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Based on the equivalent permeability tensor theory, a novel carbonate acidizing mathematical model is developed, which can be adopted to simulate the simulation process for carbonate matrix-acidizing and vuggy carbonate acidizing. Wormhole-channel in the core BC21 and BC10 was simulated based on the Omer Izgec experimental data. The comparative result between the experiment and the simulation shows that the two dominating wormhole-channels, the variation of pressure-drop curve and the pressure values at the inlet of the core are almost the same. The acid-flowing channel is significantly influenced by mineral distribution heterogeneity and macropore distribution pattern in the core. The acid breakthrough volume is defined as 1.5 PV divided by pore volume of core (PV is the acid volume consumed to form the dominating wormhole channel.). If the acid is injected continuously while the wormhole has completely broken through the core, the wormhole shape only changes a little, but the end-face dissolution will be formed at the end of the core. And the conclusion could be used to optimize the acid consumption volume.

It is crucial for vuggy carbonate formation to enhance near wellbore permeability to connect formation, release damage and improve productivity. Carbonate minerals can be dissolved by hydrochloric acid and the conductivity path will be formed between formation and wellbore [

∂ ϕ ∂ t + ∂ v x ∂ x + ∂ v y ∂ y = 0 (1)

Since the acid flow pattern conforms to Darcy laws, the velocity can be described by equivalent permeability [

{ v x = − 1 μ ( K x x ∂ P ∂ x + K x y ∂ P ∂ y ) v y = − 1 μ ( K y x ∂ P ∂ x + K y y ∂ P ∂ y ) (2)

Then, the pressure field equation is:

∂ ϕ ∂ t + ∂ ∂ x [ ( K x x ∂ P ∂ x + K x y ∂ P ∂ y ) ] + ∂ ∂ y [ − 1 μ ( K y x ∂ P ∂ x + K y y ∂ P ∂ y ) ] = 0 (3)

Assuming porosity will not change per time step, then (3) is expressed as a steady-state equation:

∂ ∂ x [ ( K x x ∂ P ∂ x + K x y ∂ P ∂ y ) ] + ∂ ∂ y [ − 1 μ ( K y x ∂ P ∂ x + K y y ∂ P ∂ y ) ] = 0 (4)

Based on the microelement analysis (

∂ ∂ t ( Δ x Δ y ϕ C f ) = Δ y ( v C f | x − v C f | x + Δ x ) − Δ y ( ϕ D e ∂ C f ∂ x | x − ϕ D e ∂ C f ∂ x | x + Δ x ) + Δ x ( v C f | y x − v C f | y + Δ y ) − Δ x ( ϕ D e ∂ C f ∂ y | y − ϕ D e ∂ C f ∂ y | y + Δ y ) − Δ x Δ y k c a v ( C f − C s ) (5)

When (5) is divided by, then：

∂ ∂ t ( ϕ C f ) = v C f | x − v C f | x + Δ x Δ x − ϕ D e ∂ C f ∂ x | x − ϕ D e ∂ C f ∂ x | x + Δ x Δ x + v C f | y x − v C f | y + Δ y Δ y − ϕ D e ∂ C f ∂ y | y − ϕ D e ∂ C f ∂ y | y + Δ y Δ y − k c a v ( C f − C s ) (6)

Assuming Δx and Δy approach zero, then (6) is expressed as:

∂ ∂ t ( ϕ C f ) = − ∂ ( v x C f ) ∂ x + ∂ ∂ x ( ϕ D e x ∂ C f ∂ x ) − ∂ ( v y C f ) ∂ y + ∂ ∂ y ( ϕ D e y ∂ C f ∂ y ) − k c a v ( C f − C s ) (7)

Diffusion coefficient D_{e}, which is composed of mass transfer and convection diffusion, can be described by Peclet coefficient Pe [

{ D e x D m = 1 F ϕ + a x P e m D e y D m = 1 F ϕ + a y P e m (8)

Then the convection diffusion equation can be simplified as:

