Experimental Investigation of Acid-Etched Creep Behavior and Mechanical Constitutive Modeling of Carbonate Rocks

Abstract

Deep and ultra-deep carbonate reservoirs commonly experience fracture closure and conductivity reduction under high-temperature and high-stress conditions. In this study, triaxial creep tests were conducted on unacid-etched and acid-etched carbonate cores under different stress levels to investigate their time-dependent deformation behavior and the influence of acid etching on rock rheology. The results indicate that carbonate rocks exhibit pronounced creep behavior, including instantaneous elastic deformation, primary creep, and steady-state creep. Acid etching significantly altered the creep characteristics and rheological parameters of carbonate rocks, leading to distinct time-dependent deformation responses compared with the unacid-etched core. The Burgers constitutive model was employed to characterize the creep behavior, and all fitting correlation coefficients exceeded 0.9. Finite element simulations based on the fitted parameters successfully reproduced the experimental creep curves, verifying the reliability of the constitutive model. This study provides a theoretical and numerical basis for evaluating the long-term deformation behavior of acid-etched carbonate rocks and its implications for fracture closure and conductivity evolution.

1. Introduction

As the exploration and development of oil and gas continues to advance towards deep and ultra-deep formations, carbonate reservoirs have become an important energy resource due to their large reserves and wide distribution [1,2,3,4]. In practical development, acid fracturing is a key technique for improving the productivity of carbonate reservoirs. It creates high conductivity fracture channels through acid etching, thereby significantly enhancing the reservoir productivity [5,6,7]. However, as the development depth increases, reservoir conditions gradually shift from low-temperature and low-stress environments in shallow formations to high-temperature and high-stress conditions in deep formations. Under such conditions, the stability of fractures becomes increasingly critical. Fractures are subjected to high closure stress, which leads to a continuous reduction in fracture width and even complete closure, resulting in a significant decline in fracture conductivity [8,9]. Therefore, accurately characterizing the fracture closure process and its associated mechanical behavior under high stress has become a key scientific problem in the development of deep carbonate reservoirs.

Under deep reservoir conditions, rocks exhibit significant time-dependent deformation under long-term high stress, known as creep behavior [10,11]. Creep is essentially a rheological process. The deformation rate depends not only on the stress level but also on temperature, rock anisotropy, and loading duration [12,13]. For acid-etched fractures in carbonate rocks, fracture closure is not instantaneous. Instead, it evolves gradually over time. This process is closely related to the creep properties. Therefore, investigating rock rheology from a time-dependent perspective is essential for understanding the evolution mechanism of fracture closure. At present, studies on rock creep behavior have established a research framework based on laboratory tests, empirical models, and constitutive theories. Various constitutive models have been proposed, including the Maxwell model, the Kelvin model, and their combined forms [14,15,16,17]. However, most of these studies focus on conventional rocks or shallow conditions. The creep response of deep carbonate rocks under complex treatment conditions remains insufficiently investigated. During the formation of fractures by acid fracturing in carbonate reservoirs, acid etching significantly alters the internal pore structure and microcrack development. This change has a strong impact on the macroscopic mechanical behavior of the rock, which further affects fracture closure and well productivity. Several studies have investigated the mechanical properties of carbonate rocks. Xie et al. [18] conducted conventional triaxial compression tests, triaxial creep tests, and coupled chemical–mechanical tests on porous limestone samples. Their results show that acid etching reduces mechanical parameters such as elastic modulus and cohesion, and also enhances time-dependent deformation. Gou et al. [19] studied the mechanical properties of dolomite after immersion in supercritical CO₂ and formation water under high temperature and pressure conditions. They found that chemical dissolution enlarges pore size with increasing immersion time, leading to a significant reduction in rock strength. Li et al. [20] investigated the effects of acid etching time and temperature on the elastic modulus, Poisson’s ratio, and tensile strength of carbonate rocks. Their results indicate that these parameters decrease with increasing treatment time and temperature, which significantly influences the fracture initiation pressure. Nevertheless, most existing studies mainly focus on instantaneous mechanical properties. The time-dependent deformation behavior of carbonate rocks under high temperature and high stress conditions remains insufficiently understood [21].

