## Abstract

Although a large volume of mudcake filtration test data is available in the literature, effects of mudcake on wellbore strengthening cannot be quantified without incorporating the data into a stress-analysis model. Traditional models for determining fracture initiation pressure (FIP) either consider a wellbore with an impermeable mudcake or with no mudcake at all. An analytical model considering permeable mudcake is proposed in this paper. The model can predict pore pressure and stress profiles around the wellbore, and consequently the FIP, for different mudcake thickness, permeability, and strength.

#### Authors

Yongcun Feng, Xiaorong Li, K. E. Gray

The University of Texas at Austin, Austin, TX, USA

Received: 5 September 2017 / Published online: 9 March 2018

©The Author(s) 2018

Numerical examples are provided to illustrate the effects of these mudcake parameters. The results show that a low-permeability mudcake enhances FIP, mainly through restricting fluid seepage and pore pressure increase in the near-wellbore region, rather than by mudcake strength. Fluid loss pressure (FLP) should be distinguished from FIP when a mudcake is present on the wellbore wall. Fracture may occur behind the mudcake at FIP without mudcake rupture. The small effect of mudcake strength on FIP does not mean its effect on FLP is small too. Mudcake strength may play an important role in maintaining integrity of the wellbore once a fracture has initiated behind the mudcake.

## 1 Introduction

As a bit drills through a permeable formation in overbalance drilling, the base fluid of drilling mud permeates into the formation while solid components are filtered out, leaving a mudcake on the wellbore wall. Both laboratory tests and field practices have shown that the presence of a mudcake can effectively inhibit fracture creation on the wellbore and thus prevent lost circulation (Cook et al. 2016; Guo et al. 2014; Song and Rojas 2006). Researchers also found that with some additives, such as lost circulation materials (LCMs), the drilling mud can improve the quality of the mudcake and better enhance wellbore strength (Ewy and Morton 2009; Song and Rojas 2006).

An optimal mudcake should have low permeability and thin thickness (Amanullah and Tan 2001). It is well known that fluid permeation through the wellbore wall increases the local pore pressure and decreases the effective hoop stress around the wellbore, resulting in a reduced fracture initiation pressure (FIP), i.e., the minimum wellbore pressure at which a fracture is created on the wellbore wall. Therefore, low permeability is required for the mudcake to prevent or mitigate base fluid permeation. On the other hand, although a thick mudcake can also help restrict fluid permeation, it may cause other drilling problems, such as stuck pipe and excessive torque and drag (Hashemzadeh and Hajidavalloo 2016; Ottesen et al. 1999; Outmans 1958). So a thick mudcake is usually not recommended.

The bottom hole conditions under which a mudcake forms are very complex. The development of the mudcake depends on a number of factors such as drilling mud constituents, wellbore pressure and temperature, pore pressure, formation permeability and porosity, annulus flow regime, and time (Hashemzadeh and Hajidavalloo 2016; Salehi and Kiran 2016). During dynamic drilling, the jets of the drilling bit may generate very turbulent flow and the rate of mudcake formation is controlled by two opposite actions—deposition and erosion (Cook et al. 2016; Mostafavi et al. 2010). Mudcake buildup stops when the deposition and erosion rates become equal. However, the point at which the two rates equilibrate is difficult to determine. In this paper, it is assumed that the mudcake forms before the creation of any fracture.

Three important parameters dictating the effectiveness of mudcake on wellbore strengthening are mudcake thickness, permeability, and strength. Numerous researchers have experimentally investigated the development of these mudcake parameters under various borehole conditions. For example, Jaffal et al. (2017) and Griffith and Osisanya (1999) performed filtration tests and studied the effects of a number of factors (e.g., differential pressure, solids content, filtration control agent) on mudcake thickness and permeability. Yield strength of the mudcake has been extensively measured under different loading conditions (Bailey et al. 1998; Cerasi et al. 2001; Cook et al. 2016). Although a large volume of mudcake data is available in the literature, the effects of different mudcake parameters on wellbore strengthening cannot be quantified without incorporating the data into a stress-analysis model.

It is well known that a fracture initiates once the stress on the wellbore wall overcomes the concentrated hoop stress and the tensile strength of the rock (Chuanliang et al. 2015; Guo et al. 2017a, b; Zhu et al. 2014). Therefore, FIP (i.e., the wellbore pressure at the critical moment of fracture initiation) is determined by the hoop stress and the tensile strength. The tensile strength of rock is a measurable parameter. However, the hoop stress can be greatly altered by the formation of mudcake on the wellbore wall. Therefore, a model relating wellbore stress profile and mudcake properties is required for evaluating the effects of mudcake on FIP and optimizing mud design (Feng et al. 2018).

