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3 Initial Conditions, Boundary Conditions and Model Parameters

[513.4.1] Consider a cylindrical column containing a homogeneous, isotropic and incompressible porous medium. [513.4.2] The column is closed at both ends and filled with two immiscible fluids. [513.4.3] Assume that the surface and interfacial tensions are such that the capillary fringe is much thicker than the column diameter so that a onedimensional description is appropriate.

[513.5.1] The experimental situation considered here is that of raising a closed column as described in [26, p. 223]. [513.5.2] It is illustrated in Figure 1. [513.5.3] Initially, at instant t=0, both fluids are at rest. [513.5.4] The column is oriented horizontally (ϑ=0), i.e., perpendicular to the direction of gravity. [513.5.5] The displacement processes are initiated by rotating the column into a vertical position. [513.5.6] The time protocol for rotating the column may generally be written as

ϑ⁢t=arcsin⁡12⁢tanh⁡t-t*t*⁣*+1(14)

where t* is the instant of most rapid rotation and t*⁣* is the inverse rate of the rotation.

Figure 1: Raising a closed column according to the protocol of (14). Initially the column is oriented horizontally and the saturations are constant. Regions of high water saturation are indicated in darker shade, regions of high oil saturation are indicated in lighter shade. The last two columns illustrate the simultaneous imbibition (lower part of column) and drainage (upper part of column) processes that finally result in the formation of a capillary fringe.

[513.6.1] The constitutive equations yield the following system of 10 coupled nonlinear partial differential equations

∂⁡S1∂⁡t+∂⁡S1⁢v1∂⁡x=η2⁢S2-S2*SW*-SW⁢∂⁡SW∂⁡t(15a)
∂⁡S2∂⁡t+∂⁡S2⁢v2∂⁡x=-η2⁢S2-S2*SW*-SW⁢∂⁡SW∂⁡t(15b)
∂⁡S3∂⁡t+∂⁡S3⁢v3∂⁡x=η4⁢S4-S4*SO*-SO⁢∂⁡SO∂⁡t(15c)
∂⁡S4∂⁡t+∂⁡S4⁢v4∂⁡x=-η4⁢S4-S4*SO*-SO⁢∂⁡SO∂⁡t(15d)
ϱW⁢D1D⁢t⁢v1+∂⁡P1∂⁡x-ϱW⁢g⁢sin⁡ϑ
=∑j=15R1⁢jϕ⁢S1⁢vj-v1-η2⁢v1S1⁢S2-S2*SW*-SW⁢∂⁡SW∂⁡t(15e)
ϱW⁢D2D⁢t⁢v2+∂∂⁡x⁢P3-γ⁢P2*⁢S2γ-1-Πa⁢S1-α-ϱW⁢g⁢sin⁡ϑ
=∑j=15R2⁢jϕ⁢S2⁢vj-v2+η2⁢v2S2⁢S2-S2*SW*-SW⁢∂⁡SW∂⁡t(15f)
ϱO⁢D3D⁢t⁢v3+∂⁡P3∂⁡x-ϱO⁢g⁢sin⁡ϑ
=∑j=15R3⁢jϕ⁢S3⁢vj-v3-η4⁢v3S3⁢S4-S4*SO*-SO⁢∂⁡SO∂⁡t(15g)
ϱO⁢D4D⁢t⁢v4+∂∂⁡x⁢P1-δ⁢P4*⁢S4δ-1-Πb⁢S3-β-ϱO⁢g⁢sin⁡ϑ
=∑j=15R4⁢jϕ⁢S4⁢vj-v4+η4⁢v4S4⁢S4-S4*SO*-SO⁢∂⁡SO∂⁡t(15h)
S1+S2+S3+S4=1(15i)
∂⁡P3∂⁡x=∂⁡P1∂⁡x+∂2⁢∂⁡x⁢Πa⁢S1-α-Πb⁢S3-β+γ⁢P2*⁢S2γ-1-δ⁢P4*⁢S4δ-1(15j)

[page 514, §0]    where v5=0,R12=0,R34=0, and the quantities SW*,SO*, S2*,S4* are defined in eq. (7). [514.0.1] This system of 10 equations is reduced to a system of only 9 equations by inserting eq. (15j) into eqs. (15) and (15) to eliminate ∂⁡P3/∂⁡x. [514.0.2] The remaining 9 unknowns are Si,vi,(i=1,2,3,4) and P1.

[514.1.1] The system (15) has to be solved subject to initial and boundary data. [514.1.2] No flow boundary conditions at both ends require

vi⁢0,t=0,i=1,2,3,4,(16a)
vi⁢L,t=0,i=1,2,3,4.(16b)

[514.1.3] The fluids are incompressible. [514.1.4] Hence the reference pressure can be fixed to zero at the left boundary

P1⁢0,t=0.(17)

[514.1.5] The saturations remain free at the boundaries of the column.

