Modeling Fracture Propagation in a Dual-Porosity System: Pseudo-3D-Carter-Dual-Porosity Model

Abstract

Despite the significant advancements in geomodelling techniques over the past few decades, it is still quite challenging to obtain accurate assessments of hydraulic fracture propagation. This work investigates the effect of fluid leak-off in a dual-porosity system on the hydraulic fracture propagation geometry, which, in turn, affects hydrocarbon recovery from tight and unconventional reservoirs. Fracture propagation within tight reservoirs was analyzed using the Pseudo Three-Dimensional-Carter II model for single- (P3D-C) and dual-porosity systems (P3D-C-DP). Previous studies have accounted for leak-off in single-porosity models; however, studies within dual-porosity systems are still quite limited. We present a novel approach to coupling fluid leak-off in a dual-porosity system along with a fracture-height growth mechanism. Our findings provide important insights into the complexities within hydraulic fracturing treatment design using our new and pragmatic modeling approach. The simulation results illustrate that fluid leak-off in dual-porosity systems contributes to a confined fracture half-length ($x_f)$, that is 31% smaller using the P3D-C-DP model as opposed to the single-porosity model (P3D-C). As for the fracture height growth ($h_f)$, the P3D-C-DP model resulted in a 40% shorter fracture height compared to the single-porosity model.

1. Introduction

Hydraulic fracturing is considered the most successful well stimulation technique [1]. The first fracturing experiment took place in 1947. In 1950, hydraulic fracturing was deemed commercial. Furthermore, 2.5 million fracs have been estimated worldwide by the Society of Petroleum Engineers, and more than 1 million of these fracs have taken place in the USA [2]. Fracture propagation predictions are non-trivial and require a careful and well-thought-out plan of the hydraulic fracturing design elements. It is essential to have ample information on the reservoir rock properties, fracturing fluids, and the magnitude and direction of the in situ stresses in order to reach an accurate prediction of fracture propagation in terms of the opening width, length, and height [3]. Furthermore, a set of parameters for the hydraulic fracturing treatment, such as time, injection rate, proppant type and concentration, and fracturing fluid viscosity with power law parameters, must be considered at the design stage [4].

Tight reservoirs are challenging in terms of their heterogeneity and extreme low permeabilities. In 1992, Rahim and Holditch [5] emphasized the importance of using 3D hydraulic fracture propagation models for obtaining accurate results; however, most of the fracture treatments in the past were designed using 2D fracture propagation models, especially for optimization processes [6]. Several introductory and key papers were published between late 1950s and early 1970s that set the foundational development for hydraulic fracture modeling approaches. In 1957, Carter [7] developed a model based on a volumetric balance, which assumes a constant fracture width. Then, in 1961 and 1972, the Perkins–Kern–Nordgren (PKN) fracture geometry model was developed by combining the fracture width model of Perkins and Kern [8] with the closed-form equations that account for fluid leak-off and storage [9]. In addition, the P3D model for fracture height propagation was developed for two categories; the first was developed for a symmetric three-layer case where the stress barrier contrasts above and below the pay-zone are assumed to be constant [10], while the second one was established in 1987 by Fung et al. [11], where they derived a generic solution for the P3D model for non-symmetric multilayer cases. In 2010, Rahman and Rahman [12] coupled fluid leak-off equations with the P3D model formulation. In 2018, Wang et al. [13] extended the application of the PKN-C model by coupling fluid leak-off equations with dual-porosity systems, where an equivalent Carter leak-off coefficient was introduced by coupling the matrix and fracture leak-off coefficients. In this work, we build a P3D-C-DP model that involves computing the equivalent leak-off coefficient for the matrix and the fracture systems.

