Quasi-implicit treatment of velocity-dependent mobilities in underground porous media flow simulation

Abstract

Quasi-implicit schemes for treating velocity-dependent mobilities in underground porous media flow simulation, occurring when modeling non-Newtonian and non-Darcy effects as well as capillary desaturation, are presented. With low-order finite-volume discretizations, the principle is to evaluate mobilities at cell edges using normal velocity components calculated implicitly, and transverse velocity components calculated explicitly (i.e., based on the previously converged time-step); the pressure gradient driving the flow is, as usual, treated implicitly. On 3D hexahedral meshes, the proposed schemes require the same 7-point stencil as that of common semi-implicit schemes where mobilities are evaluated with an entirely explicit velocity argument. When formulated appropriately, their higher level of implicitness however places them, in terms of numerical stability, closer to “real” fully implicit schemes requiring at least a 19-point stencil. A von Neumann stability analysis of these proposed schemes is performed on a simplified pressure equation, representative of both single-phase and multiphase flows, following an approach previously used by the authors to study semi-implicit schemes. Whereas the latter are subject to stability constraints which limit their usage in certain cases where the logarithmic derivative of mobility with respect to velocity is large in magnitude, the former are unconditionally stable for 1D and 2D flows, and only subject to weak restrictionsfor 3D flows.

Appendices

Appendix 1. Analytic derivation of the stability criterion of the EC-PBVR-TL-QI scheme

The extrema and saddle points of f (Eq. 60) are located where ∂f/∂ℓ = 0, ∂f/∂m = 0 and ∂f/∂q = 0, i.e.,

$$ \left\{\begin{array}{ll} \nu_{x}\left( 1+\tilde{\eta}_{x}\right)s_{\ell} & = -\delta\left( \sqrt{\nu_{x}\nu_{y}\tilde{\eta}_{x}\tilde{\eta}_{y}}s_{m} +\sqrt{\nu_{x}\nu_{z}\tilde{\eta}_{x}\tilde{\eta}_{z}}s_{q}\right)c_{\ell} \\ \nu_{y}\left( 1+\tilde{\eta}_{y}\right)s_{m} & = -\delta\left( \sqrt{\nu_{x}\nu_{y}\tilde{\eta}_{x}\tilde{\eta}_{y}}s_{\ell} + \sqrt{\nu_{y}\nu_{z}\tilde{\eta}_{y}\tilde{\eta}_{z}}s_{q}\right)c_{m} \\ \nu_{z}\left( 1+\tilde{\eta}_{z}\right)s_{q} & = -\delta\left( \sqrt{\nu_{x}\nu_{z}\tilde{\eta}_{x}\tilde{\eta}_{z}}s_{\ell} + \sqrt{\nu_{y}\nu_{z}\tilde{\eta}_{y}\tilde{\eta}_{z}}s_{m}\right)c_{q} \end{array} \right. \ . $$

(96)

A solution to Eq. 96 is ℓ = m = q = 0, where f = 0 and the Hessian of f with respect to ℓΔx, mΔy and qΔz writes:

$$ H_{f} = \left( \begin{array}{ccc} \nu_{x}\left( 1+\tilde{\eta}_{x}\right) & \delta\sqrt{\nu_{x}\nu_{y}\tilde{\eta}_{x}\tilde{\eta}_{y}} & \delta\sqrt{\nu_{x}\nu_{z}\tilde{\eta}_{x}\tilde{\eta}_{z}} \\ \delta\sqrt{\nu_{x}\nu_{y}\tilde{\eta}_{x}\tilde{\eta}_{y}} & \nu_{y}\left( 1+\tilde{\eta}_{y}\right) & \delta\sqrt{\nu_{y}\nu_{z}\tilde{\eta}_{y}\tilde{\eta}_{z}} \\ \delta\sqrt{\nu_{x}\nu_{z}\tilde{\eta}_{x}\tilde{\eta}_{z}} & \delta\sqrt{\nu_{y}\nu_{z}\tilde{\eta}_{y}\tilde{\eta}_{z}} & \nu_{z}\left( 1+\tilde{\eta}_{z}\right) \end{array} \right) \ . $$

