In this paper, the RMS error was used to check the validity of numerical solution against the analytical one, as shown in Tables 1 and 2. The two parameters, and ⁠, play an important role to investigate the performance of the numerical solution. In the explicit finite difference scheme is restricted under the stability condition. Thus, in this present study, the accuracy was investigated by selecting different mesh sizes. The RMS error was investigated for ⁠, 0.05, and 0.07 for the particular time period 20 years in the time domain for the exponential decreasing and the asymptotic form of the velocity patterns, and 30 years with within the sinusoidal and algebraic sigmoid form of the velocity patterns in sand and clay media, respectively.

Table 1

The RMS values for the sand medium at particular 20 year in 0 < T ≤ T_P

Distance .	Analytical result .	Numerical result .
		ΔZ = 0.02 .	ΔZ = 0.05 .	ΔZ = 0.07 .
Case i: For exponential decreasing form of the velocity pattern
0.0019	0.5125	0.0119	0.0149	0.0169
0.0055	0.3188	0.0159	0.0248	0.0308
0.0093	0.1736	0.0198	0.0348	0.0448
0.0130	0.0837	0.0238	0.0448	0.0588
0.0167	0.0376	0.0278	0.0548	0.0727
RMS error		0.2719	0.2665	0.2634
Case ii: For asymptotic form of the velocity pattern
0.0019	0.3613	0.0120	0.0150	0.0170
0.0055	0.0687	0.0160	0.0250	0.0310
0.0093	0.0130	0.0200	0.0350	0.0449
0.0130	0.0094	0.0240	0.0449	0.0589
0.0167	0.0091	0.0280	0.0549	0.0729
RMS error		0.1583	0.1585	0.1596

Distance .	Analytical result .	Numerical result .
		ΔZ = 0.02 .	ΔZ = 0.05 .	ΔZ = 0.07 .
Case i: For exponential decreasing form of the velocity pattern
0.0019	0.5125	0.0119	0.0149	0.0169
0.0055	0.3188	0.0159	0.0248	0.0308
0.0093	0.1736	0.0198	0.0348	0.0448
0.0130	0.0837	0.0238	0.0448	0.0588
0.0167	0.0376	0.0278	0.0548	0.0727
RMS error		0.2719	0.2665	0.2634
Case ii: For asymptotic form of the velocity pattern
0.0019	0.3613	0.0120	0.0150	0.0170
0.0055	0.0687	0.0160	0.0250	0.0310
0.0093	0.0130	0.0200	0.0350	0.0449
0.0130	0.0094	0.0240	0.0449	0.0589
0.0167	0.0091	0.0280	0.0549	0.0729
RMS error		0.1583	0.1585	0.1596

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Table 2

RMS values for the clay medium with their averaging porosity at particular 30 years for T > T_p

Distance .	Analytical result .	Numerical result .
		ΔZ = 0.02 .	ΔZ = 0.05 .	ΔZ = 0.07 .
Case i: For the sinusoidal form of the velocity pattern
0.0028	0.1094	0.0118	0.0148	0.0168
0.0083	0.2425	0.0158	0.0248	0.0307
0.0138	0.2524	0.0198	0.0347	0.0447
0.0193	0.1869	0.0238	0.0447	0.0586
0.0248	0.1135	0.0278	0.0546	0.0725
RMS error		0.1726	0.1596	0.1514
Case ii: For the algebraic sigmoid form of the velocity pattern
0.0028	0.1001	0.0119	0.0149	0.0169
0.0083	0.1608	0.0159	0.0249	0.0309
0.0138	0.0868	0.0199	0.0349	0.0449
0.0193	0.0301	0.0239	0.0449	0.0589
0.0248	0.0126	0.0279	0.0549	0.0728
RMS error		0.0818	0.0780	0.0774

Distance .	Analytical result .	Numerical result .
		ΔZ = 0.02 .	ΔZ = 0.05 .	ΔZ = 0.07 .
Case i: For the sinusoidal form of the velocity pattern
0.0028	0.1094	0.0118	0.0148	0.0168
0.0083	0.2425	0.0158	0.0248	0.0307
0.0138	0.2524	0.0198	0.0347	0.0447
0.0193	0.1869	0.0238	0.0447	0.0586
0.0248	0.1135	0.0278	0.0546	0.0725
RMS error		0.1726	0.1596	0.1514
Case ii: For the algebraic sigmoid form of the velocity pattern
0.0028	0.1001	0.0119	0.0149	0.0169
0.0083	0.1608	0.0159	0.0249	0.0309
0.0138	0.0868	0.0199	0.0349	0.0449
0.0193	0.0301	0.0239	0.0449	0.0589
0.0248	0.0126	0.0279	0.0549	0.0728
RMS error		0.0818	0.0780	0.0774

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