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LaTeX Math Block Reference

anchor	OF

can be rewritten in terms of Surface flowrates

LaTeX Math Inline

body	\{ q^{\uparrow}_L, q^{\downarrow}_W, q^{\downarrow}_G \}

and usual subject to enginerring engineering restrictions:

LaTeX Math Block

anchor	OF_L
alignment	left

G_t = \sum_{p=1}^{N^{\uparrow}_P} C^{\uparrow}_{OGW}(t)  \cdot q^{\uparrow}_{L, p}(t)
- \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,i} \cdot q^{\downarrow}_{W, i}(t) 
- \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}(t)
- C_{WS} \cdot q_{WS}(t)
- C_{GS} \cdot q_{GS}(t)

LaTeX Math Block

anchor	KPBIE
alignment	left

C^{\uparrow}_{OGW}(t) = \left[  (R_O(t) -  C^{\uparrow}_{O,p}) + (R_G(t) - C^{\uparrow}_{G,p}) \cdot  Y_{gG,p}(t) \right]  \cdot (1- Y_{wW,p}(t)) 
- C^{\uparrow}_{L,p} - C^{\uparrow}_{W,p} \cdot Y_{wW,p}(t)

LaTeX Math Block

anchor	RateLimit
alignment	left

0 \leq q^{\uparrow}_{L, p}(t) \leq  q^{\uparrow}_{LMAX, p}(t)

LaTeX Math Block

anchor	RateLimit
alignment	left

0 \leq q^{\downarrow}_{W, i}(t) \leq  q^{\downarrow}_{WMAX, i}(t)

LaTeX Math Block

anchor	RateLimit
alignment	left

0 \leq q^{\downarrow}_{G, j}(t) \leq q^{\downarrow}_{GMAX, j}(t)

LaTeX Math Block

anchor	RateLimit
alignment	left

\sum_{p=1}^{N^{\uparrow}_P} q^{\uparrow}_{L, p}(t) \leq q^{\uparrow}_{LMAX}

LaTeX Math Block

anchor	RateLimit
alignment	left

q^{\downarrow}_{WMIN}(t) \leq 
\sum_{i=1}^{N^{\downarrow}_W} q^{\downarrow}_{W, i}(t) 
\leq q^{\downarrow}_{WMAX}

LaTeX Math Block

anchor	RateLimit
alignment	left

\sum_{j=1}^{N^{\downarrow}_G} q^{\downarrow}_{G, j}(t)
\leq q^{\downarrow}_{GMAX}

LaTeX Math Block

anchor	RateLimit
alignment	left

q^{\downarrow}_{WMIN}(t) = \sum_{p=1}^{N^{\uparrow}_P} q^{\uparrow}_{W, p}(t)

...

LaTeX Math Inline

body	--uriencoded--Y_%7Bw%7BW,k%7D(t) = q_%7BW,k%7D / q_%7BL,k%7D

Watercut in

LaTeX Math Inline

body	k

-th well

LaTeX Math Inline

body	--uriencoded--Y_%7Bg%7BG,k%7D(t) = q_%7BG,k%7D / q_%7BO,k%7D

Gas-Oil Ratio in

LaTeX Math Inline

body	k

-th well

...

Expand

title	Sandface Formalism

The objective function

LaTeX Math Block Reference

anchor	OF_L

can be further rewritten in terms of Sandface flowrates

LaTeX Math Inline

body	\{ q^{\uparrow}_t, q^{\downarrow}_w, q^{\downarrow}_g \}

:

LaTeX Math Block

anchor	G_sf
alignment	left

G = \sum_{k=1}^{N^{\uparrow}_P} G^{\uparrow}_{t,k} \cdot q^{\uparrow}_{t, k}
- \sum_{i=1}^{N^{\downarrow}_W} G^{\downarrow}_{w,i}  \cdot 
q^{\downarrow}_{w, i}
- \sum_{j=1}^{N^{\downarrow}_G} G^{\downarrow}_{g,j}  \cdot q^{\downarrow}_{g, j} - 
- C_{WS} \cdot q_{WS}(t)
- C_{GS} \cdot q_{GS}(t)
 \rightarrow \rm max

LaTeX Math Block

anchor	Gtp
alignment	left

G^{\uparrow}_{t,k} = \frac{\left[  (R_O -  C^{\uparrow}_{O,k}) + (R_G - C^{\uparrow}_{G,k}) \cdot  Y_{gG,k} \right]  \cdot (1- Y_{wW,k}) 
- C^{\uparrow}_{L,k} - C^{\uparrow}_{W,k} \cdot Y_{wW,k} }
{B_{w,k} Y_{wW,k} + \left[ (B_{o,k} - R_{s,k} B_{g,k}) + (B_{g,k} - R_{v,k} B_{o,k}) \, Y_{gG,k} \right] \cdot (1-Y_{w,k})}

LaTeX Math Block

anchor	Gw
alignment	left

G^{\downarrow}_{w,i} = B_{w,i} C^{\downarrow}_{W,i}

LaTeX Math Block

anchor	Gg
alignment	left

G^{\downarrow}_{g,i} = B_{g,i} \cdot C^{\downarrow}_{G,i}

where

LaTeX Math Inline

body	B_{w,k} = B_w(p_{wf,k}(t))