Traditional models for determining FIP assume two extreme conditions of the wellbore: a wellbore with an impermeable mudcake, e.g., the Hubbert–Willis model (Hubbert and Willis 1957), and a wellbore with no mudcake, e.g., the Haimson–Fairhurst model (Haimson and Fairhurst 1969). Obviously, these models cannot capture the effects of various parameters of a permeable mudcake on FIP, resulting in an overestimated FIP while assuming impermeable mudcake and underestimated FIP if the mudcake is not considered (Tran et al. 2011).

Therefore, an analytical model considering a permeable mudcake is proposed in this paper. The model is derived based on the assumptions of Darcy’s fluid flow through the mudcake and formation and the superposition principle of elasticity. The model can be used to predict pore pressure and stress profiles around the wellbore, and consequently the FIP, for different mudcake thickness, permeability, and strength. Numerical examples are provided to illustrate the effects of these mudcake parameters.

It should be noted that FIP and fluid loss pressure (FLP—the critical wellbore pressure causing mud loss into the fracture) should be distinguished from each other. Because at the moment of fracture initiation on wellbore wall at FIP, the mudcake may remain unbroken and wellbore fluids cannot enter the fracture; thus, no fluid loss can be observed at FIP (Feng et al. 2016). Only when the mudcake eventually ruptures as the wellbore pressure increases, will significant fluid loss occur. However, from a conservative point of view, it is better to maintain wellbore pressure below FIP during drilling to avoid lost circulation. The difference between FIP and FLP is discussed in more detail in this paper.

## 2 Analytical mudcake model

In this section, an analytical mudcake model is derived, which takes into account the effects of mudcake thickness, permeability and strength on near-wellbore pore pressure and stress states, and thus FIP of the wellbore. A thin layer of mudcake is assumed on the inner surface of a wellbore subjected to non-uniform far-field stresses and pore pressure, as schematically depicted in Fig. 1.

**Fig. 1** Schematic of the cross section of wellbore, mudcake, and formation (not to scale)

For deriving the model, the following assumptions are made:

-The wellbore, mudcake, and formation are in a plane-strain condition.

-The mudcake and wellbore rock bond perfectly. The inner mudcake radius, outer mudcake radius (i.e., wellbore radius), and outer formation radius are Ri, Ro, and Re, respectively. The mudcake thickness is w (w=Ro−Ri).

-The permeabilities of the mudcake and formation are K1 and K2 (K2>K1), respectively.

-Pore pressure at the outer formation boundary is not perturbed by the wellbore fluid and maintains constant at Pe. A constant mud pressure Pi (Pi>Pe) is applied on the inner surface of the mudcake.

-Fluid flow from the wellbore to the formation under the differential pressure is steady state and obeys Darcy’s law.

-The formation rock is isotropic, homogeneous, and poroelastic material.

-The mudcake is very soft/flexible compared to the rock and has a very small yield strength so that it undergoes perfectly plastic yielding under wellbore pressure. Poisson’s ratio for post-yield deformation of mudcake is assumed to be 0.5.

-Mudcake thickness and properties do not change with time.

-The maximum and minimum far-field total stresses are σ_{H} and σ_{h}, respectively.

-The sign convention for stress is compression as positive and tension as negative.

Pore pressure varies with radial distance in this problem due to fluid flow from the wellbore to the formation under differential pressure. The pore pressure distribution can be determined with Darcy’s law. The total stress distribution around the wellbore is determined by four stress components induced by: (1) varying pore pressure around the wellbore, (2) far-field stresses, (3) wellbore pressure acting on the inner mudcake surface, and (4) the presence of mudcake. To simplify the problem, it is assumed that these terms are uncoupled and the final expression for total stress can be obtained by the superposition principal. The following subsections describe the detailed derivation processes for determining the total stress around the wellbore with the presence of a mudcake.

### 2.1 Total stresses around the wellbore induced by varying pore pressure

#### 2.1.1 Varying pore pressure distribution around the wellbore

In order to determine the total stress induced by varying pore pressure, the pore pressure distribution around the wellbore should be determined first. Assuming the pore pressure at the wellbore wall (interface between mudcake and formation) is Po, then according to Darcy’s law for radial flow from the wellbore to the formation, one can obtain

where μ_{1} and μ_{2} are viscosities of fluids in the mudcake and rock, respectively. Assuming μ_{1} = μ_{2}, the pore pressure at the wellbore wall can be obtain by rearranging Eq. (1) as

where ΔP_{o} is the differential pressure between the wellbore pressure and the far-field (undisturbed) pore pressure; B is a factor determined by the dimensions and permeabilities of the mud-formation system. They are expressed as

