Model Input/Output

• Input:   N^↑ , P_wf,  q_W^↓,  Q_W^↓, Q_O^↑ , Q_G^↑ , Q_W^↑
  
• Output:  q_O , q_G , q_W , P

Model Parameters

Model Parameters:

• M = {  V_O ,  y_W , x_W , x_G , y_W ,   x_J , J_W0 , k*_ro, n_o,  k*_rw, n_w }  – totally 11 constants  

Restrictions:

• all constants are positive
  
  
• 0 < k*_ro < 1, 0 < k*_rw < 1
  
  

Model fitting

Source data corrections:

• one time-array P_wf(t)

Restrictions:

• 0 < P_{wf, min} < P_wf(t) < P_{wf, max}

Goal function:

• E[M] =  w_O ‧ Σ δ[q_O] +  w_W ‧  Σ δ[q_W] +  w_G ‧ Σ δ[q_G] + w_P ‧ Σ δ[P] +  w_f ‧ Σ δ[P_wf]    →  min
  
  

δ[A] = [2 ‧ (A-A*)/(A+A*) ] ² 

A(t) – model prediction

A*(t) – historical value (if any)

Weights: 

• w_O + w_W + w_G + w_P + w_f = 1  (although it is not necessary)
  
  

Default settings:  w_O = 0.5 ,  w_W  = 0.1,  w_G = 0.1 , w_P = 0.2 ,  w_f = 0.1

Model function

q_O^↑ = q_O1^↑ ‧ N^↑  |   q_O1^↑= J_O ‧ ( P - P_wf )

q_W^↑ = q_W1^↑ ‧ N^↑  |   q_W1^↑ = J_W(t) ‧ ( P - P_wf )

q_G^↑ = J_G ‧ N^↑  |   q_G1^↑= J_G ‧ ( P - P_wf ) 

P = P_i +   ( y_W ‧  Bw ‧ ΣQ_W^↓ + Bo ‧ ΣQ_O^↑ + x_W ‧  Bw ‧ ΣQ_W^↑ + x_G ‧  Bg ‧ ΣQ_G^↑  )‧ P_i ‧ x_P /V_O 

s =  ΣQ_O^↑ / V_O

k*_ro(s) = k*_ro  ‧ (1 - s)^n_o

k*_rw(s) = k*_rw  ‧ s^n_w

J_O = k*_ro(s) ‧ x_J

J_W = J_W0 + k*_rw(s) ‧ x_J

J_G = R_s ‧ J_O

See Also

Petroleum Industry / Upstream /  Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM) / Capacitance-Resistivity Model (CRM) @model

[ Slightly compressible Material Balance Pressure @model ]

[ Dynamic fluid properties ]