Page 364 - Fundamentals of Reservoir Engineering
P. 364

NATURAL WATER INFLUX                                   299

                                     qµ
                           qt                                                                        (9.3)
                             () =
                               D
                            D
                                   2kh p
                                    π
                                        ∆
                     where q D (t D) is the dimensionless influx rate evaluated at r D = 1 and describes the
                     change in rate from zero to q due to a pressure drop ∆p applied at the outer reservoir
                     boundary r o at time t = 0. These functions can be generated from constant terminal rate
                     solutions and vice-versa. It is generally more convenient to express this solution in
                     terms of cumulative water influx rather than rate of influx. Thus integrating equ. (9.3)
                     with respect to time

                                   t      D t
                              µ            q (t ) dt  dt
                           2kh p     qdt =    D  D  dt D  D
                            π
                                ∆
                                   o      o
                     which gives

                             W µ   = Wt     φµ cr  2 0
                                        ()
                               e
                           2kh p      D  D    k
                                ∆
                            π
                     and therefore
                                            2
                                                   ()
                           W e     =  2πφ hcr ∆ pW t D                                               (9.4)
                                                  D
                                            0
                     in which, since Darcy units are being employed
                                   = cumulative water influx (ccs) due to a pressure drop ∆p (atm)
                           W e
                                      imposed at r o at t = 0

                     and W D (t D) = dimensionless, cumulative water influx function giving the
                                      dimensionless influx per unit pressure drop imposed at the reservoir
                                      aquifer boundary at t=0.

                     Equation (9.4) is frequently expressed as

                           W e     =  U p W (t )                                                     (9.5)
                                       ∆
                                            D
                                               D
                     where
                           U       =  2 f hcr o 2                                                    (9.6)
                                       πφ

                     which is the aquifer constant for radial geometry

                     and

                                      (encroachment angle) °
                           f       =
                                               360 °

                     which is to be used for aquifers which subtend angles of less than 360° at the centre of
                     the reservoir-aquifer system.

                     The dimensionless water influx W D (t D) is frequently presented in tabular form or as a
                     set of polynomial expressions giving W D as a function of t D for a range of ratios of the
   359   360   361   362   363   364   365   366   367   368   369