Shape factor for regular and irregular matrix blocks in fractured porous media
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Abstract
Describing matrix–fracture interaction is one of the most important factors for modeling natural fractured reservoirs. A common approach for simulation of naturally fractured reservoirs is dual-porosity modeling where the degree of communication between the low-permeability medium (matrix) and high-permeability medium (fracture) is usually determined by a transfer function. Most of the proposed matrix–fracture functions depend on the geometry of the matrix and fractures that are lumped to a factor called shape factor. Unfortunately, there is no unique solution for calculating the shape factor even for symmetric cases. Conducting fine-scale modeling is a tool for calculating the shape factor and validating the current solutions in the literature. In this study, the shape factor is calculated based on the numerical simulation of fine-grid simulations for single-phase flow using finite element method. To the best of the author’s knowledge, this is the first study to calculate the shape factors for multidimensional irregular bodies in a systematic approach. Several models were used, and shape factors were calculated for both transient and pseudo-steady-state (PSS) cases, although in some cases they were not clarified and assumptions were not clear. The boundary condition dependency of the shape factor was also investigated, and the obtained results were compared with the results of other studies. Results show that some of the most popular formulas cannot capture the exact physics of matrix–fracture interaction. The obtained results also show that both PSS and transient approaches for describing matrix–fracture transfer lead to constant shape factors that are not unique and depend on the fracture pressure (boundary condition) and how it changes with time.
Keywords
Fractured reservoirs Shape factor Matrix–fracture boundary conditions Computational fluid dynamic (CFD)1 Introduction
Peaceman (1976) evaluated gas and oil transfer between the fracture and rock matrix. They obtained a shape factor of 12/L^{2}, 14.23/L^{2} and 16.53/L^{2} for one-, two- and three-dimensional cubic matrix blocks and concluded that values of shape factors do not increase much with an increase in the number of normal sets of fractures. Thomas et al. (1983) presented an expression for the shape factor using another version of a fully implicit three-dimensional, multiphase naturally fractured simulator based on the dual-porosity approach that was validated by multiphase flow numerical simulations. They multiplied the matrix phase relative permeability values by the fracture phase saturations to include the effect of block coverage and developed a 3D, three-phase model for the shape factor by simulating the flow of fluids in a naturally fractured reservoir.
Ueda et al. (1989) used Kazemi’s model to calculate the shape factor and compared their results with fine-grid simulation results. They believed that Kazemi’s shape factor values for one- and two-dimensional matrix blocks are not suitable and need to be adjusted by a factor of 2 and 3, respectively. Coats (1989) solved the diffusivity equation to give a shape factor of 12/L^{2}, 28.45/L^{2} and 49.58/L^{2} for one-, two- and three-dimensional cubic matrix block, respectively, for single-phase flow based on the PSS assumption.
Several other studies in the area of naturally fractured reservoir simulation are devoted to analytically and numerically representing an accurate matrix–fracture transfer function (Bourbiaux et al. 1999; Coats 1989; Noetinger and Estebenet 2000; Noetinger and Estébenet 1998; Penuela et al. 2002a, b; Quintard and Whitaker 1996; Sarda et al. 2001).
Quintard and Whitaker (1996) also used the volume-averaging technique to determine the shape factor values for 1D, 2D and 3D matrix–fracture transfers in a single-phase flow. Their results were exactly the values obtained by Coats (1989). Bourbiaux et al. (1999) present different approaches to determine matrix–fracture transfer behavior. They used three methods involving either (1) local-scale flow computations, (2) the application of upscaling theory and (3) implementation of the particle random walk method to obtain the best approximate expression of the shape factor. They obtained the same expression for the shape factor, 20/a^{2} based on a PSS formulation of 2D matrix–fracture transfers in a single-phase fine-grid simulation where “a” is the lateral dimension of the matrix block. Noetinger and Estebenet (2000) used the continuous-time random walk technique (CTRW) to calculate the shape factor between the matrix and the fracture and compared the obtained shape factor from this technique with numerical simulation and found a good agreement.
