Effect of nanoparticle behaviour on mud cake buildup for directional and horizontal wells: mathematical modelling and experimental study

Figures & data

Figure 1. Forces exerted on particles in drilling fluid and on the cake surface.

Figure 2. Critical deposition zone in front of porous media and its reduction with the filtration time.

Figure 3. Designed cross-flow filtration and mud cake buildup simulator set-up. Filtration cell and O-ring along with sealant is depicted at the top right of the picture. At the bottom right of the picture, mud cake formed after circulation is illustrated; (a) Fresh drilling fluid mud cake, (b) MWCNTs fluid mud cake, (c) nano-SiO₂ fluid mud cake and (d) nano CaCO₃ fluid mud cake.

Figure 4. Rheological properties of different drilling fluid samples based on power law model.

Table 1. Drilling fluid samples characteristics.

Parameter	Fresh mud	Nano-CaCO3 mud	Nano-SiO2 mud	Nano-MWCNT mud
ρf (gr.cm^−^3)	1.02	1.02	1.02	1.02
µ (cp)	4	4	3.5	5
Ɵ600 (lb/100 ft^2)	10	11	7.5	12
Ɵ300 (lb/100 ft^2)	6	7	4	7
dbent. (nm)	226	417	374	308
dn (nm)	–	80	20	15
dagg (nm)	–	78	83	194
Nagg	1	1	32	15
Ψ	1	0.806	0.72	1
df	–	–	2.43	2.47
Uf (cm.s^−1)	8	8	7	10
ϒ (1/s)	59.7	62.83	48.27	73.04
Acore (cm^2)	10	10	10	10
ρp (gr.cm^−^3)	2.35	2.71	2.4	2.1
Cb (gr.cm^−)	0.0428	0.0428	0.0428	0.0428
Cn (gr.cm^−)	0	8.57 × 10^−4	8.57 × 10^−4	8.57 × 10^−4
Shear stress (dyne.cm^−^2)	9.18	12.73	3.82	11.73
Φcake	0.38	0.36	0.22	0.26
f	0.4	0.4	0.4	0.4
s (nm)	1.0	1.0	1.0	1.0

Table 2. Solid particles characteristics obtained by DLS and literature.

Parameters	Bentonite	Nano-CaCO3	Nano-SiO2	Nano-MWCNT
H (10^−20 J)	11.6	10.1	6.6	60
z (mV)	29.1	14	25	29
E (10^11 dyne.cm^−2)	1.0	7.0	4.0	91.0
Υ	0.4	0.21	0.15	0.11

Table 3. Deposition and erosion intensity indexes measured for solid particles.

Solid	Mst. (gr) 100 sec	Mdyn. (gr) 100 sec	Idep	Ier
Bentonite	0.8	0.72	0.011	0.00008
CaCO3	0.81	0.7	0.025	0.000235
SiO2	0.9	0.75	0.0055	0.00014
MWCNT	0.82	0.69	0.038	0.00003

Figure 5. Particle size distribution of drilling fluid samples measured by DLS technique.

Figure 6. Aggregated structure of nanoparticles in drilling fluids proposed by fractal dimension analysis.

Figure 7. FESEM images taken from mud cake after dynamic filtration test. (a) fresh mud cake with bentonite particles, (b) nano-SiO₂ mud cake, (c) nano-CaCO₃ mud cake and (d) MWCNTs mud cake.

Figure 8. Filtration rate reduction during test and dynamic equilibrium achievement for four drilling fluid samples at horizontal and vertical position. (a) Fresh mud, (b) nano-SiO₂ mud, (c) nano-CaCO₃ mud and (d) MWCNTs mud.

