Investigating the Role of Plastic and Poroelastoplastic Effects in Wellbore Strengthening Using a Fully Coupled Hydro-Mechanical Model

Abstract

Wellbore instability during drilling in soft formations often leads to unwanted hydraulic fractures and lost circulation, resulting in non-productive time and elevated costs. The fracture initiation pressure (FIP) and fracture propagation pressure (FPP) are critical for managing these risks, particularly in narrow mud weight windows, yet industrial models overlook post-plugging stress behaviors at plug locations, where changes in stress concentration may initiate secondary fractures. This study introduces a fully coupled hydro-mechanical plane-strain (KGD) finite element model to examine fluid diffusion and deformation in fractured formations, emphasizing plastic and poroelastoplastic effects for wellbore strengthening. Fluid flow in the fracture follows lubrication theory for incompressible Newtonian fluids, while Darcy’s law governs porous media diffusion. Rock deformation adheres to Biot’s effective stress principle, extended to poroelastoplasticity via the Mohr–Coulomb criterion with associative flow. Simulations yield fracture dimensions, fluid pressures, in situ stress changes, and principal stresses during propagation and plugging, for both plastic and poroplastic cases. A new yield factor is proposed, derived from the Mohr–Coulomb criterion, that quantifies the risk of failure and reveals that fracture tips resist propagation through plastic and poroelastoplastic deformation, with the poroelastoplastic coupling amplifying back-stresses and dilation after plugging. Pore pressure evolution critically influences the fracture growth and plugging efficiency. These findings advance wellbore strengthening by optimizing lost circulation material plugs, bridging the gaps from elastic and poroelastic models, and offer practical tools for safer and more efficient plugging in soft rocks through modeling.

1. Introduction

When drilling operations penetrate through soft formations, it is not uncommon to encounter natural cracks or even create unwanted hydraulic fractures, which both types of discontinuities compromise wellbore integrity. This situation is responsible for creating significant operational challenges, leading to the loss of circulation, which is the case where drilling fluids escape into the formation, resulting in non-productive time, elevated costs, stuck pipes, kicks, and, in extreme cases, blowouts that should be avoided at all costs due to health and safety issues. Globally, these issues contribute to annual industry losses estimated at EUR 2–4 billion, underscoring the economic imperative for improved mitigation [1,2]. Industry-wise, a blowout may be responsible for complete well abandonment [3,4,5]. The unwanted hydraulic fractures often initiate when the wellbore pressure exceeds the formation’s fracture initiation pressure (FIP) and fracture propagation pressure (FPP), particularly in environments with narrow drilling fluid windows [6,7]. This condition is reached when the drill-string penetrates soft formations, which are characterized by low strength and high deformations, and intensifies these issues due to their tendency for plastic deformation and diffusion effects, e.g., poroelastoplastic behavior [8]. Furthermore, in soft formations, the irreversible strains and pore pressure changes interact with each other to alter stress distributions around the wellbore, delineating the fully coupled character of the problem. Traditional models for wellbore strengthening, which typically rely on elastic or poroelastic assumptions, fail to adequately capture these phenomena because they overlook the plastic zones that develop ahead of fracture tips and associated diffusional effects, potentially leading to inaccurate predictions of fracture growth, closure stresses after plugging, and the effectiveness of lost circulation materials (LCM). To address these vulnerabilities, wellbore strengthening techniques have emerged as essential strategies to tackle this geomechanical problem [4,6,7].

Wellbore strengthening is a term used for a critical set of techniques in drilling engineering designed to enhance the pressure-bearing capacity of formations and mitigate lost circulation by artificially widening the drilling fluid window, thereby preventing unwanted hydraulic fractures from emanating and propagating, resulting in fluid losses during operations in soft rock formations. These methods typically involve the strategic addition of lost circulation materials (LCM) to drilling fluids with the aim of forming plugs or seals within induced unwanted fractures, thereby altering stress distributions near the wellbore to increase fracture propagation resistance. Wellbore strengthening is based on common mechanisms, which include (i) the stress cage approach, where LCM particles bridge fractures to redistribute hoop stresses, (ii) the fracture closure stress enhancement through plug-induced compression, (iii) tip isolation to limit fluid pressure transmission to the fracture tip [6,7,9,10]. In soft formations, plastic deformation and diffusional effects are coupled and dominate due to low rock strength and high fluid diffusivity. Traditional wellbore strengthening strategies are not always successful in resolving the LCM problem because elastic-based models underestimate the role of irreversible strains ahead and around the main body of the fracture and its tip, leading to inaccurate predictions of plug efficiency, closure stresses, and risks, like fracture reopening or LCM flow-back [6,7,11]. Such flow-back risks, exacerbated by closure stresses in plastic zones, highlight the need for models that integrate these effects [8,9]. For instance, a recent study by [6] utilized hydrodynamic models demonstrating that plug placement and pressure drop significantly influence closure stresses in poroelastic media, with diffusion effects creating back-stresses that amplify stability in permeable rocks. However, in soft formations where poroelastoplastic behavior is expected, the effective stresses can be reduced and change yielding behavior under high-pressure conditions [8]. Consequently, advancing wellbore strengthening modeling requires integrated hydro-mechanical simulations that account for poroelastoplastic effects. In such models, the influence of the yield zone can be quantified with the use of standard yield criteria, like the Mohr–Coulomb criterion, enabling the optimization of LCM design, drilling fluid formulations, and operational parameters that minimize economic losses [12].

Extensive research on wellbore strengthening has evolved from early elastic models that simplified fracture propagation and stress distributions in idealized formations [13,14] to more advanced poroelastic frameworks incorporating fluid diffusion and Biot’s theory for permeable media [15,16,17], yet these often overlook irreversible plastic deformation in soft rocks, leading to overestimated stability and inaccurate predictions of lost circulation in narrow drilling fluid windows [18,19]. For instance, studies on LCM bridging and stress cage mechanisms have demonstrated effective hoop stress redistribution in laboratory settings [4,18,20,21,22], while numerical simulations using finite element methods have analyzed closure stresses and plug efficiency in elastic and poroelastic contexts, revealing diffusion-induced back-stresses that enhance fracture arrest but neglect plasticity’s role in energy dissipation ahead of tips and at the open part of the fracture [6,23,24,25,26,27]. Experimental studies further underscore gaps, as they often prioritize material testing over coupled simulations in real poroelastoplastic conditions [20,21,22]. Plastic models have advanced the field by integrating elastoplastic relations and yield criteria, like Mohr–Coulomb, to capture shear failure and strain localization [28,29,30,31], yet they rarely couple with poromechanics, missing critical interactions between pore pressure evolution and plastic zones development in saturated formations. Works related to the creation of a fluid-driven fracture that can be applied to wellbore strengthening, addressing poroelastic coupling, are in the context of hydraulic fracturing for soft formations. This poroelastic coupling was found to significantly influence FIP, FPP, and LCM flowback risks, and it was studied both numerically and analytically in [32,33,34,35,36,37,38,39,40,41,42].

