system hydromechanical coupling effects between fracture-wall and fracture-inner systems. The dual-system between the formation and fracture is controlled by the flowing net pressure inside the fracture, which determines the dynamic width of the natural fracture and leak-off velocity. In this study, first, the leak-off velocity under the hydromechanical coupling of the fracture-wall system was obtained based on the coupled governing equations of the solid and liquid phases of the natural fracture-wall, as well as Darcy’s law. Second, the leakage-front invasion velocity, leakage rate, and leakage volume under the hydromechanical coupling of the fracture-inner system were clarified according to the geometric governing of the natural fracture morphology. Finally, the dual-system coupling leakage model was developed considering the continuous equation, while the numerical solution was obtained through a time-step deduction. Results show that at a given time, a greater formation permeability leads to a greater leakage rate and volume, with a smaller leakage front distance. The leakage rate increases with an increase in formation permeability, well bottom differential pressure, and initial width of the natural fracture, while it decreases with an increase in the fracture normal stiffness, yield stress, and plastic viscosity. The new leakage model and numerical method concerning time-step deduction are assessed by solving the issues of fully coupled fracture-wall and fracture-inner systems considering drilling fluid leak-off. The new model may be utilized to simulate various problems related to the invasion of drilling fluids into the fractures, including predicting the dynamic width of natural fracture and borehole ballooning/breathing phenomena.


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## Body

## 1. Introduction

According to statistics, the economic loss caused by lost circulation in the global oil industry exceeds hundreds of millions of dollars every year [1], and the leakage plugging cost of fractured formations accounts for more than 90% of the total cost of lost circulation [2]. For example, the time-loss caused by lost circulation during the drilling of ultradeep wells in western Sichuan is 8%. By analyzing the drilling fluid leakage data and understanding drilling mud leakage, we can judge and evaluate the fracture development characteristics in the formation and provide the required parameters for subsequent leakage plugging operations [3–8]. When drilling mud loss occurs in naturally fractured formations, Sanfillippo et al. proposed an analytical model using Newtonian fluid to describe the leakage, neglecting the effect of the rheological properties of drilling mud [9]. Lietard et al. and Majidi et al. developed models for the radial flow of non-Newtonian fluid into a single natural fracture, neglecting the effect of drilling fluid leak-off and fracture deformability [10–12]. Other researchers have proposed an elastic equation to incorporate the impact of the fracture deformability, which relates the fracture width variation to the local pressure through the fracture normal stiffness [13–19]. To analyze fluid leak-off, Bychina conducted threshold processing for this leak-off process; that is, when the well-bottom differential pressure is greater than a fixed value, the fluid seeps into the formation at a certain leak-off velocity [20]. Razavi et al. described this with the Cater model [21]. Their research shows that for permeable fractured formations, considering the impact of the fluid, the leak-off effect is necessary for a realistic characterization of the leakage. However, the above model does not consider the porous elastic media characteristics of naturally permeable fractures and the hydromechanical coupling effect of the fracture-wall. When drilling mud loss occurs in a naturally fractured formation, the initial flowing net pressure field inside the fracture causes fracture deformation and leak-off, and reaction pressure is generated. Due to the flowing net pressure, the leak-off of the drilling fluid inside the fracture permeates into the formation region close to the fracture-wall, changing the formation pore pressure and redistributing the corresponding stress field. The time-dependent disturbed stress field arising from the coupled hydromechanical process changes the natural fracture width. The dynamic width of natural fractures is determined by changes in the deformed stress field of the fracture-wall and correspondingly influences the variation of the flowing net-pressure field inside the fracture. This is the joint effect of dual-system hydromechanical coupling between the fracture-wall and fracture-inner systems. In this work, it develops a new leakage model based on the poroelastic characteristics of the fracture-wall system and geometric governing equations of the fracture-inner system. The model considers the hydromechanical coupling effects between the aforementioned dual systems and provides a realistic characterization of the drilling mud loss in a naturally fractured formation. Besides, it also delineates the variations concerning the influence of multiple factors on the leakage state.