It is showed that the shape factor also depends on the way the pressure changes in the fracture. They compared several methodologies presented by these and other authors (Hassanzadeh and Pooladi-Darvish 2006; Hassanzadeh et al. 2009). Mora and Wattenbarger (2009) used numerical simulation to obtain a correct shape factor formula for various cases. They concluded that some of the most popular formulas do not seem to be correct. Hatiboglu and Babadagli (2007) studied the effect of several rocks and fluid properties such as matrix shape factor, wettability, oil viscosity and some other parameters on the rate of capillary imbibition and development of residual non-wetting phase saturation, experimentally. They used several different core samples with different shape factors to evaluate the rate of imbibition. The effect of the fracture pressure depletion regime on the shape factor for single-phase flow of a compressible fluid was investigated by Ranjbar et al. (2011). Their investigations demonstrated that the shape factor is a function of the imposed boundary conditions in the fracture and its variability with time. They used single-porosity, fine-grid, numerical simulations to verify their presented semi-analytical model for estimating the shape factor. The dependency of shape factor on the other parameters such as gas specific gravity and temperature is investigated in their study. Saboorian-Jooybari et al. (2012) developed a new time-dependent matrix–fracture shape factor to diagnose different states of the imbibition process. They obtained an analytical solution for fluid saturation distribution within a matrix block by solving capillary-diffusion equation under different boundary conditions. They used the single-porosity fine-grid simulations and the previous experimental data presented by other authors to verify their solutions and concluded that the shape factor is completely phase sensitive that is the important parameter in diagnosing different states of imbibition process. A time-dependent matrix–fracture shape factor formulation is analytically derived for two-phase flow in a three-dimensional matrix block in the imbibition process which considers both capillary and gravity forces on matrix–fracture coupling. They verified their results by a fine-grid simulation model (Saboorian-Jooybari et al. 2015). Wang et al. (2018) developed a time-dependent shape factor for single-phase flow by considering the stress sensitivity in the matrix system. They performed a fine-grid finite element numerical model to validate the accuracy of the new analytical model. Their results showed that the stress sensitivity coefficient of permeability has a great influence on the stabilized value of the matrix–fracture shape factor for a compressible formation.
Summary of the shape factor constants σL^{2} found in the literature based on the solution of the diffusivity equation with constant fracture pressure
(updated after Hassanzadeh and Pooladi-Darvish 2006)
References | N = 1 | N = 2 | N = 3 | Approach | PSS/Transient |
---|---|---|---|---|---|
Warren and Root (1963) | 12 | 32 | 60 | Numerical | PSS |
Kazemi et al. (1976) | 4 | 8 | 12 | Numerical | PSS |
Peaceman (1976) | 12 | 14.23 | 16.53 | Numerical | PSS |
Thomas et al. (1983) | – | – | 25 | Numerical | Transient |
Ueda et al. (1989) | 8 | 24 | – | Numerical | PSS |
Coats (1989) | 12 | 28.45 | 49.58 | Analytical | PSS |
de Swaan (1990) | 12 | – | 60 | Numerical | PSS |
Zimmerman et al. (1993) | 9.87 | 19.74 | 29.61 | Numerical | PSS |
Kazemi and Gilman (1993) | 9.87 | 19.74 | 29.61 | Analytical | Transient |
Chang (1993) | 9.87 | 19.74 | 29.61 | Numerical | PSS |
Lim and Aziz (1995) | 9.87 | 19.74 | 29.61 | Analytical | Transient |
Quintard and Whitaker (1996) | 12 | 28.4 | 49.6 | Averaging | Transient |
Bourbiaux et al. (1999) | – | 20 | – | Numerical | PSS |
Noetinger and Estebenet (2000) | 11.5 | 27.1 | – | Random walk technique | Transient |
Sarda et al. (2001) | 8 | 24 | 48 | Numerical | Transient |
9.87 | – | – | Numerical | Transient | |
Hassanzadeh and Pooladi-Darvish (2006) | 9.87 | 18.2 | 25.56 | Analytical | PSS |
Mora and Wattenbarger (2009) | 9.87 | 18.17 | 25.67 | Numerical | PSS |
Hassanzadeh et al. (2009) (constant rate) | 12 | 25.13 | 38.9 | Analytical | PSS |
In this study, the behavior of the shape factor between a matrix and fractures for different boundary conditions of fracture pressure for regular and irregular shaped blocks under assumptions of PSS and the transient flow regime is investigated. As most of the studies in the literature are based on single-phase flow, this study focuses on this type of flow to compare the observed trend with other studies in terms of shape factor calculation. For this purpose, we first clarified what we mean by PSS and transient approach. For both cases, several simulations were conducted in several sections. At first, the shape factor was calculated for the standard shaped blocks (symmetric and regular shapes) for a case where the fracture pressure is constant which is normally used in the literature. Then, some pressure depletion schemes were used as the fracture boundary condition on the standard shape. Finally, different boundary conditions were performed for irregular shapes.
