GLOSSARY

Page tree

You are viewing an old version of this page. View the current version.

Compare with Current View Page History

« Previous Version 49 Next »


Consider a well-reservoir system consisting of:

  • producing well W1 draining the reservoir volume V_{\phi, 1}
  • water injecting well W2 supporting pressure in reservoir volume V_{\phi, 2} which includes the drainage volume  V_{\phi, 1} of producer W1 and potentially other producers. 

The drainage volume difference \delta V_{\phi} = V_{\phi, 2} - V_{\phi, 1} >0 may be related to the fact that water injection W2 is shared between V_{\phi, 1} and another reservoir  or with another producer. 

Case #1 –  Constant flowrate production  q_1 = \rm const >0


The pressure response  \delta p_1 in producer W1 to the flowrate variation  \delta q_2 in injector W2:

(1) \delta p_1 = - p_{u,\rm 21}(t) \cdot \delta q_2

where

t

time since the water injection rate has changed by the  \delta q_2 value.

p_{u,\rm 21}(t)

cross-well pressure transient response in producer W1 to the unit-rate production in injector W2


Consider a pressure convolution equation for the BHP in producer W1 with constant flowrate production at producer W1  q_1 = \rm const:

(2) p_1(t) = p_i - \int_0^t p_{u,\rm 11}(t-\tau) dq_1(\tau) - \int_0^t p_{u,\rm 21}(t-\tau) dq_2(\tau) = p_i - \int_0^t p_{u,\rm 21}(t-\tau) dq_2(\tau)

Consider a step-change in flowrate variation in injector W2:  dq_2(\tau) = \delta q_2 \cdot \delta(\tau) \, d\tau so that responding pressure variation  \delta p_1 in producer W1 will be:

(3) \delta p_1(t) = p_1(t)-p_i = - \int_0^t p_{u,\rm 21}(t-\tau) \delta q_2 \cdot \delta(\tau) \, d\tau = - p_{u,\rm 21}(t) \delta q_2
(4) p_{u,\rm 11}(0) \cdot q_1(t) + \int_0^t \dot p_{u,\rm 11}(t-\tau) dq_1(\tau) = - p_{u,\rm 21}(0) \cdot q_2(t) - \int_0^t \dot p_{u,\rm 21}(t-\tau) dq_2(\tau)


Case #2 – Constant BHP  p_1 = \rm const


The flowrate response  \delta q_1 in producer W1 to the flowrate variation  \delta q_2 in injector W2:

(5) \delta q_1 = - \frac{p_{u,\rm 21}(t)}{p_{u,\rm 11}(t)} \cdot \delta q_2

where

t

time since the water injection rate has changed by the  \delta q_2 in injector W2.

p_{u,\rm 21}(t)

cross-well pressure transient response (CTR) in producer W1 to the unit-rate production in injector W2

p_{u,\rm 11}(t)

drawdown pressure transient response (DTR) in producer W1 to the unit-rate production in the same well


Consider a pressure convolution equation for the above 2-wells system with constant BHP:

(6) p_1(t) = p_i - \int_0^t p_{u,\rm 11}(t-\tau) dq_1(\tau) - \int_0^t p_{u,\rm 21}(t-\tau) dq_2(\tau) = \rm const

The time derivative is going to be 

(7) \dot p_1(t) = - \left( \int_0^t p_{u,\rm 11}(t-\tau) dq_1(\tau) \right)^{\cdot} - \left( \int_0^t p_{u,\rm 21}(t-\tau) dq_2(\tau) \right)^{\cdot} = 0
(8) p_{u,\rm 11}(0) \cdot q_1(t) + \int_0^t \dot p_{u,\rm 11}(t-\tau) dq_1(\tau) = - p_{u,\rm 21}(0) \cdot q_2(t) - \int_0^t \dot p_{u,\rm 21}(t-\tau) dq_2(\tau)



For the finite-volume drain  V_{\phi,1} \leq V_{\phi,2} < \infty the flowrate response factor  \delta q_1 / \delta q_2 is getting stabilised over time as:

(9) \delta q_1 / \delta q_2 = f_{21} = \frac{c_{t,2} V_{\phi, 2}}{c_{t,1} V_{\phi, 1}} = \rm const

For the finite-volume reservoir  V_{\phi,1} \leq V_{\phi,2} < \infty the DTR and CTR are both going through the PSS flow regime at late transient times:

(10) p_{u,\rm 11}(t \rightarrow \infty) \rightarrow \frac{t}{c_{t,1} V_{\phi, 1}}
(11) p_{u,\rm 21}(t \rightarrow \infty) \rightarrow \frac{t}{c_{t,2} V_{\phi,2}}

where

c_{t,1}

average drain-area  total compressibility of formation around producer W1

c_{t,2}

average drain-area  total compressibility of formation around injector W2

Substituting  (Case2_PSS_p11) and  (Case2_PSS_p21) in  (Case2) one arrives to (Case2_PSS).

If pressure in producer W1 is supported by several injectors then:

(12) \delta q_1 =\sum_k f_{k1} \delta q_k

which makes one of the key assumptions in Capacitance Resistance Model (CRM).

See also

Capacitance Resistance Model (CRM)





  • No labels