Calibrating and Scaling Semi-empirical Foam Flow Models for the Assessment of Foam-Based EOR Processes (in Heterogeneous Reservoirs)

Abstract

Models for simulating foam-based displacements fall into two categories: population-balance (PB) models that derive explicitly foam texture or bubble size from pore-level mechanisms related to lamellas generation and coalescence, and steady-state semi-empirical (SE) models that account implicitly for foam texture effects through a gas mobility reduction factor. This mobility reduction factor has to be calibrated from a large number of experiments on a case-by-case basis in order to match the physical effect of parameters impacting foam flow behaviour such as fluids saturation and velocity. This paper proposes a methodology to set up steady-state SE models of foam flow on the basis of an equivalence between SE model and PB model under steady-state flow conditions. The underlying approach consists in linking foam mobility and foam lamellas density (or texture) data inferred from foam corefloods performed with different foam qualities and velocities on a series of sandstones of different permeabilities. Its advantages lie in a deterministic non-iterative transcription of flow measurements into texture data and in a separation of texture effects and shear-thinning (velocity) effects. Then, scaling of foam flow parameters with porous medium permeability is established from the analysis of calibrated foam model parameters on cores of different permeability, with the help of theoretical representations of foam flow in a confined medium. Although they remain to be further confirmed from other well-documented experimental data sets, the significance of those scaling laws is great for the assessment of foam-based enhanced oil recovery (EOR) processes because foam EOR addresses heterogeneous reservoirs. Simulations of foam displacement in a reservoir cross section demonstrate the necessity to scale foam SE models with respect to facies heterogeneity for reliable evaluation.

Appendices

Determination of Saturation at Steady State in the Presence of Foam

The water saturation is determined by taking into account the invariance of relative permeability to water whether foam is present or not. At steady state, the measured pressure drop during foam flow can be written using Darcy’s law applied to foam considered as a single fluid or using generalized Darcy’s law applied to water phase, that is:

$$\begin{aligned} \varDelta P = \frac{\mu _\mathrm{f}u \; L}{k} = \frac{\mu _\mathrm{w} (1- f_\mathrm{g}) u \; L}{k k_\mathrm{rw}}, \end{aligned}$$

Hence, one infers

$$\begin{aligned} k_\mathrm{rw}= \frac{\mu _\mathrm{w} (1- f_\mathrm{g})}{\mu _\mathrm{f}}. \end{aligned}$$

Power law functions are considered to model relative permeability functions versus saturation in the framework of this paper, that is $k_\mathrm{rw}= k_\mathrm{rw}^0 S^{n_\mathrm{w}}$ with $S = (S_\mathrm{w}-S_\mathrm{wr})/(1-S_\mathrm{wr}-S_\mathrm{gc})$ the normalized water saturation, $S_\mathrm{wr}$ the residual water saturation and $S_\mathrm{gc}$ the critical gas saturation. $k_\mathrm{rw}^0$ is the maximum relative permeability to water and $n_\mathrm{w}$ the relative permeability exponent. Water saturation can then be determined as:

$$\begin{aligned} S_\mathrm{w} = S_\mathrm{wr} + (1-S_\mathrm{wr}-S_\mathrm{gc}) \left( \frac{ \mu _\mathrm{w} (1- f_\mathrm{g}) }{ \mu _\mathrm{f}k_\mathrm{rw}^0} \right) ^{1/n_\mathrm{w}}, \end{aligned}$$

or more directly from measured steady-state pressure drop as:

$$\begin{aligned} S_\mathrm{w} = S_\mathrm{wr} + (1-S_\mathrm{wr}-S_\mathrm{gc}) \left( \frac{ \mu _\mathrm{w} (1- f_\mathrm{g}) u \; L }{ k_\mathrm{rw}^0 \; k \varDelta P} \right) ^{1/n_\mathrm{w}}. \end{aligned}$$

Evolution of Transition Parameter $\varTheta$ with k According to Static and Dynamic Viewpoints

The evolution of transition parameter $\varTheta$ with permeability is discussed, considering either of the two (quasi-) static and dynamic viewpoints of foam flow within porous media.

First, the static assumption implies that coalescence of foam occurs over a $P_\mathrm{c}$ interval centred around the $P_\mathrm{c}^*$ value corresponding to the disjoining pressure $\varPi$ of the foam. That $P_\mathrm{c}$ interval, denoted $[\varPi ^-, \varPi ^+]$, is the same whatever the permeability of the porous medium. For a porous medium of permeability $k_\text {ref}$, the width of the saturation transition, i.e. $(S_\mathrm{w}^{\text {opt}})_\text {ref} - (S_\mathrm{w}^{\text {min}})_\text {ref}$, can then be expressed as $P_{\mathrm{c},\text {ref}}^{-1}(\varPi ^-) - P_{\mathrm{c},\text {ref}}^{-1}(\varPi ^+)$ that is also equal to $\pi / \varTheta _\text {ref}$.