∂ ∂ t ( ϕ C f ) = − ∇ ( v x C f ) + ∇ ( ( D m F + ϕ a v r p ) ⋅ ∇ C f ) − k c a v ( C f − C s ) (9)

The porosity variation can be calculated by acid dissolving capacity number α:

− ∂ ∂ t [ Δ x Δ y ( 1 − ϕ ) ρ s ] = k c a v ( C f − C s ) Δ x Δ y (10)

By simplifying (10), the porosity equation is expressed as:

∂ ϕ ∂ t = k c a v ( C f − C s ) α ρ s (11)

(2), (4), (9), (11) constitute the numerical model of matrix acidizing in carbonate formation:

{ v x = − 1 μ ( K x x ∂ P ∂ x + K x y ∂ P ∂ y ) ; v y = − 1 μ ( K y x ∂ P ∂ x + K y y ∂ P ∂ y ) ∂ ∂ x [ − 1 μ ( K x x ∂ P ∂ x + K x y ∂ P ∂ y ) ] + ∂ ∂ y [ − 1 μ ( K y x ∂ P ∂ x + K y y ∂ P ∂ y ) ] = ∂ ϕ ∂ t ∂ ∂ t ( ϕ C f ) = − ∇ ( v x C f ) + ∇ ( ( D m F + ϕ a v r p ) ⋅ ∇ C f ) − k c a v ( C f − C s ) ∂ ϕ ∂ t = k c a v ( C f − C s ) α ρ s (12)

The acid concentration in rock is zero before being acidized. The pressure boundary and concentration boundary are closed when acid is injected into the entry end of rock with the constant volume. And the pressure boundary of the outlet end is a constant and the acid concentration is zero.

Initial conditions are:

{ P ( x , y ) = 0 t > 0 C f ( x , y ) = 0 t > 0

Boundary conditions are:

{ ∫ 0 h Δ Z K μ ∂ P ∂ x | x = 0 d y = Q i n j t > 0 P ( l , y ) = P e t > 0 ∂ P ∂ y | y = 0 = 0 t > 0 ∂ P ∂ y | y = h = 0 t > 0 C f ( 0 , y ) = C 0 t > 0 C f ( l , y ) = 0 t > 0 ∂ C f ∂ y | y = 0 = 0 t > 0 ∂ C f ∂ y | y = h = 0 t > 0

Modan et al. [

{ K K 0 = ϕ ϕ 0 [ ϕ ( 1 − ϕ 0 ) ϕ 0 ( 1 − ϕ ) ] 2 r r p = K ϕ 0 K 0 ϕ a v a 0 = ϕ r 0 ϕ 0 r P (13)

S h = 2 k c r P D m = S h ∞ + 0.35 ( r P x ) 0.5 R e P 1 / 2 S c 1 / 3 (14)

P e = ν ⋅ r p D m (15)

K―equivalent permeability, can describe heterogeneity and anisotropy;

v_{x}―Darcy velocity (x - z boundary), m/min;

v_{Y}―Darcy velocity (y - z boundary), m/min;

a_{v}―Pore specific surface, m^{2}/m^{3};

α―acid dissolving capacity number, mol/mol;

ρ_{s}―Rock density, kg/m^{3};

F―formation resistivity coefficient;

Q_{inj}―acid injection volume, m^{3}/min;

Δz―core thickness, m;

r_{p}―throat pore diameter, m;

Sh_{∞}―approximate Sherwood number, Sh_{∞} = 2;

Re_{p}―Reynolds number, Re_{p} ≈ 0.2;

S_{c}―Schmidt number, S_{c} = v/D_{m};_{ }

k_{c}―mass transfer coefficient, m/min;

C_{f}―acid mass concentration, kg/m^{3};

C_{s}―acid mass concentration in vuggy surface, kg/m^{3}.