Hydraulic fracturing design requires a full treatment analysis to carefully predict the fracture geometry parameters such as the length, height, and width of the fracture. Numerous analytical models developed over the years have provided several important aspects of hydraulic fracturing processes. Paluszny et al. [14] showed how these models are very useful in terms of highlighting the important factors influencing the growth of the hydraulic fracture. However, they also argued that such models exhibit a limitation when accounting for multiple factors regarding the matrix’s permeability and thermal effects. Wang [15] added that the existence of pre-existing natural fractures may influence hydraulic fracture propagation and the related flow capacity depending on their density, size, direction, and cementation as well as the area where the hydraulic fracturing is carried out. A number of factors, such as the in situ stresses, treatment parameters (such as injection rate and fluid viscosity), and the geometric and mechanical features of the natural fractures, all have the potential to have a major impact on the hydraulic fracture and natural fracture interactions [16,17,18,19]. A recent study by Oyarhossein and Dusseault [20] presented how geomechanical properties and injection parameters affect the fracture height and aperture, while taking into account all technical and environmental constraints. Interesting findings from a pulse-decay experiment performed by Iia and Xian [21] on a fractured shale core and a history matching of the matrix-fracture system demonstrate that, for matrix-only cores, the permeability estimated from the pulse-decay experiment precisely follows Darcy’s law, which varies proportionately with the fluid viscosity. However, the flow in the fracture and the matrix interacts with one another in the fracture–matrix system. Jia et al.’s [22] investigation into the CO₂ storage capacity has shed more light into the flow behavior in tight formations. In this study, the fractures serve as the primary flow conduit for fluid transport in a tight porous medium, which has been stimulated using a dual porosity and permeability model.

Limitations of analytical solutions in terms of handling geometries and heterogeneities can be addressed using robust numerical approaches. Kim and Um [23] described the various analytical and numerical approaches used in hydraulic fracturing propagation. Analytical hydraulic fracturing models are classified into two main categories with regards to their dimensional form: 2D and 3D models. These models include the 2D PKN and KGD (Khristianovic and Zeltova-Geertsma and de Klerk) models, where different assumptions are applied for each model. For 3D, the P3D model is mostly used. This model was introduced back in the 1980s, and it is considered an extension to the classical 2D PKN model. The 2D PKN model assumes a fixed fracture height, and the fracture half-length is a variable that propagates with time to capture the extended fracture length ($x_f\gg h_f$). In contrast to the PKN model, the P3D model allows the fracture height to vary and grow vertically into adjacent layers, and it is not limited to the thickness of the reservoir. It is important to note that both PKN and P3D models are used when the fracture height remains relatively small compared to the fracture half-length, which are incorporated in modeling the production of oil and gas from a hydraulically fractured well [24]. Figure 1 depicts the conventional P3D fracture propagation model, and the symbols are described in the relevant equations in Section 2.

2. Mathematical Formulation

2.1. Fracture Height Growth at the Wellbore

The P3D-C model allows for a varying fracture height, and the letter “C” in the name denotes the Carter Equation II as a solution for the material balance [12]. Moreover, it accounts for the fluid leak-off test as a result of the fracture propagation. The term pseudo refers to the model’s notion of not accounting for fracture geometry change in a fully three-dimensional space, while the P3D model alters the two-dimensional PKN model by including the height variation concept [25]. In other words, the height is considered to change over time and is computed as a function of the lateral position at each time step. The expansion of the aforementioned height is governed by pressure, which functions solely in response to the lateral position [26]. In this model, the fracture half-length, $x_f$, is fixed with a preassumed target length value, and the fracture height, $h_f$, is allowed to vary and propagate along the fracture length; the shape can be linear or parabolic [12].

Using a non-Newtonian fracturing fluid, the power law parameters that are correlated with the fluid’s viscosity are utilized to derive the fracture geometry parameters from the following mathematical correlations [4]:

$$n=0.1756{\mu }^{-0.1233},$$

(1)

and

$$k=\left(0.457\mu -0.0159\right)47.880,$$

(2)

where $n$ is a dimensionless exponent, $k$ is the consistency index in (Pa.sⁿ), and $\mu$ is the fluid viscosity (Pa.s). The model is initialized with a set of treatment variables, which consist of the injection rate, Newtonian fluid viscosity, target length, and fracture height. The fracture height growth at the wellbore is then computed by neglecting the material property variations in each stress layer in order to reach an equilibrium height [12]. The governing equations required for the P3D model begin with computing the stress intensity factors at the top and bottom layers, $K_{lct}$ and $K_{lcb}$ (psi.$\sqrt{\mathrm{inch}}$), respectively [27]:

$$K_{lct}=\frac{1}{\sqrt{\pi a}}{{\displaystyle \int }}_{-a}^{a}p\left(y\right)\sqrt{\frac{a+y}{a-y}}dy,$$

(3)

and

$$a=\frac{h_f}{2},$$

(4)

where $K_{lct}$ is the stress-intensity factor or fracture toughness at the top of the fracture (psi.$\sqrt{\mathrm{inch}}$), $a$ is the fracture half-height (ft), $p\left(y\right)$ is the net fracture pressure distribution opening the fracture (psi), and $y$ is the vertical position from the center of the fracture.