This work attempts to close this gap by extending the elastic and poroelastic approaches of [6,7] through a novel fully coupled hydro-mechanical model in Abaqus that integrates initially plastic and later poroelastoplastic effects. The model utilizes lubrication theory for simulating the unwanted hydraulic fracture, incorporates Darcy’s law for addressing the fluid movement from the fracture in the formation, and further analyzes the filtrate movement in the surrounding porous formation incorporated into the Reynolds diffusion equation. Mohr–Coulomb yield criterion is incorporated to simulate the plastic and poroelastoplastic stress changes in the porous formation, which is especially important at the fracture tip, thereby closing the gap between coupled plastic and poroelastoplastic modeling while upgrading the knowledge from fully coupled poroelastic simulators [6,7]. The unwanted fracture onset and propagation till plugged is performed by the cohesive zone approach, which is considered a robust fracture mechanics method for extending fluid-driven fractures. The novelty of this work lies in the understanding of the role of plastic and poroelastoplastic effects in wellbore strengthening in soft formations, like sandstones, while providing predictive insights into LCM stability and flowback prevention by using a yield factor for quantifying rock failure under plastic conditions [41] and then deriving the upgraded form for poroplastic rock conditions. Simulating coupled interactions between plasticity, pore pressure evolution, and mechanical deformation, we numerically obtain the fracture dimensions, pressure evolution, in situ stress changes, and plastic strains resulting from the creation of the unwanted hydraulic fracture and after plugging. A key objective of this work includes the evaluation of how poroelastoplastic coupling may affect fracture reopening after plugging, which is useful in assessing the implications for LCM plug stability to prevent flowback. This advancement offers actionable tools for reducing drilling risks in ultradeep environments, potentially saving costs in losses and promoting sustainable practices.

This study builds upon the author’s previous development of a poroelastoplastic fracture propagation model [8]. To ensure the numerical accuracy and computational efficiency of their model, the authors conducted a rigorous mesh dependency analysis. This study involved comparing the results from the ‘coarse’ mesh used throughout the paper (approximately 15 k elements) against a significantly more refined ‘fine’ mesh (approximately 350 k elements) for mesh dependency controls. The computational cost highlighted the importance of this check, with the coarse mesh simulation requiring a few hours while the fine mesh took a few days to complete for the same test case of a fracture propagated to 6 m. The comparison revealed that the results were not sensitive to the increased refinement by more than 20 times. However, the difference in the final fracture width profile was merely ~2%, and the variation in the fluid pressure profile was even more negligible at just 0.09%. Based on these findings, the authors concluded that the ‘coarse’ mesh provided excellent accuracy with a substantial gain in computational efficiency, thereby justifying its use for the comprehensive parametric studies presented in the work. While that foundational work focused on the mechanics of continuous fracture growth, the present work addresses the distinct physical problem of wellbore strengthening by simulating fracture arrest and analyzing the critical post-plugging dynamics. The introduction of a new yield factor to assess reopening risk represents a key novel contribution aimed at bridging theoretical gaps with field-applicable tools. Finally, it is intended to partially bridge theoretical gaps with field-applicable tools, which is also an open research question.

The paper is organized as follows: Section 2 presents the modeling methodology that was followed to simulate the problem. Section 3 details the results and new findings from the computational analysis performed. Finally, in Section 4, we outline the main contributions of this work. This structure allows for a comprehensive exploration of theoretical advancements and their practical implications for future research in geomechanics and drilling engineering.

2. Modeling Methodology

To investigate the role of plastic and poroelastoplastic effects in wellbore strengthening amid unwanted hydraulic fractures in soft formations, this study employs a fully coupled hydro-mechanical finite element model implemented in Abaqus. The model is capable of simulating the onset, propagation, and plugging of fractures, assuming plane-strain geometry for computational efficiency and focusing on saturated porous media with permeability typical of soft sandstones. This section presents the governing equations, numerical implementation, and the methods used to set up the computational model, providing a foundation for the simulation results of Section 3.

The “plug” in wellbore strengthening can be effectively simulated as a pressure difference between the wellbore fluid pressure (P_w) and the fracture fluid pressure (P_f), mimicking the isolation effect from the lost circulation materials (LCM) that restrict fluid transmission toward the fracture tip [6,7]. In impermeable formations, this pressure drop (d_p = P_w − P_f) constitutes the primary pressure differential that drives the fracture to extend because no fluid exchange occurs between the surrounding rock and the fracture, allowing for a simplified analysis focused on internal pressure creation and evolution associated with the fracture dynamics (dimensions and evolution). However, in permeable poroplastic formations, the presence of an additional formation pressure (P_p) complicates the scenario by generating two supplementary differentials. One between the wellbore pressure extending up to the plug and the formation pressure (d_p1 = P_w − P_p), which influences fluid leak-off near the wellbore, and another between the formation pressure and the residual fluid pressure within the fracture (d_p2 = P_p − P_f), which affects pressure equilibration downstream of the plug. In both impermeable and permeable cases, fluids can exchange between the plug of the fracture, but only the permeable model allows for fluid exchange between the fracture and the surrounding formation. This induces diffusion phenomena within a poroplastic framework, where pore pressure gradients lead to back-stresses that alter the effective stress field, which affects the overall stability.

The pressure difference as explained to simulate the plug, is applied at the fracture inlet for the models used in this work, which causes the reduction in stresses at the fracture tip by redistributing compressive forces, thereby ensuring that the mode-I stress intensity factor (K_I) remains less than or equal to the rock’s mode-I fracture toughness (K_Ic) ensuring that (K_I ≤ K_Ic). This mechanism provides effective resistance to further fracture propagation by reducing plastic strains, thereby mitigating lost circulation risks and improving wellbore integrity. Figure 1 shows the schematic representation of the hydromechanical plugs and the different pressure drops in the near area of the plug and the fracture tip.

2.1. Coupled Finite Element Model

To investigate the role of plastic and poroelastoplastic effects in wellbore strengthening, this study develops a fully coupled hydro-mechanical finite element model, extending prior frameworks by integrating irreversible deformation, fluid diffusion, and fracture arrest mechanisms in soft, impermeable, and permeable formations prone to unwanted hydraulic fractures. Building on analyses of cohesive process zones that demonstrate how elastic-softening behaviors create larger zones resistant to propagation and enhance shut-in stability, the model simulates the initiation, limited growth, and arrest of unwanted fractures after allowing the fracture to reach a certain length (5 m) by changing the boundary condition at the fracture inlet. The analysis is performed at plane-strain conditions (KGD fracture), focusing on plug-induced strengthening in saturated reservoir rocks like sandstones. Key components of the coupled finite element model include the lubrication theory for incompressible Newtonian fluid flow within the fracture, enabling accurate simulation of pressure drops across plugs and leak-off via Darcy’s law for porous media, also accounting for diffusion effects. The Biot’s effective stress principle is utilized for coupling the pore pressure with the mechanical deformation of the rock, enabling the capturing of back-stresses that amplify closure stresses and prevent reopening during plugging. Finally, the Mohr–Coulomb yield criterion, with an associative flow rule, is used to model the plastic zones ahead of the tip but also above the fracture face. Dilation influences fracture resistance, and with the help of a yield coefficient, we quantify the failure risk after plugging the unwanted hydraulic fracture. Fracture arrest is governed by the cohesive zone model, which relates traction to separation through parameters like fracture energy. If this fracture energy is reached during any process, the fracture propagates in a predefined path. On the other hand, if it drops, then the fracture effectively stops propagation. During plugging, the fracture energy criterion ceases to satisfy the critical energy, and thus the fracture effectively stops propagating. Initial conditions incorporate anisotropic in situ stress, formation pore pressures, and realistic perforation lengths (e.g., 0.1 m). The underlying numerical framework for the elastic and poroelastic components of this model has been rigorously validated in our previous work. Specifically, the model’s predictions for fracture aperture and pressure profiles under poroelastic conditions were compared against established analytical solutions for KGD-type fractures in toughness-dominated regimes, demonstrating excellent agreement [6,7]. The focus of the present study is to build upon this validated foundation by extending the analysis into the plastic and poroelastoplastic regimes, for which analytical solutions are generally not available, thereby underscoring the necessity of the computational approach detailed herein. While direct validation of the fully coupled poroelastoplastic model is challenging due to the lack of established benchmarks, the model utilizes standard constitutive laws built upon the validated poroelastic framework. The study’s focus is therefore on a robust comparative analysis of the governing physical mechanisms rather than a quantitative prediction for a specific case. The next subsections present the governing equations for fluid flow, rock deformation, and fracture onset and arrest, followed by the numerical implementation and the yield factor to analyze the risk of fracture reopening, laying the groundwork for the results.

