GLOSSARY

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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 bottom-hole 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 and varying injection rate at injector W2  q_2(t):

(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 injector's W2 flowrate  \delta q_2 at zero time  \tau = 0, which can be written as:  dq_2(\tau) = \delta q_2 \cdot \delta(\tau) \, d\tau.

The 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) \cdot \delta q_2

which leads to  (1).


Case #2 – Constant BHP:  p_1 = \rm const


Assume that the flowrate  \delta q_1(t) in producer W1 is being adjusted to compensate the bottom-hole pressure variation  \delta p_1(t) in response to the flowrate variation  \delta q_2 in injector W2 so that bottom-hole pressure in producer W1 stays constant at all times \delta p_1(t) = \delta p_1 = \rm const.

In this case, flowrate response  \delta q_1 in producer W1 to the flowrate variation  \delta q_2 in injector W2 is going to be:

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

where

t

time since injector's W2 rate has changed by  \delta q_2.

\dot p_{u,\rm 21}(t)

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

\dot p_{u,\rm 11}(t)

time derivative of 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:

(5) 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 zero as the bottom-hole pressure in producer W1 stays constant at all times:

(6) \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
(7) 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)

The zero-time value of DTR / CTR is zero by definition  p_{u,\rm 11}(0) = 0, \, p_{u,\rm 21}(0) = 0 which leads to:

(8) \int_0^t \dot p_{u,\rm 11}(t-\tau) dq_1(\tau) = - \int_0^t \dot p_{u,\rm 21}(t-\tau) dq_2(\tau)

Consider a step-change in producer's W1 flowrate  \delta q_1 and injector's W2 flowrate  \delta q_2 at zero time  \tau = 0, which can be written as  dq_1(\tau) = \delta q_1 \cdot \delta(\tau) \, d\tau and  dq_2(\tau) = \delta q_2 \cdot \delta(\tau) \, d\tau. Substituting this to  (8) leads to:

(9) \int_0^t \dot p_{u,\rm 11}(t-\tau) \delta q_1 \cdot \delta(\tau) \, d\tau = - \int_0^t \dot p_{u,\rm 21}(t-\tau) \delta q_2 \cdot \delta(\tau) \, d\tau
(10) \dot p_{u,\rm 11}(t) \delta q_1 = - \dot p_{u,\rm 21}(t) \delta q_2

which leads to  (4).


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:

(11) \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:

(12) p_{u,\rm 11}(t \rightarrow \infty) \rightarrow \frac{t}{c_{t,1} V_{\phi, 1}}
(13) 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:

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

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

See also

[ DTR ] [ CTR ] [ Capacitance Resistance Model (CRM) ]





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