For a porous medium of arbitrary permeability k, we can also write $S_\mathrm{w}^{\text {opt}}- S_\mathrm{w}^{\text {min}}= P_\mathrm{c}^{-1}(\varPi ^-) - P_\mathrm{c}^{-1}(\varPi ^+)$ that is also equal to $\pi / \varTheta$. From these two equalities, we infer the scaling relationship as follows:

$$\begin{aligned} \frac{S_\mathrm{w}^{\text {opt}}- S_\mathrm{w}^{\text {min}}}{(S_\mathrm{w}^{\text {opt}})_\text {ref} - (S_\mathrm{w}^{\text {min}})_\text {ref}} = \frac{\varTheta _\text {ref}}{\varTheta } = \frac{P_\mathrm{c}^{-1}(\varPi ^-) - P_\mathrm{c}^{-1}(\varPi ^+)}{P_{\mathrm{c},\text {ref}}^{-1}(\varPi ^-) - P_{\mathrm{c},\text {ref}}^{-1}(\varPi ^+)}. \end{aligned}$$

That is,

$$\begin{aligned} \varTheta = \varTheta _\text {ref} \frac{P_{\mathrm{c},\text {ref}}^{-1}(\varPi ^-) - P_{\mathrm{c},\text {ref}}^{-1}(\varPi ^+)}{P_\mathrm{c}^{-1}(\varPi ^-) - P_\mathrm{c}^{-1}(\varPi ^+)}. \end{aligned}$$

If one considers a narrow disjoining pressure interval, then the variation of inverse $P_\mathrm{c}$ and $P_{\mathrm{c},\text {ref}}$ functions over $\varPi$ interval can be, respectively, approximated as $P_\mathrm{c}^{-1}(\varPi ^-) - P_\mathrm{c}^{-1}(\varPi ^+) = S_\mathrm{w}^{\text {opt}}- S_\mathrm{w}^{\text {min}}\approx \bigl ( \frac{\mathrm {d}S_\mathrm{w}}{\mathrm {d}P_\mathrm{c}} \bigr )_\varPi (\varPi ^- - \varPi ^+)$ and $P_{\mathrm{c},\text {ref}}^{-1}(\varPi ^-) - P_{\mathrm{c},\text {ref}}^{-1}(\varPi ^+) = (S_\mathrm{w}^{\text {opt}})_\text {ref} - (S_\mathrm{w}^{\text {min}})_\text {ref} \approx \bigl ( \frac{\mathrm {d}S_\mathrm{w}}{\mathrm {d}P_{\mathrm{c},\text {ref}}} \bigr )_\varPi (\varPi ^- - \varPi ^+)$. This leads to:

$$\begin{aligned} \varTheta = \varTheta _\text {ref} \frac{\left( \frac{\mathrm {d}S_\mathrm{w}}{\mathrm {d}P_{\mathrm{c},\text {ref}}} \right) _\varPi }{\left( \frac{\mathrm {d}S_\mathrm{w}}{\mathrm {d}P_\mathrm{c}} \right) _\varPi }. \end{aligned}$$

That evolution of $\varTheta$ as a function of k cannot be predicted a priori from this relationship, even for homothetic porous media, because it depends on the shape of the $P_\mathrm{c}$ and $P_{\mathrm{c},\text {ref}}$ curves with respect to the value of $\varPi$.

Considering now the dynamic assumption, the collapse of foam films occurs over an invariant interval of the transverse velocity gradient within pores, denoted $[w_\text {max}^{\prime -}, w_\text {max}^{\prime +}]$. As explained before, to that fixed transverse velocity interval corresponded a pressure gradient interval $[2 \mu w_\text {max}^{\prime -} / r, 2 \mu w_\text {max}^{\prime +} / r]$ along flow direction and a critical capillary pressure interval $[P_\mathrm{c}^{*-}, P_\mathrm{c}^{*+}]$ for a porous medium with given characteristic pore radius r and permeability k ($r \sim \sqrt{k}$). The invariance of $[w_\text {max}^{\prime -}, w_\text {max}^{\prime +}]$ interval for different porous media implies that the interval $[r P_\mathrm{c}^{*-} , r P_\mathrm{c}^{*+} ]$ is constant since the pressure gradient and capillary pressure are in proportion one another. The scaling relationship for coalescence interval can then be expressed as an invariance of the product $rP_\mathrm{c}^*$ that we develop hereafter.