A geometry constraint term (in ft) is added for the heights ($b_2$ and $b_3$) straddling the payzone height (h):

$$b_3=h-b_2.$$

(5)

The fracture height at the wellbore can be solved for by integrating Equation (3) and using a similar equation for the bottom layer giving the following two sets of equations:

$$\frac{\sqrt{\pi }}{2\sqrt{a}}\left(K_{Icb}+K_{Ict}\right)=\left({\sigma }_2-{\sigma }_1\right){\sin }^{-1}\left(\frac{b_2}{a}\right)+\left({\sigma }_3-{\sigma }_1\right)\times {\sin }^{-1}\left(\frac{b_3}{a}\right)-\left({\sigma }_2+{\sigma }_3-2p_w\right)\frac{\pi }{2},$$

(6)

and

$$\frac{\sqrt{\pi }}{2}\left(K_{Icb}-K_{Ict}\right)=\left({\sigma }_2-{\sigma }_1\right)\sqrt{a^2-b_2^2}-\left({\sigma }_3-{\sigma }_1\right)\sqrt{a^2-b_2^2},$$

(7)

where $p_w$ is the treatment pressure at the wellbore (psi), ${\sigma }_1$ is the minimum horizontal in situ stress (psi), ${\sigma }_2$ is the intermediate principal stress (psi), and ${\sigma }_3$ is the minimum principal stress (psi). The treatment pressure at the wellbore is estimated by rearranging Equation (6). The height growth of the fracture can be computed using an iterative algorithm, which will be described in the following section after coupling the P3D model with the Carter Equation II.

2.2. Pseudo 3D Model Coupled with Carter Equation II

Rahim and Holditch [5] and Rahim [6] emphasized the importance of using 3D hydraulic propagation models. However, the majority of the fracture treatments are designed using 2D fracture propagation models. To be able to determine the fracture width, it is required to have an equilibrium condition between the pressure in the wellbore and the fracture using closed-form equations [12]. It is recommended to use the 2D PKN-C equations to compute for the maximum and average fracture width at the wellbore [25]:

$$w_f={9.15}^{\left(\frac{1}{2n+2}\right)}{3.98}^{\left(\frac{n}{2n+2}\right)}{\left[\frac{1+2.14n}{n}\right]}^{\left(\frac{n}{2n+2}\right)}{k}^{\left(\frac{1}{2n+2}\right)}{\left({\left(\frac{q_i}{2}\right)}^n\frac{h_f^{1-n}{t}_l}{E^{\prime }}\right)}^{\left(\frac{1}{2n+2}\right)},$$

(8)

and

$$\overline{w}=\frac{\mathsf{\pi }}{5}{w}_f,$$

(9)

where $E^{\prime }$ is the plane strain modulus (Pa), $q_i$ is the injection rate (m³/s), and $t_l$ is the target length (m). The fracture height growth estimation requires knowing the in situ stresses, fracture toughness above and below the payzone height, and thickness of the layer. In addition, net pressure can be estimated either from previous fracture treatments or from the output obtained from the P3D model [28]. The net fracture pressure is the difference between the fracturing fluid pressure and the closure pressure, and it refers to the driving mechanism behind the fracture growth [29]. The net fracture pressure is obtained from the following equation [26]:

$$p_{n,w}=\frac{E^{\prime }}{2h_f}{w}_f.$$

(10)

Within the proposed implementation of the P3D-C model in this study, the default fracture height was set to 101 ft. The initial estimate of the fracture half-height at the wellbore can be computed from Equation (4). $b_2$ can be computed numerically from Equation (7) using a root-finding method, such as the bisection method. In Equation (10), the net pressure was computed with the changes of the fracture height growth. Rahim and Holditch [28] presented typical net pressure values from field experience ranging between 300 to 2000 psi. It is important to note that the fracture height at any point in the fracture acts as a function of the net pressure. Following Rahman and Rahman [12], a user-defined tolerance can be set for the fracture height computations as:

$$Tolerance=\frac{Abs\left(p_{treat}-p_w\right)}{p_{treat}}\times 100,$$

(11)

where the treatment pressure of the fracturing fluid is computed from the following equation:

$$P_{treat}={\sigma }_1+P_{net}.$$

(12)