2.2. Fluid Flow in the Unwanted Fracture

Fluid flow within the fracture is modeled using the Reynolds lubrication equation, which is derived from lubrication theory for a laminar, incompressible Newtonian fluid. This approach couples the rate of fracture opening with both the tangential flow along the fracture and the transverse flow (leak-off) into the surrounding formation. The one-dimensional continuity equation, which imposes mass conservation, is given by [27]:

$${\displaystyle \frac{\partial w}{\partial t}}+{\displaystyle \frac{\partial q_L}{\partial x}}+q_T=0$$

(1)

where q_L is the local volumetric flow rate along the fracture axis x, q_T is the fluid loss rate into the rock formation (leak-off), and w is the fracture opening.

For tangential flow between two parallel plates, q_L is related to the pressure gradient by the well-known “cubic law”. This equation determines the local flow rate while incorporating leak-off from the fracture surfaces. For a Newtonian fluid with viscosity $\mu$ yields the Darcy equation relating the pressure gradient to the fracture width [27]:

$$q_L=-{\displaystyle \frac{k}{\mu }}\left(\nabla p-f\right)$$

(2)

where $k$ is the intrinsic permeability and is a direct function of the opening: k = w³/12. Substituting this definition for k into Equation ($2$) recovers the cubic law as:

$$q_L=-{\displaystyle \frac{w^3}{12\mu }}{\displaystyle \frac{\partial p}{\partial x}}$$

(3)

The use of this model is justified as the flow regime in such fractures is characterized by very low Reynolds numbers, making the assumption of laminar flow highly accurate. The remaining terms include the nabla, $\nabla$ which is the gradient operator and $f$ are fluid volume forces.

The transverse leak-off embodied in q_T is modeled as a pressure-driven flow from the fracture into the formation. By combining Equations ($1$) and ($3$), we obtain the full Reynolds equation that governs the pressure p within the fracture. Equation (4) accounts for diffusion effects, creating back-stresses that influence fracture stability, particularly in poroelastic media.

$${\displaystyle \frac{\partial w}{\partial t}}-{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{w^3}{12\mu }}{\displaystyle \frac{\partial p}{\partial x}}\right)+q_T=0$$

(4)

This system is discretized within the 6-node cohesive elements (COH2D4P). These elements calculate the tangential pressure gradient from nodal pressure values to solve the cubic flow term, and they model leak-off based on the pressure differential between the fracture’s internal nodes and the formation’s nodes at the element boundary. The use of this model is justified as the flow regime is characterized by very low Reynolds numbers (Re << 1). This places the flow deep within the laminar regime, making the lubrication theory a highly accurate and standard approach in the field.

Furthermore, the transverse leak-off term, q_T, is calculated directly at the element level based on Darcy’s law. It is driven by the pressure differential between the fluid inside the fracture (p_f) and the pore pressure at the fracture wall (p_p). This is implemented within the 6-node cohesive elements, which use their central nodes to track p_f and their outer nodes (shared with the continuum elements) to track p_p. The leak-off fluxes from the upper (q_T^u) and lower (q_T^d) faces are thus given by:

$$q_T^u=C_T^u(p_f-p_p^u),\hspace{1em}{q}_T^d=C_T^d(p_f-p_p^d),$$

(5)

where C_T is the transverse permeability coefficient of the fracture wall. The total leak-off is q_T = q_T^u + q_T^d. This direct, physics-based coupling is more accurate than simplified analytical models and is essential for capturing the transient poromechanical interactions during propagation and arrest. Numerically, the coefficient C_T is implemented in Abaqus using the keyword *Leak-off for cohesive elements. The value for this coefficient is listed in

Table 1. Summary of input parameters and material properties for the coupled hydro-mechanical model [8,27,42,43].

Parameter Group	Variable Name	Value	Units
Mechanical Properties	Young’s Modulus, E	16.2	GPa
	Poisson’s Ratio, ν	0.3	[−]
	Material Cohesion, c	1515	kPa
	Material Friction Angle, φ	28	(°)
	Material Dilation Angle, ψ	28	(°)
Cohesive Zone Model	Constitutive Thickness	1	m
	Maximum Traction, σt	0.5	MPa
	Cohesive Stiffness, Kn	3.24 × 105	MPa
	Cohesive Energy, GIC	0.112 or 112	kPa.m J/m2
	Leak-off Coefficients, qTu/qTd	2.42 × 10−10	m/(Pa·s)
Hydraulic Parameters	Fluid Viscosity, μ	0.0001	kPa.s
	Injection Rate, q	5.00 × 10−4	m3/s·m
	Domain Permeability, k	2.42 × 10−10	m/s
Initial Petrophysical State	In Situ Stress, σ1	14	MPa
	In Situ Stress, σ2	9	MPa
	In Situ Stress, σ3	3.7	MPa
	Pore Pressure, P	1.85	MPa
	Void Ratio, e	0.333	[−]

as “cake permeability coefficients” and represents the hydraulic conductivity of the fracture face. The standard units of (m/s) are normalized with the specific weight of the fluid (ρg), and, assuming a unit effective thickness of the fracture wall, it becomes m/(Pa·s).

2.3. Rock Deformation Around the Unwanted Fracture

Rock deformation is modeled as plastic and poroelastoplastic for weak formations, using Biot’s theory to couple the skeleton response in dilation with pore fluid diffusion. The total stress ${\sigma }_{ij}$ relates to effective stress ${\sigma }_{ij}^{\prime }$ through the effective stress principle:

$${\sigma }_{ij}={{\sigma }^{\prime }}_{ij}-\alpha p{\delta }_{ij}$$

(6)

where $\alpha$ is Biot’s coefficient (typically equal with unity for incompressible rocks), $p$ is pore pressure, and ${\delta }_{ij}$ is the Kronecker delta [27]. Effective stresses govern deformation and failure. The inelastic behavior follows the Mohr–Coulomb yield criterion with an associative flow rule, suitable for frictional materials like rocks [8]:

$$f=g={\displaystyle \frac{1}{2}}\left({\sigma }_1-{\sigma }_3\right)+{\displaystyle \frac{1}{2}}\left({\sigma }_1+{\sigma }_3\right)\sin \varphi -c\cos \varphi =0$$

(7)

where $c$ is cohesion, $\varphi$ is the friction angle, and ${\sigma }_1,{\sigma }_3$ are principal stresses (compression positive). The plastic potential $g$ equals $f$ for associativity, with dilation angle $\psi =\varphi$. The rock’s inelastic behavior is modeled using the Mohr–Coulomb yield criterion with an associated flow rule. The Mohr–Coulomb criterion was selected as it is a well-established and robust framework for describing the failure of frictional materials like sandstone, effectively capturing the fundamental dependency of shear strength on normal stress [8,28,29]. Within this framework, we employ an associated flow rule (ψ = φ). We acknowledge that this is a common idealization, as a non-associated rule (ψ < φ) is generally more physically representative of dense sandstones. However, this choice was made deliberately to align with the study’s primary objective: to conduct a fundamental, comparative analysis. The associated flow rule investigates the maximum potential influence of shear-induced dilation on the poroelastoplastic coupling. This approach amplifies the dilative mechanism and the resulting back-stresses, making the distinction between the plastic and poroelastoplastic models as clear as possible. Therefore, this constitutive model provides a clear and well-understood basis for analyzing the fundamental mechanisms of wellbore strengthening, rather than aiming for a precise calibration to a specific rock.

