Optimizing the rheological properties of silica nano-modified bentonite mud using overlaid contour plot and estimation of maximum or upper shear stress limit

Figures & data

Table 1. Bentonite physical requirements

Requirements	Specification	Mud sample
Viscometer dial reading at 600 RPM*	30, minimum	30
Viscometer dial reading at 300 RPM	–	23
Plastic viscosity	–	7
Yield point	–	16
Yield point/Plastic viscosity ratio*	3, maximum	2.29
Mud weight	–	8.6 ppg
Specific gravity	–	1.3
Filtrate volume*	15 cm^3, maximum	14 cm^3

* API requirements.

Table 2. Factor setting for response surface design

Variable	Notation	Unit	Level
			Low	High
Bentonite content	X	wt.%	6.3	15
Silica nanoparticles	Y	wt.%	0	1.5

Table 3. The response surface design and the measured rheological properties (responses)

Run order	Bentonite content (wt.%)*	Silica nanoparticles (wt.%)*	Plastic viscosity (Pas)	Yield point (Pa)	Apparent viscosity (Pas)
	X	Y
1	6.3	0.0	0.0034	2.1449	0.0055
2	6.3	0.5	0.0035	2.5535	0.0060
3	6.3	1.0	0.0036	2.5535	0.0061
4	6.3	1.5	0.0037	2.6046	0.0063
5	13.0	0.0	0.0120	22.982	0.0350
6	13.0	0.5	0.0130	25.535	0.0375
7	13.0	1.0	0.0130	28.089	0.0405
8	13.0	1.5	0.0130	31.663	0.0440
9	15.0	0.0	0.0170	62.816	0.0785
10	15.0	0.5	0.0230	68.945	0.0905
11	15.0	1.0	0.0280	66.391	0.0940
12	15.0	1.5	0.0300	67.923	0.0955

* Variables in un-coded levels.

Table 4. Effects and regression coefficients for plastic viscosity, yield point and apparent viscosity

Variable	Plastic viscosity			Yield point			Apparent viscosity
	Estimated effect	Regression coefficient	p-value	Estimated effect	Regression coefficient	p-value	Estimated effect	Regression coefficient	p-value
Constant	–	0.00423	0.041	–	−0.430	0.009	–	0.00382	0.051
X	0.02095	0.01048	0.000	64.06	32.03	0.000	0.08365	0.04183	0.000
Y	0.00391	0.00196	0.049	4.275	2.137	0.025	0.00741	0.00370	0.004
X²	0.01958	0.00979	0.002	69.83	34.92	0.000	0.08796	0.04398	0.000
XY	0.00492	0.00246	0.043	–	–	–	0.00718	0.00359	0.009

Table 5. Points along the path for steepest ascent and descent and observed plastic viscosity, yield point and apparent viscosity of sample at the points

Run No.		Bentonite content (wt.%)	Silica nanoparticles (wt.%)	Plastic viscosity (Pas)	Yield point (Pa)	Apparent viscosity (Pas)
	Base	10.65	0.750	0.00413	−0.40496	0.00373
1	Base - 6Δi	36.75	−0.390	0.39348	1,445.60	1.79922
2	Base - 5Δi	32.40	−0.200	0.28279	1,030.01	1.28464
3	Base - 4Δi	28.05	−0.010	0.19042	684.261	0.85619
4	Base - 3Δi	23.70	0.180	0.11637	408.341	0.51388
5	Base - 2Δi	19.35	0.370	0.06064	202.257	0.25769
6	Base - Δi	15.00	0.560	0.02322	66.0085	0.08765
	Δi	−4.350	0.190
7	Base + Δi	6.300	0.940	0.00336	3.01695	0.00594
8	Base + 2Δi	1.950	1.130	0.02090	76.2742	0.09429
9	Base + 3Δi	−2.400	1.320	0.05676	219.366	0.26876
10	Base + 4Δi	−6.750	1.510	0.11095	432.295	0.52937
11	Base + 5Δi	−11.10	1.700	0.18346	715.058	0.87611
12	Base + 6Δi	−62.50	7.440	0.27428	1,067.65	1.30899