In numerical simulations of fracture closure, some models are based on elastic or elastoplastic assumptions. These models do not fully consider the rheological behavior of rocks. As a result, they cannot accurately capture the long-term evolution of fractures under sustained stress. Lu et al. [22] employed the Drucker–Prager elastoplastic model to describe rock yielding. They conducted simulations of fracture contact and closure based on three-dimensional scanned fracture surfaces. Their results showed that stress state, mechanical properties, fracture roughness, and slip all significantly influence the closure process. Hou et al. [23] investigated fracture permeability evolution and closure behavior in granite using steady-state flow experiments, high-resolution laser scanning, and three-dimensional numerical modeling. They found that fracture permeability decreases following a quadratic exponential trend under confining stresses of 5–40 MPa, with an overall reduction of 82.1%. Meanwhile, fracture aperture decreases exponentially by 60.4%. However, under deep high-stress conditions, neglecting time-dependent effects may significantly underestimate fracture closure. This limitation prevents numerical models from reproducing the actual long-term evolution of fractures. Wei et al. [24] used a Maxwell model to characterize the creep behavior of shale. They conducted coupled simulations of fracture surfaces, proppant, and fluid flow. Their results indicate that both shale creep and proppant embedment increase fracture closure over production time. Zhang et al. [6] introduced a power-law model to describe rock rheology and investigated the evolution of acid-etched fracture morphology under stress. They found that surface strain and roughness vary significantly with time. To better account for the long-term closure behavior of fractures in deep carbonate reservoirs, recent studies have increasingly incorporated rock rheology into numerical simulations. Traditional elastic and elastoplastic fracture closure models are widely used to describe instantaneous deformation and stress redistribution; however, they generally neglect time-dependent creep effects, which are critical under sustained high-stress conditions. Recent developments in nonlinear creep constitutive modeling, including fractional derivative models, damage-coupled formulations, and nonlinear viscoelastic–plastic frameworks, have significantly improved the description of time-dependent deformation and progressive failure in rocks [25,26,27]. In addition, chemical processes such as acidification and dissolution have been shown to substantially alter the microstructure and mechanical behavior of carbonate rocks, leading to reduced stiffness, strength degradation, and enhanced time-dependent deformation response under stress [28]. These chemically induced changes further influence fracture surface evolution and contact behavior during long-term loading. Therefore, it is necessary to incorporate rock rheology and time-dependent effects into numerical simulations. This approach enables a more accurate investigation of the creep behavior of carbonate rocks and their impact on fracture closure.

To investigate the time-dependent deformation behavior of deep carbonate rocks under sustained high-stress conditions, triaxial creep experiments were conducted on unacid-etched and acid-etched carbonate core samples under varying stress levels. The creep behavior was characterized through analysis of the strain–time evolution during different creep stages. The primary objective of this study is to quantify how acid etching influences the creep behavior and rheological parameters of carbonate rocks. To this end, the Burgers constitutive model was employed to describe the complete creep process, including instantaneous elastic deformation, primary creep, and steady-state creep. The model parameters were determined through nonlinear fitting and inverse analysis of the experimental creep curves. Subsequently, the calibrated constitutive parameters were implemented into a finite element framework. Numerical simulations were performed under the same loading conditions as the laboratory experiments to validate the constitutive model and to evaluate the predictive capability of the identified rheological parameters. This study focuses on the evolution of time-dependent constitutive behavior induced by acid etching in carbonate rocks, providing a theoretical and numerical basis for characterizing their long-term deformation response under complex stress environments.

2. Experimental Materials and Methods

2.1. Experimental Sample Preparation

In this study, we selected limestone outcrop samples as the rock material. Prior to mechanical testing, we performed mineralogical analysis on the samples. The rock fragments were ground into powder and analyzed using X-ray diffraction (XRD) to determine the mineral composition (Figure 1). Quantitative phase analysis was conducted to obtain the relative content of each mineral. As shown in

Table 1. Results of rock mineral composition analysis.

Category	Calcite	Quartz	Aragonite	Clay Minerals
Content/%	96.35	2.09	0.99	0.57

, the samples mainly consist of calcite (96.35%), with minor amounts of quartz (2.09%), aragonite (0.99%), and clay minerals (0.57%). These results confirm the lithological characteristics of the selected samples and provide a basis for subsequent acoustic and mechanical analyses.