2 Numerical model and methodology
There are a lot of debates on how to calculate shape factors in the literature. The two most commonly used equations originate from Warren and Root (1963) and Kazemi et al. (1976). In this study, several different shapes for three-dimensional matrix blocks were considered to calculate the shape factor. Different boundary conditions including (1) constant fracture pressure, (2) linearly declining fracture pressure and (3) exponentially declining fracture pressure were used in the models. The finite element method was used to generate and simulate all models. The calculated shape factors from fine-scale numerical models were compared with the known analytical and numerical values that were reported by others.
To simulate matrix–fracture drainage flow, the matrix blocks are surrounded by fractures. Then, the pressure difference between the matrix and the fracture is assigned based on the boundary conditions. Thus, the reservoir fluid will flow from the matrix into the fracture. Sufficiently large simulation time is selected for simulations until the matrix and fracture systems reach pressure equilibration. Finally, for all models, the dimensionless shape factors are plotted against dimensionless time.
A general numerical technique (finite element method) was proposed to calculate the shape factor for any arbitrary shape of the matrix block (i.e., non-orthogonal fractures) for both transient and PSS considering different boundary conditions. Using the finite element method and by defining irregular shapes, we were able to implement different pressure trends as boundary conditions. Therefore, linear and exponential forms of pressure as a function of time for the boundary conditions were implemented for different models.
2.1 Mathematical method
- For constant fracture pressure,$$P = P_{\text{f}} \quad {\text{at}}\;x = 0\;{\text{and}}\;y = 0\;{\text{and}}\;z = 0$$(8)
- For constant fracture pressure followed by exponentially declining pressure,$$P_{\text{f}} = \left( {P_{\text{i}} -\Delta P_{0} } \right)\exp \left( { - \alpha t} \right)$$(9)
- For constant fracture pressure followed by linearly declining pressure,$$P_{\text{f}} = \left( {P_{\text{i}} -\Delta P_{0} } \right)\left( {1 - \beta t} \right)$$(10)
2.2 Pseudo-steady-state (PSS) shape factor
Equation (14) was used by Warren and Root (1963) to model the transfer function between the matrix and the fracture in a naturally fractured reservoir.
Equation (17) was used to find the shape factor from the numerical simulation by the computational fluid dynamic (CFD) simulation. For this purpose, all parameters are known in Eq. (17) except the matrix pressure. Using the calculated pressure profile in the matrix, one can calculate σ.
2.3 Transient shape factor
2.4 Mesh independency study
Mesh independency results for the stabilized shape factor
Grid No. | Mesh size | Number of elements | Average growth rate | Stabilized shape factor | Relative error E_{r}, % |
---|---|---|---|---|---|
1 | Extra fine | 397,005 | 1.606 | 4.23 | – |
2 | Fine | 32,189 | 1.626 | 4.17 | 1.5 |
3 | Coarse | 4733 | 1.647 | 5.09 | 18.2 |
3 Results and discussion
The rate of mass transfer from the matrix to the fracture is directly proportional to the shape factor. For modeling naturally fractured reservoirs, an accurate value of the shape factor is required for both the transient and PSS behavior and also the geometry of the matrix–fracture system.
