Fractured reservoir system holds a major oil resource of global oil reserve, which makes fractured reservoir study a hot and exciting topic in oil-gas field development. Fractured reservoirs consist of two continuums: matrix and fractures. The matrixcontains higher pore/fluid volume with low permeability while fractures have high permeability with less pore/fluid volume. The difference between the matrix and fracture permeabilityleads to early water breakthrough and poor oil recovery. In water-wet fractured reservoirs, counter-current imbibition by the capillary pressure can give significant rise to oil recoveries. Counter-current imbibition is a process in which the wetting phase spontaneously imbibes into the matrix and displaces the non-wetting phase, and the ultimate recovery for the process is related to the irreducible oil saturation. For fractured reservoirs, the displacement of oil by water occurs rapidly in fractures leading to an early water breakthrough in production wells. However, in a condition of shut wells, the oil in the matrix is displaced by water from the fractures under the imbibition process. Thus counter-current imbibition can improve the recovery of the fractured reservoir. The matrix’s geometry shape and size, matrix’s porosity, viscosity, surface tension have a significant influence on imbibition velocity. Fractures divide and form the matrix block’s geometry shape, and size. However, isolated fractures may occur within the block.The study of counter-current imbibition has more of significance in fractured reservoirsdue to high heterogeneity variation in the matrix blocks.
Three kinds of research methods in counter-current imbibition are predominant conducted, physical experiment, analytical solution analysis, and numerical solution. However, the most common research focuses on the laboratory core scale in determining the curve of the recovery changes with the dimensionless time [1-3]. Zhang et al.  analyzed the experimental results under the different boundary conditions followed by the empiricalfunction and concluded the recovery curves to be the same despite the boundary condition. Ma et al.  carried out experiments with different water-wet cores and showed the recovery to be a function of time. The experiment falls underthe universal curve of the recovery with the dimensionless time (tD) for all samples/core used. Cil et al.  obtained the dimensionless time representing the wettability of oil-water system at various contact angle distributions. Some researchers also analyzed the recovery during the counter-current imbibition through the analytical solution. Handy et al.  derived a model for one-dimensional water/gas counter-current imbibition and showed the recovery to be linear with the square root of time. Zimmerman et al. [7-9] studied the imbibition in one-dimension and found an exponential variation of recovery with time. Pooladi-Darvish et al.  used the power-law forms of the equation and the relative permeability to construct the numerical solutions of one-dimension imbibition process. Cil and Reis  studied the imbibition based on the closed-form model under the assumption that the model was one-dimension, and the capillary pressure was the linear function of water saturation. Different researchers conducted analysis on imbibition process under numerical simulation approach [17, 18]. Analysis from other scholarsinvolved simulation of the imbibition process using the dual-porosity model, but the transfer of fluid between fracture and matrix in this model comes from a transfer function and the transfer factor was difficult to approximate or obtain. Douglas et al.  described a numerical simulation of combined co-/counter-current spontaneous imbibition of water into the core sample. However, the model was too simple with no consideration on the core sample’s heterogeneity and the isolated fractures.
The experimental results on core analysis during the counter-current imbibition process need to be extrapolated to field reservoir scale through the dimensionless time. The analytical and numerical solution employs dual-porosity model and assumes homogeneous media within matrix blocks. In the fractured reservoir, fractures divide the matrix into different blocks of shapes and size. Also, micro-scale fractures or isolated fracture may exist in these matrix blocks. These fractures and matrix’s block shape and size make the media heterogeneous. Thus theexperimental and analytical solution of the imbibition has some level of limitation and uncertainty. The experiment analysis on core scale model fails to describe matrix block behavior at reservoir scale while the analytical solution assumes homogeneous model. In this paper, both the isolated-fractures in the matrix block and the counter-current imbibition process are simulated based on the discrete fracture model (DFM).
2 Numerical simulation methods
2.1 Governing equations of counter-current imbibition
In this work, we assume the fractured reservoir is water-wet, and the two immiscible and incompressibility fluids (oil and water) occupy the entire model space. The wetting and non-wetting fluids are water and oil respectively. Eq. (1) describes the fluid flow within the media. (1) where φ is porosity of the matrix, Sw is the water saturation, t is the time, vw is the seepage velocity of water. The seepage velocity of the water can be represented according to Darcy’s law as: (2) where vw is the velocity of water, K is the absolute permeability of the matrix, krw and kro are relative permeability of the water and oil, respectively, pcis capillary pressure. μw and μo are the constant kinematic viscosity of water and oil, respectively. Eq. (2) can be rewritten as: (3) where fw is the water fractional flow, which can be defined as follows: (4) The term “De’’ defines the capillary diffusion coefficient and is expressed as: (5) Substituting De in Eq. (3) gives a simplified form of equation below: (6) The dependency of the relative permeability, fractional water flow and capillary pressure on water saturation makes De a nonlinear function of the water saturation. While some researchers assumed the diffusion coefficient was constant for analytical solution, Bech et al.  showed the constant diffusivity equals to Max (De) and matched the result of his experimental data to that of Bourbiaux . Saboorian-Jooybari  assumed De was constant and developed an analytical time-dependent matrix-fracture shape factor for counter-current imbibition, then matched his experimental data with previous researchers.