Following the standard procedures recommended by the International Society for Rock Mechanics (ISRM), the samples were processed into cylindrical cores with a diameter of 25 mm and a length of 50 mm [29,30]. After cutting, both ends of the samples were carefully polished. The flatness and parallelism of the end surfaces were measured using a digital micrometer and a dial indicator to ensure compliance with testing standards. After dimensional and visual screening, the qualified samples were subjected to ultrasonic testing to evaluate internal defects and integrity [31]. The tests were conducted using a high-temperature and high-pressure triaxial testing system equipped with ultrasonic measurement capability. Before testing, a small amount of coupling agent was applied to the loading platens to improve signal transmission and ensure stable contact between the platens and the sample ends. Ultrasonic signals were recorded to determine the travel time of the samples in the axial direction (Figure 2a). The P-wave velocity was calculated based on the travel time and sample length. Each sample was tested twice, and the average value was taken as the final result. The test results for different samples are shown in Figure 2b. Samples with significant deviations were excluded, and those with similar physical properties were selected for subsequent triaxial mechanical tests.

2.2. Experimental Methods

2.2.1. Rock Acid Etching Experiments

We adopted a static acid-etching method, and the experimental procedure is shown in Figure 3. First, a 5% HCl solution was prepared in a sufficient volume to fully immerse the core samples in the reaction vessel. The target temperature and pressure were then set in the control system, and the conditions inside the vessel were monitored in real time through the digital display panel. The acid solution was preheated to the target temperature in a preparation container and then injected into the reaction vessel through the pipeline. The temperature and pressure in the vessel were maintained constant during the reaction. After reacting with the cores for 30 min, the residual liquid and gas were discharged from the vessel. The cores were then removed, rinsed briefly, and dried in an oven until their mass remained constant. To reproduce the temperature and pressure conditions of deep carbonate reservoirs at a burial depth of approximately 6000 m, the experimental temperature and pressure were set to 120 °C and 7.5 MPa, respectively. The selected temperature and pressure conditions exceed the critical point of CO₂. Therefore, these conditions satisfy the thermodynamic requirements for CO₂ to potentially reach a supercritical state during the acid–rock reaction process. However, the actual phase behavior of CO₂ depends on multiple factors, including CO₂ generation, partial pressure, and gas–liquid equilibrium, and was not directly monitored in this study. Previous studies have reported that supercritical CO₂ may significantly influence fluid–rock interactions, pore structure evolution, and the long-term rheological behavior of carbonate rocks under high-temperature and high-pressure conditions [32,33].

2.2.2. Triaxial Creep Mechanical Experiments

Creep tests are typically conducted under constant stress conditions to evaluate the rheological behavior of rocks over time, during which the evolution of strain with time is recorded. In this study, we performed mechanical experiments using the rock triaxial testing system (Figure 4a). This apparatus is a closed-loop digital servo-controlled system that enables various types of triaxial tests on rock specimens. It can simulate the mechanical behavior of rocks under in situ conditions, including compression and creep. An intelligent data acquisition system allows real-time monitoring and recording of experimental data.

In our study, we adopted a single-core stepwise continuous loading method to apply the load. This approach ensures loading stability and allows us to more clearly reveal the deformation characteristics of rocks under different stress levels. Meanwhile, it effectively avoids variability caused by using multiple specimens, thereby improving the reliability of the measured mechanical parameters [34,35,36]. The experimental procedure consists of two stages. First, we conducted triaxial compression tests (Figure 4b) on both acid-etched and unacid-etched cores. The obtained data were used to plot stress–strain curves. From these curves, we can clearly observe the evolution of strain as stress increases, as well as the entire process in which the rock reaches its peak compressive strength (σ_m) and eventually fails. Subsequently, triaxial creep tests were carried out (Figure 4c). Based on the peak compressive strength (σ_m) under the corresponding confining pressure, five stress levels were sequentially applied: 0.5 σ_m, 0.6 σ_m, 0.7 σ_m, 0.8 σ_m, and 0.9 σ_m. Each stress level was maintained for 24 h. After completing one loading stage, the creep test at the next stress level was initiated immediately. During the experiment, stress and strain data were automatically recorded every 5 s until all five loading stages were completed. The detailed experimental conditions are listed in

Table 2. Detailed specifications for stepwise continuous-load creep experiment conditions.

Sample Type	No.	Confining Load/MPa	Axial Load/MPa	Loading Time/h
unetched core	A-1	60	0.5 σm(A)	24
			0.6 σm(A)	24
			0.7 σm(A)	24
			0.8 σm(A)	24
			0.9 σm(A)	24
acid-etched core	B-1	60	0.5 σm(B)	24
			0.6 σm(B)	24
			0.7 σm(B)	24
			0.8 σm(B)	24
			0.9 σm(B)	24

.