[514.2.1] Initially the fluids are at rest and their velocities vanish. [514.2.2] The initial conditions are (i=1,2,3,4)

vi⁢x,0=vi0(x)=0(18a)
P1=P10(x)=0(18b)
Si⁢x,0=Si0(x)=Si0.(18c)

[514.2.3] In the present study the initial saturations will be taken as constants, i.e. independent of x.

[514.3.1] The model parameters are chosen largely identical to the parameters in [26]. [514.3.2] They describe experimental data obtained at the Versuchseinrichtung zur Grundwasser- und Altlastensanierung (VEGAS) at the Universität Stuttgart [40]. [514.3.3] The model parameters are ϱW=1000⁢kg⁢m-3, ϱO=800⁢kg⁢m-3, ϕ=0.34, SW⁢i=0.15, SO⁢r=0.19, η2=4, η4=3, α=0.52, β=0.90, γ=1.5, δ=3.5, Πa=1620⁢Pa, Πb=25⁢Pa, P2*=2500⁢Pa and P4*=400⁢Pa. [514.3.4] In [26] only stationary and quasistationary solutions were considered, and the viscous resistance coefficients remained unspecified. [514.3.5] In order to [page 515, §0]    find realistic values remember that R31+R41+R15=2⁢μW⁢ϕ2/k [25, 26]. [515.0.1] Realistic values for the viscosity and permeability are μW=0.001kgm-1s-1 and k=10-12m2. [515.0.2] Based on these orders of magnitude the viscous resistance coefficients are specified as R13=R31=R14=R41=R23=R32=R24=R42=R15=R35=1.7×108 kg m-3s-1, and R25=R45=1.7×1016 kg m-3s-1.

[515.1.1] The column is filled with water having total saturation SW=0.45 and oil with saturation SO=0.55. [515.1.2] Two different initial conditions will be investigated that differ in relative abundance of the nonpercolating phase. [515.1.3] The saturations for initial condition A and B are in obvious notation given as

S10⁢A=0.449,S10⁢B=0.302(19a)
S20⁢A=0.001,S20⁢B=0.148(19b)
S30⁢A=0.377,S30⁢B=0.549(19c)
S40⁢A=0.173,S40⁢B=0.001.(19d)

[515.2.1] These phase distributions can be prepared experimentally by an imbibition process for A or by a drainage process for inital condition B. [515.2.2] The values for the nonpercolating saturations were chosen from the nonpercolating saturations predicted within the residual decoupling approximation. [515.2.3] They can be read off from Figure 5 in [26]. [515.2.4] The values for the percolating phases follow from the requirement that the total water saturation is 0.45. [515.2.5] The time scales for raising the column are chosen as t*=50000⁢s t*⁣*=10000⁢s corresponding to roughly 3 hours.

Figure 2: Approximate numerical solutions of eqs. (15) showing time evolution of saturation profiles S2⁢x,t, SW⁢x,t, 1-S4⁢x,t at times t=0s, t=6×106s (solid lines) and t=105,2.5×105,5×105,7.5×105s (dashed lines) when raising a closed column of length L=4m from a horizontal to a vertical orientation. The left figure shows the time evolution starting from initial condition A in eq. (19), the right figure for initial condition B. Solid vertical lines correspond to t=0 when the column is horizontal. Dashed lines correspond to intermediate times. The first dashed line corresponding to instant t=105s is the time when the column has just reached a vertical orientation. Solid curves show the quasistationary solution for t=6×106s. While the upper part of the column is drained, imbibition takes place simultaneously in the lower part. Near x=2.5m drainage takes place initially but is later followed by imbibition.

Figure 3: Initial (t=0) and quasistationary (t=6×106) saturation profiles Si⁢x,t as a function of height x after raising a closed column of length L=4m. Two triples of curves (solid and dashed) are shown. In each triple the leftmost curve shows S2⁢x, the center curve shows SW⁢x and the rightmost curve shows 1-S4⁢x. The triple of dashed vertical straight lines represents the saturation profile at t=0 for initial condition A in eq. (19), while the triple of solid vertical lines represents initial condition B (the rightmost solid line and leftmost dashed line coincide almost with the bounding box). Note that the dashed and solid lines at S=0.45 coincide. The triple of dashed curves represents the quasistationary saturation profile for t→∞ resulting from initial condition A, while the triple of solid curves represents the stationary saturation profile for t→∞ resulting from initial condition B. The figure illustrates that the quasistationary profiles depend strongly on the initial condition.