Abstract
We study a heuristic, core-scale model for the transport of polymer particles in a two-phase (oil and water) porous medium. We are motivated by recent experimental observations which report increased oil recovery when polymers are injected after the initial waterflood. We propose the recovery mechanism to be microscopic diversion of the flow, where injected particles can accumulate in narrow pore throats and clog it, in a process known as a log-jamming effect. The blockage of the narrow pore channels leads to a microscopic diversion of the water flow, causing a redistribution of the local pressure, which again can lead to the mobilization of trapped oil, enhancing its recovery. Our objective herein is to develop a core-scale model that is consistent with the observed production profiles. We show that previously obtained experimental results can be qualitatively explained by a simple two-phase flow model with an additional transport equation for the polymer particles. A key aspect of the formulation is that the microscopic heterogeneity of the rock and a dynamic altering of the permeability must be taken into account in the rate equations.
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Notes
For the other two cores: one was not stabilized during polymer particle injection and the other achieved pressure drops one order of magnitude higher than the other cores, such that we considered those two cores to be outside the parametric range for comparison considered here.
Abbreviations
- \(A_1, A_2\) :
-
Constants for residual oil saturation change
- \(c_l\) :
-
Mass concentration of polymer particles (\(\hbox {kg}/\hbox {m}^3\))
- \({\varvec{D}}_l\) :
-
Diffusion matrix of polymer particles in water (\(\hbox {m}^2/\hbox {s}\))
- \(d_\mathrm{p}\) :
-
Polymer particle diameter (m)
- \(\mathbf {g}\) :
-
Gravity vector (\(\hbox {m}/\hbox {s}^2\))
- \(\mathbf{K}\) :
-
Absolute permeability tensor (\(\hbox {m}^2\))
- \(K_\mathrm{B}\) :
-
Boltzmann constant (\(\hbox {m}^2\,\hbox {kg}/\hbox {s}^2\,\hbox {K}\))
- \(k_l\) :
-
Constant rate of clogging (\(\hbox {m}^{-1}\))
- \(k_r\) :
-
Constant rate of unclogging (\(\hbox {s}^{-1}\))
- \(k_{r\alpha }\) :
-
Relative permeability of phase \(\alpha \)
- \(n_o,n_w\) :
-
Exponents for the relative permeabilities
- \(p_\alpha \) :
-
Pressure of phase \(\alpha \) (Pa)
- R :
-
Reaction rate \((\hbox {kg}/\hbox {m}^3\,\hbox {s}\))
- \(s_\alpha \) :
-
Saturation of phase \(\alpha \)
- T :
-
Temperature (K)
- \(\mathbf {u}_\alpha \) :
-
Velocity of phase \(\alpha \, (\hbox {m}/\hbox {s})\)
- \(\gamma \) :
-
Constant for permeability change
- \(\delta \) :
-
Heterogeneity factor
- \(\lambda _\alpha \) :
-
Mobility of phase \(\alpha \, (\hbox {m s}/\hbox {kg})\)
- \(\mu _\alpha \) :
-
Viscosity of phase \(\alpha \, (\hbox {kg}/\hbox {m s})\)
- \(\rho _\alpha \) :
-
Mass density of phase \(\alpha \, (\hbox {kg}/\hbox {m}^3)\)
- \(\sigma \) :
-
Volumetric concentration of accumulated particles
- \(\tau \) :
-
Tortuosity
- \(\phi \) :
-
Porosity
- \(\varphi \) :
-
Clogging rate (\(\hbox {kg}/\hbox {m}^3\,\hbox {s}\))
- \(\psi \) :
-
Unclogging rate (\(\hbox {kg}/\hbox {m}^3\,\hbox {s}\))
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This work was partially supported by Statoil, through Akademiaavtalen, the Norwegian Academy of Science and Letters and Statoil through the VISTA AdaSim Project No. 6367.
A Heterogeneity Factor
A Heterogeneity Factor
As mentioned previously, a key aspect for the accumulation of particles is the heterogeneity of the core. In Fig. 8a, we show the normalized pore size distributions and in (b) their respective normalized cumulative distribution functions, for cores A, B and C. We also plot the distribution functions for the Berea core, which is the homogeneous core utilized for comparison. Experiments with polymer particle injection have shown negligible increase in oil recovery for Berea cores (Skauge et al. 2010). Since Berea is a fairly homogeneous core, this supports our claim that heterogeneity plays a significant role in microscopic diversion. Therefore, we chose Berea as a representative homogeneous core for the calculation of the heterogeneity factor \(\delta \).
In Fig. 9a–c, we present the comparison between cores A to C and the Berea, with their respective values for \(\delta \). In essence, \(\delta \) is a measurement of how far from a homogeneous core (Berea) the given sample is. Larger values of \(\delta \) gives a high heterogeneity, indicating that the core is more suitable for oil recovery enhancement through polymer particle injection. For convention, pores with radius \(r < 1~\upmu m\) are considered in the microscopic region, whereas pores with radius \(> 1~\upmu m\) are at the macroscopic region. We consider the maximum separation between distributions \(\delta \) in the microscopic region. In other words, we neglect the variations in pore sizes in the macroscopic region (\(r>1~\upmu m\)). This is justified by the fact that particle-carrying water flow only clogs a pore when the large-to-narrow pore throat ratio is large. In other words, a flow change from two pores with different throat radius does not permit clogging if both pores are in the macroscopic region.
In essence, the heterogeneity factor compares distributions at the microscale. Nevertheless, the parameter \(\delta \) is dependent on the base distribution used for comparison (Berea, in the present case). If instead of Berea we considered another distribution with a very different pore throat radius range as a base, the different scales for the compared cores would render a wrong calculation for the parameter \(\delta \). In fact, a more precise definition and derivation of \(\delta \) would have to be independent on considering a base core. This will be done in a future work.
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Endo Kokubun, M.A., Radu, F.A., Keilegavlen, E. et al. Transport of Polymer Particles in Oil–Water Flow in Porous Media: Enhancing Oil Recovery. Transp Porous Med 126, 501–519 (2019). https://doi.org/10.1007/s11242-018-1175-2
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DOI: https://doi.org/10.1007/s11242-018-1175-2