3. Experimental Results and Analysis

3.1. Triaxial Mechanical Compression Characteristics Under Acid Etching

Triaxial compression tests were conducted on both unacid-etched and acid-etched cores under a confining pressure of 60 MPa. In all subsequent graph plots and result analyses in this paper, the axial loads were calculated using deviatoric stress. The corresponding stress–strain curves are shown in Figure 5. Taking the axial strain curve of the unacid-etched core as an example (Figure 6), we observe that the rock initially undergoes elastic deformation as the applied stress increases. This stage appears as a linear segment with a constant slope on the stress–strain curve. After the elastic stage, the stress–strain curve exhibits a noticeable change, and the slope gradually decreases, indicating that the rock enters the yielding stage and begins to develop irreversible plastic deformation. With further increase in stress, the curve transitions into a relatively flat plateau. At this stage, the rock enters a hardening state, internal damage accumulates, the stress no longer increases significantly, while the strain continues to grow, eventually leading to compressive failure.

By comparing the stress–strain curves of the acid-etched core, we find that its peak strength is significantly lower than that of the unacid-etched core. In addition, during loading, the acid-etched rock fails within a short period after the elastic stage. In contrast to the unacid-etched rock, no distinct yielding stage is observed.

Based on the stress–strain results obtained from the triaxial compression tests, we determined that the peak compressive strength of the unacid-etched core under a confining pressure of 60 MPa is 310 MPa, whereas that of the acid-etched core is 71 MPa. These results indicate that acid etching has a significant impact on the overall strength of the rock, leading to a pronounced reduction in compressive strength. Furthermore, the variations in mechanical properties can be clearly observed through the calculation and analysis of the elastic modulus and Poisson’s ratio (

Table 3. Comparison of mechanical properties parameters between unacid-etched and acid-etched cores.

Sample Type	No.	Modulus of Elasticity/GPa	Poisson Ratio	Maximum Compressive Strength σm /MPa
unacid-etched core	A-2	67	0.16	310
acid-etched core	B-2	35	0.15	71

). The unacid-etched core exhibits an elastic modulus of 67 GPa and a Poisson’s ratio of 0.16, while the acid-etched core shows an elastic modulus of 35 GPa and a Poisson’s ratio of 0.15. The reduction in elastic modulus indicates that the rock’s resistance to deformation decreases, resulting in larger strain under the same applied stress. Similarly, the change in Poisson’s ratio further reflects the influence of acid etching on the rock structure, making the rock more susceptible to deformation under stress [37,38,39].

3.2. Triaxial Creep Mechanical Characteristics Under Acid Etching

3.2.1. Strain–Time Results

To further clarify the influence of acid etching on the creep behavior, we compare the creep responses of unacid-etched and acid-etched cores under identical loading conditions (Figure 7). The results show that the cumulative creep strain of the unacid-etched rock increases significantly with stress level, reaching 0.907% at the fifth loading stage. In contrast, the acid-etched rock exhibits a much lower cumulative strain, with a maximum value of only 0.217%.

A similar trend is observed for the incremental creep strain at each loading stage. For the unacid-etched core, the creep strain generally increases with increasing stress level, and a pronounced acceleration is observed at higher stress levels. In contrast, the acid-etched core exhibits relatively small incremental creep strains, with the maximum value of 0.107% occurring at the first loading stage and decreasing thereafter. It should be noted that the stress levels applied to the unacid-etched and acid-etched cores were defined as fractions of their respective peak strengths (0.5σ_m–0.9σ_m). Therefore, although both groups were tested under comparable normalized stress levels, the absolute stresses applied to the acid-etched core were substantially lower due to its reduced peak strength. The results indicate that acid etching significantly modifies the creep response of the limestone. Acid-induced dissolution weakens the load-bearing skeleton and reduces both the strength and stiffness of the rock, resulting in failure at lower absolute stress levels. Consequently, the acid-etched core does not accumulate large creep strains before failure, whereas the unacid-etched core is capable of sustaining higher absolute stresses and developing greater cumulative creep deformation. These results suggest that acid etching reduces the long-term deformation resistance of limestone and changes its creep deformation mechanism under sustained loading.

3.2.2. Creep Rate Evolution and Stage Characteristics

•

unacid-etched core

To further characterize the time-dependent evolution of creep strain under stress, we plotted the creep strain and instantaneous strain rate curves at different loading stages. The corresponding results are shown in Figure 8. The strain rate curve can more sensitively capture the transition of the rock from primary creep to steady-state, thereby revealing the characteristic features and rate evolution of each stage [40,41,42,43]. In the figure, the strain-time curve is divided into a decelerating creep stage (I) and a steady-state creep stage (II) based on changes in the strain rate.