## 2. Hydromechanical Coupling of Fracture-Wall System

When drilling mud loss occurs in a naturally fractured formation, owing to drilling fluid leak-off, the hydromechanical coupling effect of the natural fracture-wall should be analyzed to understand the characteristics of the porous elastic medium and meet the governing equations of solid and fluid phases.(1)
Solid governing equationsStress equilibrium equation ignoring the body force.(1)∑j=13∂σij∂xj=0;σij=σji,where σij is the component of total stress tensor in MPa.Strain-displacement equation (positive strain indicates compression).(2)εij=−12ui,j+uj,i,where εij is the component of the bulk strain tensor and ui,j is the component of the deformation tensor in m.Strain-stress-pressure equation(3)εij=1+υEσij−υEσKKδij−α3Kpδij,where σKK=σxx+σyy+σzz and δij, E, υ, and p are the Kronecker symbol, Young’s modulus in MPa, Poisson’s ratio, and fluid pressure in MPa, respectively. Other constants include the bulk modulus K=E/31−2v in MPa, Biot coefficient α=1−K/Ks, and Ks is Particle volume modulus in MPa.From the above equation, the stress-strain-pressure is(4)σij=2Gεij+λεeδij+αpδij,where εe is the volumetric strain defined by εe=εx+εy+εz, the shear modulus G is G=E/21+v in MPa, and λ is the Lame coefficient defined by λ=K‐2/3G in MPa.Substituting Eq. (4) into Eq. (1) while considering Eq. (2) gives
(5)∑j=13∂∂xj−G∂ui∂xj+∂uj∂xi=∂−αp−λεe∂xi.Considering the relation between the volumetric strainεe and the solid displacement vector u→,
(6)εe=−∇·u→=−∑j=13∂uj∂xj.Furthermore, substituting Eq. (6) into Eq. (5) yields
(7)−G∇2ui+λ+G∂εe∂xi+α∂pi∂xi=0.(2)
Fluid governing equationsMass conservation, Darcy’s law, and the state equation of the pore fluid in the formation region next to the natural fracture-wall are porous medium and are expressed as follows:Mass conservation(8)∇·ρϕvf→+∂ϕρ∂t=0.Darcy’s law [22] is written as
(9)vr⇀=ϕvf⇀−vs→=−κμ∇p.The state equation reads as(10)cf=1ρ∂ρ∂p.In Eqs. (8) through (10), ρ is the fluid density in kg/m3, ϕ is porosity expressed as a percentage, vf→ is the fluid velocity vector in m/s, t is time in s, vs→ is solid velocity vector in m/s, μ is fluid viscosity in Pa·s, κ is the formation permeability in μm2, p is the fluid pressure in Pa, and cf is the fluid compressibility in MPa-1.Decomposingvsj→ and vrj→ (j=1,2,3) into three directions and considering Eqs. (8) and (9), one obtains
(11)∑i=13∂∂xi•ρfvrj→+ϕvsj→+∂ϕρf∂t=0.Using the material derivatived•/dt=∂•/∂t+vs→·∇•, Eq. (11) can be rearranged as
(12)ϕ1ρfdρf∂t+dϕ∂t+ϕ∑i=13∂vsi∂xi+∑i=13∂∂xi•vrj→=0.Equation (12) is further expressed as
(13)−αdεedt+ϕcf+α−ϕKsdpdt−κμ∇2p=0,when the four relations of fluid compressibility 1/ρfdρf/∂t=cfdp/dt, dϕ/∂t=−α−ϕdεe/dt+α−ϕ/Ksdp/dt, ∑i=13∂vsi/∂xi=−dεe/dt, and ∂/∂xi•−κ/μ∇p=‐κ/μ∇2p related to the fluid leak-off occurring only along the fracture-wall are considered.
(3)
Leak-off velocity equation under fracture-wall hydro-mechanical couplingFor the first term in Eq. (7), the solid displacement is a nonrotational field; therefore,
(14)∇2ui=−∂εe∂xi.Substituting Eq. (14) into Eq. (7) and integrating it obtains
(15)εe=−1K+4G/3αp.Substituting Eq. (15) into Eq. (13) yields
(16)1cΠdpdt‐∇2p=0,where cΠ=MK+4G/3κ/μ/K+4G/3+α2M in which the Biot modulus M is defined by 1/M=ϕcf+α−ϕ/Ks.As leak-off occurs only along the fracture-wall (along the Z-direction), it is described by radial coordinates; thus,(17)∂2p∂z2=1cΠ∂p∂t,p=pit=0,z>0,p=pfz=0,t>0,p=piz→∞,t>0,where ξ=z/4cΠt, by substituting it into the above equation, one obtains (c is an undetermined coefficient)
(18)∂p∂ξ=ce−ξ2.As shown in Figure1, the natural fracture leak-off is divided into two areas, namely, the drilling fluid invasion area (area 1) and original formation area (area 2). The pressure distribution in the corresponding areas is represented by p1 and p2, respectively. The fluid viscosities of the two areas are the drilling fluid viscosity μl and the formation viscosity μ, respectively. The term st indicates the contact surface of the two areas; where for area 1, cΙ=MK+4G/3κ/K+4G/3+α2Mμl, ϖ=z/4cΙt. Owing to the continuous pressure and mass distribution on the contact surface, the boundary conditions of the contact surface are given by Eqs. (19) and (20).
(19)p1s=p2s,(20)−κμl∂p1∂zz=st=−κμ∂p2∂zz=st=ϕdstdt.Figure 1
Schematic diagram of fracture-wall hydromechanical coupling.The pressurep1 of area 1 and p2 of area 2 are calculated by Eq. (18). Combining the boundary condition in Eq. (19), the pressure at the contact surface st is
(21)pf+c1π2erfϖ=pi−c2π2erfcξ,where erfξ is the error function, erfcξ is the complementary error function, and erfcξ=1−erfξ. To ensure that Eq. (21) is applicable at all times, η can be set as an undetermined coefficient. The terms η and st have the following relationship:
(22)st=ηt.c1=−ϕημlcΙ/κeη/4cΙ2 and c2=−ϕημcΠ/κeη/4cΠ2 are obtained by the boundary conditions in Eq. (20). Substituting them and Eq. (22) into Eq. (21) yields
(23)pf−pi=π2ϕηκμlcΙeη/4cΙ2erfη4cΙ+μcΠeη/4cΙ2erfcη4cΠ.Based on Eq. (23), the dichotomy method is used to solve for the undetermined coefficient η, yielding c1. Therefore, the drilling mud leak-off velocity under the fracture-wall hydromechanical coupling is
(24)ul=−κμldp1dzx=0=−κμlc14cΙt.Lettingc1=−2Δp/ς and substituting it into Eq. (24) yields
(25)ul=κμlΔpςcΙt,where ς is the harmonic coefficient of the leak-off velocity. As leak-off occurs in both the upper and lower sides of the fracture-wall, the total leak-off velocity equation is
(26)uL=2ul=2κμlΔpςcΙt,where uL is drilling fluid leak-off velocity in m/s.


