Shock saturation

This section introduces the concept of saturation shock and discusses the fractional flow curve analysis of different processes. 

Before discussing fractional flow analysis, we first need to derive the moving velocity of a saturation discontinuity or shock. Figure 2.13 shows a saturation shock from 8 2 to S i. 8 2 moves from Xj to X2 during the time interval At = t2 - h. The total injection rate, qt, is constant, but the water cut changes from f i to f 2, which corresponds to 8 i and 8 2, respectively. Therefore, during the time interval. At, the total incremental water injected into the block from Xi to X2 is (q,)(At)(f 2-f i). Meanwhile, this incremental water injected results in the increase in saturation from 8 1 to 8 2. The material balance of water gives... 

Then the velocity Vas at which saturation shock exists is... 

When a chemical solution is injected into a reservoir at interstitial water saturation, due to chemical retention, a denuded water zone is formed at the injection front, which causes a chemical shock at x 3, as shown in Figure 2.15. This chemical shock causes the saturation shock from 8 3 to 8 1 at x 3. The denuded water displaces the interstitial water. There is a boundary between the denuded water and the displaced interstitial water at x b. 

In the case of polymer flooding with a sharpening front, polymer concentration jumps from zero (its initial value) to its injection concentration Ciq. D, in Eq. 2.88 becomes Dp. In this case, the high polymer concentration solution flushes the initial zero polymer concentration solution. As discussed in Section 2.5 on types of fronts, there is a concentration shock. Corresponding to this concentration shock, there is a saturation shock from S p to Swi. The specific velocity of this saturation shock, (vd )as > is... 

The polymer concentration shock, corresponding to the saturation shock from S p to S i, moves at Vcpi... 

Pope (1980) assumes in his paper that the condition of having a negative curvature isotherm (i.e. (d Q/dC ) < 0) is sufficient to guarantee a self-sharpening polymer front and saturation shock front formation. Although in practice this normally occurs, the governing conditions are rather more complex than this, but this matter will not be discussed in this work. 

Shock velocity. We will consider the problem that arises when saturation shocks do form. (Problems with smooth but rapidly varying properties are... 

Fortran implementation. Equation 21-77 is easily programmed in Fortran. Because the implicit scheme is second-order accurate in space, thus rigidly enforcing the diffusive character of the capillary pressure effects assumed in this formulation, we do not obtain the oscillations at saturation shocks or the saturation overshoots having S > 1 often cited. The exact Fortran producing the results shown later is displayed in Figure 21-5 and in several function statements given later. For convenience, the saturation derivatives F (Sy ) and G (Sy ) are denoted FP and GP (P indicates prime for derivatives). 

In the sequence of snapshots in Figures 21-8a,b,c, the formation and movement of the saturation shocks are shown for high, very high, and very slow invasion rates, all using 0.001 sec time steps. Again, complete stability is obtained, without numerical saturation oscillations. 

The early time saturation solution shown in Figure 21-12a indicates that inertia effects are not yet strong. This is clear, since reference to our source code shows that we have initialized our pressure field to a constant value throughout, so that the flow is initially stagnant. At t = 0+, a sudden applied pressure differential is introduced (that is, PLEFT - FRIGHT > 0), and fluid movement commences. However, the saturation shock has not formed, and the flow is controlled by capillary pressure. Note how the computed pressure shows a mild slope discontinuity, not unlike that presumed in Chapter 17. 

The uppermost line represents the mudcake initial condition that is, at time t = 0, the surface of the infinitesimally thin cake coincides with the borehole radius. In Figures 21-15a,b,c, the computational parameters are identical to those in Figures 21-14a,b,c, except that the cake grows from zero thickness, as opposed to being fixed at 0.01 in. for all time. Since the mudcake considered in Figure 21-15 is typically thinner than that in Figure 21-14 for any instant in time, we expect greater relative invasion. In fact, we do observe increased water saturation and deeper penetration of the saturation shock into the rock. 

We emphasize that we have obtained stable numerical results, without saturation overshoots and local oscillations, all using second-order accurate spatial central differencing, without having to introduce special upwind operators. The methods developed are stable and require minimal computing since they are based on tridiagonal equations. Several subtle aspects of numerical simulation as they affect miscible diffusion and immiscible saturation shock formation are discussed in Chapter 13. 

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