Next, the pore pressure in the formation around the wellbore can be determined. Define Pr as the pore pressure in the formation at distance r (R_{o}≤r≤R_{e}) from the wellbore center. The following equation can be written from Darcy’s law

Inserting the pore pressure Po at the wellbore wall (Eq. (2)) into Eq. (5), the pore pressure Pr at the radial distance r from the wellbore center can be obtained as

2.1.2 Total stresses around the wellbore induced by varying pore pressure distribution

For the plane-strain circular wellbore model, the total radial and tangential (hoop) stresses around the wellbore induced by varying pressure can be determined by (Fjar et al. 2008)

where σ_{r,p} and σ_{θ,p} are the total radial and tangential stresses around the wellbore induced by varying pore pressure; η is the poroelastic stress coefficient; ΔP(r)=Pr−Pe is the pore pressure difference between location r and far field; combining Eqs. (2) and (6), ΔP(r) can be expressed as

; B is defined in Eq. (4).

Define the two integration terms appearing in Eqs. (8) and (9) as

Inserting Eq. (10) into Eqs. (11) and (12) and doing the integrations, I_{1} and I_{2} can be determined as

Inserting Eqs. (13) and (14) into Eqs. (8) and (9), the total radial and tangential stresses around the wellbore induced by varying pore pressure can be expressed as

2.2 Total stresses around the wellbore induced by wellbore pressure, far-field stresses and plastic mudcake

#### 2.2.1 Kirsch solutions

Because the mudcake is usually very soft/flexible compared to the formation, it is reasonable to assume that the mudcake does not exert any shear tractions on the wellbore wall (Tran et al. 2011). Radial mechanical pressure is the only force applied to the wellbore wall by the mudcake. Under this condition, the Kirsch solutions are still valid. Therefore, the total stress around wellbore induced by the far-field stresses and the mechanical pressure on wellbore wall can be determined as

where σ_{r,s+w} and σ_{θ,s+w} are the total radial and tangential stresses around wellbore induced by far-field stresses superposed on the mechanical pressure on the wellbore wall exerted by mudcake; θ is the circumferential angle to the direction of σ_{H}, as shown in Fig. 1; P_{iw} is the pressure exerted on wellbore wall by the mudcake. It should be noted that P_{iw} is not equal to wellbore pressure P_{i} acting on the inner surface of mudcake; rather, it is influenced by the fluid flow through the mudcake and the plastic flow of the mudcake itself.

#### 2.2.2 Stress applied on the wellbore wall by the mudcake

For the plain strain problem considered in this study, the equations of equilibrium for the mudcake can be simplified to a single equation in cylindrical coordinates (Fjar et al. 2008)

where σ_{r,c} and σ_{θ,c} are the radial and tangential stresses in the mudcake, respectively. The stress–strain relationships for the mudcake can be expressed as

where E_{c} and v_{c} are the Young’s modulus and Poisson’s ratio of the mudcake, respectively; ε_{r,c}, ε_{θ,c}, and ε_{z,c} are the radial, tangential, and axial strains of the mudcake, respectively; σ_{z,c} is the axial stress in the mudcake.

Due to its very soft/flexible features, the mudcake can be assumed to be in a perfectly plastic-yielding condition with a small yield strength under the differential pressure between the wellbore and the formation (Tran et al. 2011). Poisson’s ratio for mudcake plastic flow is assumed to be 0.5. The mudcake is considered in a plane-strain condition; thus ε_{z,c}=0. Then, from the last equation of Eq. (20), one can get

According to the von Mises yield theory (also known as maximum distortion energy theory), the following expression can be found when the mudcake yields (Aadnøy and Belayneh 2004)

where Y is the yield strength of the mudcake.

Inserting Eq. (21) into Eq. (22), one can get

Inserting Eq. (23) into the equilibrium equation (Eq. (20)), the radial stress distribution in the mudcake can be expressed as

where C is an integration constant.

Applying the boundary condition at the inner mudcake surface σ_{r,c|r}=R_{i}=P_{i} into Eq. (24), the integration constant C can be determined as

Inserting Eq. (25) into Eq. (24), the radial stress distribution in the mudcake can be determined as

where R_{i}≤r≤R_{o}.