The results indicate that the instantaneous strain rate decreases rapidly from an initially high level to a stable low range within a short period. This behavior reflects that, during the initial application of stress, the strain increases rapidly, and the instantaneous creep rate remains relatively high. After reaching a constant stress level, the rock enters the primary creep stage, during which the instantaneous strain rate continuously decreases until it eventually stabilizes. A comparison of the instantaneous strain rates under different loading levels shows that the peak values during the initial stage increase with stress (Figure 9a). Specifically, the maximum instantaneous strain rates are 0.216 mm·h⁻¹ at the first loading stage, 0.234 mm·h⁻¹ at the second stage, 0.270 mm·h⁻¹ at the third stage, 0.342 mm·h⁻¹ at the fourth stage, and 0.630 mm·h⁻¹ at the fifth stage. These results demonstrate that the instantaneous strain rate varies significantly with the applied stress level. Higher stress levels correspond to higher peak strain rates during the loading process.

We further divided the primary creep stage and the steady-state creep stage under different stress levels, as shown in Figure 9b. The results indicate that, with increasing stress, the proportion of time that the rock spends in the primary creep stage changes significantly. Specifically, the proportion of the primary creep stage is 6.22% at the first loading level, 11.24% at the second level, 31.57% at the third level, 33.37% at the fourth level, and 59.39% at the fifth level. These results show that, under low stress conditions, the primary creep stage occupies only a short duration. However, as the stress level increases, its proportion in the overall creep process increases markedly. In contrast, the proportion of the steady-state creep stage decreases progressively with increasing stress, indicating that the rock is unable to maintain a long-term stable creep deformation state under high stress conditions. Overall, these results demonstrate that the applied stress not only affects the magnitude of creep strain but also significantly controls the evolution of different creep stages, making it a key factor governing the time-dependent deformation behavior of the rock.

•

acid-etched core

We plotted the creep strain and instantaneous strain rate curves at different loading stages, as shown in Figure 10. The results indicate that the instantaneous strain rate rapidly decreases from an initially high value to a stable low range within a short period. A comparison of the instantaneous strain rates under different loading levels shows that, during the initial stage when the creep rate reaches its maximum, the peak values are 0.288 mm·h⁻¹ at the first loading stage and 0.270 mm·h⁻¹ at the fifth stage. The peak instantaneous strain rate of the acid-etched samples did not change significantly with increasing loading level. The values varied from 0.288 mm·h⁻¹ at the first stage to 0.270 mm·h⁻¹ at the fifth stage, remaining within a relatively narrow range throughout the test. In contrast, the unacid-etched samples showed a more obvious increase in peak instantaneous strain rate with loading level, from 0.216 mm·h⁻¹ to 0.630 mm·h⁻¹. This difference suggests that acid etching altered the instantaneous deformation response of the carbonate rock during graded creep loading.

As shown in Figure 10, under constant stress, the strain curve initially exhibits a rapidly increasing linear segment, reflecting the rapid growth of strain during this stage. Subsequently, the slope of the curve decreases, indicating a decelerating process, and then the curve maintains a relatively constant slope with a continuous increasing trend. Compared with the unacid-etched core, the acid-etched core shows a much less pronounced decelerating stage during loading. The transition from the initial rapid increase to the later stable growth is shorter, and under low stress conditions, the strain curve does not exhibit a near-horizontal trend indicative of stabilization. The acid-etched core exhibits a different creep response from that of the unacid-etched core, particularly during the early stages of loading. The reduction in strength and stiffness caused by acid etching decreases the ability of the rock skeleton to resist long-term deformation, leading to noticeable changes in the rheological parameters and time-dependent deformation characteristics. These results indicate that acid etching significantly modifies the rheological response of limestone and influences its long-term deformation behavior under sustained loading.

Linear regression was performed on the steady-state creep curves under different stress levels, and the corresponding steady-state creep strain rates are shown in Figure 11a. As the applied stress level increases, the steady-state creep strain rate decreases progressively. Compared with the first loading level (9 × 10⁻⁴ mm·h⁻¹), the second, third, fourth, and fifth levels decrease by approximately 32%, 73%, 84%, and 93%, respectively. To further characterize the evolution of creep stages, the durations of the primary creep stage and steady-state creep stage under different loading levels were quantified, as shown in Figure 11b. The results indicate that the proportion of time occupied by the primary creep stage decreases with increasing stress level, from 33.55% at the first level to 7.58% at the fifth level. Correspondingly, the relative contribution of the steady-state creep stage increases with increasing stress level, indicating a transition toward a more rapidly stabilized deformation process. These results suggest that higher stress levels promote faster structural adjustment and earlier stabilization of creep deformation, leading to a reduced duration of the primary creep stage and a more dominant steady-state deformation behavior.