Table 4. Vertical force ∑F_y, particle depositional acceleration a_y and possible depositional volume V_c exerted on bentonite computed by model for fresh mud at different inclinations.

	q Filtration (cm^3.s^−1)	∑Fy (10^−^12 dyne)	ay (980 cm.s^−^2)	Vc (cm^3)
Ɵ = 0°	0.06	35.7	2.56	17.27
	0.04	14.4	1.03	16.85
	0.02	1.65	0.11	14.62
	0.01	0	0	0
Ɵ = 30°	0.06	39.7	2.85	17.31
	0.02	5.59	0.4	16.16
	0.01	2.4	0.17	15.19
	0.004	1.5	0.1	14.45
Ɵ = 60°	0.06	42.6	3.05	17.33
	0.02	8.47	0.6	16.5
	0.01	5.28	0.38	16.11
	0.004	4.38	0.31	15.92
Ɵ = 90°	0.06	43.6	3.13	17.34
	0.02	9.53	0.68	16.6
	0.01	6.33	0.45	16.27
	0.006	5.65	0.4	16.17

Table 5. Vertical force ∑F_y, particle depositional acceleration a_y and possible depositional volume V_c exerted on particles computed by model for nano-SiO₂ mud at different inclinations.

		∑Fy (10^−13 dyne)		ay (980 cm.s^−^2)		Vc (cm^3)		Deposition
	q Filtration (cm^3.s^−1)	Bentonite	Nano	Bentonite	Nano	Bentonite	Nano	Bentonite	Nano
Ɵ = 0°	0.06	554	29.9	0.88	9.5	16.91	17.67	Yes	Yes
	0.04	202	12.8	0.32	4.07	16.2	17.49	Yes	Yes
	0.02	0	2.55	0	0.8	0	16.86	No	Yes
	0.01	0	0	0	0	0	0	No	No
Ɵ = 30°	0.06	733	30.9	1.16	9.8	17.05	17.67	Yes	Yes
	0.02	169	3.47	0.26	1.1	16.03	17.02	Yes	Yes
	0.01	116	0.89	0.18	0.28	15.62	16.09	Yes	Yes
	0.003	99.7	0.11	0.16	0.04	15.43	12.72	Yes	Yes
Ɵ = 60°	0.06	863	31.5	1.37	10	17.12	17.67	Yes	Yes
	0.02	299	4.14	0.47	1.31	16.52	17.11	Yes	Yes
	0.01	246	1.57	0.39	0.5	16.37	16.55	Yes	Yes
	0.003	230	0.79	0.36	0.25	16.31	15.96	Yes	Yes
Ɵ = 90°	0.06	911	31.8	1.44	10.09	17.15	17.68	Yes	Yes
	0.02	347	4.3	0.55	1.39	16.62	17.13	Yes	Yes
	0.01	294	1.81	0.46	0.57	16.5	16.65	Yes	Yes
	0.004	279	1.09	0.44	0.35	16.47	16.27	Yes	Yes

Table 6. Vertical force ∑F_y, particle depositional acceleration a_y and possible depositional volume V_c exerted on particles computed by model for nano-CaCO₃ mud at different inclinations.

		∑Fy (10^−13 dyne)		ay (980 cm.s^−^2)		Vc (cm^3)		Deposition
	q Filtration (cm^3.s^−1)	Bentonite	Nano	Bentonite	Nano	Bentonite	Nano	Bentonite	Nano
Ɵ = 0°	0.06	530	10.6	0.6	0.84	16.5	16.73	Yes	Yes
	0.04	137	4.08	0.15	0.32	15.06	0.32	Yes	Yes
	0.02	0	0.151	0	0.012	0	7.38	No	Yes
	0.01	0	0	0	0	0	0	No	No
Ɵ = 30°	0.06	778	14.6	0.89	1.15	16.76	16.9	Yes	Yes
	0.02	149	4.08	0.17	0.32	15.18	15.95	Yes	Yes
	0.01	90.2	3.1	0.1	0.25	14.37	15.65	Yes	Yes
	0.004	73.7	2.82	0.08	0.22	14	15.54	Yes	Yes
Ɵ = 60°	0.06	959	17.4	1.09	1.38	16.88	17.01	Yes	Yes
	0.02	330	6.9	0.37	0.55	16.1	16.9	Yes	Yes
	0.01	271	5.98	0.31	0.47	15.9	16.3	Yes	Yes
	0.004	255	5.7	0.29	0.45	15.84	16.27	Yes	Yes
Ɵ = 90°	0.06	1030	18.5	1.17	1.46	16.9	17.04	Yes	Yes
	0.02	397	8	0.45	0.63	16.27	16.54	Yes	Yes
	0.01	338	7.03	0.38	0.55	16.12	16.44	Yes	Yes
	0.005	323	6.78	0.37	0.53	16.08	16.41	Yes	Yes

Table 7. Vertical force ∑F_y, particle depositional acceleration a_y and possible depositional volume V_c exerted on particles computed by model for nano-MWCNTs mud at different inclinations.