The pressure equation can be simplified from (4) by the centered difference as the nine-diagonal equation:

A N i , j P i − 1 , j − 1 + A A i , j P i , j − 1 + A P i , j P i + 1 , j − 1 + B B i , j P i − 1 , j + C C i , j P i , j + D D i , j P i + 1 , j + E N i , j P i − 1 , j + 1 + E E i , j P i , j + 1 + E P i , j P i + 1 , j + 1 = 0 (16)

where:

A N i , j = K i − 1 , j x y + K i , j − 1 y x 4 Δ x i Δ y j

A A i , j = K i , j y y + K i , j − 1 y y 2 ( Δ y j ) 2

A P i , j = − K i + 1 , j x y + K i , j − 1 y x 4 Δ x i Δ y j

B B i , j = K i , j x x + K i − 1 , j x x 2 ( Δ x i ) 2

D D i , j = K i , j x x + K i + 1 , j x x 2 ( Δ x i ) 2

E N i , j = − K i − 1 , j x y + K i , j + 1 y x 4 Δ x i Δ y j

E E i , j = K i , j y y + K i , j + 1 y y 2 ( Δ y j ) 2

E P i , j = K i + 1 , j x y + K i , j + 1 y x 4 Δ x i Δ y j

C C i , j = − ( A A i , j + B B i , j + D D i , j + E E i , j )

The convection diffusion equation can be simplified from (9) by the centered difference as the five-diagonal equation:

A A i , j C f i , j − 1 n + 1 + B B i , j C f i − 1 , j n + 1 + C C i , j C f i , j n + 1 + D D i , j C f i + 1 , j n + 1 + E E i , j C f i + 1 , j n + 1 = F F i , j (17)

where:

A A i , j = − v y i , j − 1 / 2 n 2 Δ y j − ϕ i , j − 1 / 2 n ( D m F + a y v y i , j − 1 / 2 n r P i , j − 1 / 2 n ) Δ y j ⋅ Δ y j − 1 / 2

B B i , j = − v x i − 1 / 2 , j n 2 Δ x i − ϕ i − 1 / 2 , j n ( D m F + a x v x i − 1 / 2 , j n r P i − 1 / 2 , j n ) Δ x i ⋅ Δ x i − 1 / 2

D D i , j = v x i + 1 / 2 , j n 2 Δ x i − ϕ i + 1 / 2 , j n ( D m F + a x v x i + 1 / 2 , j n r P i + 1 / 2 , j n ) Δ x i ⋅ Δ x i + 1 / 2

E E i , j = v y i , j + 1 / 2 n 2 Δ y j − ϕ i , j + 1 / 2 n ( D m F + a y v y i , j + 1 / 2 n r P i , j + 1 / 2 n ) Δ y j ⋅ Δ y j + 1 / 2

C C i , j = v x i + 1 / 2 , j n − v x i − 1 / 2 , j n 2 Δ x i + v y i , j + 1 / 2 n − v y i , j − 1 / 2 n 2 Δ y i + ( ϕ i , j − 1 / 2 n ( D m F + a y v y i , j − 1 / 2 n r P i , j − 1 / 2 n ) Δ y j ⋅ Δ y j − 1 / 2 + ϕ i − 1 / 2 , j n ( D m F + a x v x i − 1 / 2 , j n r P i − 1 / 2 , j n ) Δ x i ⋅ Δ x i − 1 / 2 + ϕ i + 1 / 2 , j n ( D m F + a x v x i + 1 / 2 , j n r P i + 1 / 2 , j n ) Δ x i ⋅ Δ x i + 1 / 2 + ϕ i , j + 1 / 2 n ( D m F + a y v y i , j + 1 / 2 n r P i , j + 1 / 2 n ) Δ y j ⋅ Δ y j + 1 / 2 ) + ϕ i , j n Δ t + k c i , j n a v i , j n ( k s k s + k c i , j n )

F F i , j = ϕ i , j n C f i , j n Δ t

The step of the central point is calculated by the arithmetic method:

Δ x i + 1 / 2 = Δ x i + Δ x i + 1 2

Δ y j + 1 / 2 = Δ y j + Δ y j + 1 2

The porosity equation can be simplified from (11) by the forward difference as a linear equation:

ϕ i , j n + 1 = k c i , j n a v i , j n ( k s k s + k c i , j n ) C f i , j n + 1 α ρ s Δ t + ϕ i , j n (18)

Solving processes:

1) Calculate equivalent permeability tensors of each grid at initial time;

2) Integrate the equivalent permeability tensors into Equation (16), and solve the five-diagonal matrix to achieve pressure distribution in the core;

3) According to Equation (17) and the pressure distribution, the acid concentration in the core is received.