2.4. Unwanted Fracture Onset and Propagation

Fracture propagation uses the cohesive zone model, avoiding stress singularities by assuming a process zone ahead of the tip where traction decreases with separation until complete failure. This approach is very convenient for finite element analysis. The traction-separation law for mode-I plane strain is defined by the fracture energy $G_{IC}$ and tensile strength ${\sigma }_t$ or critical separation ${\delta }_{IC}$. For rigid-softening, the traction-separation expressions are [27,42]:

$$\sigma ={\sigma }_t\left(1-{\displaystyle \frac{\delta }{{\delta }_{IC}}}\right),\hspace{1em}{\delta }_{IC}={\displaystyle \frac{2K_{IC}^2\left(1-{\nu }^2\right)}{E{\sigma }_t}},\hspace{1em}{K}_{IC}^2={\displaystyle \frac{G_{IC}E}{1-{\nu }^2}}$$

(8)

where K_ic is the fracture toughness for mode-I fractures, E is the Young’s modulus, and ν is the Poisson ratio. Elastic-softening includes an initial linear branch with stiffness $k_n$ [27,42]:

$$\sigma ={\sigma }_t\left({\displaystyle \frac{\delta }{{\delta }_{el}}}\right),\hspace{1em}{\delta }_{el}={\displaystyle \frac{{\sigma }_t}{k_n}}$$

(9)

followed by softening. Parametric studies [27] have shown that elastic-softening yields larger process zones and higher pressures than rigid-softening, especially in poroelastic media. The cohesive zone model simulates the physical fracture process zone by defining a traction-separation law that governs crack opening. In this study, the parameters were deliberately chosen (see Table 1) to ensure the results are robust and reflect the bulk material behavior rather than artifacts of the cohesive law itself. Specifically, a high initial cohesive stiffness (K) was used to approximate a rigid-softening response. This approach minimizes the influence of the cohesive element’s initial elasticity and the specific value of the tensile strength (T_max), making the fracture propagation primarily dependent on the fracture energy (G_IC), a fundamental and physically meaningful material property related to the energy required to produce new surface. The detailed influence of cohesive law parameters has been investigated previously [27], and the current parameterization was selected based on those findings to best isolate the plastic and poroelastoplastic effects under investigation.

2.5. Coupled Numerical System

The coupled equations are discretized using finite elements and solved with Abaqus, using a 4-node plane-strain element (CPE4P) for the rock domain and 6-node cohesive elements (COH2D4P) for fracture flow and propagation. Time integration uses backward differencing ($\theta =1$) for stability, solved via Newton’s method. The system matrix is [8,27,42]:

$${\left[\begin{array}{cc}\theta K_e& \theta Q\\ Q^T& S+{\Delta }_t\theta H\end{array}\right]}_{n+\theta }{\left\{\begin{array}{c}u\\ p\end{array}\right\}}_{n+1}={\left[\begin{array}{cc}\left(\theta -1\right)K_e& \left(\theta -1\right)Q\\ Q^T& S+{\Delta }_t\left(\theta -1\right)H\end{array}\right]}_{n+\theta }{\left\{\begin{array}{c}u\\ p\end{array}\right\}}_{n+1}+{\left\{\begin{array}{c}{f}_u\\ {\Delta }_t{f}_p\end{array}\right\}}_{n+\theta }$$

(10)

This equation represents the discretized form of the coupled hydro-mechanical system, where the left side handles the current time step stiffness and flow, and the right side accounts for previous step contributions and external loads. Namely, $K_e$ is the elastic stiffness matrix, which relates nodal forces to displacements in the rock skeleton, capturing how the material resists deformation, elastically computed as ${\int }_{\mathsf{\Omega }}{B}^T{D}_{e}Bd\mathsf{\Omega }$, where $B$ is the strain operator linking nodal displacements to strains, and $D_e$ is the elastic tangent matrix. The coupling matrix $Q$ = ${\int }_{\mathsf{\Omega }}{B}^T\alpha m{N}_{p}d\mathsf{\Omega }$ links mechanical deformation to pore pressure changes, physically representing how volumetric strains induce fluid flow or pressure buildup via Biot’s coefficient $\alpha$. The permeability matrix $H$ = ${\int }_{\mathsf{\Omega }}(\nabla N_p{)}^T(k/\mu )\nabla N_{p}d\mathsf{\Omega }$ governs fluid diffusion through the porous medium, describing Darcy’s law effects, where permeability $k$ and viscosity $\mu$ control leak-off rates and back-stresses that aid fracture arrest. The compressibility matrix $S$ = ${\int }_{\mathsf{\Omega }}{N}_p^T(1/Q^{\prime })N_{p}d\mathsf{\Omega }$ accounts for fluid and solid compressibility, with (1/Q’) incorporating porosity, bulk moduli, and Biot parameters to model volume changes under pressure (e.g., dilation in plastic zones). Vectors $f_u$ and $f_p$ represent external forces from boundary tractions, and fluid fluxes, ensuring mass and momentum balance. Shape functions $N_u$ and $N_p$ interpolate displacements $u$ and pressures $p$ across elements, while the time operator $\theta =1$ ensures unconditional stability for transient diffusion in poroelastoplastic media. Plastic yielding integrated via the elastoplastic tangent matrix $D_{ep}$ to handle non-linearities during arrest. This setup enables robust simulation of plug-induced strengthening, where matrices like Q and H capture diffusion-plasticity interactions critical for yield zone scaling and fracture tip resistance.

In the hydro-mechanical model, coupling refers to the bidirectional interaction between mechanical deformation (rock skeleton) and fluid flow (pore pressure diffusion), ensuring that changes in one field affect the other iteratively for a realistic simulation of plastic zones, back-stresses, and plugging behavior. Physically, this means volumetric strains from deformation (e.g., plastic dilation ahead of the tip) induce pore pressure gradients that drive fluid flow, while diffusion alters effective stresses, influencing yield and arrest (via Biot’s principle, with $\alpha$ linking fields). Numerically, coupling is achieved through the system matrix in Abaqus, where the off-diagonal term $Q$ connects displacement $u$ and pressure $p$ equations, allowing simultaneous solution via Newton’s method for convergence. Using the staggered (iterative) scheme as shown in Figure 2, the process alternates between solving the equilibrium equation for deformation plasticity by updating stresses and strains, while the flow equation updates pressures for diffusion, iterating until residuals converge (within a tolerance like 10⁻⁶) per time step. This “2-Deck-coupling” handles non-linearities from plasticity (via $D_{ep}$) and diffusion (via H and S matrices), ensuring accurate prediction of yield coefficients and fracture resistance in soft formations.

The solution of the discretized system was controlled by specific numerical parameters to ensure accuracy and stability. The non-linear equations were solved using the Newton–Raphson method. Convergence at each time increment was achieved when the largest residuals of both the mechanical forces and the fluid fluxes fell below a strict tolerance of 10⁻⁶. An automatic time-stepping scheme was employed, with an initial time step of 0.1 s and a minimum allowed step of 10⁻⁵ s. This variable stepping is essential for efficiently handling the severe non-linearities associated with plastic yielding and the transient diffusion fronts. The solution typically converged within 5–10 iterations per increment. If convergence was not reached within a maximum of 15 iterations, the time step was automatically reduced, and the increment was re-attempted. These settings ensure a robust and reproducible simulation process.

2.6. Incorporating the Poroelastoplastic Behavior in the Coupled System

The poroelastoplastic behavior is included after modifying the elastic tangent matrix $D_e$ in $K_e$ is replaced by the elastoplastic tangent matrix $D_{ep}$, which modifies the constitutive relation to account for irreversible plastic strains and non-proportional loading in weak formations. Physically, $D_{ep}$ represents the material’s stiffness during yielding, incorporating plastic flow to simulate dilation, shear failure, and energy dissipation in the yielded zones that resist fracture reopening and enhance wellbore strengthening. It is derived from the Mohr–Coulomb criterion as shown in Equation (8) [8]. This theoretical form is implemented numerically within the native Mohr–Coulomb model provided by Abaqus. This built-in feature utilizes a consistent tangent operator derived from the software’s internal stress integration scheme, which is essential for ensuring the quadratic convergence of the Newton solver during the coupled iterations.

$$D_{ep}=D_e-{\displaystyle \frac{D_e\frac{\partial Q}{\partial {\sigma }^{\prime }}{\left(\frac{\partial f}{\partial {\sigma }^{\prime }}\right)}^T{D}_e}{{\left(\frac{\partial f}{\partial e^p}\right)}^T\frac{\partial Q}{\partial {\sigma }^{\prime }}+{\left(\frac{\partial f}{\partial {\sigma }^{\prime }}\right)}^T{D}_e\frac{\partial Q}{\partial {\sigma }^{\prime }}}}$$

(11)

where $D_e$ is the elastic tangent matrix that defines linear stress–strain relations before yielding.

