PURIFICATION WITH EARTH (AT-TAYAMMUM):
He- the Most High- says:
“And if you are ill or upon a journey, or one of you comes from going to the toilet, or if you have had sexual intercourse with women and you do not find water, then perform tayammum with clean earth: and wipe your faces and your hands.” [Sooratul-Maa.idah (5):6]
And Allaah’s Messenger (صلّى الله عليه وسلّم ) said:
“Clean earth is a purification for the Muslim even if he does not find water for ten years.” 
The causes which render it permissible:
It is permissible to perform tayammum when one is unable to use water, if he can not find it, or if he fears harm from using- it because of illness in the body or severe cold. From `Imraan ibn Husayn -radiyallaahu `anhu- who said: “We were along with Allaah’s Messenger (صلّى الله عليه وسلّم ) on a journey. So he led the people in the Prayer and then he saw a man who was keeping himself away. So he said:
“What prevented you from praying?” He said: “I was struck by janaabah and there was no water.” So he (صلّى الله عليه وسلّم ) said: “Use the earth, for it will suffice you.” 
And from Jaabir -radiyallaahu `anhu- who said: “We went out on a journey, so a man from us was struck by a rock and it caused him a head injury. Then he had a wet dream, so he asked his companions: ‘Do you find any concession for me to perform tayammum?’ So they said: “We do not find any concession for you and you, and you are able to use water.” So he took a bath and he died. So when we came to Allaah’s Messenger (صلّى الله عليه وسلّم ) he was informed of that. So he said:
“They killed him, may Allaah kill them. Why didn’t they ask when they did not know? Indeed the cure for ignorance is to ask. It would have sufficed him to perform tayammum.” 
And from `Amr ibn al-`Aas -radiyallaahu `anhu- that when he was sent on the military expedition of Dhaatas-Salaasil he said: “I had a wet dream on a night which was severely cold, so I feared that if I took a bath I would die. So I performed tayammum, and then I led my companions in the Dawn Prayer. Then when we came to Allaah’s Messenger (صلّى الله عليه وسلّم ) they mentioned that to him. So he said:
“O `Amr did you lead your companions in Prayer when you were junub?” So I said: “I remembered the Saying of Allaah- the Most High:
“And do not kill yourselves indeed Allaah is ever Merciful to you.” [Sooratun-Nisaa· (4): 29]
So therefore I performed tayammum and then I prayed.” So Allaah’s Messenger (صلّى الله عليه وسلّم ) laughed, and he did not say anything.” 
What is ‘earth’ (as-Sa’eed)?
He said in ‘Lisaanul `Arab’  : “The Sa`eed is the earth; it is said: pure earth, and it is said: it is every pure soil; and there occurs in the Revelation sent down:
“So perform tayammum with pure earth” [Sooratul Maa·idah (5): 6]
Aboo Ishaaq said: ‘as-Sa`eed is the face of the earth, and it is upon the person that he should strike his two hands upon the face of the earth, and not care whether the place has soil upon it or not. Because as-Sa`eed is not soil rather it is the face of the earth, whether that be soil or something else. He said: and if there is a land which is all rock, with no soil upon it, and the person performing tayammum strikes his hand upon that rock, then that would be purification if he wipes his face with it.’
The description of the tayammum:
From `Ammaar ibn Yaasir -radiyallaahu `anhu(maa)- who said: ‘I became junub and I did not find any water. Therefore I rolled around in the dust and I prayed. So I mentioned that to the Prophet (صلّى الله عليه وسلّم ) and he said: “It would have sufficed you to do like this”, and the Prophet (صلّى الله عليه وسلّم ) struck the ground with his two palms, and blew upon them. Then he wiped his face with them and his two hands.’ 
A point of benefit:
The principal with regard to tayammum is that it takes the place of the wudoo· So whatever is permissible to do in a state of wudoo·, then is permissible to do it with it (i.e. tayammum). And it is permissible to perform it before the time, just as that is permissible with the wudoo·; and the person can prayer whatever he wishes with it, just as he prays in a state of wudoo.
The things which break tayammum:
Tayammum is broken by that which breaks the wudoo·. It is also broken by the presence of water -for the person who couldn’t find it, and the ability to use it- for the one who was unable to use it; and whatever came before from his Prayer is correct, it is not necessary for him to repeat it.
