Fluid flow through porous media using distinct element based numerical method
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Abstract
Many analytical and numerical methods have been developed to describe and analyse fluid flow through the reservoir’s porous media. The medium considered by most of these models is continuum based homogeneous media. But if the formation is not homogenous or if there is some discontinuity in the formation, most of these models become very complex and their solutions lose their accuracy, especially when the shape or reservoir geometry and boundary conditions are complex. In this paper, distinct element method (DEM) is used to simulate fluid flow in porous media. The DEM method is independent of the initial and boundary conditions, as well as reservoir geometry and discontinuity. The DEM based model proposed in this study is appeared to be unique in nature with capability to be used for any reservoir with higher degrees of complexity associated with the shape and geometry of its porous media, conditions of fluid flow, as well as initial and boundary conditions. This model has first been developed by Itasca Consulting Company and is further improved in this paper. Since the release of the model by Itasca, it has not been validated for fluid flow application in porous media, especially in case of petroleum reservoir. In this paper, two scenarios of linear and radial fluid flow in a finite reservoir are considered. Analytical models for these two cases are developed to set a benchmark for the comparison of simulation data. It is demonstrated that the simulation results are in good agreement with analytical results. Another major improvement in the model is using the servo controlled walls instead of particles to introduce tectonic stresses on the formation to simulate more realistic situations. The proposed model is then used to analyse fluid flow and pressure behaviour for hydraulically induced fractured and naturally fractured reservoir to justify the potential application of the model.
Keywords
Distinct element method (DEM) Particle Fracture Time step Flow rate Pressure Reservoir Permeability Analytical model Numerical model Porous media DimensionlessAbbreviation
List of symbols
 DEM
Distinct element method
 PFC2D
Particle flow code  Two dimensional
Symbol/notation
 \(c_{\text{t}}\)
Total compressibility
 F
Compressive Force
 g
Gap (Distance between particles)
 \(h\)
Sample height
 \(J_{0}\)
Zeroth order of first kind Bessel function
 \(J_{1}\)
First order of first kind Bessel function
 k
Permeability
 K_{f}
Fluid bulk modulus
 L
Sample length
 L_{P}
Pipe length
 m
Calibration constant
 N
Number of pipes for each domain
 P
Pressure
 \(P_{\text{D}}\)
Dimensionless pressure
 \(P_{i}\)
Initial pressure
 \(P_{\text{w}}\)
Wellbore pressure
 q
Flow rate
 \(R\)
Radial distance from wellbore centre
 r
Particle radius
 \(R_{\text{D}}\)
Dimensionless radius
 \(R_{\text{De}}\)
External dimensionless radius
 \(R_{\text{e}}\)
External radius
 \(R_{\text{w}}\)
Wellbore Radius
 \(\bar{R}\)
Average radius of particles
 \(S.F.\)
Safety Factor
 t
Time
 \(t_{\text{D}}\)
Dimensionless time
 V
Domain volume
 w
Aperture
 \(w_{0}\)
Initial Aperture
 \(x_{\text{D}}\)
Dimensionless position
 \(Y_{0}\)
Zeroth order of second kind Bessel function
 \(Y_{1}\)
First order of second kind Bessel function
 \(\alpha^{2}\)
Hydraulic diffusivity
 β
Angle (In radians)
 \(\Delta t\)
Time step
 ϴ
Angle (In radians)
 λ
Root of characteristic equation
 \(\mu\)
Fluid Viscosity
 σ
Stress
 φ
Porosity
Introduction
Fluid flow through porous media has been the subject of interest in many areas, such as petroleum and resource engineering, geothermal energy extraction, and/or ground water hydrology, etc., for many years. Many analytical as well as numerical models have been developed to explain fluid flow through porous media. Although most of these models work well with reasonable accuracy in the case of reservoir consisting of homogenous media with simple geometry, they encourage huge uncertainty with erroneous results in the case of heterogeneous and discontinuous media, especially in presence of natural fractures and interacted induced hydraulic fractures with complex geometry.
In addition, numerical or analytical methods of solutions depend on the geometry of the reservoir, fluid flow type (Linear, Radial and Spherical), fluid flow regime (Transient, Late Transient, Steady State, SemiSteady state), as well as discontinuity (conductive discontinuity with different permeability, sealed discontinuity with low permeability). To combine all of these factors to get a solution that can describe the flow, many assumptions need to be made. The more assumptions that are integrated into the solution, the more susceptible the solution will be to incur errors in results. Another problem with these methods is that they are not unique solutions. At any time, if one of the factors that are mentioned earlier is changed, the whole solution might change. A robust model is always desirable to address all of these issues and provide better and more accurate solution. In this view, study has been focused to develop a numerical tool to analyse fluid flow in the above mentioned complex scenarios.
Particle flow code (PFC) developed by Itasca consulting group which is based on distinct element method (DEM), is considered as a numerical tool to simulate fluid flow in porous media in this study. The initial fluid flow model was developed by Itasca. Further modifications were made on the model to simulate more realistic situations and validate the simulation results. The method developed is independent of the reservoir geometry, discontinuity, fluid flow type and regime; and is found to be more appropriate to simulate the reservoir that is heterogeneous in nature in relation to both the media and complex geometry (grain shape, size, as well as pore geometry).
A numerical model to simulate the fluid flow through heterogeneous porous media, especially in presence of natural fractures interacted with hydraulic fracture is presented in this paper. In this study, two cases of laminar and radial fluid flow conditions are simulated using the numerical model developed based on distinct element method. The accuracy of the model is validated by comparing the simulation results with analytical results.
After validation, this model is used to simulate fluid flow in a reservoir that is hydraulically fractured and contains two sets of natural fractures. Based on the results, it is demonstrated that proposed DEM based model can potentially be used to analyse fluid flow through complex reservoirs.
Discrete element method