An iterative scheme is then implemented based on the following incremental change:

$$\Delta h_f=0.1\times h_f\left(i\right)$$

(13)

where $\Delta h_f$ is the deferential height change (ft). In addition, if $P_w$ < $P_{treat}$ then the new fracture height, $h_f\left(i+1\right),$ is expected to increase by:

$$h_f\left(i+1\right)=h_f\left(i\right)+\Delta h_f,$$

(14)

and if $P_w$ > $P_{treat}$ it would decrease by:

$$h_f\left(i+1\right)=h_f\left(i\right)-\Delta h_f.$$

(15)

Assuming a linear height variation along the fracture half-length, the fracture area is calculated as:

$$A_p=x_f\left(\frac{h_f+h}{2}\right).$$

(16)

For two fracture wings, the propped fracture area, $A_p,$ is given by:

$$A_p=x_f\left(h_f+h\right),$$

(17)

and the total surface area for two fracture wings, $A_f,$ would be:

$$A_f=4A_p.$$

(18)

The Carter Equation II gives a relationship between the fracture-half length, height, average fracture width, and injection time for a given injection rate [4]. This relationship can be extended further by adding the summation of the changing fracture height and the payzone height, leading to the following expression for the fracture half-length:

$$x_f=\frac{\left(\overline{w}+2S_p\right)}{4C_l^2\pi }\frac{q_i}{\left(h_f+h\right)}\left[exp\left({\beta }^2\right)erfc\left(\beta \right)+\frac{2\beta }{\sqrt{\pi }}-1\right],$$

(19)

and

$$\beta =\frac{2C_L\sqrt{\pi t_i}}{\overline{w}+2S_p},$$

(20)

where $S_p$ is the spurt loss (m), $C_l$ is the leak-off coefficient (m/$\sqrt{\mathrm{s}}$), $\beta$ is the auxiliary variable for the Carter II Equation, $t_i$ is the injection time (seconds), and $\overline{w}$ is the average fracture width (m). Finally, the fracture volume, $V_f$ (m³), is calculated from:

$$V_f=\frac{\pi }{2}\gamma \left(\frac{h_f+h}{2}\right)x_f{w}_f$$

(21)

2.3. Governing Equations for Fluid Leak-Off in a Dual-Porosity System

Fluid leak-off is investigated in this work for a dual-porosity system. In a hydraulic fracturing process, leak-off occurs when the influx of the fracturing fluid propagates to the adjacent natural fracture path layers. Ghaderi et al. [30] stated that the propagating dimension would be affected significantly by the leak-off rate. Here, we compare our P3D-C-DP model in a dual-porosity system against a single-porosity system (P3D-C model) and analyze the results in terms of the induced fracture aperture, width, fracture half-length, and height. Ghaderi et al. [30] further mentioned that the fracture and matrix permeabilities play an important role in linking the leak-off phenomena in a naturally fractured reservoir. There are three main concepts to consider for a hydraulic fracture treatment in naturally fractured reservoirs:

• fluids penetrating into the matrix;
• fluids flowing into natural fractures;
• the transfer of fluids from natural fractures into the matrix.

Wang et al. [13] presented an equivalent Carter leak-off coefficient for a dual medium implemented in the PKN-C model. Following Wang et al. [13], the equivalent Carter leak-off approach was used in this work and implemented in a P3D-C model for dual-porosity systems. The P3D-C model applied in this study accounts for fracturing fluids that exhibit a non-Newtonian behavior. The 1-D single-porosity leak-off perpendicular to the hydraulic fracture was compared to the dual-porosity leak-off. The governing equations follow the same set of equations in Wang et al. [13], with the exception of the extension to P3D-C models. During the first stage of initiating the hydraulic fracture treatment, it is assumed that the natural fractures remain sealed and unchanged. However, when the treatment pressure increases, it would lead to dilating and reactivating the natural fractures. When the natural fractures open, it is required to calculate the critical fissure opening pressure, $p_c$, given by:

$$p_c=\frac{1}{{\alpha }_f}\left[\left(2vp+{\sigma }_{max}\right){\sin }^2\theta +\left(p+{\sigma }_{min}\right){\cos }^2\theta \right],$$