• ${\displaystyle \frac{\partial Q}{\partial {\sigma }^{\prime }}}$ is the gradient of the plastic potential function $Q$ with respect to effective stress ${\sigma }^{\prime }$, which controls the direction of plastic flow and dilation. In other words, the volumetric expansion in porous rocks that influences pore pressure and back-stresses.
• ${\displaystyle \frac{\partial f}{\partial {\sigma }^{\prime }}}$ is the gradient of the yield function $f$ (Mohr–Coulomb in this case), indicating how close the stress state is to failure and triggering plasticity when $f=0$.
• ${\displaystyle \frac{\partial f}{\partial e^p}}$ is the hardening modulus, accounting for strain-dependent changes in yield strength.

The denominator ensures consistency with the yield surface, while the subtracted term reduces stiffness post-yielding, capturing irreversible deformation like plastic zones ahead of the tip that dissipate energy and promote arrest. This matrix is computed iteratively within the Newton solver, updating at each time step to handle non-linearities from the coupled plasticity and diffusion, enabling accurate prediction of onset and shut-in behavior in soft formations. The internal force vector, resisting elastoplastic deformation, integrates $D_{ep}$ as [8]:

$$P\left(u\right)={\displaystyle \underset{\Omega }{\int }{B}^{T}d{\sigma }^{\prime }d\Omega }$$

(12)

This formulation allows for the model to transition from elastic/poroelastic to plastic/poroplastic regimes, essential for simulating how poroelastoplastic effects strengthen or weaken wellbores by containing unwanted fractures through enhanced tip resistance and plug stability.

2.7. The Yield Factor as a Fracture Reopening Risk

In the context of investigating plastic and poroelastoplastic effects for wellbore strengthening, a key metric introduced in this study is the yield factor f, a dimensionless scalar derived from the Mohr–Coulomb criterion to quantify the proximity to shear failure and plastic yielding under non-porous conditions [41]. The yield factor and associated expressions for non-porous conditions yield:

$$f_{plastic}={\displaystyle \frac{\tau }{q-P}},\hspace{1em}\tau ={\displaystyle \frac{{{\sigma }^{\prime }}_1-{{\sigma }^{\prime }}_3}{2}},\hspace{1em}q=c\cot \phi ,\hspace{1em}P={\displaystyle \frac{{{\sigma }^{\prime }}_1+{{\sigma }^{\prime }}_3}{2}}$$

(13)

where ${{\sigma }^{\prime }}_1$ and ${{\sigma }^{\prime }}_3$ are the maximum and minimum effective principal stresses.

For the purposes of this work, we modified the non-porous yield factor f_plastic for saturated porous formations, thereby assessing the risks of unwanted fracture propagation and the efficacy of arrest mechanisms, like that of LCM plugs under porous and fully saturated conditions. The yield factor f_{poroelastoplastic} adapts the standard Mohr–Coulomb equation for effective stresses, incorporating pore pressure p via Biot’s principle (Equation (6)). The modified expression for the poroelastoplastic yield factor reads:

$$f_{poroelastoplastic}={\displaystyle \frac{\tau }{q-P-p}},\hspace{1em}\tau ={\displaystyle \frac{({\sigma }_1-p)-({\sigma }_3-p)}{2}},\hspace{1em}q=c\cot \phi ,\hspace{1em}P={\displaystyle \frac{({\sigma }_1-p)+({\sigma }_3-p)}{2}}$$

(14)

when, f_{poroelastoplastic} < 1 indicates safety, f_{poroelastoplastic} = 1 marks poroplastic yield onset, and f_{poroelastoplastic} > 1 signals active poroelastoplasticity. In this form, the yield factor f_{poroelastoplastic} highlights poroplastic coupling, with high p values (e.g., slow diffusion as a result of back-stress) shifts the denominator, decreases f_{poroelastoplastic,} and reduces yielding which dissipates energy through dilation (ψ = ϕ) to form larger process zones. On the other hand, when p values drop (e.g., diffusion effects at late times), f_{poroelastoplastic} increases yielding and raises the risk of fracture reopening after fracture plugging.

Physically, the yield factor encapsulates how anisotropic stresses (σ₁ > σ₃) drive shear (σ₁–σ₃) against frictional resistance, which is the (sinϕ) term and cohesion, with pore pressure effectively “weakening” the material by reducing normal stresses. This metric novelty lies in quantifying failure under coupled conditions, offering predictive insights for optimizing LCM designs, bridging theoretical gaps with field applications.

3. Computational Modeling

This section presents the outcomes from the numerical simulations using the fully coupled finite element model described in Section 2, focusing on differences between plastic and poroelastoplastic effects and how those influence the unwanted fracture arrest and wellbore strengthening in soft formations that are susceptible to plastic yielding. The results are based on (i) fracture profiles, (ii) propagation pressures, (iii) yield zones, (iv) changes in the in situ stresses, (v) quantified risk through application of the yield factor, during propagation and after plugging for both early times and late times. This evaluation is critical for evaluating the performance of the plug.

3.1. Geometry and Boundary Conditions

The numerical simulations in this study utilize a 2D plane-strain geometry, commonly known as the Khristianovic–Geertsma–de Klerk (KGD) model. This is a standard idealization appropriate for simulating a long hydraulic fracture that is vertically confined between two high-stress or high-stiffness layers, which is a common scenario in many sedimentary basins. Under such confinement, the fracture length greatly exceeds its height, and the plane-strain assumption accurately captures the dominant physics away from the upper and lower boundaries. The fracture path is predefined along a straight line, which is consistent with the principle that fractures propagate perpendicular to the minimum principal stress in a homogeneous material under anisotropic stress. While an axisymmetric model would be suitable for an uncontained, penny-shaped fracture, the KGD model provides a robust and computationally efficient framework for the primary objective of this work: to compare the fundamental effects of plastic and poroelastoplastic material behavior on wellbore strengthening.

Figure 3 illustrates a schematic finite element model domain for simulating a plane-strain (KGD geometry) hydraulic fracture in a weak consolidated formation, accounting for plastic and poroelastoplastic effects. Such domains are commonly used in studies of wellbore strengthening and fracture arrest.

The square domain measures 30 m × 30 m, representing a quarter symmetric section of rock formation with the wellbore (onset point, marked by a red dot) at the bottom-left corner, (0,0), which is also the origin of the model. The 30 m × 30 m square domain was selected to be sufficiently large to minimize boundary effects. With a final fracture length of approximately 6 m, the zone of mechanical and hydraulic perturbation is well-contained within the model domain, ensuring the results are independent of the far-field boundaries. The fracture onsets and propagates horizontally along an 11 m predefined path from the wellbore (onset point), modeled with cohesive elements embedded in a mesh of plane strain elements, which are 4-node isoparametric elements for rock deformation. Boundary conditions include X-symmetry on the left (fixed x-displacements, $dx=0$) and Y-symmetry at the bottom (fixed y-displacements, $dy=0$) to exploit the quarter-symmetry, reducing computational cost while enforcing no shear along these lines and mimicking a full domain. Applied stresses are ${\sigma }_3=3.7$ MPa (minimum horizontal stress) on the top boundary (closure stress perpendicular to fracture growth) and ${\sigma }_1=14$ MPa (maximum horizontal stress) on the right boundary, creating an anisotropic field that drives vertical confinement and in-plane deformation for fracture arrest analysis. The setup initializes equilibrium with applying the in situ stresses and pore pressures, allowing for fluid injection at the wellbore to simulate unwanted fracture onset and growth up to 5 m and then the strengthening plugs are included by changing the boundary condition at the fracture inlet, with diffusion and plasticity captured in the surrounding mesh directly supporting investigations of plastic but also poroelastoplastic effects on yield zones.