Rotation, finite displacement and complete detachment of discretised bodies are allowed

While the calculation progresses, new contacts can be automatically recognized (Morris et al. 2003)
 1.
Distinct Element Programs
 2.
Modal Methods
 3.
Discontinuous deformation analysis
 4.
Momentumexchange methods
Distinct element programs have been developed based on Distinct Element Method (DEM) which is a subclassification of Discrete Element Method. In this method, contacts are deformable and discretised elements can be either rigid or deformable (Cundall and Hart 1992). The solution scheme is based on explicit time stepping which time steps are chosen so small that the disturbances introduced by a single particle cannot propagate beyond neighbouring particles (Cundall and Strack 1979). Detailed description of the method can be found in (Cundall 1988) and (Hart et al. 1988).
More information about the formulation and analysis procedure can be found in PFC2D manual (Itasca 2008a).
DEM fluid flow
Fluid flow happens between domains through a pipe centred at the contact point between each two particles. Pipe length is the sum of two particles radius. Aperture between parallel plates is denoted as “w”. The depth of the pipe is equal to unity (Itasca 2008b).
q: Flow Rate, P _{ i }: Pressure in Domain i, P _{ j }: Pressure in Domain j, μ: Fluid Viscosity, L _{p}: Pipe Length.
Pipes can be defined between two particles, only if they are in contact. However, after they have been initialized they will exist even if particles detach from each other. When particles are in contact, the aperture of the pipe will be zero. But to take into account the macroscopic permeability of the rock and to overcome the 2D limitation of the simulation, “w” will be set to a number greater than zero (Itasca 2008b).
For simplicity of calculations, the mechanical volume change in domains is neglected as its value is very small and will not make a noticeable change in results. On the other hand, domain volumes will be updated in every time step. The solution to fluid flow alternates between flow through pipes and pressure adjustments between domains. This means, in each time step, the fluid flow through pipes is calculated and the total net flow to or from each domain will cause pressure change in domains. For stability analysis, a critical time step needs to be calculated. The procedure is to first calculate the critical time step for each domain and then take the minimum time step of all critical time steps as the global time step (Itasca 2008b).
It can be concluded from this section that the fluid flow at microscopic level is independent of the reservoir geometry because the fluid flows between the domains, can be created for any reservoir shape. This characteristic makes this method applicable for any reservoir geometry. Also, this method is independent of flow type (i.e., linear, radial, etc.) and flow regime (i.e., transient, late transient, steady state or semi steady state), and can be applied for any flow type and regime.
Analytical methods
To validate numerical model, two cases of linear fluid flow and radial fluid flow in porous medium is considered. The formation in both cases is finite. For each case, the analytical formula with its boundary and initial conditions is presented in Sects. “Linear fluid flow condition” and “Radial fluid flow condition”. Derivation of these analytical equations is presented in Appendices A and B. Solutions of both cases are used in the comparison section to compare the results of numerical model against analytical models.
Linear fluid flow condition
Equation 24 is the dimensionless form of pressure diffusion equation of laminar fluid flow in a sample with initial pore pressure of \(P_{i}\) and constant boundary pressures of \(P_{1} = P_{i}\) and \(P_{2} = 0\).
Radial fluid flow condition
Comparison of numerical and analytical models
Linear fluid flow with constant boundary pressures
To make sure that simulation results are correct, they are compared against analytical results. To do so, data in Fig. 15 are converted to dimensionless form by using Eqs. 21–23.
Parameters of simulation at steady state condition
Parameter  Metric system  Imperial system  