(22)

where $p$ is the net pressure in the hydraulic fracture, and $\theta$ is the angle between the hydraulic fracture and the natural fracture. Furthermore, the value of the critical fissure opening pressure and the treatment pressure determines which equation to use for the hydraulic aperture of the natural fracture (m),$b$, using the following criterion:

$$b=\{\begin{array}{ll}{b}_0& p_f<p_c\\ b_{0}exp\left(-\chi {\sigma }_n^e\right)& p_f>p_c\end{array},$$

(23)

where $b_0$ is the initial natural fracture aperture, $\chi$ is a relatively insignificant constant that describes the compliance of a natural fracture with regard to a change in pressure, and ${\sigma }_n^e$ is the effective normal stress. The dynamic porosity,${\varphi }_f,$ and permeability, $k_f,$ of the natural fracture system are calculated, respectively, from [31,32]:

$${\varphi }_f={\varphi }_{f0}\frac{b}{b_0},$$

(24)

and

$$k_f=k_{f0}{\left(\frac{b}{b_0}\right)}^3\left[\begin{array}{ll}{\cos }^2\theta & \sin \theta \cos \theta \\ \sin \theta \cos \theta & {\sin }^2\theta \end{array}\right].$$

(25)

Assuming a constant leak-off coefficient, the fluid leak-off velocity directly penetrating into the matrix can be written as follows:

$$u_1\left(x,t\right)=\frac{2C_l^m}{\sqrt{t-\tau \left(x\right)}},$$

(26)

and

$$C_l^m=\Delta p\sqrt{\frac{k_m{\varphi }_m}{\mu \pi K_w}},$$

(27)

where $C_l^m$ is the matrix leak-off coefficient (m/$\sqrt{sec}$), $t$ is the total injection time ($sec$), $\tau \left(x\right)$ is the arrival time ($sec$) of the fracture tip at location x, $k_m$ is the matrix permeability (m²), ${\varphi }_m$ is the matrix porosity, $K_w$ is the fluid bulk modulus (Pa), and $\Delta p$ is the pressure differential between the hydraulic fracture and the formation (Pa). The fluid bulk modulus is computed from the following mathematical expression:

$$K_w=\frac{E}{3\left(1-2v\right)}.$$

(28)

The differential pressure, $\Delta p,$ is computed as:

$$\Delta p=p_{fluid\left(x,t\right)}-p_0$$

(29)

where $p_{fluid\left(x,t\right)}$ is the treatment pressure (Pa) and $p_0$ is the initial formation pressure (Pa). In addition, the dimensionless time scaling is given by [13]:

$$\tau =\frac{t}{t_{*}},$$

(30)

and the characteristic quantity, $t_{*}$ is computed as:

$$t_{*}=\frac{{\pi }^2{H}^6{p}_{*}{}^5}{4{E^{\prime }}^4\mu \pi Q_o{}^2},$$

(31)

where $p_{*}$ is a characteristic net pressure (Pa). Assuming that the fluid transfer from the natural fractures to the matrix is ignored, then the fluid leak-off velocity flowing into the natural fractures can be approximated by [13]:

$$u_2\left(x,t\right)=\frac{2C_l^f}{\sqrt{t-\tau \left(x\right)}},$$

(32)

and

$$C_l^f=\Delta p\sqrt{\frac{k_f\left(p\right){\varphi }_f}{\mu \pi K_w}}.$$

(33)

The fracture leak-off coefficient, $C_l^f,$ is considered a varying parameter due to the fracture permeability, $k_f\left(p\right),$ being a function of net pressure, as shown in Equation (25). In addition, the net pressure varies slowly with time, $t$, and location, $x$. Therefore, the use of an average net pressure, $\overline{p}$, is adopted for simplicity [13]. Furthermore, when the fluid transfer from the natural fractures to the matrix is not ignored, then the fluid leak-off velocity can be approximated by:

$$u_3\left(x,t\right)=\frac{8v_{tr}{C}_l^f\sqrt{t-\tau \left(x\right)}}{b\left(\overline{p}\right)},$$

(34)

where the fluid leak-off velocity from the natural fractures to the matrix, $v_{tr}$ (m/s), using the fracture spacing, $s$ (m), is given by:

$$v_{tr}=\frac{k_m}{\mu }\frac{\Delta p}{s/2}.$$

(35)