Table 6a. Herschel Bulkley model parameters for silica nano-modified bentonite drilling mud

Bentonite (wt.%)	Silica nanoparticles (wt.%)	Herschel Bulkley model (Equation 11)
		τ01 (Pa)	K1 [(Pa)s^n]	n	RMSE (Pa)	R^2
6.3	0.0	1.399	0.02605	0.747	0.221	0.999
	0.5	1.869	0.01787	0.803	0.279	0.998
	1.0	1.959	0.01481	0.834	0.266	0.998
	1.5	2.217	0.01277	0.853	0.253	0.998
13	0.0	17.76	0.20683	0.664	2.408	0.998
	0.5	21.03	0.24105	0.638	2.701	0.998
	1.0	25.04	0.15423	0.696	2.730	0.999
	1.5	26.63	0.38303	0.581	3.740	0.997
15	0.0	54.29	0.66851	0.556	6.966	0.998
	0.5	58.83	0.56228	0.616	9.657	0.998
	1.0	67.36	0.17568	0.764	24.67	0.991
	1.5	68.64	0.22317	0.731	20.53	0.994

Table 6b. Hyperbolic model parameters for silica nano-modified bentonite drilling mud

Bentonite (wt.%)	Silica nanoparticles (wt.%)	Hyperbolic model (Equation 14)
		τ03 (Pa)	A [(Pa)s]^−1	B [Pa]^−1	RMSE (Pa)	R^2
6.3	0.0	1.501	39.02	0.020	0.193	0.999
	0.5	1.935	41.49	0.016	0.203	0.999
	1.0	2.025	43.78	0.013	0.178	0.999
	1.5	2.244	45.06	0.013	0.199	0.999
13	0.0	18.54	7.533	0.006	1.002	0.999
	0.5	21.88	7.349	0.007	1.419	0.999
	1.0	25.65	8.467	0.007	1.621	0.999
	1.5	27.87	6.169	0.006	3.349	0.999
15	0.0	56.28	3.932	0.005	4.301	0.999
	0.5	60.72	3.494	0.004	5.812	0.999
	1.0	67.82	5.080	0.003	24.11	0.999
	1.5	69.20	4.692	0.003	19.56	0.999

Table 6c. Casson model parameters for silica nano-modified bentonite drilling mud

Bentonite (wt.%)	Silica nanoparticles (wt.%)	Casson model (Equation 13)
		τ02 (Pa)	K4 [(Pa)s^−1]	RMSE (Pa)	R^2
6.3	0.0	1.140	0.0020	0.264	0.996
	0.5	1.463	0.0015	0.445	0.991
	1.0	1.498	0.0015	0.511	0.988
	1.5	1.715	0.0015	0.572	0.988
13	0.0	16.62	0.0041	3.467	0.994
	0.5	20.06	0.0036	3.431	0.995
	1.0	23.54	0.0031	5.133	0.991
	1.5	26.15	0.0031	3.725	0.996
15	0.0	53.83	0.0036	6.971	0.996
	0.5	57.05	0.0056	11.43	0.995
	1.0	63.58	0.0041	30.01	0.982
	1.5	65.17	0.0041	25.94	0.986

Table 6d. Sisko model parameters for silica nano-modified bentonite drilling mud

Bentonite (wt.%)	Silica nanoparticles (wt.%)	Sisko model (Equation 12)
		K2 [(Pa)s^−1]	K3 [(Pa)s^m]	m	RMSE (Pa)	R^2
6.3	0.0	0.0036	1.165	0.110	0.310	0.964
	0.5	0.0041	1.697	0.062	0.381	0.941
	1.0	0.0041	1.835	0.047	0.371	0.931
	1.5	0.0041	2.109	0.036	0.341	0.927
13	0.0	0.0143	16.48	0.053	4.930	0.828
	0.5	0.0133	19.69	0.048	5.245	0.792
	1.0	0.0143	23.92	0.032	5.075	0.742
	1.5	0.0133	25.12	0.048	6.654	0.755
15	0.0	0.0189	52.04	0.037	14.10	0.688
	0.5	0.0266	56.02	0.037	17.85	0.722
	1.0	0.0281	65.42	0.018	25.31	0.676
	1.5	0.0276	66.60	0.019	22.79	0.674