## 3. Hydromechanical Coupling of Fracture-Inner System

When drilling mud loss occurs in a naturally fractured formation, this can be a penny-shaped natural fracture with an infinitely large radius [11, 18, 20, 21] (see Figure 2(a)). The initial flowing net pressure inside the fracture pN′ causes fracture deformation and leakage and correspondingly produces the reaction pressure pS′. The superposition of pressures pN′ and pS′ further yields the flowing net pressure inside the fracture pr (see Figure 2(b)).Figure 2
Schematic diagram of fracture inner hydromechanical coupling.
(a)
Elevation view(b)
Local viewThe flowing net pressure inside the fracturepr acts on the natural fracture-wall, causing it to deform. The linear deformation rule yields Eq. (27) [16, 17, 19]. In Eq. (27), the natural fracture is assumed to have a nonzero fracture width (w0) at zero overpressure, which implies that the natural fracture remains open at the formation of pore pressure [23–25].
(27)wr=w0+prKFrac,where wr is the dynamic width of natural fracture in m, w0 is the initial width of the fracture (natural fracture width at pr=0) in m, and pr is the net pressure flowing inside the fracture in MPa. The term KFrac is the fracture normal stiffness in MPa/m. The flowing net pressure inside the fracture along the radial direction is [21]
(28)pr=Δp1−r−rwrft−rwCt,where Δp is the well bottom differential pressure in MPa. Ct is the dimensionless morphological index coefficient of the natural fracture, and rft is the distance to the leakage front in m.By analyzing Eq. (28), at the fracture opening (r=rw), pr=Δp is the well bottom differential pressure; at the leakage front distance (r=rf), pr=0 is the internal pressure in the original fracture. Both the morphologic index coefficient of natural fracture Ct and the leakage front distance rft are time functions, which are hereinafter referred to as C and rf. Substituting Eq. (28) into Eq. (27), the radial distribution of the dynamic width of the natural fracture is
(29)wr=w0+ΔpKFrac1−r−rwrf−rwC.When drilling mud loss occurs in a naturally fractured formation, the leakage front distancerf increases from rf_n−1 to rf_n (i.e., time n−1Δt→nΔt), and the increment of the drilling fluid loss volume ΔV_n within Δt is divided into two parts: the leakage volume increment ΔVF_n due to the deformation of the natural fracture and the increase in the distance to the leakage front, and the leak-off volume increment ΔVL_n of the permeable fracture-wall (see Figure 3).Figure 3
Schematic diagram of the increment of drilling mud loss in a naturally fractured formation within ∆t.According to the above analysis,ΔV_n is obtained based on mass conservation
(30)ΔV_n=ΔVF_n+ΔVL_n,where ΔVF_n is solved by the following equation
(31)ΔVF_n=∫rf_n−1rf_n2πrwdr,where rf_n is leakage front distance at t=nΔt in m.Based on the leak-off velocity Eq. (26), ΔVL_n is expressed as
(32)ΔVL_n=uL×ΔA=∫rf_n−1rf_n2πr∗uLr∗,t∗dr∗,where ΔA is the increment leak-off area within Δt, m2.The geometric governing equation of the leakage invasion velocity at any locationr (r∈rw,rf where rf is the leakage front distance) is written as
(33)vmr,t=qr,t2πrw=dVFt+VLr,t/dt2πrw.At this point, the total leakage volumeVFt due to the deformation of the natural fracture and increase in the distance to the leakage front is
(34)VFt=∫rWrf2πrwdr.Meanwhile, the leak-off volume at the leakage distancer is the total leak-off volume within t minus the leak-off volume before leakage distance r; VLr,t is expressed as
(35)VLr,t=∫0t∫rwrf2πr∗uLr∗,t∗dr∗dt−∫0t∫rwr2πr∗uLr∗,t∗dr∗dt=∫0t∫rrf2πr∗uLr∗,t∗dr∗dt.By substituting Eqs. (29), (34), and (35) into Eq. (33), the geometric governing equation of the leakage rate is
(36)qr,t=Qdrfdt+∫rrf2πr∗uLr∗,tdr∗,where the expression of Q is
(37)Q=2πw0rf+2πΔpC+1C+2KFrac2rf+Crw.Substituting the leak-off velocityuLr∗,t from Eq. (26) into Eq. (36) yields
(38)qr,t=Qdrfdt+∫rrf4πκprμlςcΙr∗t−τr∗dr∗,where τ is the time required to reach the leakage distance r∗. Substituting Eq. (38) into Eq. (33) yields the expression of the geometric governing equation of leakage invasion velocity
(39)vmr,t=Qdrf/dt+∫rrf4πκpr/μlςcΙr∗/t−τr∗dr∗2πrw.