The total fluid leak-off velocity introduced by Wang et al. [13] can then be approximated by:

$$u\left(x,t\right)=u_1\left(x,t\right)+u_2\left(x,t\right)+u_3\left(x,t\right)=\frac{2C_l{}^e}{\sqrt{t-\tau \left(x\right)}},$$

(36)

where the equivalent Carter leak-off coefficient, $C_l{}^e,$ is expressed by:

$$C_l{}^e=C_l{}^m+C_l{}^f+\frac{8k_m\Delta p{C}_l{}^f\left[t-\tau \left(x\right)\right]}{s\mu b\left(\overline{p}\right)}.$$

(37)

The Carter Equation II, Equation (19), with the equivalent leak-off coefficient from Equation (37) predicts the fracture half-length as the fracture propagates.

3. Numerical Results

3.1. Model Validation

Wang et al. [13] presented an equivalent Carter leak-off coefficient into a pressure-sensitive dual medium. The model was used to study the interaction between the natural fractures and the induced fracture, and it was treated as a dual-porosity medium. The 1-D single-porosity leak-off perpendicular to the hydraulic fracture was compared with the dual-porosity leak-off. In our work, we have developed two codes: (1) PKN-C-DP and (2) P3D-C-DP. Our work extends Wang et al.’s [13] work to the P3D model. The PKN-C-DP code was validated with the same input parameters used in Wang et al. [13]. The algorithm for the P3D-C-DP model is shown in Figure 2 and

Table 1. Input parameters for model validation.

Input Design Parameters	Value
Injection rate, $Q_i$, m3/s	$0.004$
Fracture height, $h_f$, m	$10$
Plain strain Young’s modulus, $E^{\prime }$, GPa	$25$
Poisson’s ratio, $v$	$0.2$
Fluid dynamic viscosity, $\mu$, Pa.s	$0.2$
Fluid bulk modulus, $K_w$, GPa	$2.2$
Permeability of the natural fracture system, $k_f$, m2	$1.5\times {10}^{-17}$
Initial permeability of the natural fracture system, $k_{f0}$, m2	$3.25\times {10}^{-15}$
Initial porosity of the natural fracture system, ${\varphi }_{f0}$	$0.002$
Fracture spacing, $s$, m	$0.2$
Biot coefficient, ${\alpha }_f$	$0.8$
Permeability of the matrix, $k_m$, m2	$1.48\times {10}^{-17}$
Porosity of the matrix, ${\varphi }_m$	$0.06$
Initial formation pore pressure, $p_0$, MPa	$6$
Minimum in situ principal stress, ${\sigma }_{min}$, MPa	$15$
Maximum in situ principal stress, ${\sigma }_{max}$, MPa	$16$
Angle between NFs and HF, $\theta$	$\pi /2$

presents the input parameters used for the model validation [12].

The pressure difference between the hydraulic fracture and the formation is needed to compute the fluid leak-off coefficient in both the matrix and the fracture. Using a non-dimensional time, Wang et al. [13] ensured that the simulation progressed from a small-time to a large-time solution. Comparing the small-time asymptote at the beginning of the simulation, Wang et al. [13] found the storage dominated regime occurs at the beginning of the simulation. As a result, the fracture’s development stretches to the end of the leak-off dominated regime at a large time asymptote. Equation (31) was utilized as a basis for validation, but since the resultant dimensionless time value was larger than 1, it could only be employed in the leak-off-dominated regime. This validation neglected the storage-dominated regime. The model did not analyze fracture geometry parameters after shut-in for a 3000-second injection time.

Table 2. Model validation results for the PKN-C-DP code.

Parameters	PKN-C-DP Code Results
Fracture half-length, $x_f$, m	64.75
Average fracture width, $w_{avg}$, m	0.0025
Maximum fracture width, $w_{max}$, m	0.0039
Total pumping time, $t_e$, seconds	$3000$
Net pressure, $P_{net}$, MPa	$4.90$

shows the results after a 3000-second injection time.

Figure 3a,b provide a comparison of model validation in the dual-porosity medium with the Wang et al. [13] model. Figure 3a showcases the average width (mm) changes as a function of total pumping time. The validation model and the observed average width changes at the inlet are very similar. Figure 3a also represents the fugitive fluid leak-off from the fracture into the surrounding formation, which eventually leads to a reduction in the maximum width. This is due to the interaction between the induced hydraulic fracture and the natural fracture. The average fracture width at the middle region does not exactly match Wang et al.’s [13] model, which could be due to a variety of factors including: differences in the way the time function is constructed, use of non-Newtonian equations, or variation of the target length value used. Leak-off in the dual-porosity medium is expected to impact fracture geometry during treatment by reducing the fracture length. Figure 3b shows a validated example of fracture length (m), and it closely matches Wang et al.’s [13] model.