On the wellbore wall r=R_{o}, the radial stress is

As aforementioned, it is reasonable to assume that the mudcake does not exert any shear traction on the wellbore wall because it is extremely soft/flexible compared to the formation (Tran et al. 2011). However, the mudcake can transmit part of the radial stress exerted by the wellbore fluid on the inner surface of mudcake to the wellbore wall, depending on the yield strength and thickness of the mudcake. The radial stress on the inner surface of the mudcake is equal to drilling fluid pressure P_{i}, while the radial stress on the wellbore wall is given by Eq. (27). Therefore, by simply replacing P_{iw} in the Kirsch solutions (Eqs. 17 and 18) with σ_{r,c|r}=R_{o} (Eq. (27)), the total stress around the wellbore induced by wellbore pressure, far-field stress, and plasticity of mudcake can be determined as

In Eqs. (28) and (29), the first two terms are the total stress concentration contributed by far-field stresses as a result of the creation of the wellbore; the third term denotes the stress induced by the wellbore pressure; and the last term is the contribution of the mudcake.

### 2.3 Total stresses around the wellbore

In the above sections, the total stresses induced by varying pore pressure around the wellbore due to fluid flow (Eqs. 15 and 16) and the total stress induced by wellbore pressure, far-field stress, and plasticity of mudcake (Eqs. 28 and 29) have been derived. Assuming these terms are uncoupled, the total stress solutions are therefore a superposition of them

where σ_{r} and σ_{θ} are the total radial and tangential stresses around the wellbore, respectively.

### 2.4 Effective stresses around wellbore

Effective stresses around the wellbore are equal to total stresses minus pore pressure at the corresponding locations, i.e.,

where σ′_{r} and σ′_{θ} are the effective radial and tangential stresses around the wellbore; P_{r} is the pore pressure at distance r from the wellbore center, defined in Eq. (6).

The effective stresses on the wellbore wall (r=R_{o}) can be determined as

where σ′_{r|r}=Ro and σ′_{θ|r}=Ro are the effective radial and tangential stresses on the wellbore wall, respectively; σ_{r|r}=Ro and σ_{θ|r}=Ro are the total tangential stresses on the wellbore wall which can be determined by Eqs. (32) and (33), respectively; P_{o} is the pore pressure on the wellbore wall, defined in Eq. (2).

### 2.5 Fracture initiation pressure

Fracture initiation occurs when the minimum effective tangential stress on the wellbore wall reaches the tensile strength of the rock. Under non-uniform horizontal stresses as shown in Fig. 1, the minimum effective tangential stress on wellbore wall can be found at θ=0 or π and determined by Eq. (35) as

where σ_{θ,p|r}=R_{o} is the total tangential stress on the wellbore wall induced by the varying pore pressure and can be determined by Eq. (16) as

where M and N are functions of the geometry and permeability of the wellbore-mudcake system in Fig. 1

A and B are the same as defined in Sect. 2.1.

The tensile strength of the rock is assumed to be zero in this study in order to simplify the analyses in the later sections. Therefore, fracture occurs when

Combining Eqs. (40), (36), (37), and (2), the fracture initiation pressure (i.e., the minimum wellbore pressure at which a fracture occurs on the wellbore wall) for a wellbore with a mudcake can be determined as

Besides far-field stresses and formation pore pressure, this solution also takes into account the effects of mudcake parameters on FIP, including mudcake permeability, thickness, and strength. In the following sections, the effects of mudcake parameters on FIP and distributions of near-wellbore stresses and pore pressure are illustrated through numerical examples. In addition, the implications of the analysis results on wellbore strengthening for lost circulation prevention are discussed.

It should be noted again that, due to the complexity of the mudcake problem, the current model assumes steady-state fluid flow, and thus the transient pore pressure effect on wellbore strengthening is not considered. Further improvement of the model to include transient effects can use the decomposition scheme for time-dependent poroelastic solution of a borehole without mudcake proposed by Cui et al. (1997). However, more complicated mathematical derivations will be involved due to the introduction of mudcake to the problem.

In Cui’s study, the transient problem was decomposed to three linear problems, namely a poroelastic plane-strain problem, an elastic uniaxial stress problem, and an elastic anti-plane shear problem. Due to the linearity of the problems, the final solution can be obtained by the superposition principle. Readers are referred to Cui et al. (1997, 1999) for more details of the decomposition approach.

## 3 Effects of mudcake parameters

Recent experimental studies on wellbore strengthening have revealed the important role of mudcake on inhibiting the growth of fractures that would otherwise cause lost circulation (Cook et al. 2016; Guo et al. 2014; Salehi and Kiran 2016). A thorough understanding of the effects of mudcake parameters on near-wellbore stress profiles and FIP is critical for mud design to obtain optimal mudcake during drilling. The analytical analysis in Sect. 2 indicates that mudcake alters the stress profile and FIP through its thickness, permeability, and strength. In this section, the effects of these parameters are quantitatively illustrated with a parametric study.