Table 7. Points along the path for steepest ascent and descent and observed maximum shear stress tolerance of sample at the points

Run no.		Bentonite content (wt.%)	Silica nanoparticles (wt.%)	Shear stress limit (Pa)
	Base	10.65	0.750	12.01
1	Base - 6Δi	36.75	−0.390	2,725
2	Base - 5Δi	32.40	−0.200	1,945
3	Base - 4Δi	28.05	−0.010	1,296
4	Base - 3Δi	23.70	0.180	777.8
5	Base - 2Δi	19.35	0.370	391.2
6	Base - Δi	15.00	0.560	135.9
	Δi	−4.350	0.190
7	Base + Δi	6.300	0.940	19.51
8	Base + 2Δi	1.950	1.130	158.4
9	Base + 3Δi	−2.400	1.320	428.6
10	Base + 4Δi	−6.750	1.510	830.4
11	Base + 5Δi	−11.10	1.700	12.01
12	Base + 6Δi	−62.50	7.440	2,726

Figure 1. Contour plots for (a) plastic viscosity, (b) yield point and (c) apparent viscosity.

Figure 2. (a) Overlaid contour plot for plastic viscosity, yield point and apparent viscosity, (b) effect of silica nanoparticles on plastic viscosity, (c) effect of silica nanoparticles on yield point and (d) effect of silica nanoparticles on apparent viscosity.

Figure 3. Predicted and measured shear stress-shear rate data for 6.3 wt.% bentonite mud containing different amount of silica nanoparticles (a) 0 wt.%, (b) 0.5 wt.%, (c) 1 wt.% and (d) 1.5 wt.%.

Figure 4. Predicted and measured shear stress-shear rate data for 13 wt.% bentonite mud containing different amount of silica nanoparticles (a) 0 wt.%, (b) 0.5 wt.%, (c) 1 wt.% and (d) 1.5 wt.%.

Figure 5. Predicted and measured shear stress-shear rate data for 15 wt.% bentonite mud containing different amount of silica nanoparticles (a) 0 wt.%, (b) 0.5 wt.%, (c) 1 wt.% and (d) 1.5 wt.%.

Figure 6. Residual plot for 6.3 wt.% bentonite mud containing different amount of silica nanoparticles (a) 0 wt.%, (b) 0.5 wt.%, (c) 1 wt.% and (d) 1.5 wt.%.

Table 8a. The response surface design and the estimated maximum shear stress limit from Equation (20)

Run order	Bentonite content (wt.%)	Silica nanoparticles (wt.%)	Shear stress limit (Pa)
	X	Y
1	6.3	0.0	14.60
2	6.3	0.5	18.25
3	6.3	1.0	21.37
4	6.3	1.5	22.27
5	13.0	0.0	61.74
6	13.0	0.5	62.20
7	13.0	1.0	62.66
8	13.0	1.5	72.62
9	15.0	0.0	107.4
10	15.0	0.5	133.7
11	15.0	1.0	152.9
12	15.0	1.5	171.3

Table 8b. Effects and regression coefficients for maximum shear stress tolerance

Variable	Shear stress limit
	Estimated effect	Regression coefficient	p-value
Constant	–	11.85	0.014
X	122.2	61.09	0.000
Y	23.25	11.62	0.016
X²	136.7	68.37	0.000
XY	21.09	10.54	0.041

Figure 7. (a) Contour plot for shear stress limit, (b) overlaid contour plot for shear stress limit and (c) effect of silica nanoparticles and bentonite content on maximum shear stress limit.