3.2. Simulation Input Data

The input data used in the simulations are shown in Table 1 and provide the essential parameters for simulating the unwanted hydraulic fracture arrest and wellbore strengthening in the plastic and the poroelastoplastic medium using the fully coupled hydro-mechanical model, tailored to soft formations with plastic deformation and diffusion effects. The data are from [8,27,42,43].

Elastic rock properties include a Young’s modulus $E=16.2$ GPa, representing the material stiffness under drained conditions before yielding, and Poisson’s ratio $\nu =0.3$, which governs lateral strain response and volumetric changes in poroelastic coupling. Inelastic properties incorporate Mohr–Coulomb parameters like cohesion $c=1.515$ MPa for shear strength, friction angle $\varphi ={28}^{\circ }$ for internal resistance to sliding, and dilation angle $\psi ={28}^{\circ }$ (associative with $\varphi$) to model volumetric expansion during plastic flow, critical for capturing energy dissipation and back-stresses. Cohesive zone properties define the fracture propagation criterion, with constitutive and anti-plane thicknesses of 1 m (numerical scaling factors for element integration), maximum traction ${\sigma }_t=0.5$ MPa as tensile strength (soft rock), stiffness $k_n$ = 324 GPa in order to minimize the effect of the cohesive process zone from elastic- to rigid-softening behavior. The fracture energy was $G_{IC}=0.112$ kPa·m (equivalent to 112 J/m² or $K_{IC}=1$ MPa·m^0.5) for work required to create new surfaces, and cake permeability coefficients $q_t,q_b=2.421\times {10}^{-10}$ m/(Pa·s) for leak-off across fracture walls. Fluid parameters feature a fluid viscosity $\mu =0.0001$ kPa·s equivalent to 1 Pa.s for Newtonian resistance and an injection rate $q=500\times {10}^{-6}$ m³/s·m to drive propagation at controlled velocities, enabling studies of plugging dynamics. Domain permeability $k=2.421\times {10}^{-10}$ m/s facilitates diffusion, while the effective in situ stress field (${\sigma }_1=14$ MPa maximum horizontal, ${\sigma }_2=9$ MPa intermediate, ${\sigma }_3=3.7$ MPa minimum horizontal) creates stress anisotropy to probe stress deviators affecting yield zones. Initial conditions include void ratio $e=0.333$ equivalent for 20% porosity, an initial perforation gap of 0.1 m as the starting fracture length, and pore pressures $p=1.85$ MPa to explore how pressure impacts plastic scaling and plug efficiency. It should be noted that in the cases of poroelastoplastic modeling, a total stress analysis is performed, and the pore pressure is added to the effective stresses and applied accordingly in the model.

3.3. Simulation Results

Figure 4 presents the fracture halfwidth profiles (w/2, in meters) as a function of distance from the wellbore (d, in meters), illustrating propagation and post-plugging behavior in all theories examined: elastic, plastic, poroelastic, and poroplastic models.

Figure 4a compares the elastic fracture (red dashed line) with the plastic fracture (blue continuous line) during propagation, showing the evolving widths as the fracture extends to 5 m before plugging. Figure 4b compares the poroelastic (red dashed line) and poroplastic (blue continuous line) fractures under similar propagation conditions, highlighting the diffusion and back-stress effects. Figure 4c depicts plastic fracture after plugging, showing the closure procedure of the fracture after it was left to reach 5 m. At times, t = 0 s designates the initial plugged state and is left to balance until t = 4000 s (colored lines), capturing time-dependent narrowing but also showing that the fracture continues to propagate for at least ~1 m after plugging. Finally, Figure 4d shows poroplastic fracture behavior post-plugging from t = 3 s to t = 200 s (colored lines), emphasizing on the coupled poroelastoplastic responses of the fracture.

In Figure 4a, the plastic fracture exhibits wider profiles than the elastic one during propagation, with differences most pronounced near the wellbore (up to ~20%) and cusping at the tip, indicating energy dissipation during propagation from the plastic fracture. Figure 4b reveals even greater widening in poroplastic cases compared to poroelastic (e.g., ~30%), with poroplastic profiles showing more pronounced tip opening due to combined plastic yielding and diffusion, causing back-stresses. For post-plugging shown in Figure 4c, the plastic fracture narrows progressively over time, achieving a balanced fracture by t = 4000 s, with a rapid initial reduction (e.g., 50% width loss by t = 1500 s), slowing as residual stresses stabilize. This is explained by the fact that in the elastic and plastic cases, the fracture is practically impermeable, and upon plugging the unwanted fracture, it continues to propagate until all the energy of the system is dissipated and the mass of fluid inside the fracture is trapped between the impermeable faces and the tip of the fracture. If the closure is left to dissipate more, the result will not be complete closure but rather the same as with the t = 4000 s profile. In Figure 4d, poroplastic closure is much faster and less uniform. The way that the fracture closes is also different. It is evident that the fracture closes from the tip towards the wellbore. Initially, the fracture also extends for about ~1 m until closure stresses begin to be significant. This is explained by the fact that the fracture walls are permeable and as soon as the plug is inserted at the fracture inlet, the pressure in the fracture does not dissipate, immediately forcing the fracture to propagate a little longer and then starting to close because back stresses become significant causing gradual tip narrowing, resulting in ~15% more closure than the plastic case for the same time comparison. If the pressure in the fracture is left to dissipate for longer periods of time, the fracture would close completely because the permeable walls of the fracture will cause the fluid to bleed in the formation, enhancing back-stresses, and without a continuous supply of fluid, the fracture would close completely.

The wider profiles in plastic and poroplastic propagation of Figure 4a,b arise from plastic yielding and dilation ahead of the tip, dissipating energy and increasing fracture resistance, as plastic zones shield the tip and enhance effective toughness from the Mohr–Coulomb criterion, aligning with larger process zones. Poroplastic widening is amplified by diffusion-induced back-stresses (via Darcy’s law and Biot coupling), where pore pressure gradients from leak-off create volumetric expansion, reducing effective stresses and promoting larger yield zones for arrest in high-permeability soft formations. Post-plugging closure, as shown in Figure 4c,d, is driven by stress redistribution around the fracture, with faster poroplastic narrowing due to bleeding off the fluid that created the fracture with minimizing diffusion delays, allowing for the rapid dissipation of residual pressures and effective plugging. Slower closure is observed for the plastic fracture because fluid flow is not permitted from the impermeable walls; thus, the pressure in the fracture causes the fracture to extend a little further and then traps the fluid inside the fracture. Both cases show long-term stability against reopening in saturated media if the plug is effectively applied at the fracture inlet. This result is important for LCM plug efficiency and wellbore strengthening.

Figure 5 displays the simulation outputs for pressure profiles (in MPa) and equivalent plastic strain (PEEQ), which is a dimensionless quantity as a function of distance from the wellbore. Figure 5c,d compare the plastic and poroplastic models during (i) fracture propagation, (ii) early fall-off (post-plugging transient), (iii) late fall-off (near-equilibrium plugging). Figure 5a shows representative pressure profiles for the plastic model during propagation (blue line), showing a slightly non-uniform pressure profile along the fracture. Figure 5a also shows the early fall-off (red line), which, after plugging, presents a fully flat pressure profile inside the fracture and shows a rapid drop near the tip, as expected. Finally, late fall-off (green line) drops even further while extending the fracture to about 6.1 m, approaching slowly the in-situ stress applied. Figure 5b illustrates the representative poroplastic pressure profiles with blue color at the state of propagation, which presents a non-linear pressure distribution that is gradually declining along the fracture walls until it reaches the tip and then continues asymptotically to the pore pressure field considered as initial condition (1.85 MPa).