3.2. Numerical Examples

Table 3. Reservoir and well data for a single-porosity system.

Input Design Parameters	Value
Spurt loss, $S_p$, in	$0.0$
Fluid viscosity, $\mu$, cP	$250$
Pumping rate, $q_i,$ bbl/min	$20$
Wellbore radius, $r_w$, ft.	$0.35$
Fracture height, $h_f$, ft.	$100$
Target length, $t_l$, ft.	$1968.5$
Young’s modulus, $E$, psi	$5.075\times {10}^6$
Fluid leak-off coefficient, $C_L$, ft/min0.5	$0.00025$
Max. horizontal stress, ${\sigma }_H$, psi	$7000$
Min. horizontal stress, ${\sigma }_1$, psi	$6000$
Min. horizontal stress (shale), ${\sigma }_2$, psi	$6700$
Min. horizontal stress (shale), ${\sigma }_3$, psi	$7200$
Fracture toughness, $K_{Icb}$, psi-in0.5	$1700$
Fracture toughness, $K_{Ict}$, psi-in0.5	$1500$
Porosity	$0.1$
Permeability, $k$, mD	$0.1$
Spurt loss, $S_p$, in	$0.0$

presents a reservoir well data for a single-porosity system. Here, the P3D-C and PKN-C models were used to compare single and dual-porosity systems (PKN-C-DP and P3D-C-DP) utilizing previously published numerical example data [12].

Both single- and dual-porosity system models used the same initial set of input parameters.

Table 4. Additional input parameters for the dual-porosity system.

Input Design Parameters	Value
Fluid bulk modulus, $K_w$, psi	$4.1\times {10}^{11}$
Initial permeability of the natural fracture system, $k_{f0}$, mD	$3.3$
Initial porosity of the natural fracture system, ${\varphi }_{f0}$	$0.002$
Fracture spacing, $s$, ft.	$0.66$
Biot coefficient, ${\alpha }_f$	$0.8$
Permeability of the matrix, $k_m$, mD	$0.1$
Porosity of the matrix, ${\varphi }_m$	$0.1$
Angle between NFs and HF, $\theta$	$\pi /2$

presents additional input parameters for computing the equivalent Carter leak-off coefficient [13].

3.2.1. Fracture Geometry Propagation

The fracture geometry parameters include the fracture half-length, maximum fracture width at the wellbore, and average fracture width. Figure 4a,b present the fracture half-length evolution for the single- and dual-porosity systems using PKN-C and P3D-C, respectively. The fracture half-length, in Figure 4a, for the dual-porosity system, is approximately 653 ft less (21% shorter) than it in the single-porosity system. This is expected since the fluid leak-off is much larger in the dual medium. The fracture half-length predicted by the P3D-C model for a single-porosity system in Figure 4b gives a larger fracture half-length when compared to Figure 4a. However, the fracture half-length in the P3D-C-DP is 150 ft less than that of the PKN-C-DP model and exhibits a 31% reduction when compared to the P3D-C model. This is because the fracture half-length and fracture height obtained from the P3D-C and P3D-C-DP models are inversely proportional to each other [12]. Both Figure 4a,b depict a common significant decrease when the dual-porosity-medium system is introduced. This is due to the fracture permeability, where it exerts a significant amount of fluid pressure, which in turn results in a much reduced length of the propagating fracture.

Figure 5a,b indicate that the PKN-C model in the single-porosity system predicts a slightly greater average fracture width and maximum fracture width at the wellbore. In contrast, lower values are obtained from the PKN-C-DP model. This is because a higher leak-off coefficient results in a lower build-up pressure, leading to a narrower fracture width [33]. Similarly, when comparing the single-porosity system to the dual-porosity system, we find that more fluid leak-off occurs in the dual porosity medium resulting in a smaller maximum and average fracture width. Thus, the fracture for the dual-porosity leak-off shrinks and becomes more confined in size compared to the fracture for the single-porosity leak-off.