## 4. Leakage Coupling Model for Dual Systems

When leakage occurs in a naturally fractured formation, it includes two sets of coupling systems for this process, the fracture-wall system (coupling system 1) and fracture-inner system (coupling system 2). The initial flowing net pressure field inside the fracture causes the leak-off and deformation of the fracture. Meanwhile, reaction pressure is generated. Due to the flowing net pressure, the leaked drilling fluid inside the fracture permeates into the formation close to the fracture-wall and changes the formation pore pressure; this causes the corresponding stress field to redistribute. The time-dependent disturbed stress field arising from the coupled hydromechanical process changes the natural fracture width. The dynamic width of a natural fracture is determined by changes in the deformed stress field of the fracture-wall and correspondingly reacts to variations of the flowing net-pressure field inside the fracture. The link between the two coupled systems of the formation and fracture is the flowing net pressure inside the fracture, which determines the dynamic width of the natural fracture and leak-off velocity (see Figure4).Figure 4
Schematic diagram of dual-system coupling.When drilling mud loss occurs in a naturally fractured formation, the pressure gradient inside the fracture of the Bingham fluid model is [11]
(40)dpdr=12μpvmw2+3τyw.Substituting the geometric governing equation of the leakage invasion velocity Eq. (39) into Eq. (40), integrating r from rw⟶rf, and combining the flowing net pressure inside the fracture pr Eq. (28), the well bottom differential pressure Δp is written as
(41)Δp=∫rwrf12μpw2Qdrf/dt+∫rrf4πκpr/μlςcΙr∗/t−τr∗dr∗2πrw+3τywdr.After integrating Eq. (41), the well-bottom differential pressure Δp is given by
(42)ΔP=6μpπQF1rfdrfdt+3τyF2rf+6μpπ∫rwrf1rw31rf−rwC∫rrfχΙrf−r∗Cr∗t−τr∗dr∗dr,where χΙ=4πκΔp/μlςcΙ and the expressions of F1rf and F2rf are described in Eqs. (43) and (44). Solving Eq. (42) and separating the geometric governing equation of the leakage front invasion velocity drf/dt to the left of the equation, we obtain
(43)F1r=∫rWr1rw3dr=∫rwr1rw0+Δp/Kfrac1−r−rw/rf−rwc3dr,(44)F2r=∫rwr1wdr=∫rwr1w0+Δp/Kfrac1−r−rw/rf−rwcdr,(45)drfdt=ΔP−3τyF2rf−6μp/π∫rwrf1/rw31/rf−rwC∫rrfχΙrf−r∗Cr∗/t−τr∗dr∗dr6μp/πQF1rf.The pressure distribution inside the fracture can be obtained from Eq. (45) as
(46)pr=Δp−6μpQπF1rdrfdt−3τyF2r−6μpπ∫rwr1rw31rf−rwC∫rrfχΙrf−r∗Cr∗t−τr∗dr∗dr.To solve Eqs. (45) and (46), the leakage time t is divided into nΔt, that is, if t=nΔt, then rf=rf_n. The leakage front distance increases from rf_n−1⟶rf_n within n−1Δt⟶nΔt(see Figure 5). In this way, the spatial distance increment Δrf_n corresponding to the time increment Δt has the same leak-off velocity, which allows the equation to be solved in segments. The item ∫rwr1/rw31/rf−rwC∫rrfχΙrf−r∗Cr∗/t−τr∗dr∗dr in Eq. (46) is processed by segments. At this point, the leakage front distance is rf, and the upper limit of the integral is r∈rw,rf. The integrated domain is processed in segments and expressed as ∑k=0k=m∫rf_krf_k+1frdr, where m∈0,n−1 is an integer; then, this item can be expressed as
(47)∫rwr1rw31rf−rwC∫rrfχΙrf−r∗Cr∗t−τr∗dr∗dr=∑k=0k=m∫rf_krf_k+11rw31rf_n−rwC∫rrf_nχΙrf_n−r∗Cr∗t−τr∗dr∗dr.Figure 5