Water FVF for

LaTeX Math Inline

body	k

-th well

LaTeX Math Inline

body	p_{wf,k}(t)

BHPin

LaTeX Math Inline

body	k

-th well

LaTeX Math Inline

body	B_{o,k} = B_o(p_{wf,k}(t))

Oil FVF for

LaTeX Math Inline

body	k

-th well

LaTeX Math Inline

body	R_{s,k} = R_s(p_{wf,k}(t))

Solution GOR in

LaTeX Math Inline

body	k

-th well

LaTeX Math Inline

body	B_{g,k} = B_g(p_{wf,k}(t))

Gas FVF for

LaTeX Math Inline

body	k

-th well

LaTeX Math Inline

body	R_{v,k} = R_v(p_{wf,k}(t))

Vaporized Oil Ratio in

LaTeX Math Inline

body	k

-th well

Expand

title	Derivation

Panel

borderColor	wheat
bgColor	mintcream
borderWidth	7

LaTeX Math Block

anchor	427RB
alignment	left

G(t) = \sum_{p=1}^{N^{\uparrow}_P} \left[ 
(R_O -  C^{\uparrow}_{O,p}) \cdot q^{\uparrow}_{O, p} + (R_G - C^{\uparrow}_{G,p}) \cdot  q^{\uparrow}_{G, p} 
- C^{\uparrow}_{L,p}  - C^{\uparrow}_{W,p} \cdot q^{\uparrow}_{W, p}
\right]  
- \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, i} 
- \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}

LaTeX Math Block

anchor	427RB
alignment	left

G(t) = \sum_{p=1}^{N^{\uparrow}_P} \left[ 
\left[  (R_O -  C^{\uparrow}_{O,p}) + (R_G - C^{\uparrow}_{G,p}) \cdot  Y_{gG,p} \right]  \cdot q^{\uparrow}_{O, p} 
- C^{\uparrow}_{L,p}  - C^{\uparrow}_{W,p} \cdot Y_{wW,p} \cdot q^{\uparrow}_{L, p}
\right]  
- \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, i} 
- \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}

LaTeX Math Block

anchor	427RB
alignment	left

G(t) = \sum_{p=1}^{N^{\uparrow}_P} \left[ 
\left[  (R_O -  C^{\uparrow}_{O,p}) + (R_G - C^{\uparrow}_{G,p}) \cdot  Y_{gG,p} \right]  \cdot (1- Y_{wW,p}) 
- C^{\uparrow}_{L,p} - C^{\uparrow}_{W,p} \cdot Y_{w,p} 
\right]  \cdot q^{\uparrow}_{L, p}
- \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, i} 
- \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}

Translating

LaTeX Math Inline

body	q^{\downarrow}_{W, i}

and

LaTeX Math Inline

body	q^{\downarrow}_{G, j}

to Sandface flowrates

LaTeX Math Inline

body	q^{\downarrow}_{w, i}

and

LaTeX Math Inline

body	q^{\downarrow}_{g, j}

with formation volume factor and substituting liquid production rate

LaTeX Math Inline

body	q^{\uparrow}_{L, p}

from

LaTeX Math Block Reference

anchor	qL
page	Liquid production rate = qL

one arrives to:

LaTeX Math Block

anchor	427RB
alignment	left

G(t) = \sum_{p=1}^{N^{\uparrow}_P}  \frac{\left[  (R_O -  C^{\uparrow}_{O,p}) + (R_G - C^{\uparrow}_{G,p}) \cdot  Y_{g,p} \right]  \cdot (1- Y_{w,p}) 
- C^{\uparrow}_{L,p} - C^{\uparrow}_{W,p} \cdot Y_{wW,p} }
{B_w Y_{wW,p} + \left[ (B_o - R_s B_g) + (B_g - R_v B_o) \, Y_{gG,p} \right] \cdot (1-Y_{wW,p})}
 
 \cdot q^{\uparrow}_{t, p}
- \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot B_w \cdot q^{\downarrow}_{w, i} 
- \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot B_g \cdot q^{\downarrow}_{g, j}

which is equivalent to

LaTeX Math Block Reference

anchor	G_sf

.

Depending on Lift mechanism the rates in equation

LaTeX Math Block Reference

anchor	G_sf

may be set directly or calculated from THP and formation pressure

LaTeX Math Inline

body	p_e

(which is a usual case in injection wells):

LaTeX Math Block

anchor	3WM13
alignment	left

q^{\uparrow}_{t, k} = J_{t,k} \cdot ( p_{e,k} - p_{wf,k} )

LaTeX Math Block

anchor	9HEU2
alignment	left

q^{\downarrow}_{w,i} = J_{w,i} \cdot ( p_{wf,i} - p_{e,i} )

LaTeX Math Block

anchor	FQ5IH
alignment	left

q^{\downarrow}_{g,i} = J_{g,i} \cdot ( p_{wf,i} - p_{e,i} )

Producing wells may spontaneously vary between Constant rate production: q_L = const and Constant pressure production: pwf = const (see Constant rate production: q_L = const for alternation details).

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[ Constant rate production: q_L = const ] [ Constant pressure production: pwf p_wf = const ]

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