From Aboo Sa`eed al-Khudree -radiyallaahu `anhu- who said: “Two men went out on a journey and it became time for the Prayer and they did not have any water with them. So both of them performed tayammum with pure earth and they prayed. Then they found water within the time. So one of them repeated the wudoo· and the Prayer, and the other one did not repeat. Then they came to Allaah’s Messenger (صلّى الله عليه وسلّم ) and they mentioned that to him. So he said to the one who had not repeated: “You attained the Sunnah, and your Prayer suffices you.” And he said to the one who performed wudoo· and repeated: “You have the reward twice over.”’’ 
A point of benefit:
Whoever has a wound upon which he has wrapped a bandage, or a fracture which he has set, then having to wash that place falls away from him, and he does not have to wipe over it, nor perform tayammum for it.
The proof for that is the Saying of Allaah- the Most High:
“Allaah does not place a duty on any soul except that which it can carry out.” [Sooratul-Baqarah (2): 286]
And the saying of the Messenger (صلّى الله عليه وسلّم ) “When I command you with a matter, then do what you are able to do from it.” 
So everything which the person is unable to do falls away, by the text of the Qur’aan and the Sunnah, and to declare something else to be a replacement for it is legislation, and legislation is not binding except by a text of the Qur’aan or Sunnah; and there is nothing in the Qur’aan or the Sunnah to show that wiping upon a splint or upon ointment is a replacement for washing that which the person is not able to wash. So that saying falls away. 
The permissibility of performing tayammum with a wall: 
From Ibn `Abbaas-radiyallaahu `anhu(maa)-who said: “ I came along with `Abdullaah ibn Yasaar- the slave belonging to Maymoonah the wife of the Prophet (صلّى الله عليه وسلّم ) until we entered upon Aboo Juhaym ibn al-Haarith ibn as-Simmah al-Ansaaree.
So Abul- Juhaym said: The Prophet (صلّى الله عليه وسلّم ) came from the direction of the well of Jamal , so he was met by a man who greeted him with Salaam, but the Prophet (صلّى الله عليه وسلّم ) did not respond to him until he went to the wall, and wiped his face and two hands. Then he responded to his greeting of Salaam.” 
 Saheeh: [Saheeh Sunan Abee Daawood (322); at-Tirmidhee (1/81/124), Aboo Daawood ( 1/528/329), anNasaa·ee (1/171)-with wordings which are close.
Agreed upon: al-Bukhaaree (1/477/344), Muslim (1/474/682), an-Nasaa·ee (1/171).
 Hasan: [Saheeh Sunan Abee Daawood (326)]; Aboo Daawood (1/532/332), and it contains an addition which is munkar, and that is “…and wipe or tie a piece of cloth upon his wound and then wipe over it. And wash the rest of his body.’’ Shamsul-Haqq said in ‘ `Awnul-Ma`bood’ (1/535): ‘The narration about combining tayammum and washing is not reported by anyone other than Zubayr ibn Khurayq, and he- along with the fact that he is not strong in hadeeth – has also contradicted the rest of those who report it from `Ataa· ibn Abee Rabaah. So the narration of the combination between tayammum and taking a bath is a narration which is weak, rulings cannot be established through it.’ And take note of the point of benefit which follows after one page.
 Saheeh: [Saheeh Sunan Abee Daawood (323), Aboo Daawood (1/530/330), Ahmad (2/191/16),al-Haakim (1/177).
 Agreed upon: al-Bukhaaree (1/455/347), Muslim (1/280/368), Aboo Daawood (1/514/317), an-Nasaa·ee (1/166).
 Saheeh:[Saheeh Sunan Abee Daawood (327), Aboo Daawood (1/536/334), an- Nasaa·ee (1/213).
 Saheeh: [Mukhtasar Muslim (639), Muslim (2/975/1337), an-Nasaa·ee (5/110).
 al-Muhallaa of Ibn Hazm (2/74).
 Whether it is made of clay or made of rock; whether it is painted or not painted .This verdict was given to me by our Shaikh al-Albaanee –rahimahullaah, and he said: “And your Lord is never forgetful” [Soorah Maryam (19): 64]
 A place close to al-Madeenah.
 Agreed upon: al-Bukhaaree (1/441/337), Muslim (1/281/369) -in disconnected form, Aboo Daawood (1/521/325), an-Nasaa·ee (1/165).
Source : Al-Wajeez, The Book Of Purification – Abdul-`Azeem ibn Badawee al-Khalafee, Translated by Abu Talhah Dawood Burbank . An excellent book on the rules and regulations of various acts of worship, with evidences from the Book and Sunnah. This chapter deals with water and aspects of purification, wudhoo, ghusl, the toilet etc
Related Link: https://abdurrahman.org/tahara-purification/