(97)

f is positive in the neighborhood of ℓ = m = q = 0 provided H_f is positive definite there. Using Sylvester’s criterion [28], stating that a symmetric matrix is positive definite if and only if all of its leading principal minors are positive, this implies:

$$ \begin{array}{ll} (a)\quad &1+\tilde{\eta}_{x}>0 , \\ (b)\quad &1+\tilde{\eta}_{x}+\tilde{\eta}_{y}>0 ,\\ (c)\quad & 1+\tilde{\eta}_{x}+\tilde{\eta}_{y}+\tilde{\eta}_{z}> 0\ . \end{array} $$

(98)

Similarly, the extrema and saddle points of g (Eq. 60) are located where ∂g/∂ℓ = 0, ∂g/∂m = 0 and ∂g/∂q = 0, i.e.,

$$ \left\{\begin{array}{ll} \nu_{x}\left( 1+\tilde{\eta}_{x}\right)s_{\ell} &= \delta\left( \sqrt{\nu_{x}\nu_{y}\tilde{\eta}_{x}\tilde{\eta}_{y}}s_{m} +\sqrt{\nu_{x}\nu_{z}\tilde{\eta}_{x}\tilde{\eta}_{z}}s_{q}\right)c_{\ell} \\ \nu_{y}\left( 1+\tilde{\eta}_{y}\right)s_{m} &= \delta\left( \sqrt{\nu_{x}\nu_{y}\tilde{\eta}_{x}\tilde{\eta}_{y}}s_{\ell} + \sqrt{\nu_{y}\nu_{z}\tilde{\eta}_{y}\tilde{\eta}_{z}}s_{q}\right)c_{m} \\ \nu_{z}\left( 1+\tilde{\eta}_{z}\right)s_{q} &= \delta\left( \sqrt{\nu_{x}\nu_{z}\tilde{\eta}_{x}\tilde{\eta}_{z}}s_{\ell} + \sqrt{\nu_{y}\nu_{z}\tilde{\eta}_{y}\tilde{\eta}_{z}}s_{m}\right)c_{q} \end{array} \right. \ . $$

(99)

A solution to Eq. 99 is ℓ = m = q = 0, where g = 0 and the Hessian of g with respect to ℓΔx, mΔy and qΔz writes: writes:

$$ H_{g} = \left( \begin{array}{ccc} \nu_{x}\left( 1+\tilde{\eta}_{x}\right) & -\delta\sqrt{\nu_{x}\nu_{y}\tilde{\eta}_{x}\tilde{\eta}_{y}} & -\delta\sqrt{\nu_{x}\nu_{z}\tilde{\eta}_{x}\tilde{\eta}_{z}} \\ -\delta\sqrt{\nu_{x}\nu_{y}\tilde{\eta}_{x}\tilde{\eta}_{y}} & \nu_{y}\left( 1+\tilde{\eta}_{y}\right) & -\delta\sqrt{\nu_{y}\nu_{z}\tilde{\eta}_{y}\tilde{\eta}_{z}} \\ -\delta\sqrt{\nu_{x}\nu_{z}\tilde{\eta}_{x}\tilde{\eta}_{z}} & -\delta\sqrt{\nu_{y}\nu_{z}\tilde{\eta}_{y}\tilde{\eta}_{z}} & \nu_{z}\left( 1+\tilde{\eta}_{z}\right) \end{array} \right) \ . $$

(100)

g is positive in the neighborhood of ℓ = m = q = 0 provided H_g is positive definite there, i.e., if and only if

$$ \begin{array}{ll} (a)\quad & 1+\tilde{\eta}_{x}>0 , \\ (b)\quad & 1+\tilde{\eta}_{x}+\tilde{\eta}_{y}>0 ,\\ (c)\quad & 1+\tilde{\eta}_{x}+\tilde{\eta}_{y}+\tilde{\eta}_{z}-4\tilde{\eta}_{x}\tilde{\eta}_{y}\tilde{\eta}_{z}> 0\ . \end{array} $$

(101)

The conditions of Eq. 98 are always satisfied for any physically meaningful nonlinear model (bijectivity condition of Eq. 36), and the only constraint for the solution ℓ = m = q = 0 to yield a local minimum of g is condition (c) of Eq. 101.

In addition to ℓ = m = q = 0, non-zero wavenumbers with (ℓ = 0 or ℓ = π/Δx) and (m = 0 or m = π/Δy) and (q = 0 or q = π/Δz) are also obvious solutions to Eqs. 96 and 99, where f > 0 and g > 0 (hence not yielding a global minimum of f and g).