The early fall-off pressure distribution curve is also shown in red color, showing slower dissipation at the area of the tip; however, higher pressure magnitudes verified our assumption that after plugging, the pressure in the fracture increases until bleeding-off in the formation emanates. Finally, late fall-off is shown with green color at a stabilized level, but with higher pressure magnitude due to diffusion rate effects, back-stresses were created. Figure 5c depicts the plastic strain distribution along the fracture face for the plastic model, during propagation, with blue color. The early fall-off is shown with red color and the late fall-off with green color. The plastic strains for all cases increase to a certain level (peak near 0.5 m), most probably the fracture is beginning to grow, and then as the tip of the fracture extends away from the plug location, plastic strains reduce and drop to zero ahead of the tip location. It is also worth noting that after plugging, the fracture continues to propagate until it reaches balance, and this extension has its signature on the plastic strain profiles for both early and late times. Finally, Figure 5d presents the poroplastic strains for the same cases. Propagation is shown in blue, early fall-off in red, and late fall-off in green. It is worth mentioning that the late fall-off increases at least four times compared to the plastic strains during propagation and during early fall-off.

In Figure 5a, the plastic model pressure profile during propagation remains elevated and fairly non-uniform (from 4.5 MPa to 4.1 MPa along d = 0–6 m), dropping in early fall-off and further in late fall-off, indicating rapid stabilization. Figure 5b reveals that the poroplastic pressures have lower pressure profiles and decline more gradually near the tip. The plastic strains in Figure 5c peak sharply during propagation and in a location closer to the plug, and then reduce by ~50% in early fall-off and late fall-off, near the tip, reflecting active plastic yielding when the tip extends. On the other hand, the poroplastic strains in Figure 5d also present the effect of high concentration near the location of the plug, but more uniform during fracture propagation, then falling to zero poroplastic strains ahead of the tip.

The uniform high pressures in plastic propagation of Figure 5a are the result of plastic yielding dissipating energy near the tip, requiring sustained injection to overcome resistance, while the rapid fall-off reflects sudden loading without diffusion delays, enabling quick arrest because the closure stresses take over fast. The poroplastic profiles shown in Figure 5b show gradual declines due to coupled diffusion (Darcy’s law), where leak-off creates back-stresses sustaining pressures, with negative tip values indicating suction from dilation (Biot coupling), and together with the closure stresses, increase the poroplastic strains, creating a high risk of reopening the fracture either from the plug location or from anywhere along the fracture face. The sharp strain peaks in plastic models, as seen in Figure 5c, arise from localized Mohr–Coulomb yielding, relaxing quickly post-plugging as effective stresses equilibrate without pore fluid effects, promoting efficient plugging and without additional plastic yielding. The broader and persistent strains in the poroplastic cases (Figure 5d) stem from plastic–porous interactions, where dilation increases porosity and influx at the tip location, leading to back-stresses that maintain yielding longer, and as explained earlier, together with the closure stresses cause the plastic strains to increase after plugging, increasing the risk for reopening the fracture. These observations from modeling underscore that poroelastoplasticity amplifies diffusion–plasticity coupling.

Figure 6 illustrates the pore pressure field around a poroplastic fracture post-plugging in a saturated formation, with Figure 6a showing the early fall-off (transient phase shortly after plugging) and Figure 6b the late fall-off (near-equilibrium state).

In Figure 6a,b, the fracture is depicted as a thin horizontal line extending from the wellbore (left origin), embedded in a 2D domain with contours representing pressure gradients. Early fall-off captures high-pressure buildup (up to ~5 MPa) along the fracture and near-tip diffusion, a low-pressure sink with negative values indicating suction of formation fluids from the fracture tip. Figure 6b shows that the dissipated pressures, with reduced gradients, reach broader diffusion zones and slowly diminish with time, reflecting stabilization. In Figure 6a, the early fall-off exhibits intense pressure localization along the fracture, with sharp gradients diffusing outward and a pronounced negative zone at the tip, suggesting rapid fluid redistribution. Figure 6b reveals significant dissipation by late fall-off, with pressures uniformizing to ~1.5 MPa across the fracture, while near the tip, the pressures drop to (~0 MPa), indicating equilibrium. The localized high pressures in early fall-off stem from the sudden stop of injection effects and plastic dilation from the trapped fluid, creating steep gradients that drive diffusion via Darcy’s law, with tip negativity arising from suction-induced volumetric expansion in the yield zones, pulling pore fluid toward the tip and generating back-stresses. As time progresses to late fall-off, diffusion equilibrates the fluid pressures, but the gradients of the closure stresses increase through the dissipated leaked-off filtrate, increasing the effective stresses, which in turn increase the poroplastic strains and enhance the risk of reopening the fracture. The transition from sharp contrasts to uniform fields reflects the poroelastoplastic coupling. Early suction amplifies tip resistance, while late dissipation sustains the back-stresses that increase plastic strain. From a wellbore strengthening perspective, these distribution characteristics are critical for the LCM plug’s effectiveness. The high-pressure gradients drive the fluid leak-off that generates beneficial back-stresses, helping to clamp the fracture shut. Simultaneously, the negative pressure zone at the tip acts as a powerful arrest mechanism, effectively pulling the fracture closed and significantly increasing the plug’s stability against reopening.

Figure 7 depicts the equivalent plastic strain field (PEEQ) for the poroplastic fracture post-plugging in a saturated formation, with Figure 7a showing early fall-off, which is the transient phase, and Figure 7b showing the late fall-off, which is the equilibrium phase. In both figures, the fracture is shown as a horizontal line from the wellbore extending in the 2D domain. Figure 7a,b highlight concentrated strains near the tip and along fracture walls, forming rabbit-ears extending to ~1 m, indicative of active yielding. The strain intensity is more profound by a factor of three in the case of the late fall-off.

The concentrated strains in early fall-off (Figure 7a) result from residual stresses post-plugging, inducing immediate plastic yielding via the Mohr–Coulomb criterion, creating volumetric expansion that forms rabbit-ears ahead of the tip, forcing tip resistance, a similar mechanism to the tip screen-out. As time advances to late fall-off (Figure 7b), diffusion equilibrates the pore pressures, increasing the effective stresses and causing the plastic strain magnitude to increase by a factor of three. The implications of this plastic strain evolution for the LCM plug are twofold. First, the formation of this yield zone dissipates substantial energy, which blunts the fracture tip and increases its resistance to propagation, a primary strengthening mechanism. Second, the finding that plastic strain continues to accumulate long after plugging reveals a potential long-term risk. This ongoing yielding could create a weakened zone around the arrested fracture, making it more susceptible to reopening if wellbore pressure later increases. This highlights a critical risk factor that non-plastic models cannot capture.

Figure 8 plots the changes in the in situ stress field σ_x, σ_y, σ_z as a function of distance from the initiation point post-plugging for both the plastic and poroplastic models. Stresses are in the form of total stresses. Figure 8a,b show the early fall-off for the plastic and poroplastic models, respectively, while Figure 8c,d show the late fall-off for the same models. Positive stress change indicates stress increase (tension) while negative values show a reduction in stress and designates (compression). Figure 8a–d present a cross-section along the central axis of the fracture starting from the initiation point, passing through the open part of the fracture, through the tip, and up to 1 m before the end of the cohesive element zone (see Figure 3). The black dashed lines indicate the initial in situ stress field, while the color lines show the changes in the stress field after inserting the plug at the fracture inlet when the fracture has reached 5 m. Specifically, the blue color shows the change in σ_y, green the change in σ_z, and red the change in σ_x.