3.3. Constitutive Modeling of Creep Behavior

3.3.1. Creep Constitutive Model

Based on the creep test results, we observe that the deformation curves of carbonate rocks under constant stress exhibit distinct stage characteristics. These stages mainly include instantaneous elastic deformation at the initial loading stage, a primary creep stage with gradually decreasing strain rate, and a steady-state creep stage that approaches a constant rate at later times. Each stage corresponds to different mechanical response mechanisms, indicating that the deformation process is governed by the coupled effects of elasticity and viscosity [44,45,46].

Therefore, the constitutive model must be capable of uniformly characterizing the multi-stage creep behavior. The classical Maxwell model can describe steady-state creep but fails to capture the primary creep process. In contrast, the Kelvin model can represent primary creep behavior but cannot describe instantaneous elastic response or long-term steady flow. As a result, both models are insufficient to fit the complete experimental creep curves [17,47]. In comparison, the Burgers model effectively integrates the Maxwell and Kelvin elements in series, incorporating instantaneous elastic deformation, primary creep, and steady-state creep mechanisms. This enables a comprehensive description of the entire creep process of carbonate rocks. Based on the above analysis, we adopt the Burgers model to fit the experimental curves and to achieve a unified characterization of the time-dependent deformation behavior of the rock.

The Burgers model, also referred to as the M–K model, consists of a Maxwell element connected in series with a Kelvin element, as shown in Figure 12a. At t = 0, only the elastic spring responds, yielding an initial strain of ε₀ = σ/E_M. As t → ∞, the strain ε_∞ → ∞, as illustrated in Figure 12b; however, the rate of strain approaches a constant value.

The constitutive equation and the corresponding creep equation of the Burgers model are given as follows (Equations (1) and (2)):

$$\ddot{\sigma }+\left({\displaystyle \frac{E_{\mathrm{M}}}{{\eta }_{\mathrm{K}}}}+{\displaystyle \frac{E_{\mathrm{K}}}{{\eta }_{\mathrm{K}}}}+{\displaystyle \frac{E_{\mathrm{M}}}{{\eta }_{\mathrm{M}}}}\right)\dot{\sigma }+{\displaystyle \frac{E_{\mathrm{M}}{E}_{\mathrm{K}}}{{\eta }_{\mathrm{M}}{\eta }_{\mathrm{K}}}}\sigma =E_{\mathrm{M}}\ddot{\epsilon }+{\displaystyle \frac{E_{\mathrm{M}}{E}_{\mathrm{K}}}{{\eta }_{\mathrm{K}}}}\dot{\epsilon }$$

(1)

$$\epsilon \left(t\right)={\displaystyle \frac{\sigma }{E_{\mathrm{M}}}}+{\displaystyle \frac{\sigma }{E_{\mathrm{K}}}}(1-e^{-{\displaystyle \frac{E_{\mathrm{K}}}{{\eta }_{\mathrm{K}}}}\mathit{t}}\mathrm{)}+{\displaystyle \frac{\sigma }{{\eta }_{\mathrm{M}}}}t$$

(2)

where ε(t) is the strain; σ is the applied constant stress; E_M is the instantaneous elastic modulus, reflecting the elastic response capacity of the rock at the moment of loading; η_M is the viscosity coefficient of the Maxwell element, which governs the deformation rate during the steady-state creep stage; E_K is the elastic modulus of the Kelvin element, mainly characterizing the recovery capacity during the primary creep stage; η_K is the viscosity coefficient of the Kelvin element, describing the time-dependent behavior of the primary creep process.