Schematic diagram of drilling mud loss in naturally fractured formation segments.By substituting Eq. (47) into Eqs. (46) and (45), we can obtain the expressions of the flowing net pressure inside the fracture and leakage front invasion velocity when the leakage front distance is rf_n, as follows:
(48)pr=Δp−6μpπQF1rdrf_ndt−3τyF2r−6μpπ∑k=0k=m∫rf_krf_k+11rw31rf_n−rwC∫rrf_nχΙrf_n−r∗Cr∗t−τr∗dr∗dr,(49)drf_ndt=Δp−3τyF2rf_n−6μp/π∑k=0k=n−1∫rf_krf_k+11/rw31/rf_n−rwC∫rrf_nχΙrf_n−r∗Cr∗/t−τr∗dr∗dr6μp/πQF1rf_n.As shown in Eq. (28), performing segment manipulations of the expression of flowing net pressure inside the fracture yields the following expression of t=nΔt, pr written as
(50)pr=Δp1−r−rwrf_n−rwC.By Eq. (36), at a given time, we can obtain the leakage rate at any leakage distance. The leakage rate measured on the ground is the leakage rate at the fracture opening, that is, qrw,t. Substituting the expression of ΔpL into Eq. (36) and calculating it by segments, we obtain qrw,t(51)qrw,tn=Qdrf_ndt+1rf_n−rwC×∑k=0k=n−1χΙ_kn−kΔtrf_n−rf_kC+1Crf_k+rf_k+rf_nC2+3C+2−rf_n−rf_k+1C+1Crf_k+1+rf_k+1+rf_nC2+3C+2,where χΙ_k=4πκΔp/μlς_kcΙ, ς_k is calculated by the flowing net pressure inside the fracture of each spatial distance Δrf_n in any period of time Δt (set Δt=0.001s). At a given time, the total leak-off volume is VLrw,t=∑j=1j=nqL_jΔt, calculates it by segments provides VLrw,tn(52)VLrw,tn=∑j=1j=n1rf_j−rwC∑k=0k=j−1χΙ_kΔtj−krf_j−rf_kC+1Crf_k+rf_k+rf_jC2+3C+2−rf_j−rf_k+1C+1Crf_k+1+rf_k+1+rf_jC2+3C+2.As seen from Eq. (30), the total leakage volume measured on the ground is the sum of the two parts of VLrw,tn and VFtn(53)VTotaltn=VLrw,tn+VFtn.The pressure inside the fracture solved by Eq. (48) is pIr; the pressure inside fracture solved by Eq. (50) is pIIr; the rationality of C is determined by judging whether they satisfy the solution accuracy εP=pIr−pIIr/pIr. The whole solution process is divided into two steps “initial value determination” and “recursive prediction” (see Figure 6). The specific steps are as follows:
(1)
Enter the basic parametersΔp, τy, μp, w0, KFrac, and basic formation parameters (k, μ, E, ν, Ks, and Kf) to calculate the leak-off harmonic coefficient ς(2)
Calculaterft1 (t1=Δt) by assuming that the initial value of C is C0. Calculate drft1/dt by assuming Ct1, combining it with rft1 and substituting it into Eq. (49)(3)
Substitutedrft1/dt into Eq. (48) to calculate pIr and substitute drft1/dt and rft1 into Eq. (50) to calculate pIIr until the solution accuracy εp is satisfied(4)
CompareC0 and Ct1 to judge whether the solution accuracy εC=C0−Ct1/C0 is satisfied; if not, repeat step (2) until the solution accuracy εC is satisfied(5)
Solverftn=drftn−1/dt∗Δt for the next time step, as per drftn−1/dt(n≥2), solved by the previous time step. Assuming the value of Ctn, calculate drftn/dt by combining it with rftn and substituting it into Eq. (49)(6)
Substitutedrftn/dt into Eq. (48) to calculate pIr and substitute Ctn and rftn into Eq. (50) to calculate pIIr until the solution accuracy εp is satisfied(7)
Substitute the calculatedrftn and Ctn into Eqs. (51), (34), (52), and (53) and calculate the leakage rate qtn, leakage volume inside fracture VFtn, leak-off volume VLtn, and total leakage volume VTotaltnFigure 6
Schematic diagram for solving the dual-system coupling leakage model.Among them, steps (2)–(4) represent the “initial value determination” step, and steps (5)–(7) are the “recursive prediction” step.