Injecting the second and third lines of Eq. 96 in the first and assuming s_ℓ ≠ 0, s_m ≠ 0 and s_q ≠ 0 yields

$$ 1 = \zeta_{xy}^{2}c_{\ell}c_{m} +\zeta_{xz}^{2}c_{\ell}c_{q} +\zeta_{yz}^{2}c_{m}c_{q} -2\zeta_{xy}\zeta_{xz}\zeta_{yz}c_{\ell}c_{m} c_{q} , $$

(102)

where

$$ \zeta_{xy} = \sqrt{\frac{\tilde{\eta}_{x}\tilde{\eta}_{y}}{\left( 1+\tilde{\eta}_{x}\right)\left( 1+\tilde{\eta}_{y}\right)}}, $$

(103)

and similarly for ζ_xz and ζ_yz. A similar manipulation of Eq. 99 yields

$$ 1 = \zeta_{xy}^{2}c_{\ell}c_{m} +\zeta_{xz}^{2}c_{\ell}c_{q} +\zeta_{yz}^{2}c_{m}c_{q} +2\zeta_{xy}\zeta_{xz}\zeta_{yz}c_{\ell}c_{m}c_{q} \ . $$

(104)

Clearly, if condition (101c) is not satisfied, the scheme is not strictly stable. Otherwise, we can rewrite this condition as $1 > \zeta _{xy}^{2}+\zeta _{xz}^{2}+\zeta _{yz}^{2}+2\zeta _{xy}\zeta _{xz}\zeta _{yz}$, which implies that solutions to Eqs. 102 and 104 do not exist. In this case, both f and g admit a global minimum at ℓ = m = q = 0 where f = g = 0, and the scheme is stable.

In addition to the bijectivity condition of Eq. 36, a necessary and sufficient criterion for stability in the general 3D case is therefore given by condition (101c), which can be rewritten as

$$ 1+\tilde{\eta}-4\tilde{\eta}_{x}\tilde{\eta}_{y}\tilde{\eta}_{z}>0 , $$

(105)

or equivalently

$$ \left( 1-\eta\right)^{2}-4\eta_{x}\eta_{y}\eta_{z}>0 \ . $$

(106)

Appendix 2. Numerical experiments

Our in-house research reservoir simulator (IHRRS) is a general-purpose reservoir simulator based on a natural-variables formulation [21]. By default, it uses a first-order accurate finite-volume discretization with backward Euler time integration.

The analysis presented in the present paper is based on simplified model equations, as discussed in Section 6.2. In a companion paper [22], we describe the implementation of the EC-PBVR-TL-QI scheme in IHRRS to treat non-Newtonian polymer flows, accounting for model nonlinearities and heterogenous permeability fields.

Different numerical experiments have been performed, to validate our implementation and to illustrate the advantages of the proposed scheme. The benchmark includes two accuracy tests in 1D radial and 2D 5-spot geometries (test #1 and test #2), two stability tests in 1D linear and 3D quarter 5-spot geometries (test #3 and test #4), one monotonicity test in 1D radial geometry (test #5), and finally one performance test in a 3D heterogeneous sector model with horizontal wells (test #6).

Appendix 3. Practical significance of the stability criteria

3.1 A3.1 Non-Newtonian flows

Shear thinning polymer solutions typically exhibit a Newtonian viscosity plateau at low shear rate and a low viscosity plateau at high shear rate; in between, their viscosity can be fitted to a power-law form

$$ \mu_{\varphi}=C \dot{\gamma}_{\varphi}^{n_{s}-1} , $$

(107)

where n_s is the power-law index and the shear rate $\dot {\gamma }_{\varphi }$ is given by Eq. 6 or a close variant. In this parameter region,

$$ \eta_{\varphi}=1-n_{s} , $$

(108)

independent of shear rate (hence Darcy velocity). Values of n_s for Xanthan biopolymer at concentrations relevant to EOR applications have been measured by Cannella and coauthors [29] in the range 0.25–0.8 (yielding η ∈ [0.2,0.75]).

For strong velocities typical of near-wellbore conditions, synthetic polymers such as HPAM can exhibit shear thickening behavior, which can formally be modeled with a power-law viscosity (Eq. 107 with n_s > 1), although such “effective” viscosity would be different from that measured in a rheometer. Seright and coauthors [30] report a set of experiments where the logarithmic slope of high molecular weight HPAM resistance vs. Darcy velocity can be fitted with n_s up to $\sim 2.8$, i.e., η_w down to $\simeq -1.8$.

