Page 366 - Fundamentals of Reservoir Engineering
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NATURAL WATER INFLUX                                   301


                                   Darcy Units                                Field Units

                                      kt                                                 kt
                                t =      2  (t-sec)                      t D = constant ×   2        (9.9)
                                 D
                                    φµ cL                                              φµ cL
                                                              constant, same as for equ. (9.7)


                               U = wLhφc  (cc/atm)                       U = .1781 wLhφc  (bbl/psi) (9.10)

                     Other characteristic features of the plots of W D(t D) versus t D depend upon whether the
                     aquifer is bounded or infinite in extent.

                     Bounded Aquifers

                     Irrespective of the geometry there is a value of t D for which the dimensionless water
                     influx reaches a constant maximum value. This value is, however, dependent upon the
                     geometry as follows


                                                 2
                           Radial W D  (max =  1 2  ( eD  −  ) 1                                    (9.11)
                                          )
                                                r
                           Linear W D  (max ) 1=                                                    (9.12)

                     Note that if W D in equ. (9.11) is used in equ. (9.4), for a full radial aquifer (f = 1), the
                     result is

                                                     2
                                                        2
                                          2
                                                        0
                                                     e
                           W      = 2πφ hcr ×∆ p×  1 2  (r − r )
                                          0
                             e
                                                      r 0 2
                                          2
                                       2
                                  = π (r − r )h c p
                                             φ ∆
                                          0
                                       e
                     But this latter expression is also equivalent to the total influx occurring, assuming that
                     the ∆p is instantaneously transmitted throughout the aquifer. A similar result can be
                     obtained using equ. (9.12) for linear geometry. Therefore, once the plateau level of
                     W D (t D) has been reached, it means that the minimum value of t D at which this occurs
                     has been sufficiently large for the instantaneous pressure drop ∆p to be felt throughout
                     the aquifer. The plateau level of W D(t D) is then the maximum dimensionless water influx
                     resulting from such a pressure drop.
                     Infinite Aquifer

                     Naturally, no maximum value of W D (t D) is reached in this case since the water influx is
                     always governed by transient flow conditions. For radial geometry, values of W D (t D)
                     can be obtained from the graphs for r eD = ∞. There is no plot of W D (t D) for an infinite
                     linear aquifer. Instead, the cumulative water influx can be calculated directly using the
                     following equation


                                      φ kct
                           W =  2hw        ×∆ p (ccs)                                               (9.13)
                             e
                                       πµ
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