## 5. Leakage Simulation and Analysis of Influence Factors

### 5.1. Method Verification and Leakage Simulation

To analyze the leakage state at different formation permeabilities,κ was set as 0, 1, 5, and 10 md. Other parameters are shown in Table 1. When κ=0md, the calculation results were compared with Bychina’s [20] finite difference method to verify the correctness of the new model.Table 1
Calculation parameters.
ParameterSymbolValueParameterSymbolValueFormation permeability (md)κ0/1/5/10Yield stress (Pa)τy15Formation fluid viscosity (Pa·S)μ10-3Plastic viscosity (mPa·S)μp39Elasticity modulus (MPa)E104Well bottom differential pressure (MPa)Δp5Poisson’s ratio (dimensionless)v0.22Initial width of a natural fracture (μm)w0800Particle volume modulus (MPa)Ks36∗103Fracture normal stiffness (GPa/m)KFrac30Pore fluid volume modulus (MPa)Kf2.25∗103Wellbore radius (mm)rw155.6Formation porosity (dimensionless)ϕ0.1As shown in Figures7–9, when κ=0md, the calculation results of the proposed model are consistent with those of Bychina’s finite difference method, which verifies the reliability of the model. At any given time, a greater formation permeability causes a greater leakage rate and volume, as well as a smaller leakage front distance. Although there is little difference in the leakage rate and leakage volume under different formation permeabilities, they have different physical meanings. With an increase in formation permeability κ, the leakage front distance decreases, which means that the leakage volume VFt decreases and the leak-off volume VLt increases. In addition, as shown in Eq. (38), with the increase of rf, the effect of κ on the leakage rate is evident. Moreover, the lower formation permeability produces a higher leakage invasion speed, which increases the difficulty of solid plugging [26].Figure 7
Leakage front distance graph under different permeability.Figure 8
Leakage rate graph under different formation permeability.Figure 9
Leakage volume graph under different formation permeability.


### 5.2. Analysis of Multiple Influence Factors

To analyze the influence of multiple influence factors under different formation permeabilities, fixing the formation permeabilityκ and setting κ=1 and 10 md can allow a comparative analysis of the leakage front distance, leakage rate, and leakage volume under the influence of multiple influence factors.
(1)
Fracture normal stiffnessAs shown in Figures10–12, at any given time, a greater fracture normal stiffness causes a smaller leakage front distance, leakage rate, and leakage volume under the same formation permeability. Based on Table 2, it can be seen that the formation permeability and fracture normal stiffness have the same effect on the leakage front distance, while having the opposite effect on the leakage rate and leakage volume. In other words, the leakage front distance decreases with an increase in the formation permeability and fracture normal stiffness. However, the leakage rate and leakage volume increase with increases in formation permeability and decrease with increasing fracture normal stiffness.
(2)
Rheological properties of drilling fluidFigure 10
Leakage front distance under different formation permeability and fracture normal stiffness.Figure 11
Leakage rate under different formation permeability and fracture normal stiffness.Figure 12
Leakage volume under different formation permeability and fracture normal stiffness.Table 2
Influence of multiple factors.
Affected objectLeakage front distancerfLeakage rateqLeakage volumeVTotalInfluence factorFixed influencing factor↑κ−++Multiple influence factors↑kf−−−↑τy−−−↑μp−−−↑∆p+++↑w0+++As shown in Figures13–15, at any given time, a greater normal yield stress causes a smaller leakage front distance, leakage rate, and leakage volume under the same formation permeability. Based on Table 2, it can be seen that the formation permeability and yield stress have the same effect on the leakage front distance while having the opposite effect on the leakage rate and leakage volume. In other words, the leakage front distance decreases with an increase in the formation permeability and yield stress. However, the leakage rate and leakage volume increase with increasing the formation permeability and decrease with increasing yield stress.Figure 13
Leakage front distance under different formation permeability and yield stress.Figure 14
Leakage rate under different formation permeability and yield stress.Figure 15
Leakage volume under different formation permeability and yield stress.As shown in Figures16–18, at any given time, a greater plastic viscosity causes a smaller leakage front distance, leakage rate, and leakage volume under the same formation permeability. Based on Table 2, it can be seen that the formation permeability and plastic viscosity have the same effect on the leakage front distance while having the opposite effect on the leakage rate and leakage volume. In other words, the leakage front distance decreases with an increase in the formation permeability and plastic viscosity. However, the leakage rate and leakage volume increase with increasing formation permeability and decrease with increasing plastic viscosity.
(3)
Well bottom differential pressureFigure 16
Leakage front distance under different formation permeability and plastic viscosity.Figure 17
Leakage rate under different formation permeability and plastic viscosity.Figure 18
Leakage volume under different formation permeability and plastic viscosity.As shown in Figures19–21, at any given time, a greater well bottom differential pressure causes a greater leakage front distance, leakage rate, and leakage volume under the same formation permeability. Based on Table 2, it can be seen that the formation permeability and well bottom differential pressure have opposite effects on the leakage front distance, while having the same effect on the leakage rate and leakage volume. In other words, the leakage front distance decreases with an increase in the formation permeability and increases with an increasing well bottom differential pressure. However, the leakage rate and leakage volume increase with both an increasing formation permeability and well bottom differential pressure.
(4)
Initial width of natural fractureFigure 19
Leakage front distance under different formation permeability and well bottom differential pressure.Figure 20
Leakage rate under different formation permeability and well bottom differential pressure.Figure 21
Leakage volume under different formation permeability and well bottom differential pressure.Figure 22
Leakage front distance under different formation permeability and initial width of natural fracture.Figure 23
Leakage rate under different formation permeability and initial width of natural fracture.As shown in Figures21–24, at any given time, a greater initial width of the natural fracture causes a greater leakage front distance, leakage rate, and leakage volume under the same formation permeability. Based on Table 2, it can be seen that the formation permeability and initial width of the natural fracture have opposite effects on the leakage front distance while having the same effect on the leakage rate and leakage volume. In other words, the leakage front distance decreases with an increase in the formation permeability and increases with an increasing initial width of the natural fracture. However, the leakage rate and leakage volume increase with an increase in both formation permeability and initial width of the natural fracture.Figure 24
Leakage volume under different formation permeability and initial width of natural fracture.In summary, the leakage front distance decreases with an increase in the formation permeability, fracture normal stiffness, yield stress, and plastic viscosity and increases with an increasing well bottom differential pressure and initial width of natural fracture. The leakage rate and leakage volume increase with increases in formation permeability, well bottom differential pressure, and initial width of the natural fracture; they decrease with increasing fracture normal stiffness, yield stress, and plastic viscosity. As shown in Table2, the combined impact of an arbitrary combination of two or three sets of influencing factors on leakage front distance, leakage rate, and leakage volume can be obtained by multiplying the single influencing factor. When the impacts have opposite effects, the final result depends on the degree of influence of each factor.