The FI and the TL-SI schemes can therefore accommodate any shear thickening strength of practical interest to EOR, whereas the R-SI scheme is limited to weak shear thickening. Conversely, the FI and R-SI schemes can accommodate any shear thinning strength, whereas the TL-SI schemes are limited.

The advantage of the QI schemes are here clearly apparent. Both EC-PBVR-TL-QI and EC-FBVR-R-QI schemes remove any shear thinning or shear thickening strength limit in 1D and 2D, and sufficiently extend the limits of the corresponding SI schemes in 3D to handle any model of practical interest.

3.2 A3.2 Non-Darcy flows

Let us consider a non-Darcean mobility model based on Forchheimer’s law, that we write for phase φ [24]:

$$ \lambda_{\varphi}=\frac{1}{\mu_{\varphi}\cdot\left( 1+\beta\sqrt{\tilde{k}}\text{Re}_{\varphi}\right)} , $$

(109)

where β is Forchheimer’s parameter, and the Reynolds number is as in Eq. 8. Equation 30 then yields

$$ \eta_{\varphi}=-\frac{\beta\sqrt{\tilde{k}}\text{Re}_{\varphi}}{1+\beta\sqrt{\tilde{k}}\text{Re}_{\varphi}}\in ]-1,0] \ . $$

(110)

$\beta \sqrt {\tilde {k}}\text {Re}_{\varphi }$ can be seen as a rescaled Reynolds number sometimes referred to as the Forchheimer number. Regardless of its value, η_φ > − 1; therefore, any semi-implicit scheme would be stable. When $\beta \sqrt {\tilde {k}}\text {Re}_{\varphi }\gg 1$ however, $\eta _{\varphi }\simeq -1$ and the R-SI schemes are close to their stability boundary, hence will respond with delay to a pressure signal. TL-SI are therefore appropriate in this case, although the EC-PBVR-TL-QI (and perhaps even the EC-FBVR-R-QI) would presumably bring improved monotonicity.

3.3 A3.3 Capillary desaturation

In a reservoir engineering context, capillary desaturation of phase φ by phase $\varphi ^{\prime }$ typically occurs in two circumstances: (a) during surfactant flooding of oil reservoirs, in which case φ is oil and $\varphi ^{\prime }$ is the aqueous phase, and (b) in the near-wellbore region where gas flows past banked condensate, in which case φ is oil/condensate and $\varphi ^{\prime }$ is gas.

We consider a two-phase model where the mobility factor of phase φ is given by

$$ \lambda_{\varphi}=\frac{k_{r\varphi}^{0}}{\mu_{\varphi}}\left( \frac{S_{\varphi}-S_{r\varphi}}{1-S_{r\varphi}-S_{r\varphi^{\prime}}}\right)^{n_{\varphi}} , $$

(111)

$k_{r\varphi }^{0}$, n_φ and S_rφ being phase φ’s permeability endpoint, Corey exponent and residual saturation, respectively. The mobility factor of phase $\varphi ^{\prime }$ is formally obtained by swapping φ and $\varphi ^{\prime }$ indices in Eq. 111, with $S_{\varphi }+S_{\varphi ^{\prime }}=1$.

We further consider a simplification of the desaturation model of Ref. [7], whereby S_rφ and n_φ are interpolated between $S_{r\varphi }^{\text {low}}\geq S_{r\varphi }^{\text {high}}$ and $n_{\varphi }^{\text {low}}\geq n_{\varphi }^{\text {high}}$, the residual saturations and Corey exponents at low and high trapping number, respectively, as

$$ S_{r\varphi}=S_{r\varphi}^{\text{high}}+\frac{S_{r\varphi}^{\text{low}}-S_{r\varphi}^{\text{high}}}{1+T_{\varphi}{N}_{t\varphi}} $$