In general, there are two models to consider the matrix and fracture interaction including PSS and transient transfer. The former model ignores the pressure transient in the matrix while the latter model accounts for the pressure transient in the matrix. In this study, both PSS and transient transfer have been evaluated.
3.1 PSS shape factor, standard shaped matrix blocks
3.1.1 Constant fracture pressure
Figure 4 indicates new stabilized values of 4.17, 9.7 and 16.17 for the shape factor of different matrix block shapes of the slab, cylindrical and spherical, respectively, at large dimensionless times and shows that the results are in the same range that other researchers reported (i.e., Kazemi et al. 1976; Peaceman 1976). It is shown that the matrix–fracture transfer shape factor depends on the matrix block shape and how it changes with time. All of these are derived with an assumption of PSS.
3.1.2 Variable fracture pressure
Results show that the shape factor depends on the fracture pressure and how it changes with time. It is found that the linearly declining pressure depletion scheme leads to 4.53, 11 and 20.2 for the slab, cylindrical and spherical shapes, respectively. By comparison of Figs. 4 and 5, it can be seen that the matrix–fracture transfer shape factor depends on the matrix block shape, the pressure regime in the fracture and how it changes with time.
These figures demonstrate that the presented model can reproduce the slightly compressible fluid shape factor with acceptable accuracy.
In this section, the behavior of the shape factor for different fracture pressure depletion in different shapes of matrix blocks (i.e., slab, cylindrical and spherical) is also described.
Shape factor constants for different geometry matrix blocks subject to different boundary conditions under the PSS assumption
Boundary conditions | Shape factor constants σL^{2} under the PSS assumption | ||
---|---|---|---|
1D flow (slab) | 2D flow (cylindrical) | 3D flow (spherical) | |
Constant fracture pressure | 4.17 | 9.70 | 16.70 |
Exponential, α = 1 | 4.18 | 9.74 | 16.60 |
Exponential, α = 0.0001 | 4.98 | 12.10 | 22.10 |
Exponential, α = 0.01 | 4.22 | 9.90 | 17.20 |
Linear decline β = 0.001 | 4.53 | 11.00 | 20.20 |
3.2 Transient shape factor, standard shaped matrix blocks
A precise value of the shape factor at the transient state is essential to consider the performance of the matrix–fracture interaction. To more precisely understand the physics of flow behavior, the shape factor was also evaluated using the transfer function (Eq. 21).
Shape factor constants for different geometry matrix blocks subject to different boundary conditions under the transient transfer assumption
Boundary conditions | Shape factor constants σL^{2} under the transient transfer assumption | ||
---|---|---|---|
1D flow (slab) | 2D flow (cylindrical) | 3D flow (spherical) | |
Constant fracture pressure | 10.0 | 23.3 | 39.3 |
Exponential, a = 1 | 10.0 | 23.3 | 39.9 |
Exponential, a = 0.0001 | 11.9 | 28.8 | 52.5 |
Exponential, a = 0.01 | 10.1 | 23.7 | 40.9 |
Linear decline β = 0.001 | 10.0 | 23.3 | 48.2 |
The above results show that both the transient and PSS values of the single-phase shape factor depend on the geometry and how the fracture pressure changes with time.
3.3 Shape factor for 3D irregular shapes and three-dimensional flow
Shape factor constants for irregular geometry matrix blocks subject to different boundary conditions under the PSS and transient transfer assumption
Boundary condition | Shape factor constants σL^{2} | |||
---|---|---|---|---|
Prism shape, 3D flow | Complex shape, 3D flow | |||
PSS | Transient | PSS | Transient | |
Constant fracture pressure | 11.4 | 26.4 | 10.5 | 25.6 |
Exponential, a = 1 | 11.5 | 27.7 | 11.1 | 26.1 |
Exponential, a = 0.0001 | 17.4 | 41.4 | 16.5 | 38.3 |
Exponential, a = 0.01 | 12.6 | 29.8 | 12.0 | 28.1 |
Linear decline β = 0.001 | 16.6 | 38.9 | 15.6 | 36.0 |
3.4 Shape factor for three-dimensional flow in regular and irregular shaped blocks
In this section, the results of the calculated shape factor for different boundary conditions of pressure of the fractures are shown. Results are for both regular and irregular shaped matrix blocks.