## 5.1. Method Verification and Leakage Simulation

To analyze the leakage state at different formation permeabilities,κ was set as 0, 1, 5, and 10 md. Other parameters are shown in Table 1. When κ=0md, the calculation results were compared with Bychina’s [20] finite difference method to verify the correctness of the new model.Table 1
Calculation parameters.
ParameterSymbolValueParameterSymbolValueFormation permeability (md)κ0/1/5/10Yield stress (Pa)τy15Formation fluid viscosity (Pa·S)μ10-3Plastic viscosity (mPa·S)μp39Elasticity modulus (MPa)E104Well bottom differential pressure (MPa)Δp5Poisson’s ratio (dimensionless)v0.22Initial width of a natural fracture (μm)w0800Particle volume modulus (MPa)Ks36∗103Fracture normal stiffness (GPa/m)KFrac30Pore fluid volume modulus (MPa)Kf2.25∗103Wellbore radius (mm)rw155.6Formation porosity (dimensionless)ϕ0.1As shown in Figures7–9, when κ=0md, the calculation results of the proposed model are consistent with those of Bychina’s finite difference method, which verifies the reliability of the model. At any given time, a greater formation permeability causes a greater leakage rate and volume, as well as a smaller leakage front distance. Although there is little difference in the leakage rate and leakage volume under different formation permeabilities, they have different physical meanings. With an increase in formation permeability κ, the leakage front distance decreases, which means that the leakage volume VFt decreases and the leak-off volume VLt increases. In addition, as shown in Eq. (38), with the increase of rf, the effect of κ on the leakage rate is evident. Moreover, the lower formation permeability produces a higher leakage invasion speed, which increases the difficulty of solid plugging [26].Figure 7
Leakage front distance graph under different permeability.Figure 8
Leakage rate graph under different formation permeability.Figure 9
Leakage volume graph under different formation permeability.


