Linear Aquifer Behavior

Abstract

Introduction Linear aquifers, either limited or essentially infinite, may be encountered in reservoir engineering practice. In areas where faulting fixes reservoir boundaries, the fault block reservoir may have an aquifer of limited extent whose geometry is best approximated as linear. An infinite linear aquifer can occur as a regional feature whenever water movement through the aquifer member is constrained to one direction. Such constraints can arise from major faults. facies changes or pinchout of the member. Miller* pointed out that linear aquifers have received only meager attention in the past. He analyzed the performance of finite and infinite aquifers, developed working equations and curves, and presented examples. While Miller's curves may be used fairly easily, a separate one is required for each size of aquifer. In this paper, Miller's equations have been used as a starting point. By modifying them slightly, they can be reduced to a form which yields a single a working curve, applicable to any size of aquifer. Thus, interpolation between curves is eliminated and accuracy is improved. Miller's results for finite aquifers covered only the boundary condition of no flow across the outer aquifer boundary. This paper also includes the case of constant pressure at the outer aquifer boundary. DEVELOPMENT OF EQUATIONS FOR LINEAR AQUIFERS Miller's equations give pressure drop or cumulative influx at the linear aquifer-reservoir boundary as a function of time for the boundary conditions of an infinite aquifer and a finite aquifer with sealed outer boundary. In addition to these equations, those appropriate for the boundary condition of a finite aquifer with constant pressure at the outer boundary have been developed. The approach used in developing these equations was the same as that used by Miller. BOUNDARY CONDITION 1: CONSTANT RATE OF INFLUX ACROSS AQUIFER-RESERVOIR BOUNDARY Infinite Linear Aquifer (1) Finite Linear Aquifer, Constant Pressure at Outer Boundary (2) BOUNDARY CONDITION 2: CONSTANT PRESSURE AT AQUIFER-RESERVOIR BOUNDARY Infinite Linear Aquifer (4) Finite Linear Aquifer, Sealed Outer Boundary (5) Finite Linear Aquifer, Constant Pressure at Outer Boundary (6) These equations are usually put in a form where dimensionless time is defined by (7) Here, x is a reference distance and is usually taken to be a unit distance. However, the choice is really arbitrary, as long as consistency is maintained. We choose x = L; then (8) For finite aquifers, L is the length of aquifer; for infinite cases, it may be considered as an arbitrarily chosen length. The reason for this choice will be clear later when the performances of finite and infinite aquifers are compared. JPT P. 561ˆ

Coarse Scale Simulation in Tight Gas Reservoirs

“…The % error in p D is calculated using the following formula: The % error in J D is calculated using the following formula: The previous formula was deduced from the productivity index from the linear flow case (11,12) : We conclude from Table 4 that a match with the analytical solution before reaching the pseudo steady state period is not possible given that using a coarse grid block can cause a significant error in the transient period, whereas Table 5 shows that there is not so much of an error in the pseudo steady state period.…”

Water Influx, and Its Effect on Oil Recovery: Part 1. Aquifer Flow, SUPRI TR-103

“…The mathematics of the problem were first published by Miller in 1962, but shortly after that a quite elegant piece of work by Nabor and Barham (1964) presented the entire linear aqtier equations and curves in a three-page paper in the Journal of Petroleum…”

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“…Linear flow solutions have been discussed extensively by many authors (Miller, 1962;Nabor, 1964 where p i is initial reservoir pressure, pwf is wellbore pressure, 4p is flow rate, w is reservoir width, h is the reservoir thickness, and C , is the total system compressibility.…”

“…Transient influx considering both radial (van Everdingen and Hurst, 1949) and linear (Miller, 1962;and Nabor, 1964) aquifer geometries.…”