Summary.
This paper develops the conditions for gravity-stable displacement in stratified reservoirs. Results are given for the shape of a gravity-stabilized displacement front and for the asymptotic behavior of nonstable segregated fronts. Partial stability may prevail in the latter case. The analysis assumes piston-like displacement and incompressible fluids and is based on the vertical-equilibrium (VE) approximation.
Introduction
The vertical sweep efficiency of a reservoir flooding process may be influenced strongly by gravity if a density contrast exists between displacing and displaced fluids. Dietz's classic paper gives a criterion for gravity-stable displacement in homogeneous reservoirs and a simple expression for the shape of a stabe displacement front (linear). Dietz's theory also allows determination of the asymptotic behavior of nonstable fronts.
Dietz's results are not applicable in stratified reservoirs. Later authors who used similar techniques tended to ignore gravity or to retain the assumption of homogeneity. With stratification it has been customary to rearrange layers to obtain a monotonous distribution of permeabilities, a procedure which is strictly not valid. An alternative is offered by van Daalen and van Domselaar, who studied displacement in stratified reservoirs without rearranging layers. They neglect the dip-normal component of gravity, however, with the consequence that gravity-stable displacement is generally no longer possible.
This paper generalizes Dietz's results to cover stratified reservoirs. Unavoidably, results become more complex. More fundamentally, stratified reservoirs exhibit effects not seen in homogeneous reservoirs, the most important of which are breakdown of gravity segregation (i.e., internal tonguing in the reservoir) and partial gravity stability.
system
A 2D (linear) dipping reservoir is considered (Fig. 1). General stratification is allowed; i.e., all reservoir parameters are constant in the along-dip direction, x, but may vary in the dip-normal direction, y. Upper and lower boundaries are impermeable. Constantrate injection takes place along the left boundary, and corresponding production along the right boundary. Continuous injection of one single fluid is assumed. Fluids and the rock matrix are incompressible. Displacing and displaced fluids may have different densities. The displacement is assumed to be piston-like in the sense that a sharp interface (front) exists between the fluids, on either side of which only one of the fluids is mobile. The front is not.
VE Approximation
Appendix A gives the governing equations for the present problem. The VE approximation is used to solve the equations. The central point of the VE approximation is the neglect of dip-normal potential gradients. 1-11 In Appendix B, zero dip-normal potential gradients are shown to result in the limit as dip-normal mobility (permeability) components approach infinity.
For a given front shape, the VE approximation allows analytical solution for velocity and pressure fields and the front velocity. Appendix B gives the mathematical details. Example results in Fig. 2 display an arbitrary front in a case with favorable mobility ratio and a displacing fluid denser than the displaced fluid. Fig. 2 illusirates some properties of the VE approximation.The VE (pressure and velocity) solution along a given dipnormal (constant-x) line will depend only on the front configuration along that line itself. This accounts for the abrupt stream-lineslope (and position) changes at front-slope discontinuities.Fluid velocities are continuous and finite, except per-naps at the along-dip position of front-slope discontinuities.
The last statement addresses a point of some confusion m the literature. Statements on crossflow range from "all flow takes place in the horizontal direction only" to "requires the transfer of material-to be instantaneous," to "infinite vertical flow rate." As indicated by the continuity of stream lines, crossflow is finite in the VE approximation, except perhaps at the along-dip position of front-slope dicontinuities. At these positions, along-dip velocities may be discontinuous and dip-normal velocities may the infinite. With negligible gravity, infinite dip-normal velocities are produced only by infinite front slopes. in Fig. 2, lines have been drawn across such "irregular" points by shifting their dip-normal positions in such a manner that total flow between adjacent stream lines is identical on either side of the point.
Gravity-Stable Displacement
Appendix B offers a level of generality with respect to front shapes that will not be used fully in tills work. Rather, attention is restricted to segregated fronts- -1. e., fronts where the displacing fluid is mono-tonicallnderriding (or overriding) the displaced fluid (such that front height h is a single-valued monotonous function of x,). In the interest of brevity, only the underride case (see Fig. 1) is treated. If, for a segregated front, the denser fluid is underlying the less dense fluid, we will call the front gravity-segregated. Similarly, n displacement process is called gravity-segregated if it proceeds through a succession of gravity-segregated fronts.
If a gravity-segregated displacement proceeds with a finite-length front of stable stationary shape, we will call the process a gravity-stable displacement. A front of stationary shape (a stationary front) is one for which the (along-dip) front velocity is constant along the front. If stable, such a front will move through the reservoir without changing shape.
By continuity, the front velocity of a stationary front must be
v= Q/hRO phi e dy............................... (1)
In the VE approximation, the front velocity, v(h), of an arbitrary front can be inferred from (see Eq. B-14)
delta xh phi e (h)=delta xq(h)...................... (2)
In these equations, Q is the total flow rate, hR is the reservoir height, phi e is the effective porosity, and h = h(x) is the front height (see Fig. 1). q(h) is the flow rate through the interval (x, O) to [x, h(x)].
Combining Eqs. 1 and 2 and integrating from the tip of the tongue gives
ho phi e dy Q-----------------=q(h)−q(0)=q(h)......................(3) hr0 phi e dy
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