## 5.2. Analysis of Multiple Influence Factors

To analyze the influence of multiple influence factors under different formation permeabilities, fixing the formation permeabilityκ and setting κ=1 and 10 md can allow a comparative analysis of the leakage front distance, leakage rate, and leakage volume under the influence of multiple influence factors.
(1)
Fracture normal stiffnessAs shown in Figures10–12, at any given time, a greater fracture normal stiffness causes a smaller leakage front distance, leakage rate, and leakage volume under the same formation permeability. Based on Table 2, it can be seen that the formation permeability and fracture normal stiffness have the same effect on the leakage front distance, while having the opposite effect on the leakage rate and leakage volume. In other words, the leakage front distance decreases with an increase in the formation permeability and fracture normal stiffness. However, the leakage rate and leakage volume increase with increases in formation permeability and decrease with increasing fracture normal stiffness.
(2)
Rheological properties of drilling fluidFigure 10
Leakage front distance under different formation permeability and fracture normal stiffness.Figure 11
Leakage rate under different formation permeability and fracture normal stiffness.Figure 12
Leakage volume under different formation permeability and fracture normal stiffness.Table 2
Influence of multiple factors.
Affected objectLeakage front distancerfLeakage rateqLeakage volumeVTotalInfluence factorFixed influencing factor↑κ−++Multiple influence factors↑kf−−−↑τy−−−↑μp−−−↑∆p+++↑w0+++As shown in Figures13–15, at any given time, a greater normal yield stress causes a smaller leakage front distance, leakage rate, and leakage volume under the same formation permeability. Based on Table 2, it can be seen that the formation permeability and yield stress have the same effect on the leakage front distance while having the opposite effect on the leakage rate and leakage volume. In other words, the leakage front distance decreases with an increase in the formation permeability and yield stress. However, the leakage rate and leakage volume increase with increasing the formation permeability and decrease with increasing yield stress.Figure 13
Leakage front distance under different formation permeability and yield stress.Figure 14
Leakage rate under different formation permeability and yield stress.Figure 15
Leakage volume under different formation permeability and yield stress.As shown in Figures16–18, at any given time, a greater plastic viscosity causes a smaller leakage front distance, leakage rate, and leakage volume under the same formation permeability. Based on Table 2, it can be seen that the formation permeability and plastic viscosity have the same effect on the leakage front distance while having the opposite effect on the leakage rate and leakage volume. In other words, the leakage front distance decreases with an increase in the formation permeability and plastic viscosity. However, the leakage rate and leakage volume increase with increasing formation permeability and decrease with increasing plastic viscosity.
(3)
Well bottom differential pressureFigure 16
Leakage front distance under different formation permeability and plastic viscosity.Figure 17
Leakage rate under different formation permeability and plastic viscosity.Figure 18
Leakage volume under different formation permeability and plastic viscosity.As shown in Figures19–21, at any given time, a greater well bottom differential pressure causes a greater leakage front distance, leakage rate, and leakage volume under the same formation permeability. Based on Table 2, it can be seen that the formation permeability and well bottom differential pressure have opposite effects on the leakage front distance, while having the same effect on the leakage rate and leakage volume. In other words, the leakage front distance decreases with an increase in the formation permeability and increases with an increasing well bottom differential pressure. However, the leakage rate and leakage volume increase with both an increasing formation permeability and well bottom differential pressure.
(4)
Initial width of natural fractureFigure 19
Leakage front distance under different formation permeability and well bottom differential pressure.Figure 20
Leakage rate under different formation permeability and well bottom differential pressure.Figure 21
Leakage volume under different formation permeability and well bottom differential pressure.Figure 22
Leakage front distance under different formation permeability and initial width of natural fracture.Figure 23
Leakage rate under different formation permeability and initial width of natural fracture.As shown in Figures21–24, at any given time, a greater initial width of the natural fracture causes a greater leakage front distance, leakage rate, and leakage volume under the same formation permeability. Based on Table 2, it can be seen that the formation permeability and initial width of the natural fracture have opposite effects on the leakage front distance while having the same effect on the leakage rate and leakage volume. In other words, the leakage front distance decreases with an increase in the formation permeability and increases with an increasing initial width of the natural fracture. However, the leakage rate and leakage volume increase with an increase in both formation permeability and initial width of the natural fracture.Figure 24
Leakage volume under different formation permeability and initial width of natural fracture.In summary, the leakage front distance decreases with an increase in the formation permeability, fracture normal stiffness, yield stress, and plastic viscosity and increases with an increasing well bottom differential pressure and initial width of natural fracture. The leakage rate and leakage volume increase with increases in formation permeability, well bottom differential pressure, and initial width of the natural fracture; they decrease with increasing fracture normal stiffness, yield stress, and plastic viscosity. As shown in Table2, the combined impact of an arbitrary combination of two or three sets of influencing factors on leakage front distance, leakage rate, and leakage volume can be obtained by multiplying the single influencing factor. When the impacts have opposite effects, the final result depends on the degree of influence of each factor.


## 6. Conclusions

(1)
The new leakage model clarifies the influence of dual-system hydromechanical coupling effects between fracture-wall and fracture-inner systems when drilling mud loss occurs in a naturally fractured formation. The model can account for various problems related to the invasion of drilling fluids into the fractures, including predicting the dynamic width of natural fracture and borehole ballooning/breathing phenomena(2)
When drilling mud loss occurs in a naturally fractured formation, the leakage rate increases with an increase in formation permeability, well-bottom differential pressure, and initial width of the natural fracture, while it decreases with an increase in the fracture normal stiffness, yield stress, and plastic viscosity(3)
When drilling mud loss occurs in a naturally fractured formation, at any given time, a greater formation permeability causes a larger leakage rate and leakage volume, whereas a smaller leakage front distance.



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