For two porous media with respective characteristic pore radii r and $r_\text {ref}$, the scaling relationship can be written over $[w_\text {max}^{\prime -}, w_\text {max}^{\prime +}]$ interval as $r P_\mathrm{c}(S_\mathrm{w}^{\text {opt}}) = r_\text {ref} P_{\mathrm{c},\text {ref}} [ (S_\mathrm{w}^{\text {opt}})_\text {ref} ]$ and $r P_\mathrm{c}(S_\mathrm{w}^{\text {min}}) = r_\text {ref} P_{\mathrm{c},\text {ref}} [ (S_\mathrm{w}^{\text {min}})_\text {ref} ]$, that is,

$$\begin{aligned} r \left[ P_\mathrm{c}(S_\mathrm{w}^{\text {opt}}) - P_\mathrm{c}(S_\mathrm{w}^{\text {min}}) \right] = r_\text {ref} \left[ P_{\mathrm{c},\text {ref}}((S_\mathrm{w}^{\text {opt}})_\text {ref}) - P_{\mathrm{c},\text {ref}}((S_\mathrm{w}^{\text {min}})_\text {ref}) \right] . \end{aligned}$$

For a generally abrupt transition, the previous relationship can be written as:

$$\begin{aligned} r \left( S_\mathrm{w}^{\text {opt}}- S_\mathrm{w}^{\text {min}}\right) \left( \frac{\mathrm {d}P_\mathrm{c}}{\mathrm {d}S_\mathrm{w}} \right) _{S_\mathrm{w}^{*}} = r_\text {ref} \left[ (S_\mathrm{w}^{\text {opt}})_\text {ref} - (S_\mathrm{w}^{\text {min}})_\text {ref} \right] \left( \frac{\mathrm {d}P_{\mathrm{c},\text {ref}}}{\mathrm {d}S_\mathrm{w}} \right) _{(S_\mathrm{w}^{*})_\text {ref}}. \end{aligned}$$

Taking into account the relationship established above between $\varTheta$ and saturation interval (Eq. 28), we infer the following general relation between $\varTheta$ and $\varTheta _\text {ref}$ that holds for any couple of porous media:

$$\begin{aligned} \varTheta = \varTheta _\text {ref} \frac{(S_\mathrm{w}^{\text {opt}})_\text {ref} - (S_\mathrm{w}^{\text {min}})_\text {ref}}{S_\mathrm{w}^{\text {opt}}- S_\mathrm{w}^{\text {min}}} = \varTheta _\text {ref} \frac{r \left( \frac{\mathrm {d}P_\mathrm{c}}{\mathrm {d}S_\mathrm{w}} \right) _{S_\mathrm{w}^{*}}}{r_\text {ref} \left( \frac{\mathrm {d}P_{\mathrm{c},\text {ref}}}{\mathrm {d}S_\mathrm{w}} \right) _{(S_\mathrm{w}^{*})_\text {ref}}}. \end{aligned}$$

For two homothetic porous media of characteristic pore radii r and $r_\text {ref}$, $P_\mathrm{c}(S_\mathrm{w}) = \frac{r_\text {ref}}{r} P_{\mathrm{c},\text {ref}} (S_\mathrm{w})$, then the invariance of $r P_\mathrm{c}^*$, i.e. $r P_\mathrm{c}(S_\mathrm{w}^{*}) = r_\text {ref} P_{\mathrm{c},\text {ref}}[(S_\mathrm{w}^{*})_\text {ref}]$, implies that $P_{\mathrm{c},\text {ref}}(S_\mathrm{w}^{*}) = P_{\mathrm{c},\text {ref}}[(S_\mathrm{w}^{*})_\text {ref}]$, i.e. that $S_\mathrm{w}^{*}= (S_\mathrm{w}^{*})_\text {ref}$, which is also verified for the optimal and minimal saturation. Hence, $S_\mathrm{w}^{\text {opt}}- S_\mathrm{w}^{\text {min}}= (S_\mathrm{w}^{\text {opt}})_\text {ref} - (S_\mathrm{w}^{\text {min}})_\text {ref}$ and $\varTheta = \varTheta _\text {ref}$.

To end with, the transition saturation interval and the transition parameter $\varTheta$ are invariant for homothetic porous media under dynamic assumption.