4) The reservoir porosity is solved by combining the reaction equation Equation (17) and the porosity equation Equation (18).

5) Return to step. 1 to calculate the equivalent permeability tensor of each grid next time, recycling the above steps until the end of the injection.

Omer Izgec et al. [

In this paper, the acid consumption for wormhole breakthrough and differential pressure varying with time is calculated by the proposed model. The following simulation results are based on the parameters in

Items/Units | Value | |
---|---|---|

BC21 | BC10 | |

Acid injection volume Q_{inj}/(m^{3}/min) | 2 × 10^{−5} | 1.2× 10^{−5} |

Acid mass concentration C_{0}/(kg/m^{3}) | 1070 | 1070 |

Acid dissolving capacity number α/(m^{3}(rock)/m^{3} (acid)) | 0.082 | 0.082 |

Rock density ρ_{s}/(kg/m^{3}) | 2500 | 2500 |

Mass transfer coefficient k_{s}/(m/min) | 0.12 | 0.12 |

Diffusion coefficient D_{m}/(m^{2}/min) | 1.8× 10^{−7} | 1.8× 10^{−7} |

Average pore throat radius r_{p}/m | 7.65× 10^{−}^{5} | 6.82× 10^{−}^{5} |

Average pore specific surface a_{v}/(m^{2}/m^{3}) | 9.15× 10^{5} | 1.17 × 10^{5} |

Porosity φ | 0.3 ~ 0.4 | 0.35 ~ 0.45 |

Average permeability K/(mD) | 25.6 | 23.3 |

Core size (cm * cm) | 50.8 × 10.18 | 50.8 × 10.18 |

By the CT scanning and computer simulation, the porosity distribution of BC21 and BC10 before and after acidizing is shown in

Porosity distribution of BC21 and BC10 before and after acidizing by simulation is shown in

The differential pressure drops of experimental and simulation result are similar (^{3} (BC21) and 0.048 m^{3} (BC10), which is different from Mohan’s viewpoint [

Furthermore, the acid type, concentration and injection velocity influence the wormhole development. More intensive studies will be developed in the follow-up work.

1) In this paper, a novel numerical model for carbonate formation and vuggy carbonate formation acidizing is deduced based on equivalent permeability, which avoids distractions of pressure field and seepage variation between fracture-vuggy and matrix. The model is verified to be correct by the computer simulation.

2) According to the experimental and simulation results, the acid flow channel is depended on the rock, minerals and the macropores distribution. If continuing to inject acid after the wormhole breaking through the core completely, the formed wormhole will not change significantly but the end will be dissolved.

3) However, the main reason for the difference between experimental results and simulations is that, the simulation results calculated through our models are based on 2-D vision. However, the experimental results come from 3-D core flooding experiments.

4) PV is defined as the acid consumption when the major wormhole channel is formed while 1.5 times of PV divided by pore volume of core is defined as the acid breakthrough volume.

5) Vug size and its distribution are the main factors for the formed wormhole and for the wormhole to break through the rock. The simulation result can provide an important guidance to optimize the operation parameters of matrix acidizing in the vuggy carbonate formation.

Huang, J., Zhang, B.H., Liu, S.J., Cao, Y.F. and Xue, H. (2018) A Novel Model for Carbonate Formation Acidizing Based on Equivalent Permeability. Open Journal of Yangtze Gas and Oil, 3, 167-178. https://doi.org/10.4236/ojogas.2018.33015