From the conservation law of mass and momentum: (7) Supposed the fluid flow in the fractures obeys Darcy’ law, the fracture aperture is the function of linear relative permeability curve and the capillary pressure.
2.2 Hybrid mixed-finite element approximation
Galerkin finite element, finite volume, and finite difference are various methods used to solve equations related to fluid flow . However, they have their limitation in applicability . In the hybrid-mixed-finite element (hybridized MFE) method, the approximation is made simultaneously for the two unknown variables of water saturation and flux . For fracture pattern approximation, the unstructured triangular grids are used to discretize the model. We assume a constant saturation along the width of the fracture grid-cells and cross-flow equilibrium across the fractures.
Eq. (6) and (7) are the governing equations in the matrix, which are discretized by the hybridized MFE method with the lowest-order Raviart-Thomas space. We plug the base function ω into the Eq. (6), and use the integration by parts for the integral in the grid-cell. (8)
According to the characteristics of the base function ω. We can get the flux across the boundaries on the grid-cells. Eq. (8) can be written as: (9) where
During the counter-current imbibition process, the fluid obeys the conservation law of the mass, and the fluid volume does not change in the grid-cell. Eq. (9) is integrated in the grid-cell and expressed as: (10) The velocity vw in the Eq. (10) can be expressed by the base function and flux, adding or substituting the Eq. (9) into the Eq. (10), we get the following Eq. (11) below: (11)
In the same way, we can get the equations on the fracture grid-cell: (12) where Swfl,l is the water saturation on the fracture line-cell, and Swfd1,l and Swfd2,l are the water saturation on the two points of the fracture line-cell.
Eq. (9), (11) and (12) are described locally within each matrix grid-cell or fracture line-cell. The two boundary conditions must be satisfied first in order to couple the finite elements together. The flux and saturation are continuous between the interface of the adjacent element A and B. That is, the flux flowing out from the element A across the interface into the element B is equal to the flux flowing into the element B from element A. Moreover, the total flux within the finite element is conserved. After satisfying the conditions, we can get the finite element equations of coupling the matrix-matrix, matrix-fracture, and the fracture-fracture. At last, we can build the whole finite element equation of counter-current imbibition in the water-wet fractured media. (13) where M is an inverse-matrix, Sw is the matrix water saturation.
3 Analysis of numerical examples
3.1 Validation of numerical simulation
We test and check the validity of the numerical simulation algorithm using the model from Ref. . The model has a complete set of experimental imbibition data including rock properties, capillary pressure, relative permeability curves, fluid properties and imbibition oil recovery. The model is 8 cm ×28 cm in dimension and initially oil saturated with the one-end-open-free (OEO-free) boundary geometry. Figure 1 shows the relative permeability and the capillary pressure curves. The relative permeability of the fractures are assumed to be linear functions of saturation but not at irreducible residual saturation, and the capillary pressure in the fractures is calculatedfrom the fracture aperture. Table 1 shows the matrix and fluid properties. Bourbiaux  and Behbahani  simulated counter-current imbibition using the 1D numerical model. Now, we use a 2D model to simulate the counter-current imbibition of the model based on the discrete-fracture model. Firstly, we mesh the model using the triangulation and get 572 grid blocks, and then achieve the simulation by the hybrid-mixed-finite element.
The greenline in the Figure 2 shows the result of 2D counter-current imbibition simulations using the discrete-fracture model derived data. The simulations accurately predict Bourbiaux  and Behbahani  results. Figure 3 shows the relative errors between simulation and Bouriaux experimentresult. In the early times, tD ≤ 1, the relative errors are relatively large just for the small measured data in the experment, while in the middle and late times, the relative errors are very small and within the permissible range. The simulation results are identical to both experiments and simulations performed by Bourbiaux and Behbahani using the same parameters. Therefore, the numerical simulation of a hybrid-mixed-finite element based on the discrete-fracture model is proved to be sufficient and accurate as it matches the results of the experimental data and simulations, and this simulation method is reliable to describe the counter-current imbibition.
