Exact Solution for Tertiary Polymer Flooding with Polymer Mechanical Entrapment and Adsorption

Abstract

Two-phase transport with simultaneous component adsorption and mechanical entrapment occurs in numerous suspension and nano-colloidal flows in porous media, including polymer flooding in oilfields. To the contrary of one-dimensional (1D) self-similar solution for polymer flooding that ignores the mechanical entrapment, accounting for this phenomenon breaks self-similarity. However, this non-self-similar problem allows for an exact integration. For the first time, the present work derives an exact solution for polymer injection into a rock saturated by oil and water with simultaneous adsorption and mechanical entrapment. The splitting method that uses Lagrangian coordinate as an independent variable instead of time, splits the 3 × 3 governing system into 2 × 2 auxiliary system that determines suspended and captured polymer concentrations, and the scalar equation for unknown phase saturation. The exact solution reveals two fingerprints of the component entrapment under small entrapped concentrations: stabilisation of the breakthrough concentration at the value below the injected one, and non-stabilised pressure drop growth. Three sets of laboratory polymer core-floodings exhibit a close match by the analytical model.

Appendix 1: Shock Waves in Governing System

Following Wagner (1987), Pires et al. (2006), Khorsandi et al. (2017) and Borazjani and Bedrikovetsky 2017 and Borazjani et al. (2019), here we present a stream function φ(x, t) associated with the conservation law Eq. (20)

$$s = - \frac{\partial \varphi }{\partial x},\quad f = \frac{\partial \varphi }{\partial t}.$$

(62)

The corresponding differential form for two-phase flux is

$${\text{d}}\varphi = f{\text{d}}t - s{\text{d}}x.$$

(63)

which determines stream function

$$\varphi \left( {x,t} \right) = \int\limits_{00}^{{\left( {x,t} \right)}} {f{\text{d}}t - s{\text{d}}x} .$$

(64)

Differential dt can be found form Eq. (63)

$${\text{d}}t = \frac{{{\text{d}}\varphi }}{f} + \frac{{s{\text{d}}x}}{f}.$$

(65)

Changing the independent variables from (x, t) to (x, φ) maps system (20)–(22) from variables (x, t) to (x,φ) plane

$${M:(x,t) \to (x,\varphi )\,}.$$

(66)

Figure 2 shows the mapping M.

The equality of second mixed derivatives of function t = t(x, φ) in Eq. (65) yields

$${\frac{{\partial F\left( {G,c,a,\sigma } \right)}}{\partial \varphi } + \frac{\partial G}{\partial x} = 0,\quad F = - \frac{s}{{f\left( {s,c,a,\sigma } \right)}},\quad G = \frac{1}{{f\left( {s,c,a,\sigma } \right)}} .}$$

(67)

Equation (67) is called the lifting equation, variables F and G are called density and flux functions, respectively.

Consider a curve of x(t) that is transformed to the trajectory x(φ) by mapping M, see Fig. 2. Substitution of the curve x(t) and x(φ), into the differential form of Eq. (63) yields

$$\frac{{{\text{d}}\varphi }}{{{\text{d}}x}} = f\frac{{{\text{d}}t}}{{{\text{d}}x}} - s,$$

(68)

Resulting in the following expression between the Lagrangian and Eulerian speeds V and D

$$\frac{1}{V} = \frac{f}{D} - s,\;\;\frac{{{\text{d}}x\left( \varphi \right)}}{{{\text{d}}\varphi }} = V,\;\;\frac{{{\text{d}}x\left( t \right)}}{{{\text{d}}t}} = D.$$

(69)

The Hugoniot–Rankine conditions on the shock waves for Eqs. (20)–(22) are formulated as equalities of the incoming and outgoing fluxes for the discontinuity trajectories for water, suspended and entrapped polymer (Lake et al. 2014)

$$\left[ s \right]D = [f(s,a,\sigma )],$$

(70)

$$[cs + a + \sigma ]D = [cf(s,a,\sigma )],$$

(71)

$$[\sigma ]D = 0.$$

(72)

where the square brackets [ ] indicate the difference between the two states across the discontinuity.

From Eq. (72), the entrapped concentration is continuous for D ≠ 0. Therefore from Eq. (71), the velocity of a saturation and concentration c-shock is

$$D = \frac{[cf(s,a,\sigma )]}{[cs + a]}.$$

(73)

The velocity of saturation shock from Eq. (70) with continuous c is

$$D = \frac{[f(s,a,\sigma )]}{\left[ s \right]}.$$

(74)

The Hugoniot–Rankine condition for auxiliary system Eqs. (28, 29) are

$$[a + \sigma ]V = [c],$$

(75)

$$[\sigma ]V = 0.$$

(76)

Equation (76) shows that the entrapped concentration is continuous for V ≠ 0. Therefore, Eq. (75) becomes

$$[a]V = [c].$$

(77)

and the velocity of the c shock is

$$V = \frac{[c]}{[a]} = \frac{{c^{ - } - c^{ + } }}{{\varGamma c^{ - } - \varGamma c^{ + } }} = \frac{1}{\varGamma }.$$

(78)

Equation (78) corresponds to the shock trajectory φ = Гx.

Consider the shock condition for the lifting equation Eq. (67)

$$\left[ F \right] = V\left[ G \right].$$

(79)

Equation (79) obtains the speed for the saturation shock with continuous c. for the c-shock, s-shock propagated with the Г⁻¹ speed.

$$\frac{\left[ F \right]}{[G]} = \varGamma .$$

(80)

The c- and s-shocks fulfil the Lax’s stability conditions (Gelfand 1959; Bedrikovetsky 1993).