Stabilized shape factors in three-dimensional flow under the PSS assumption
Boundary conditions | Shape factor constants σL^{2} in 3D flow | ||||
---|---|---|---|---|---|
Cube | Cylindrical | Spherical | Prism | Complex | |
Constant fracture pressure | 12.4 | 13.8 | 16.7 | 11.4 | 10.5 |
Exponential, α = 1 | 12.3 | 13.9 | 16.6 | 11.5 | 11.1 |
Exponential, α = 0.0001 | 17.8 | 19.2 | 22.1 | 17.4 | 16.5 |
Exponential, α = 0.01 | 12.9 | 14.4 | 17.2 | 12.6 | 12.0 |
Linear decline β = 0.001 | 15.6 | 17.2 | 20.2 | 16.6 | 15.6 |
Stabilized shape factors in three-dimensional flow under the transient transfer assumption
Boundary conditions | Shape factor constants σL^{2} in 3D flow | ||||
---|---|---|---|---|---|
Slab | Cylindrical | Spherical | Prism | Complex | |
Constant fracture pressure | 29.9 | 33.0 | 39.3 | 26.4 | 25.6 |
Exponential, α = 1 | 30.0 | 33.4 | 39.9 | 27.7 | 26.1 |
Exponential, a = 0.0001 | 41.8 | 45.5 | 52.5 | 41.4 | 38.3 |
Exponential, α = 0.01 | 30.7 | 34.1 | 40.9 | 29.8 | 28.1 |
Linear decline β = 0.001 | 36.8 | 40.6 | 48.2 | 38.9 | 36.0 |
The results show that the shape factor is different for regular and irregular shapes and flow regimes as it is a function of time. However, having an idea about the shape and size of matrix blocks (i.e., from FMI log, geomechanical study, etc.) can help reservoir engineers to estimate a realistic range of shape factor. Both transient and PSS models can help us to determine the upper and lower limits for shape factor when the shape factor is considered as a matching parameter.
4 Comparison with existing models
5 Conclusions
- 1.
In this paper, the value of the shape factor is calculated for different geometries and under transient and PSS assumptions in a systematic approach. Using fine-scale numerical simulation, it has been shown that the matrix–fracture shape factor for a single-phase flow of slightly compressible fluid illustrates a period with a decreasing value and then stabilizes to a stable value. This is true for both regular and irregular shapes.
- 2.
Based on the pressure depletion regime in the fracture, the stabilized value of the shape factor varies between two limits. The upper limit is obtained for an exponentially declining fracture pressure with a decline exponent of 0.0001 which corresponds to a slow pressure depletion regime. The lower limit is derived for the constant fracture pressure boundary conditions where depletion takes place faster.
- 3.
The shape factor values calculated from PSS pressure behavior for the one- and two-dimensional blocks have a good agreement with those proposed by Kazemi et al. (1976). Reasonable agreement is also found between the obtained values for the three-dimensional block and that presented by Peaceman (1976).
- 4.
It has also been shown that the depletion time of a matrix block is a function of the fracture pressure depletion regimes. In the case of constant fracture pressure or exponential decline with a large exponent, the block is depleted faster than that in the linear decline and the exponential decline with a small exponent. The same behavior has been reported for a slightly compressible fluid by Chang (1993) and Hassanzadeh and Pooladi-Darvish (2006).
- 5.
The shape factor is estimated and made dimensionless with characteristic length, σL^{2}, for irregular shapes where there is no analytical solution for them. It is shown that dimensionless shape factors for irregular shapes are closer to those of the slab case. This solution facilitates accurate simulation of oil transfer between the matrix and fracture in fractured reservoirs.
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