In the Burgers model, four parameters must be identified, among which E_M denotes the elastic modulus. Under a constant applied stress σ, the elastic strain related to E_M occurs instantaneously. Thus, at t = 0, E_M can be determined from the ratio of stress to strain. The other three rheological parameters (E_K, η_K, η_M) are obtained using the least squares approach [48]. For each time point t_i, a theoretical strain value ${\overline{\epsilon }}_i$ can be calculated from Equation (3), corresponding to the measured strain ε_i. The least squares method aims to identify the optimal parameter combination by minimizing the objective function P, which is defined as the sum of the squared deviations between the measured and predicted strain values, as expressed in Equation (4).

$${\overline{\epsilon }}_i={\displaystyle \frac{\sigma }{E_{\mathrm{M}}}}+{\displaystyle \frac{\sigma }{E_{\mathrm{K}}}}(1-e^{-{\displaystyle \frac{E_{\mathrm{K}}}{{\eta }_{\mathrm{K}}}}{t}_i})+{\displaystyle \frac{\sigma }{{\eta }_{\mathrm{M}}}}{t}_i$$

(3)

$$P={\displaystyle \sum _{i=1}^N{\left({\overline{\epsilon }}_i-{\epsilon }_i\right)}^2}$$

(4)

$$\left.\begin{array}{l}{\displaystyle \frac{\partial P}{\partial E_{\mathrm{K}}}}=0\\ {\displaystyle \frac{\partial P}{\partial {\eta }_{\mathrm{K}}}}=0\\ {\displaystyle \frac{\partial P}{\partial {\eta }_{\mathrm{M}}}}=0\end{array}\right\}$$

(5)

An initial estimate of the rheological parameter vector P (E_K, η_K, η_M), namely (E_K0, η_K0, η_M0), is first assigned. The partial derivatives of strain with respect to each rheological parameter are then calculated to determine the corresponding parameter corrections (ΔE_K, Δη_K, Δη_M). These corrections are applied to update the parameter set to (E_K1, η_K1, η_M1). The updated parameters are subsequently used as the starting values for the next iteration. This iterative process continues until the three rheological parameters meet the convergence criterion specified in Equation (5) within the prescribed error range. The data were fitted using the least squares method, and the resulting parameter values and the coefficient of determination (R²) are shown in

Table 4. Results of the Burgers model fit under different stress loads.

Type	Stress Load	EM/MPa	EK/MPa	ηM/MPa·s	ηK/MPa·s	R2
Unacid-etched core	Stage 1	81,365	4.56 × 106	3.27 × 1010	2.57 × 109	0.91
	Stage 2	74,519	3.23 × 106	2.99 × 1011	5.14 × 109	0.99
	Stage 3	67,982	1.11 × 106	1.69 × 1011	4.53 × 109	0.98
	Stage 4	59,932	5.35 × 105	7.58 × 1010	1.63 × 109	0.98
	Stage 5	47,889	1.47 × 105	1.65 × 1010	6.25 × 108	0.98
Acid-etched core	Stage 1	69,606	8.56 × 104	1.77 × 1010	2.46 × 109	0.99
	Stage 2	35,702	1.69 × 105	6.86 × 1011	6.28 × 109	0.99
	Stage 3	31,847	4.54 × 105	1.12 × 1011	6.38 × 109	0.97
	Stage 4	31,063	4.00 × 105	1.99 × 1011	1.38 × 1010	0.99
	Stage 5	30,317	1.16 × 106	3.31 × 1011	2.53 × 1010	0.99

.

The fitting results show that the Burgers model parameters vary with stress level. Although several viscosity-related parameters exhibit a decreasing trend as the stress increases, the changes are not completely monotonic, and local fluctuations can be observed at certain loading stages. In contrast, the contribution of the viscous deformation component gradually increases under higher stress levels, indicating that the time-dependent deformation behavior becomes more pronounced. These results suggest that the rheological characteristics of carbonate rocks are affected by stress level, but the evolution of individual model parameters does not follow a simple linear or monotonic pattern. Compared with conventional elastoplastic models, the Burgers model can effectively describe the instantaneous deformation, decelerating creep, and steady-state creep observed in the experiments, making it suitable for characterizing the long-term deformation behavior of carbonate rocks under sustained loading conditions.

3.3.2. Finite Element Simulation and Validation

To validate the rationality of the established creep constitutive model, we developed a numerical model (20 mm × 20 mm × 20 mm) in finite element software (Abaqus version 2022), as shown in Figure 13. The numerical model was discretized using hexahedral elements (C3D8) and analyzed as a three-dimensional solid body. Boundary conditions were applied to the bottom surface with fully fixed constraints to simulate the fixed-end condition in the experiments. Meanwhile, a uniformly distributed constant axial stress was applied to the top surface. The same loading conditions and creep durations as those used in the laboratory tests were adopted in the numerical simulations. The Burgers constitutive parameters obtained from the inversion of the experimental creep curves were implemented into Abaqus through a user-defined creep subroutine. During the analysis, set the time step to 1 s and set the stress and strain output fields as a function of time (Figure 14). The simulated creep results were subsequently compared with the experimental results to evaluate the ability of the Burgers model to reproduce the time-dependent deformation behavior of carbonate rocks before and after acid etching.