\(q\)  1.37E−05  \({\text{m}}^{ 3} / {\text{s}}\)  7.42E+00  \({\text{bbl/day}}\) 
\(P_{1}\)  5.00E+06  \({\text{Pa}}\)  7.25E+02  \({\text{psia}}\) 
\(P_{2}\)  0  \({\text{Pa}}\)  0.00E+00  \({\text{psia}}\) 
\(A\)  14  \({\text{m}}^{ 2}\)  \(150.69\)  \({\text{ft}}^{ 2}\) 
\(\mu\)  1  \({\text{Pa}} . {\text{Sec}}\)  \(1000\)  \({\text{cp}}\) 
\(L\)  13.5  \({\text{m}}\)  \(42.29\)  \({\text{ft}}\) 
\(c\)  1.00E−09  \({\text{Pa}}^{  1}\)  6.09E−06  \({\text{psia}}^{  1}\) 
\(\varPhi\)  \(0.2\)  –  \(0.2\)  – 
Simulation time (t) and dimensionless time (t _{D})
t (seconds)  t (day)  t _{D} 

67  0.00078  0.00484 
267  0.00309  0.01931 
667  0.00772  0.04823 
1467  0.01698  0.10608 
3067  0.03550  0.22178 
15067  0.17439  1.08951 
The units for different parameters are:
k: md, t: day, μ: cp, c _{ t}: psi^{−1}, L: ft.
Radial fluid flow
Parameters of Simulation
Parameter  Metric system  Imperial system  

\(q\)  2.00E−05  \({\text{m}}^{ 3} / {\text{s}}\)  7.42E+00  \({\text{bbl}}/{\text{day}}\) 
\(P_{i}\)  7.00E+06  \({\text{Pa}}\)  1.02E+03  \({\text{psia}}\) 
\(P_{\text{e}}\)  7.00E+06  \({\text{Pa}}\)  1.02E+03  \({\text{psia}}\) 
\(P_{{{\text{w}}_{\text{steady state}} }}\)  3.49E+06  \({\text{Pa}}\)  5.06E+02  \({\text{psia}}\) 
\(r_{\text{w}}\)  9.64E−01  \({\text{m}}\)  \(3.16\)  \({\text{ft}}\) 
\(r_{\text{e}}\)  1.38E+01  \({\text{m}}\)  \(45.11\)  \({\text{ft}}\) 
\(\mu\)  \(1\)  \({\text{Pa}} . {\text{Sec}}\)  \(1000\)  \({\text{cp}}\) 
\(h\)  \(1\)  \({\text{m}}\)  \(3.28\)  \({\text{ft}}\) 
\(c\)  1.00E−09  \({\text{Pa}}^{  1}\)  6.90E−06  \({\text{psia}}^{  1}\) 
\(\varPhi\)  \(0.2\)  –  \(0.2\)  – 
k: md, P: psia, μ: cp, h: ft, R: ft.
Using Eq. 36, permeability is found to be equal to \(2.44\; \times \;10^{3} \;{\text{md}}\).
The units for different parameters in Eq. 37 are:
k: md, t: hr, μ: cp, c _{ t}: psi^{−1}, r _{ w}: ft.
Simulation time (t) and dimensionless time (t _{D})
t (seconds)  t _{D} 