(112)

and

$$ \begin{array}{@{}rcl@{}} n_{\varphi}&=&n_{\varphi}^{\text{low}}+\frac{S_{r\varphi}^{\text{low}}-S_{r\varphi}}{S_{r\varphi}^{\text{low}}-S_{r\varphi}^{\text{high}}}\left( n_{\varphi}^{\text{high}}-n_{\varphi}^{\text{low}}\right)\\ n_{\varphi^{\prime}}&=&n_{\varphi^{\prime}}^{\text{low}}+\frac{S_{r\varphi}^{\text{low}}-S_{r\varphi}}{S_{r\varphi}^{\text{low}}-S_{r\varphi}^{\text{high}}}\left( n_{\varphi^{\prime}}^{\text{high}}-n_{\varphi^{\prime}}^{\text{low}}\right) , \end{array} $$

(113)

In the above, T_φ is a model parameter and N_tφ is the trapping number of the displaced phase. Note that in the original model [7], $k_{r\varphi }^{0}$, $k_{r\varphi ^{\prime }}^{0}$ and $S_{r\varphi ^{\prime }}$ can vary as well (they are here kept constant), and n_φ in Eq. 113 would be interpolated based on the value of $S_{r\varphi ^{\prime }}$ (rather than S_rφ).

Considering a trapping number expressed as the norm of a linear function of the potential gradient (e.g., given by Eq. 10), Eq. 34 writes

$$ \tilde{\eta}_{T}= \frac{\lambda_{\varphi}}{\lambda_{T}} {N}_{t\varphi} \frac{\partial \ln(\lambda_{\varphi})}{\partial{N}_{t\varphi}} + \frac{\lambda_{\varphi^{\prime}}}{\lambda_{T}} {N}_{t\varphi}\frac{\partial\ln(\lambda_{\varphi^{\prime}})}{\partial{N}_{t\varphi}}\ ; $$

(114)

if needed, η_T can then be obtained from Eq. 35.

Figure 7 shows the dependence of $\tilde {\eta }_{T}$ on $S_{\varphi ^{\prime }}$ for a case where $S_{r\varphi }^{\text {high}}=0$ (i.e., complete desaturation), $n_{\varphi }^{\text {low}}=n_{\varphi ^{\prime }}^{\text {low}}=2$, $n_{\varphi }^{\text {high}}=n_{\varphi ^{\prime }}^{\text {high}}=1$ (i.e., linear relative permeability curves at strongly desaturating conditions), and $S_{r\varphi }^{\text {low}}=S_{r\varphi ^{\prime }}=0.2$. Figure 7a considers an endpoint mobility ratio $M_{_{\varphi ^{\prime }\varphi }}=k_{r\varphi ^{\prime }}^{0}\mu _{\varphi }/k_{r\varphi }^{0}\mu _{\varphi ^{\prime }}=0.1$ characteristic, for example, of water sweeping light oil. At low $S_{\varphi ^{\prime }}$ (i.e., when the first term of the RHS in Eq. 114 dominates) $\tilde {\eta }_{T}$ increases to a maximum before dropping to a negative value when approaching residual saturation of the displaced phase (i.e., the second term of the RHS in Eq. 114 dominates). With all other parameters set, the maximum of $\tilde {\eta }_{T}$ occurs when $T_{\varphi }{N}_{t\varphi }\simeq 1$, i.e., at the desaturation midpoint (see Eq. 112). Figure 7b considers T_φN_tφ = 1, and shows that with all other parameters set, the maximum of $\tilde {\eta }_{T}$ increases as $M_{_{\varphi ^{\prime }\varphi }}$ decreases. While the variation of mobility with potential gradient is mild when $M_{_{\varphi ^{\prime }\varphi }}=10$, characteristic of gas sweeping condensate, it becomes strong when $M_{_{\varphi ^{\prime }\varphi }}=0.01$, characteristic of viscous surfactant sweeping oil.

Fig. 7

(a) $\tilde {\eta }_{T}$ dependence on the trapping number, and (b) $\tilde {\eta }_{T}$ dependence on the mobility ratio, for a case with $S_{r\varphi }^{\text {low}}=S_{r\varphi ^{\prime }}=0.2$, $S_{r\varphi }^{\text {high}}=0$, $n_{\varphi }^{\text {low}}=n_{\varphi ^{\prime }}^{\text {low}}=2$ and $n_{\varphi }^{\text {high}}=n_{\varphi ^{\prime }}^{\text {high}}=1$

In case of very small endpoint mobility ratio, $\tilde {\eta }_{T}$ can significantly exceed unity. In this case, fully implicit as well as recursive methods (semi-implicit or quasi-implicit) would guarantee numerical stability; two-level methods (particularly if semi-implicit) are less appropriate.