3.2 Counter-current imbibition simulation of matrix with isolated fracture
Based on the model and parameters of the Ref. , we test the influence of the isolated fracture in the model. Suppose there is one fracture in the model with a length of 0.20 m and aperture of 0.001 m as shown in (Figure 4). We perform simulation based on two scenarios of existence, and non-existence of the isolated fracture in the matrix block.
Figure 5 shows comparisons of the simulation results of both cases. For the case of not-isolated fracture in the model, the oil recovery reaches 100% at the tD = 100. While, for the case of presence of isolated fracture in the model, the oil recovery reaches 100% at the tD = 4000. Large discrepancy is seen in the simulation results as the recoveries do not fall exactly on the same recovery curve. The result in the Figure 5 confirms that the isolated fracture changes the recovery curve of counter-current imbibition and it is inaccurate to use the recovery curve of no-isolated fracture-matrix model to predict the recovery of a model containing isolated fracture.
Figure 6 shows the saturation distributions of different models at the different dimensionless times. At each of the indicated dimensionless time in Figure 6, the above model is a homogeneous media, and below model corresponds to the existence of the isolated fracture. Waterfront moves smoothly in the model without isolated fracture, while for the case of isolated fracture the waterfront is distorted and moves quickly on both sides of the model compared to regions around the fracture. Usually, the capillary pressure in the isolated fracture is far less than that in the matrix; this weakens the whole matrix model’s imbibition capacity. Therefore, the difference in the capillary pressure between the matrix and isolated fracture affects dynamics of the counter-current imbibition and changes the recovery curves. It is unsuitable to develop the recovery curves based on the empirical formula without considering the influence of the isolated fracture.
4 Analysis of counter-current imbibition with isolated fracture
In this part, we study the effect of isolated fracture under counter-current imbibition using the parameters from Ref. , assuming the fracture aperture remains at 0.001 m.
4.1 Effect of fracture’s length for imbibition
The isolated fracture affects the dynamics of the counter-current imbibition and a series of 2D simulations are performed. Thus, the effect of the isolated fracture’s length and the fracture position is further analyzed. Using Bourbiaux model  with a range of the fracture’s length from 0.20 m to 0.01 m with the various placement positions of the fracture (Figure 7).
Figure 8 shows that all recoveries do not fall exactly on the same universal curve and there is a little scatter in the results. The change of the fracture’s length and position leads to the much larger scatter in the oil recovery curves. Before the waterfront arrives the fracture position, the oil recovery curves arecoincided with each other; when the water arrive the fracture, the oil recovery becomes slowly changing for the small capillary force in the fracture, and the oil recovery can’t be fitted by a simple exponential function of time.The longer the fracture, the greater the effect of it. For the same matrix rock, the isolated fracture positions also change the recovery curves. In general, the isolated fracture’s length and position affect the fluid flow under the capillary pressure in the fractured media. Therefore, the expression based on the exponential function of time no longer matches the oil recovery curves of the matrix with isolated fracture, and the results confirm that it is important to simulate the imbibition based on the discrete-fracture model.
4.2 Analysis of complex model
Suppose the matrix block is of the irregular pentagon and two isolated fractures intersect in the matrix block. The matrix block is initially oil saturated with the five-end-open-free boundary geometry (Figure 9).
From Figure 10, the isolated fractures make the matrix rock to behave more heterogeneous. Along the fractures, the imbibition process is slow hence the oil recovery is also low. Figure 11 shows the oil recovery curve is rising with the time in the semi-logarithm scale, and the oil recovery reaches up to 70% for about 70 days and takes about 500 days to reach 100%. It is very hard to predict oil recovery curve using the exponential function of time for the irregular fractured rock.For the fractured reservoir, it is possible to build the fractured models according to the information of the fractures, then estimate the oil recovery curves by numerical simulation for different fracture models and finally predict the best time for opening and shutting wells of the open-shut well of the reservoir.
The above studycan be summarized into following four points as outlined below:
Simulation of two-dimensional counter-current imbibition matches the results of the experimental data and simulations, which confirms that this simulation method is adequate to describe the counter-current imbibition.
The effect of the isolated fracture is test by performing simulation using the discrete-fracture model and the recoveries does not fall exactly on the same curve. For the water-wet fractured media, the isolated fractures not only strengthen the heterogeneity of the matrix blocks but also change the distribution of oil and water under the counter-current imbibition. Usually, the capillary pressure in the fracture is smaller than that in the matrix, and the isolated fractures weaken the effects of counter-current imbibition of the whole matrix block. So it is inaccurate to use the recovery curve of no-isolated fracture-matrix model to predict the recovery of a model containing isolated fractures
The analyses of fractures’ influences are performed by simulation of different length and position using discrete-fracture model. The results indicate that the recoveries do not fall exactly on the same universal curve based on the empirical formula. The large discrepancy confirms that the isolated fracture changes the recovery curve of counter-current imbibition and it is inaccurate to neglect fractures in the simulation of counter-current imbibition.