120.03  1.67 
420.03  5.84 
1420.03  19.75 
2753.36  38.29 
24,086.70  334.97 
Values of first 50 λ _{ n }
n  λ _{ n }  n  λ _{ n }  N  λ _{ n }  n  λ _{ n }  n  λ _{ n } 

1  0.170  11  2.499  21  4.863  31  7.230  41  9.598 
2  0.395  12  2.735  22  5.099  32  7.467  42  9.835 
3  0.624  13  2.971  23  5.336  33  7.704  43  10.072 
4  0.855  14  3.207  24  5.573  34  7.941  44  10.309 
5  1.088  15  3.444  25  5.810  35  8.177  45  10.546 
6  1.322  16  3.680  26  6.046  36  8.414  46  10.783 
7  1.557  17  3.916  27  6.283  37  8.651  47  11.020 
8  1.792  18  4.153  28  6.520  38  8.888  48  11.256 
9  2.027  19  4.390  29  6.757  39  9.125  49  11.493 
10  2.263  20  4.626  30  6.993  40  9.362  50  11.730 
Fractured reservoir
This section showed how easily this model was modified from simple circular reservoir to a hydraulically and naturally fractured reservoir.
Potential application of the model
 1.
Oil and gas flow in oil and gas reservoirs.
 2.
Injection of water or surfactants into oil and gas reservoirs.
 3.
Both the injection and production simulation in oil and gas reservoirs.
 4.
Single stage as well as multi stage hydraulic fracturing of the oil and gas reservoirs such as shale gas reservoirs.
 5.
Water flow in mines both in rock matrix as well as in joints, faults and fractures for the application in mining industry.
 6.
Water flow underground in soil to be used by civil engineers for simulating water flow into tunnels.
 7.
Water movement in soil to be used by agricultural engineers to simulate the rate of hydration, dehydration or draining of soil.
Conclusion
Deriving analytical expressions to describe fluid flow in porous medium is a complex task. This is because for any change in the reservoir geometry or any change in the condition of fluid flow (e.g., transient, late transient, semisteady state or steady state) a new analytic expression needs to be developed. In this view, a DEM based numerical model is proposed to analyse the fluid flow through reservoir’s porous media with complex characteristics, especially in the case of existence of natural fractures, hydraulic fractures and interaction of hydraulic and natural fractures for any condition of fluid flow.
Proposed model is used to simulate and analyse some of these field representative cases as an example case studies. Both simple and complex cases are considered in this study. The simple case is used to validate the accuracy of the model. The DEM model that was used in this study is observed to be independent of reservoir geometry as well as the condition of fluid flow since it was shown that it worked for both linear and radial flow without modification. The simulation results are found to be in good agreement with analytical results. It is also demonstrated that the model can potentially be used as a powerful numerical simulation tool to handle both simple and complex reservoir conditions such as complex formations with irregular shapes, and complex set of discontinuity and fluid flow regime.
Notes
Acknowledgments
I would like to express my appreciation to the Australian and Western Australian Governments and the North West Shelf Joint Venture Partners, as well as the Western Australian Energy Research Alliance (WA:ERA) for their financial support to undertake my research toward completion of my PHD.
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