In Figure 8a, the early fall-off in the plastic fracture shows a sharp stress drop for σ_x (red line), to −12 MPa from fracture opening reaching the location of the tip at d = 5 m. Stress distribution recovers to the in situ stress field ahead of the tip. The intermediate stress σ_z (green line) and the minimum stress σ_y exhibit similar behavior with σ_x. The biggest stress change, as expected, is observed at the tip location, expressing a stress softening because of the fracture process. Figure 8b presents the poroplastic fracture at early fall-off, which reveals a broader and deeper drop behind the tip and recovers to the initial in situ ahead of it. What is important to note is that even at early fall-off, the fracture easily propagated to nearly 6 m before it stopped growing. By the late fall-off in Figure 8c, the stresses in the plastic fracture stabilize, indicating full relaxation (equilibrium) because the plug is applied in the fracture inlet, thereby trapping the fluid inside the fracture. In Figure 8d, the poroplastic late fall-off causes the stresses to decrease significantly (up to 3 MPa) with complete recovery ahead of the tip. This is a direct result of the diffusion process, wherein the fluid inside the fracture bleeds in the formation, creating back-stresses and further decreasing the total in situ stress field. Overall, the poroplastic case exhibits wider stress perturbations with persistent changes, while plastic models appear to recover faster after plugging.

The stress changes in the plastic model and at early fall-off are the result of rapid elastic unloading post-plugging, reducing the minimum stress and promoting immediate closure via compressive rebound because of the impermeable fracture walls. The fracture fluid is trapped inside the fracture, and plastic yielding dissipates energy without diffusion delays. Poroplastic stresses show a broader change that arises from coupled diffusion, where the pore pressure gradients sustain effective stress reductions longer, enhancing back-stresses for gradual arrest, but delaying full stabilization. The late recovery in plastic fracture (Figure 8c) reflects undrained equilibrium, enabling effective plugging and fracture arrest. The persistent late-time poroplastic stress changes after plugging (Figure 8d) as a result of ongoing fluid redistribution through permeable fracture walls, which enhances dilation, as well as sustains and possibly expands yield zones, increasing the risk of fracture reopening. These stress changes underscore the role of plastic yielding and diffusion in wellbore strengthening by optimizing closure stresses and preventing flowback by incorporating coupled mechanisms in the analysis.

Figure 9 plots the dimensionless yield factor f as a function of distance from the wellbore for the plastic (Equation (10)) and poroplastic (Equation (11)) fractures. In Figure 9a,b, the blue line represents the fracture at the state of propagation, the red represents the early fall-off, and the green represents the late fall-off. In Figure 9a, plastic fracture propagation shows a sharp peak at the fracture tip location, indicating active yielding while reducing to ~0.75 behind the fracture tip and ~0.8 ahead of the tip, suggesting effective fracture arrest after plugging. However, at early and late fall-off, the behavior is retained, indicating a high risk of fracture reopening at the tip location. In Figure 9b, the poroplastic propagation peaks only when the fracture is in the state of propagation and drops significantly after plugging. What is important to note is that the yield factor after plugging drops at the tip but increases significantly from propagation (~0.68) to early fall-off (~0.76) and (~0.79) at late fall-off, and then stabilizes. This is a direct result of the diffusion effects that increase the risk for reopening the fracture behind the tip and not from the tip, as in the case of the plastic fractures.

The sharp yield peak in the plastic fracture propagation (Figure 9a) arises from localized Mohr–Coulomb failure ahead of the tip, where stress concentrations trigger yielding (f = 1) that dissipates energy for initial resistance, with a rapid fall-off reflecting elastic unloading without diffusion sustaining strains, enabling swift arrest from reduced effective stresses. The results from the poroplastic solution result from coupled dilation and diffusion, where plastic expansion increases porosity, drawing fluid from the domain and maintaining a high yield factor via back-stresses, with fluctuations in early and late fall-off due to transient pore pressure gradients delaying relaxation. In contrast, the plastic solution presents a softer yield factor behind and ahead of the fracture tip. At any rate, both cases illustrate that plugging affects the overall plug stability. These patterns highlight how poroplastic coupling extends yield zones, increasing the risk for reopening plugged fractures and rendering the wellbore strengthening insufficient in soft rocks.

4. Conclusions

This research investigated the role of plastic and poroelastoplastic effects in wellbore strengthening through a fully coupled hydro-mechanical finite element model, focusing on arresting unwanted hydraulic fractures in soft, saturated formations through plugging. By integrating lubrication theory for fracture flow, Darcy’s law for diffusion, Biot’s poroelasticity, Mohr–Coulomb plasticity with associative flow, and a cohesive zone propagation criterion, the model simulated fracture initiation, limited growth, and post-plugging under plane-strain conditions, revealing key differences between plastic and poroplastic behaviors. From the fully coupled hydro-mechanical simulations of this study, the following findings can be outlined:

• Both plastic and poroplastic fractures are wider than their elastic counterparts during propagation due to energy dissipation in the yield zones. The poroplastic cases show even greater widening (~30%) from diffusion-induced back-stresses.
• Post-plugging, the plastic fractures narrow rapidly (~50% width loss early) but extend about ~1 m before stabilizing due to trapped fluid in impermeable walls. The poroplastic fractures close faster from the tip inward, with permeable leak-off promoting complete arrest via back-stresses, though risking reopening.
• Poroplastic propagation demands ~20% higher pressures with gradual declines and negative tip values from suction. Plastic equivalent strains are broader and persist longer (~3× increase late fall-off), driven by coupled dilation and diffusion sustaining yielding even after plugging.
• Early and late fall-offs show sharp stress drops in plastic, recovering ahead of the tip. However, the poroplastic fracture exhibits broader reductions with slower equilibration, amplifying closure stresses sustaining back-stress effects.
• The normalized plastic and the newly derived poroplastic yield factor f_poroplastic peaks at tips during propagation (f = 1), dropping post-plugging, but remaining high in the poroplastic case, quantifying failure risks and highlighting the role of diffusion in elevating reopening potential behind plugs.

The novelty of this study lies in its integration of plastic and poroelastoplastic effects into a fully coupled hydro-mechanical finite element model specifically tailored for wellbore strengthening, extending beyond the traditional elastic and poroelastic frameworks to focus on arresting unwanted fractures in soft, saturated formations through plugs. With this work, we address a gap in the existing literature that largely overlooks post-plugging shut-in dynamics and reopening risks under coupled diffusion-yielding conditions. By introducing a poroelastoplastic Mohr–Coulomb-based yield factor f_{poroelastoplastic} to quantify formation failure during propagation and fall-off, the work provides a new predictive tool for assessing plug stability and flowback prevention, revealing how poroplastic back-stresses influence the plug stability as compared to plastic models, offering actionable advancements for optimizing plug design by bridging computational models with field applications. Future research should build upon these findings by investigating the sensitivity of the results to non-associated flow rules, which would enable more precise, quantitative predictions for specific geological formations.

It is important to acknowledge the limitations imposed by the model’s idealizations, specifically for the assumption that a predefined, straight fracture path is appropriate for a homogeneous, anisotropic stress field, which does not account for path curvature or branching that could occur in heterogeneous formations. Such complex geometries would undoubtedly alter the quantitative aspects of stress concentration and plastic zone development. However, the fundamental physical mechanisms investigated in this work, shear-induced dilation and poro-mechanical coupling, are local to the fracture tip. We therefore posit that our central conclusion, namely that poroelastoplastic effects provide distinct and significant strengthening mechanisms compared to purely plastic behavior, would remain valid even for more complex fracture network geometries.