Figure 6a,b depict the average fracture width and the maximum fracture width at the wellbore for the single and dual mediums, respectively. The average fracture width in the single porosity medium is 12% greater than that in the dual medium. Similarly, the maximum fracture width at the wellbore in the single medium is 11% greater than in the dual medium. In the dual-medium setting, the fluid leak-off has a particularly strong impact and as fluid leak-off increases, the fracture width decreases. The average and maximum values for fracture width in the P3D-C and P3D-C-DP models are similar to the findings obtained from the PKN-C and PKN-DP models. It is important to note that the intersection angle between the natural fractures and the hydraulic fracture affects the fracture length and width; a larger angle causes the fracture width and length to be reduced [13]. In this work, however, the angle was kept constant at $\pi /2$.

3.2.2. Fracture Height Growth

The fracture height in the single-porosity system in Figure 7 tends to grow linearly with time as it reaches 188 ft. However, there is a significant contrast in the P3D-C dual-porosity model, where the fracture height is 40% shorter compared to the single-porosity system. In contrast to the P3D-C model, where the fracture height propagation grows more slowly due to the constant leak-off coefficient used in the single-porosity system, the P3D-C-DP coupled with fluid leak-off equation for the dual-porosity system displays a steady increase in terms of the fracture height propagation. However, in the dual-porosity system, when the equivalent Carter leak-off equation is introduced, both the matrix and fracture leak-off coefficients influence fracture height growth and since the equivalent Carter leak-off velocity is substantial, the resulting fracture height growth is limited. This is because a higher fluid leak-off contributes to more confined fracture height and limits its growth.

3.2.3. Net Fracture Pressure

The net fracture pressure is required to provide the energy needed for fracture propagation. Figure 8a,b present the net fracture pressure changes for the single and dual mediums using the PKN-C and P3D-C models, respectively. As a consequence of fugitive fluid leak-off into the surrounding formation, the net fracture pressure in the dual media decreases, which, in turn, decreases the maximum fracture width [13]. This is because the maximal fracture width at the wellbore is directly proportional to the net fracture pressure. When compared to the 2D PKN model, the P3D model’s net fracture pressure values are lower, since they are based on the average fracture height. In all models, the net fracture pressure in the single-porosity system is greater than that in the dual-porosity medium.

3.3. Sensitivity Analysis

A series of sensitivity analyses were performed to examine the effect of varying the fracture height and half-length. Figure 9a,b illustrate the effect of changing the fracture height on the fracture half-length in the PKN-C and PKN-C-DP models, respectively. Naturally, a larger fracture height contributes to a reduced fracture half-length.

Figure 10a,b present the impact of changing the fracture half-length on the fracture height growth; a higher fracture height growth is achieved in the single porosity P3D-C model in Figure 10a, whereas the fracture height growth is more confined in the dual-porosity system due to the fugitive fluid leak-off that limits the fracture height growth and results in a more confined fracture height in Figure 10b.

4. Conclusions

In this work, a semi-analytical model to evaluate and interpret the hydraulic fracture propagation in dual-porosity reservoirs was developed. The model entailed the coupling of the so-called equivalent Carter leak-off coefficient for dual-porosity models in a P3D approach. The specific findings from this work are listed below.

• The model was successfully validated against a case study conducted on fluid leak-off in a dual-porosity system.
• Generally, the fracture geometry parameters ($x_f,h_f,w_f,\overline{w}$) in the single-porosity models are significantly larger when compared to the dual-porosity models, and this is primarily due to the effect of fluid leak-off into the natural fractures.
• The equivalent Carter leak-off coefficient in a dual-porosity system is directly related to the permeability of the natural fractures; higher permeability values accelerate the fluid leak-off velocity, which ultimately affects the fracture geometry propagation and limits the fracture growth.
• For a given injection time in the PKN-C-DP model, the effect of increasing the fracture height yields shorter fracture half-lengths. Furthermore, the confinement of fracture height in the P3D-C-DP model increases the fracture half-lengths.
• The simulation findings show that fluid leak-off in a dual-porosity system contributes to a constrained fracture half-length ($x_f$), with $x_f$ being 21% and 31% smaller in PKN-C-DP and P3D-C-DP models, respectively, compared to the single-porosity model. The P3D-C-DP model shows a 40% reduction in fracture height growth ($h_f$) compared to the single-porosity model (P3D-C).