		∑Fy (10^−13 dyne)		ay (980 cm.s^−^2)		Vc (cm^3)		Deposition
	q Filtration (cm^3.s^−1)	Bentonite	Nano	Bentonite	Nano	Bentonite	Nano	Bentonite	Nano
Ɵ = 0°	0.06	423	87.5	1.2	1.6	16.67	16.85	Yes	Yes
	0.04	132	25	0.37	0.46	15.62	15.85	Yes	Yes
	0.03	30.7	3.17	0.08	0.06	13.07	11.96	Yes	Yes
	0.02	0	0	0	0	0	0	No	No
Ɵ = 30°	0.06	522	111	1.48	2.04	16.8	16.98	Yes	Yes
	0.02	57.9	11.3	0.16	0.2	14.4	14.8	Yes	Yes
	0.01	14.3	1.88	0.04	0.03	10.78	10.17	Yes	Yes
	0.006	5.04	0	0.01	0	5.84	0	Yes	No
Ɵ = 60°	0.06	595	129	1.69	2.35	16.88	17.05	Yes	Yes
	0.02	131	28.6	0.37	0.52	15.6	16	Yes	Yes
	0.01	87.3	19.2	0.24	0.35	15.08	15.55	Yes	Yes
	0.005	76.4	16.9	0.21	0.31	14.87	15.38	Yes	Yes
Ɵ = 90°	0.06	622	135	1.76	2.48	16.9	17.07	Yes	Yes
	0.02	158	35	0.44	0.64	15.82	16.18	Yes	Yes
	0.01	114	25.6	0.32	0.47	15.44	15.87	Yes	Yes
	0.004	102	23	0.29	0.42	15.3	15.76	Yes	Yes

Table 8. Erosion parameters (vertical force ∑F_y, shear stress τ, critical shear stress τ_cr and mud cake surface occupied by particle S_p) exerted on aggregated particles on the cake surface for four drilling fluid samples.

	Sp (%)		∑Fy (10^−5 dyne)			τcr (dyne.cm^−2)		Erosion τ > τcr
Mud type	Bent.	Nano	Bent.	Nano	τ (dyne.cm^−2)	Bent.	Nano	Bent.
Fresh mud	100	–	3.94	–	9.18	3.22	–	Yes
Nano-CaCO3	87.6	12.4	7.3	1.14	12.73	5.02	10.3	Yes
Nano-SiO2	97.2	2.8	6.54	0.59	3.82	1.33	2.36	Yes
Nano-MWCNT	82.3	7.7	5.38	7.92	11.73	2.74	7.92	Yes

Table 9. Mud cake thickness, h_c, weight ratio of nanoparticle and bentonite on the cake, M_r, comparison between model and experimental outputs after 300 sec circulation for studied drilling fluids.

Mud type	Angle (deg.)	hc (mm) (model)	hc (mm) (Lab)	Error (%)	Mr Model (%)	Mr DLS (%)	Error (%)
Fresh mud	0	0.6	0.74	18.9
	30	1.38	1.51	8.6
	60	1.44	1.58	8.8	–	–	–
	90	1.45	1.6	9.4
Nano-CaCo3	0	0.07	0.1	30
	30	1.24	1.33	6.7
	60	1.35	1.42	4.9	1.82	1.95	6.67
	90	1.37	1.49	8.1
	0	0.16	0.19	15.8
Nano-SiO2	30	1.16	1.23	5.7
	60	1.21	1.27	4.7	0.753	0.817	7.83
	90	1.22	1.3	6.2
	0	0.2	0.24	16.6
Nano-MWCNT	30	0.84	0.97	13.4
	60	1.07	1.2	10.8	7.31	7.39	1.08
	90	1.14	1.24	8.1

Figure 9. Mud cake buildup during dynamic filtration test for different drilling fluid samples at vertical position predicted by the model.

Figure 10. Variation of mud cake thickness at different well inclinations for four drilling fluid samples predicted by the model to analyse hole problems related to mud cake buildup.