Figure 13. Schematic diagram of stress-load boundary conditions and mesh generation for the finite element model.

Figure 13. Schematic diagram of stress-load boundary conditions and mesh generation for the finite element model.

Figure 14. Cloud map of strain results over time during the stress loading process (unacid-etched core stage 5: σ = 0.9 σ_m(A)).

Figure 14. Cloud map of strain results over time during the stress loading process (unacid-etched core stage 5: σ = 0.9 σ_m(A)).

The comparison curves between the experimental data and the simulation results are shown below (Figure 15 and Figure 16). The comparison demonstrates that the simulation results are in good agreement with the experimental data, with a high degree of consistency in both curve morphology and strain magnitude. The simulated strain–time curves successfully reproduce the creep characteristics at different stages, including the instantaneous elastic stage, primary creep stage, and steady-state creep stage. These results indicate that the established finite element model can effectively characterize the time-dependent rheological behavior of carbonate rocks under long-term high-stress conditions.

Moreover, the numerical simulation successfully reproduces the characteristic deformation behavior of the different creep stages and captures the overall time-dependent strain accumulation of carbonate rocks under sustained loading conditions. However, some discrepancies between the simulation results and the experimental data are observed in the later stage of creep, where the model tends to slightly underestimate the strain evolution. These deviations can be attributed to the progressive development of microstructural damage, including microcrack initiation and propagation, as well as local heterogeneity and pore structure evolution induced by acid etching, which are not explicitly considered in the present model formulation. In this study, the rock is idealized as a homogeneous viscoelastic medium within the Burgers framework, and the coupling between time-dependent deformation and damage evolution is not incorporated. Despite these limitations, the model still provides a reasonable representation of the overall creep response and captures the essential features of long-term deformation. Therefore, it can be considered effective for describing the macroscopic rheological behavior of acid-etched carbonate rocks within the scope of the present assumptions. Future work should focus on incorporating damage-coupled or microstructure-dependent constitutive formulations to better represent the late-stage creep behavior and improve the predictive capability under complex stress and chemically altered conditions.

4. Conclusions

Triaxial creep experiments and numerical simulations were conducted to investigate the long-term deformation behavior of carbonate rocks before and after acid etching under deep high-stress conditions. Based on the experimental results and constitutive modeling analysis, the conclusions are as follows:

(1) Carbonate rocks exhibit significant rheological behavior under constant stress conditions, including instantaneous elastic deformation, primary creep, and steady-state creep. With increasing stress level, the creep stages become more distinct and the strain accumulation becomes more pronounced. Compared with the unacid-etched core, acid etching significantly altered the creep characteristics and rheological response of carbonate rocks. Although the acid-etched cores were subjected to lower absolute stress levels due to their reduced peak strength, they still exhibited evident time-dependent deformation behavior under comparable normalized stress levels. These results indicate that acid etching has a significant influence on the long-term deformation characteristics of carbonate rocks.

(2) Acid etching significantly reduced the mechanical properties of carbonate rocks, particularly the elastic modulus and compressive strength. In addition, noticeable changes were observed in the creep response and Burgers model parameters after acid treatment. These variations are likely associated with acid-induced alterations in the internal rock structure reported in previous studies. The results demonstrate that acid etching weakens the mechanical strength of carbonate rocks and modifies their long-term rheological behavior.

(3) The Burgers constitutive model effectively describes the full creep deformation process of carbonate rocks from initial loading to the steady state. The fitting results demonstrate a high level of accuracy, with all correlation coefficients (R²) exceeding 0.9. The identified constitutive parameters indicate that acid etching leads to significant changes in the rheological characteristics of carbonate rocks, including reduced elastic parameters and altered stress-dependent deformation responses.

(4) After incorporating the inverted constitutive parameters into the finite element model, the numerical simulations reproduced the experimental creep curves and stage-dependent deformation characteristics with good agreement. The results verify the applicability of the Burgers constitutive model for describing the creep behavior of carbonate rocks before and after acid etching. The established numerical model can provide a basis for further studies on the long-term deformation of acid-etched carbonate formations under deep reservoir conditions.