Fractures are described explicitly and accurately based on the discrete-fracture model, which makes the oil recovery curve more accurate. Application of the hybrid-mixed-finite element method makes it easy to study or analyze theirregular matrix block and broadens the applicability of the simulation model. The hybrid-mixed-finite element method is suitable for the counter-current imbibition simulation study of the water-wet fractured reservoirs based on the discrete-fracture model.
This work was supported by the National Natural Science Foundation of China (Grant No.11102237,51404291), the Important National Science and Technology Project of China (Grant No.2015ZX05014-003), the Fundamental Research Funds for the Central Universities (Grant No.14CX02045A), and the Natural Science Foundation of Shandong Province (ZR2014EL017).
Mattax C.C., Kyte J.R., Imbibition oil recovery from fractured water drive reservoirs, Society of Petroleum Engineers Journal, 1962, 7, 177-184. Google Scholar
Cil M., Reis J.C., Miller M.A., Misra D., An examination of counter current capillary imbibition recovery from single matrixblocks and recovery predictions by analytical matrix/fracture transfer functions, Annual Technical Conference (SPE, New Orleans, Louisiana, USA, 1998) Google Scholar
Handy L.L., Determination of effective capillary pressures for porous media from imbibition data, Petr. Trans., AIME, 1960, 219, 75-80 Google Scholar
Zimmerman R.W., Chen G., Hadgu T., et al., A numerical dualporosity model with the semi-analytical treatment of fracture/matrix flow, Water Resources Research, 1993, 29, 2127-2137. CrossrefGoogle Scholar
Tavassoli Z., Zimmerman R.W., Blunt M.J., Analytic analysis for oil recovery during counter-current imbibition in strongly water-wet systems, Porous Media Transp, 2005, 58, 173-189.CrossrefGoogle Scholar
Li Y., Morrow N.R., Ruth D., Similarity solution for linear counter-current spontaneous imbibition, Journal of Petroleum Science and Engineering, 2003, 1077, 309-326.Google Scholar
Gupta A., Civan F., An improved model for laboratory measurement of the matrix to fracture transfer function parameters in immiscible displacement, Annual Technical Conference (SPE, New Orleans, Louisiana, USA, 1994)Google Scholar
Zhou D., Jia L., Kamath J., et al., Scaling of counter-current imbibition processes in low permeability porous media, Journal of Petroleum Science and Engineering, 2002, 33, 61-74.CrossrefGoogle Scholar
Pooladi-Darvish M., Firoozabadi A., Experiments, and modeling of water injection in water-wet fractured porous media, Journal of Canadian Petroleum Technology, 1998, 39, 31-42.Google Scholar
Cil M., Reis J.C., Miller M.A., et al., An examination of countercurrent capillary imbibition recovery from single matrix blocks and recovery predictions by analytical matrix/fracture transfer functions. SPE annual technical conference and exhibition (SPE, New Orleans, LA, USA, 1998) Google Scholar
Behbahani H.S., Di Donato G., Blunt M.J., Simulation of countercurrent imbibition in water-wet fractured reservoirs, Journal of Petroleum Science and Engineering, 2006, 50, 21-39. CrossrefGoogle Scholar
Bagherinezhad A., Pishvaie M. R., A new approach to counter-current spontaneous imbibition simulation using Green element method. Journal of Petroleum Science and Engineering, 2014, 119, 163-168. CrossrefWeb of ScienceGoogle Scholar
Douglas W.R., Mason G., Martin A.F., et al., Numerical simulation of combined co-current/counter-current spontaneous imbibition, International Symposium of the Society of Core Analysts (SCA, St. John, Newfoundland and Labrdor, Canada, 2015) Google Scholar
Bech N., Jensen O.K., Nielson B., Modeling of gravity-imbibition and gravity-drainage process: analytical and numerical solutions, SPE Reservoir Engineering, 1991, 277-290. Google Scholar
Saboorian-Jooybari H., Ashoori S., Mowazi G., Development of an Analytical Time-Dependent Matrix/Fracture Shape Factor for Countercurrent Imbibition in Simulation of Fractured Reservoirs, Transport in Porous Media, 2012, 92, 687-708. Web of ScienceCrossrefGoogle Scholar
Yang X.J., Machado J.A., Srivastava H.M., A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach, Applied Mathematics and Computation, 2016, 274,143-151. CrossrefWeb of ScienceGoogle Scholar
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Published Online: 2017-08-03